THE NETWORK PROPERTIES OF
A SIMULATED POLYMERIC GEL
A Thesis
Presented to the
Faculty of
San Diego State University
In Partial Fulfillment
of the Requirements for the Degree
Master of Science
in
Physics
by
Mark Allen Wilson
Fall 2008
SAN DIEGO STATE UNIVERSITY
The Undersigned Faculty Committee Approves the
Thesis of Mark Allen Wilson:
The Network Properties of
a Simulated Polymeric Gel
Arlette Baljon, Chair
Department of Physics
Michael Bromley
Department of Physics
Peter Salamon
Department of Mathematics and Statistics
Approval Date
iii
Copyright 2008
by
Mark Allen Wilson
iv
ABSTRACT OF THE THESIS
The Network Properties of
a Simulated Polymeric Gel
by
Mark Allen Wilson
Master of Science in Physics
San Diego State University, 2008
Complex network structure can be used to describe a large range of real-world
systems. The pages which comprise the World Wide Web, social interactions between friends,
and the highway system across the nation are just a few examples of this type of network
structure, consisting of a large number of components, dynamically evolving, and highly
interconnected. Within this study, a molecular dynamic simulation of a polymeric system will
be described in terms of a complex network. An analysis of two characteristic properties of
the polymeric network system will be performed. Through an analysis of the degree
distribution and clustering coefficient, information about the network topology will be gained.
Ensembles of Erdös-Rényi random models will then be created to investigate to what degree
the polymeric network can be described by a random network.
v
TABLE OF CONTENTS
PAGE
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
LIST OF TABLES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
CHAPTER
1
2
3
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Real-World Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
A Polymer System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Purpose of Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
THE SIMULATED POLYMER SYSTEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
System Configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
Earlier Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
A Polymer Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
NETWORK BASICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Definitions of Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Network Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Degree Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Clustering Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Erdös-Rényi Random Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Network Properties of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4
NETWORK PROPERTIES OF THE POLYMER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Degree Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Aggregate Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Multiple Bridge Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Single Bridge Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Least-Squares Curve Fits of the Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Tails of the Polymer Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
vi
Clustering Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Proximity Within the Polymer Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5
DEVELOPMENT OF THE MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Bimodal ER Random Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Adding Proximity to the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6
DISCUSSION AND CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Ongoing and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
APPENDIX
DESCRIPTIONS OF FORTRAN CODES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
vii
LIST OF TABLES
Table 1. The Number of Nodes n(k) at a Specific Degree k and the Degree
Distribution p(k) for the Simple Graph in Figure 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Table 2. Least Squares Fitting Parameters of the Aggregate Distribution for the
Polymer Network Using a Two Poisson Fitting Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Table 3. Least Squares Fitting Parameters of the Multiple Bridge Distribution
for the Polymer Network Using a Two Poisson Fitting Function. . . . . . . . . . . . . . . . . . . . 33
Table 4. Least Squares Fitting Parameters of the Single Bridge Distribution for
the Polymer Network Using a Two Poisson Fitting Function. . . . . . . . . . . . . . . . . . . . . . . . 34
Table 5. The Average Clustering Coefficient C for the Polymer Network and as
Calculated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Table 6. Three Average Clustering Coefficients for the Polymer Network: Simulated, Calculated, and After Rewiring. The Rewired Network Produces
a Similar Value for C as to the Calculated Value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Table 7. The Parameters Used to Create the Three Bimodal Models at T = 0.5. . . . . . . . . . . . 49
Table 8. Clustering Coefficient of Three Differing Bimodal Models at T = 0.5
for Several Values of Cut-off Distances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
viii
LIST OF FIGURES
Figure 1. A depiction of five polymer chains forming a micelle or aggregate. . . . . . . . . . . . . . .
4
Figure 2. A representation of a polymeric system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
Figure 3. A depiction of a small polymer network containing several aggregates. . . . . . . . . .
9
Figure 4. Three different types of network structure: (a) a wheel structure, (b) a
star structure, (c) a random structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Figure 5. A small example of a network with eight nodes and thirteen links. . . . . . . . . . . . . . . . 12
Figure 6. A small graph consisting of four nodes and five links. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Figure 7. An example of how to calculate the clustering coefficient ci . . . . . . . . . . . . . . . . . . . . . . 15
Figure 8. Probability degree distribution of four ER random graphs with differing values for λ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Figure 9. The clustering coefficient C as a function of hki for the ER random
model and as calculated using Equation 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Figure 10. The high temperature aggregate size probability distributions of the
polymer network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Figure 11. A closer view of the high temperature aggregate size probability
distributions of the polymer network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Figure 12. The low temperature aggregate size probability distributions of the
polymer network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Figure 13. A closer view of the low temperature aggregate size probability
distributions of the polymer network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Figure 14. The high temperature multiple bridge probability distributions of the
polymer network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Figure 15. A closer view of the high temperature multiple bridge probability
distributions of the polymer network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Figure 16. The low temperature multiple bridge probability distributions of the
polymer network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Figure 17. A closer view of the low temperature multiple bridge probability
distributions of the polymer network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Figure 18. The high temperature single bridge probability distributions of the
polymer network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Figure 19. A closer view of the high temperature single bridge probability distributions of the polymer network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
ix
Figure 20. The low temperature single bridge probability distributions of the
polymer network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Figure 21. A closer view of the low temperature single bridge probability distributions of the polymer network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Figure 22. Least squares fit of several temperatures for the multiple bridge distribution. . . 32
Figure 23. The fraction within the A-community to the entire distribution as a
function of temperature for the three types of distributions. . . . . . . . . . . . . . . . . . . . . . . . . . 35
Figure 24. The derivative of the A-community fraction as a function of temperature shows the largest rate of change occurring at approximately T = 0.5. . . . . . . . 35
Figure 25. The multiple bridge distribution in the high temperature range displayed on a log-log scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Figure 26. The multiple bridge distribution in the low temperature range displayed on a log-log scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Figure 27. The clustering coefficient C as a function of temperature for the
polymer network and as calculated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Figure 28. The average clustering coefficient c̄(k) as a function of k on a log-log scale. . . 39
Figure 29. An example of how the rewiring process occurs within a small network. . . . . . . . 41
Figure 30. The clustering coefficient C as a function of number of rewires per
time step. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Figure 31. c̄(k) of the polymer network after being rewired 5000 times per time
step on a log-log scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Figure 32. The clustering coefficient C as a function of temperature. . . . . . . . . . . . . . . . . . . . . . . . 44
Figure 33. An example of a two community network structure consisting of `A
and `B links. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Figure 34. An example of a star-like network structure consisting of two communities with `A and `AB links. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Figure 35. An example of a star-like network structure consisting of two communities with `A , `B , and `AB links. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Figure 36. A representative degree distribution for the three types of bimodal
ER random models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Figure 37. c̄(k) of a bimodal model for T = 0.5 with several differing cutoff
distances shown on a log-log scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
x
ACKNOWLEDGMENTS
There are many people that I need to thank for getting me to this point of my
education. I would like to acknowledge all of the instructors within the department, who have
inspired me through their continuous display of enthusiasm for the respective topic. I would
like to acknowledge my mother for not only helping to fund my education, but also inspiring
me to continue to this level and possibly beyond. I would like to thank my sister for reading
this thesis and helping me to understand how to make it better. To each of these people, I say a
deep hearted, “Thank you.”
Further more, there are three people that directly influenced this research in terms of
my comprehension, contribution, and inspiration. I would like to dedicate this section to those
three people: Dr. Arlette Baljon, Department of Physics, San Diego State University, Dr.
Avinoam Rabinovitch, Department of Physics, Ben Gurion University of the Negev, Israel and
Joris Billen, Ph.D. candidate, Department of Computational Science.
Working on this project has been an invaluable learning experience for me. I can not
express how much I appreciate each of your contributions, suggestions, and directions within
the project. Without you, I would not be here today. Thank You!
1
CHAPTER 1
INTRODUCTION
Many systems throughout nature can be described in terms of complex networks. The
electrical power grid can be described as a complex network of power plants and substations
connected by transmission lines. The Internet is a complex network of computers and routers
connected by wires. We, as individuals, are participants within the complex network of
various social relationships. These systems are just a few examples that have recently
prompted research in the field of network theory. Traditionally studied mathematically within
graph theory, the network properties associated with these systems give insight into the
mechanisms that determine the arrangement and evolution of connections within each of these
networks.
In this chapter, we will give an introduction of the development of studying networks.
We will look at several classes of real-world networks and examples therein. This will then be
followed by a description of a polymer system. We conclude with the purpose of this study.
BACKGROUND
Recently, the focus of network studies has been moving away from small networks to
complex network systems. One of the principle works, which pushed this trend, was a paper
by Watts and Strogatz [1] in 1998. Within this early work, they characterized two global
network properties, which identified and defined a “small-world” network. Following their
work, a flurry of studies on complex networks in a variety of disciplines began.
Complex networks differ from small graphs in several ways. They typically have a
large number of members and dynamically evolve with time. Also, they differ in their
composition, being highly irregular and interconnected. The World Wide Web is just one
example of a highly interconnected, continually growing network with millions of nodes. Due
to these differences, the analytic approaches previously used to study the properties of small
graphs can only describe the local properties of complex graphs. These approaches are no
longer sufficient when the subject of interest is large. For complex networks, consideration of
large-scale statistical properties is necessary.
One of the driving forces behind this new trend of complex network analysis is the
vast availability of computational resources. Modern computers have provided the ability to
analyze data on a scale far greater than previously possible. Only with the resources available
now, can systems of extremely large sizes be studied. The properties of a specific node within
2
a small network can be analyzed visually in graph form. Yet, with these complex networks,
due to their high interconnectivity and multitudes of nodes, it is not possible to visually depict
these graphs. Hence, computers are necessary to analyze the properties of these networks.
Computational resources have also allowed for large-scale data acquisition.
Computers have only recently provided the means to store and gather large quantities of data
in all fields of study. Now, large databases of real-world networks exist and studying the
properties of these physical systems only recently became a realization. With the combination
of computational resources and large, real-world databases, the study of network properties is
heading toward large-scale complex network structure.
R EAL -W ORLD N ETWORKS
Ranging from the Internet, to the electrical power grid, to social relationships, to
protein interactions, there are a multitude of real-world systems that can be described through
a network type of structure. These networks can be constructed of physical connections
between participants, as is the case with the Internet being connected with wires. Or, networks
can be comprised of non-physical connections between participants, as in the friendship
interactions between high school students.
Network structure is everywhere and the desire to describe, characterize and model the
properties associated with these networks has driven studies in a variety of fields. Here, three
categories of real-world networks and studies therein are briefly introduced: transportation,
social, and information networks.
Transportation networks consist of various destinations as nodes within the network.
Modes of transportation from one destination to the next serve as connections between these
nodes. Studying transportation networks can assist in the understanding of the movement of
people around the world resulting in the spread of infectious diseases, the flow of vehicles
resulting in travel times and transportation cost, and the movement of commodities world
wide. These networks have a direct impact on local, national and international economies. Of
the extensive collection of studies based upon transportation networks, a few examples consist
of airports [2], railways [3], highways [4], subways [5] and urban streets [6].
A social network is comprised of individuals or groups of individuals, as participants
within the network. Various relationships or interactions between these individuals serve as
connections between the participants. A few of these social networks that have been studied
in the past include acquaintances [7], sexual interaction [8], professional actors participating
in a common movie [9] and players within a college football season [10]. Within a social
network, information such as trends, diseases, beliefs or fads is passed throughout.
Of the large variety within social networks, acquaintance networks are just one type
that has been highly studied. Acquaintance networks evolve. The number of nodes and
3
connections within the network grows as new friendships form through an individual being
introduced by common friends. Likewise, friendships disappear resulting in a decrease in the
number of nodes and connections within the network. Two examples of studies on this topic
include friendships within high school students [9] and acquaintances of Utah Mormons [11].
One example of an information network is the World Wide Web [12, 13]. Built on top
of the existing Internet network, the documents that comprise the Web form a complex
network structure. Each web page, acting as a node, possesses a number of incoming and also
outgoing links pointing to other web pages. Hyperlinks, directing the user to new documents,
serve as links connecting these nodes.
The World Wide Web represents the largest network for which topological information
is currently available [14]. Estimated at 8 × 108 nodes [15] in July of 1999, this number
continues to grow with the addition of new documents and hyperlinks. Due to this particularly
large number of nodes, the Web has the ability to produce extremely accurate statistical
results when analyzing its network properties.
A few other examples of studies based upon information networks include the network
of citations between academic publications [16], the Internet [17] and a file sharing
peer-to-peer network [18].
A P OLYMER S YSTEM
The polymer system of interest for this study is defined as a telechelic associating
polymer. This type of polymer consists of repeating molecules, comprising the polymer chain.
The ends of the polymer chain are functionalized, and hence possess the ability to attach to
the end of other molecules. This type of polymer has been studied for its ability to form
flower-like micelle or aggregates (Figure 1). Typically, these types of polymers are added to
an aqueous solution. With an increasing concentration of polymers, the viscosity increases
and the solution thickens. These types of polymers have a number of applications and have
been used as thickeners within water-based coatings such as paints, adhesives, and sealants.
The rheological properties of the resultant solution are extremely sensitive to the chemical
composition of the polymer. This fact allows experimentalists to tune the properties of a
polymer solution, and hence broadens the number of potential applications.
The general properties of this type of polymeric system is directly affected by the
concentration of polymers within the solution. At low concentrations, isolated chains exist
without connection to others (Figure 2(a) ). With increasing concentration, the polymeric
chains begin to form flower micelles by making connections at their ends (Figure 2(b) ). As
the concentration of the polymer within the solution continues to increase, the system begins
to self-assemble, forming a polymeric network (Figure 2(c) ). As the number of connections
between micelles increases, the solution finally forms a gel-like structure. This phase
4
Figure 1. A depiction of five polymer chains forming a micelle
or aggregate. Functionalized end groups are the dark color
beads, while the light colored beads construct the polymer
chain joining the end groups.
transition from a fluid-like solution to a gel is regarded as the gel transition [19]. At this
concentration, there is a dramatic slowing of the polymer diffusion and significant increase in
the viscosity, indicating a limited dynamics and flow of the polymer system.
Although many of the experimental studies involving polymers investigate rheological
properties as a function of the concentration of polymers within the system [19, 20], the same
properties are exhibited as a function of temperature while maintaining a specific
concentration. Within one study of a telechelic polymer solution [21], the thermodynamic
property of the specific heat was used to investigate phase transitions within the polymeric
system. With decreasing temperature, they identified the micelle forming phase transition at
T = 31.4 ◦ C, from a rise in the specific heat. At this temperature functionalized end groups
begin forming aggregates. At higher temperatures, the polymer system was shown to be
fluid-like in that unconnected, isolated polymeric chains were suspended throughout the
solution.
P URPOSE OF S TUDY
In this study, we will describe a polymer system in terms of a complex network. To
this end, data from a molecular dynamic simulation of a polymeric system is analyzed. These
polymers have end groups that can attach to each other. Hence, small aggregates can form.
5
Figure 2. A representation of a polymeric system. (a) A low concentration of
polymers within the solution produces isolated chains without connections to
others. (b) With increased polymer concentrations, the polymer system
begins to form flower-like micelle. (c) At high concentrations of polymers, the
system forms a polymer network resulting in a gelation transition.
The end groups also have the ability to detach over time. Within the polymer network,
aggregates serve as nodes and the polymer chains serve as links connecting these nodes. Quite
similar to the social networks described above, the quantity of nodes and links within the
polymer system changes with time as end groups attach and detach.
The polymer system is studied at a range of temperatures. Through an analysis of the
characteristic network properties of the polymer system at differing temperatures, topological
information about the network will be gained. To acquire further insight into this polymer
network structure, a model that mimics these network properties will be created.
We will soon discover that with decreasing temperature, the simulated polymer
network is comprised of two independent communities, as indicated by a bimodal degree
distribution. This property will allow us to describe the polymer network in terms of two
networks with differing hki. Through an analysis of the clustering coefficient C, we will find
that the polymer network is more highly connected than predicted by a theoretical expression
for C. We will then see, by developing our network model, so as to include these two
topologically defining properties, the random network model can not only predict the degree
distribution of the polymer network but also, with the addition of a proximity constraint,
predict the high value for the clustering coefficient.
This study begins with a brief description of the simulated polymer system. This is
followed by a general introduction to network structure and a defining of the characteristic
properties that will be used to investigate the topology of the polymer network: the degree
6
distribution p(k) and the clustering coefficient C. Algorithms to create the Erdös-Rényi (ER)
random model will then be introduced. From here, an analysis of the p(k) and C for the
polymer network will be presented. Based upon these results, the model will then be
developed to mimic these two properties of the polymer network.
7
CHAPTER 2
THE SIMULATED POLYMER SYSTEM
Within this chapter, only a brief description of the molecular dynamic simulation of
the polymer system will be provided. The simulation was implemented and performed in the
group of Dr. Arlette Baljon, Department of Physics, San Diego State University. Further
information regarding the details behind the simulation can be found in [22] and in the
publications of Kremer and Grest [23]. The chapter will continue with previous simulation
results, defining some of the simulated polymer’s known characteristics. This will then be
followed by an explanation of how the polymer system can be defined in terms of a network.
S YSTEM C ONFIGURATION
The molecular dynamic simulation of the polymer system is comprised of a
well-tested model, as developed by Kremer and Grest [23]. The simulation consists of 1000
polymer chains, each eight beads long. Modeled as a bead-spring system, the beads within a
chain are joined under the influence a strong anharmonic potential. Each bead within the
system is subject to a truncated Lennard-Jones (LJ) potential,
UijLJ (r) = 4ε
"
σ
rij
12
−
σ
rij
6
12 6 #
σ
σ
−
+
, r < rc ,
rc
rc
(1)
and zero otherwise. Within the LJ potential, rc = 21/6 , producing only the repulsive
component. This potential provides excluded volume interactions between beads, assuring
that two beads do not occupy the same space at the same time. The interaction energy
between the beads and the temperature provide the dynamics of the system. With each time
step, the radial forces between beads are calculated. The spatial locations of every bead within
the system are then updated according to Newton’s laws of motion. Figure 1 shows a
representation of five polymer chains and how they could possibly arrange spatially.
Within the simulation, polymer chains have the ability to form and break junctions at
their ends. These junction beads, at both ends of a polymer chain, are known as end groups.
Only end groups, never the beads that comprise the chain, can be involved in junctions. At
each twentieth time step, following a Monte Carlo method, the forming and breaking of
junctions occurs with a given probability. The probability is calculated based upon the energy
difference between the old and potentially new state of the polymer.
8
The joining process between end groups creates aggregates within the system.
Defined in terms of the number of end groups joined together, aggregates can form in a
variety of sizes. As an example, within Figure 1, the six dark colored end groups in the center
of the figure form an aggregate. Every end group within the system has the ability to be
involved in multiple junctions, allowing large sized aggregates to form.
The simulation takes place within a cell that maintains a constant volume. The cell
consists of periodic boundary conditions in the two horizontal directions and is confined in the
vertical direction with two solid surfaces.
The polymer system is analyzed through a range of temperatures by thermally cooling
from a high temperature. The system is coupled to thermal bath using the fluctuation
dissipation theorem as described by Grest et al. [23], setting the temperature. At each desired
temperature, the system is allowed to equilibrate for a period of time prior to acquiring data.
Within this study, the temperature is reported in reduced LJ units, making temperature
unitless. The spatial locations of each bead within the system, along with end group junction
data is then gathered for a desired number of time steps.
Gathering data from the molecular dynamic simulation is a time intensive process. For
a given temperature, the time necessary to acquire a statistically accurate quantity of data is on
the order of several months. The data used within this study was provided by Dr. Arlette
Baljon, following a post processing of the resulting spatial bead and junction data.
E ARLIER S TUDIES
Previous work with this simulation [22] has shown that a high temperature fluidity of
the polymer system transitions to an aggregate forming gel in lower temperatures. With
temperature decreasing from the onset at T = 0.75, the properties of the system began to
deviate from those of a liquid. An increased number of junctions between end groups were
observed with lower temperatures, with the maximum rate of change occurring at T = 0.51.
Following an analysis of the energy within the system, it was found that a rapid rise in the
specific heat also occurred at this temperature. With this indication of a phase transition from
a liquid type behavior to a junction forming system, T = 0.51 marked the temperature at
which aggregates most rapidly form. Earlier within Chapter 1, we had seen that, for this type
of polymeric system, this aggregate forming transition occurs at approximately T = 31.4 ◦ C.
Next, an investigation into relaxation times gave indications of a second phase
transition for the simulated polymer system. Occurring at T = 0.3, the dynamics of the
polymer system became limited, in that polymer chains were not mobile. T = 0.3 defined the
temperature at which the system freezes and ceases to move. This transition from a dynamic
to a gel-like behavior is defined as the gelation transition within polymeric systems. For this
9
and lower temperatures, end groups were confined to specific aggregates for long periods of
time, limiting the dynamics within the system.
A P OLYMER N ETWORK
When describing the simulated polymer system in terms of a network, the aggregates
that form through a joining of end groups are considered to be nodes. The polymer chains,
which comprise the entire simulation, serve as links connecting two aggregates. Multiple
connections between two aggregates can occur within the polymer system. A multiple bridge
is therefore defined when two or more polymer chains join the same two aggregates together.
Polymer chains also have the ability to loop back onto themselves, forming a junction
between the end groups of the chain. Loops are defined as a polymer chain, of which its two
end groups are contained within the same aggregate. Figure 3 shows how this polymer
network might arrange spatially. The figure contains examples of single bridges, multiple
bridges and loops.
Figure 3. A depiction of a small polymer network containing
several aggregates. Among the polymer chains are examples of
single bridges, multiple bridges and loops.
The topology of the network associated with the polymer system changes with time,
through the forming and breaking of junctions between end groups. This leads to one of the
difficulties of describing this molecular dynamic simulation in terms of a network. Since the
topology of the system is dynamic, the properties of the network must be analyzed over a
10
multitude of specific configurations. For a given temperature, the data gathered from the
simulation contains a large number of time steps, during which the polymer network changes
its topology from one time step to the next. With each new time step, the properties of the
network have the ability to change considerably. Therefore, all of the network properties
analyzed within this study are statistical averages over multiple configurations of the polymer
network at a given temperature.
11
CHAPTER 3
NETWORK BASICS
A network can take on a multitude of different structures. Ranging from a wheel, a
star, or random, the orientation of links connecting nodes can produce a variety of differing
network properties. Figure 4 is an example of how network structure can vary with the same
number and placement of nine nodes. The links, connecting these nodes, not only define the
topology, they also provide the characteristic properties associated with these types of graphs.
A small set of nodes joined by links is only the simplest type of a network. There are many
ways in which network structure may be more complex than this. This example provides just
three of the possible different structures a network can take on. With variations there of,
networks can quickly become highly complex with size.
Figure 4. Three different types of network structure: (a) a wheel structure,
(b) a star structure, (c) a random structure.
This chapter will begin by introducing the terminology which will be used throughout
the rest of this study. Two characteristic network properties, the degree distribution and the
clustering coefficient will then be presented. This will then be followed by an introduction to
the well-studied Erdös and Rényi (ER) random model [24], including the algorithm for
producing this network model. We will then look at the degree distribution and the clustering
coefficient of this ER random model.
12
D EFINITIONS OF T ERMS
• N ode : A node is an element of which graphs and networks are formed. Nodes
represent points or objects within graphs where connections can be made. In terms of a
transportation network, an intersection on a city street would be considered a node.
Nodes have other synonymous names: vertex, site, or point.
• Link : A link is a connection between two nodes. Links can either be physical or
non-physical connections. The wires connecting computers within the Internet is one
example of a physical link, where as social interactions between people are considered
non-physical links. A link also has other synonymous names: line, connection, or edge.
• M ultipleLinks : Two or more links connecting the same pair of nodes is defined as a
multiple link.
• Loop : A loop is a link, of which its two ends are contained within the same node.
• Degree : The degree of a node is the total number of links incident on that node.
• IsolatedN ode : An isolated node is a node with degree zero. These nodes do not have
connections or links shared with other nodes within the system.
• N etwork : Consisting of nodes and links, a network refers to any interconnected group
or system. In mathematical terms a network is represented by a graph (Figure 5).
Figure 5. A small example of a network with eight nodes and
thirteen links. This figure provides examples of loops, links,
multiple links and isolated nodes.
13
N ETWORK P ROPERTIES
Of the multitude of network properties, two characteristic properties will be used to
compare and contrast the polymer network and the model. Within this study, the degree
distribution and the clustering coefficient will be used to analyze these two networks.
Degree Distribution
The degree distribution is one of the network properties that gives insight into the
topology of the network. This is due to the fact that the arrangement of the links within the
network defines its characteristics. The degree distribution describes how the links throughout
the network are dispersed on a global scale.
The functional form of the degree distribution can be used to give insight into what
class of networks the graph of interest belongs to. For example, as we will soon see, random
graphs possess a degree distribution that is Poisson distributed. For many large real-world
networks as introduced earlier in Chapter 1, their defining characteristic is a significant
deviation from a Poisson distributed degree distribution. Of these real-world network
examples, many of them exhibit a power law degree distribution. Identifying the functional
form of the degree distribution gives insight not only to the topology, but also to how the
network of interest might evolve.
To achieve the degree distribution, we first need to investigate the degree of each node
within a graph. The degree of a node is defined as the number of links incident upon that
node. As an example of this definition, Figure 6 shows a small graph consisting of four nodes
and five links. Node 1 has one link incident upon it, therefore the degree of node 1 is one.
Similarly, node 2 has three links incident upon it, so its degree is three. The determination of
the remainder of the nodes’ degrees within the graph would follow the same process.
Figure 6. A small graph consisting of four nodes and five links.
This graph is used to describe the degree distribution.
14
The degree distribution p(k) is then defined as the probability of finding a node with
degree k within the network. If n(k) is the number of nodes with degree k, then p(k) is
normalized such that the degree distribution is a probability,
p(k) =
n(k)
.
∞
X
n(k)
(2)
k=0
The resulting number of nodes at a specific degree n(k) and the degree distribution
p(k) of our simple graph example can be found in Table 1.
Table 1. The Number of Nodes n(k) at a Specific Degree
k and the Degree Distribution p(k) for the Simple Graph
in Figure 6.
k
1
2
3
4
5
n(k)
1
1
1
1
0
p(k)
0.25
0.25
0.25
0.25
0
Clustering Coefficient
The clustering coefficient is another characteristic property that can give insight into
the topology of a network. The clustering coefficient defines a fractional measure of how
connected the nodes within the network are. A large value for the clustering coefficient is an
indication that the nodes within the network are highly interconnected.
In the context of an acquaintance network, the clustering coefficient can be considered
a measure of cliquishness, in that, it defines the fraction of ones’ friends who are also friends
with each other. Of the real-world examples introduced earlier, the majority show tendencies
of being highly clustered networks, resulting in a relatively large clustering coefficient. This
large value for the clustering coefficient is consistent with the majority of real-world networks.
As first introduced by Watts and Strogatz [1], the clustering coefficient varies in a
range of 0 ≤ ci ≤ 1. The clustering coefficient ci of a node, whose neighbors are fully
connected, has a value of 1, while if none of these neighbors are connected the resulting value
is 0.
Since the clustering coefficient is a measure of the connectedness of the network,
multiple connections between nodes need not be considered. Within the process of analyzing
15
the clustering coefficient for the polymer network, single bridge distributions are only
considered.
The clustering coefficient can be defined as follows. For a distinct node i the
clustering coefficient is given by the ratio of existing links Ei between its ki neighbors to the
possible number of such connections ki (ki − 1)/2, [25] such that the clustering coefficient of
an individual node is defined as
ci =
2Ei
.
ki (ki − 1)
(3)
Figure 7 provides an example of how to calculate the clustering coefficient ci for an
individual node. Within the figure, the node for which ci is being calculated is the solid filled
node. In (a), the filled node possesses three neighbors, as indicated by the links to the unfilled
nodes. These neighbors have no links between them, therefore the clustering coefficient for
the filled node is equal to zero. In (b), again the filled node has three neighbors. Now, two
links exist between these neighbors. Using Equation 3, the clustering coefficient for this
arrangement of links is equal to 2/3. In (c), all possible links between the three neighbors
exist. The clustering coefficient for this arrangement is equal to 1.
Figure 7. An example of how to calculate the clustering coefficient ci . Within
the figure, the filled node is the node for which ci is being calculated. (a)
ci = 0, (b) ci = 2/3, (c) ci = 1.
The local clustering coefficient ci provides information only on a local perspective and
solely describes the properties of single nodes. This local measurement can be extended to
include the global properties of the entire network by defining an average clustering
coefficient as a function of degree
16
c̄(k) =
1 X
ci .
Nk
(4)
i∈Y (k)
Within the definition for c̄(k), Nk are the number nodes with degree k and Y (k) is the set of
all such nodes. An average clustering coefficient c̄ of the network is then related to the degree
distribution p(k) by
c̄ =
X
p(k) c̄(k).
(5)
k
The global property of the clustering coefficient C of the entire network only considers the set
of nodes with degree k > 1, such that
C=
c̄
.
1 − p(0) − p(1)
(6)
The resulting clustering coefficient C is therefore an average of all nodes contained within the
network of degree two and higher.
For an uncorrelated network, in the sense that connection between nodes are not
dependent upon their degree, a theoretical calculation of C can be performed, given hki and
hk 2 i of the degree distribution, where
hki =
∞
X
k p(k)
(7)
k 2 p(k)
(8)
k=0
and
2
hk i =
∞
X
k=0
A strict definition for a network being uncorrelated is when c̄(k) has no dependence on k. In
this case, the theoretical expression for the clustering coefficient is as follows
2
hki hk 2 i − hki
,
c̄(k) = C =
N
hki2
for k > 1 [14, 26, 27].
(9)
17
E RD ÖS -R ÉNYI R ANDOM M ODEL
Within this study, comparisons between the polymer network and a random network
model will be made. The model of interest for this work is an Erdös and Rényi random
network. Two equivalent algorithms for producing this model will be introduced within this
section.
The realization of the following network model is accomplished through an adjacency
matrix. Defined as a N × N matrix, the adjacency matrix describes the connectivity of N
nodes within a network. The entries aij contain the number of links between node i and j,
where the diagonal elements aii are representative of a link looping back onto the same node.
The adjacency matrix is symmetric. With the existence of a link between nodes i and j, it
follows that there is a link between nodes j and i, thus aij = aji . When investigating the
properties of these models, adjacency matrices make the assembly relatively simple, in that
large ensembles of networks can be conceived.
With the intention of studying the properties of graphs with increasing number of
random connections, Erdös and Rényi proposed a model to generate a random graph with N
nodes and ` links. The procedure they used to generate this model begins with a fixed number
of disconnected nodes N . Then, connections between pairs of randomly selected nodes are
formed. Multiple links between nodes are prevented, such that within the adjacency matrix
aij = 0 or 1. Loops are also prevented from forming, such that aii = ajj = 0. The random
connection process continues until the total number of links within the graph equals `. Since
their original work in 1959 [24], this model has been known as an Erdös-Rényi (ER) random
graph. Within this study, this method for producing an ER random network will be used.
The equivalent and alternative method for creating an ER random graph consists of
connecting pairs of nodes with a given probability. Beginning with a fixed number of
disconnected nodes N , a link connecting a pair of nodes is independently formed with a
probability P and chosen from the N (N − 1)/2 possible links within the graph. Likewise,
within this method, multiple links and loops are not allowed to form.
The input parameter to the model is the average degree hki of the resulting network.
Starting with a arbitrary number of nodes N , the number of links ` is calculated, such that the
model produces the desired hki by
2` = hkiN.
(10)
The probability of a connection forming between two nodes can then be determined, such that
18
P =
2`
hki
=
.
N (N − 1)
N −1
(11)
After probabilistically attempting to form connections through all the N (N − 1)/2
possible links, the total number of links ` and average degree hki of the model will vary. Due
to the probabilistic nature of making connections, only within an average of an ensemble of
ER random graphs will the number of existing links be equal to `. Therefore, within the
statistical average of an ensemble of graphs hki will be equal to the input parameter.
For both methods of producing an ER random network, in the limit of large N , the
degree distribution p(k) of the resulting model is distributed according to a Poisson
distribution [26] with a peak at λ = hki [14], such that
p(k) =
λk e−λ
.
k!
(12)
Network Properties of the Model
Using the methodology for creating an ER random network as described in the
previous section, verification that they possess Poisson distributed degree distributions with
their peak hki = λ was performed. Ensembles of ER random networks were created for four
values of hki. Figure 8 shows the resulting degree distributions of the random networks
plotted as symbols. Normalized Poisson distributions with corresponding values of λ are
plotted as dashed lines. It is observed that the degree distributions of the models are
well-predicted by a Poisson distribution with a corresponding value for λ.
One of the most important features to notice about the degree distribution of the
random networks is the change in the peak location hki. A low ratio of links to nodes is
necessary to produce a model possessing a small hki. Likewise, a large ratio of links to nodes
results in a larger value for hki. As this ratio of links to nodes decreases, the resulting degree
distribution shifts to lower values of k.
When investigating the clustering coefficient C for the random model, it is observed
(Figure 9) that with increasing hki the network model increases in connectivity, as indicated
by increasing values of C. When using Equation 9 to calculate the value of C, it is observed
that the clustering coefficient of the ER random model is well-predicted by the calculation for
a range of hki. By definition, ER random graphs are uncorrelated, since links between nodes
are formed regardless of their degree [28], and are therefore subject to Equation 9.
19
0.8
λ=6
λ=4
λ=2
λ = 0.5
0.7
0.6
p(k)
0.5
0.4
0.3
0.2
0.1
0
0
2
4
8
6
10
12
14
k
Figure 8. Probability degree distribution of four ER random
graphs with differing values for λ.
0.08
ModelSimulated
Calculated
C
0.06
0.04
0.02
0
1
2
3
4
5
6
7
<k>
Figure 9. The clustering coefficient C as a function of hki for
the ER random model and as calculated using Equation 9.
8
20
Using the results from the degree distribution and the clustering coefficient of the
model, generalizations about network topology can be made. We have found that a network
consisting of a small ratio of links to nodes produces a degree distribution with a low valued
hki and a low valued clustering coefficient. With an increasing ratio of links to nodes the
degree distribution shifts to higher values of hki. Also, as indicated by the increasing values
for C, a network becomes more highly connected with increasing values of hki.
21
CHAPTER 4
NETWORK PROPERTIES OF THE POLYMER
In this chapter, the degree distribution and the clustering coefficient of the simulated
polymer network will be investigated. These two defining network properties will then be
compared to that of the random model.
D EGREE D ISTRIBUTION
Once the polymer system reaches a statistically stationary state for a given
temperature, one of the characteristic measures is the degree distribution. Within the system,
not all the nodes have the same number of links. This variation in the degree of nodes is
characterized by the degree distribution p(k). Of the real-world networks introduced earlier in
Chapter 1, many possess degree distributions that differ greatly from a random Poissonian
distribution. It is therefore, not only of interest to identify the p(k) of the polymer network,
but also to compare it to that of the random model’s degree distribution. This comparison will
give insight to what degree the polymer network can be described by a random model.
Three different types of distributions are analyzed: aggregate, multiple bridge, and
single bridge. Aggregate distributions are defined in terms of the number of end groups
contained within the aggregate. The distribution n(k) is a count of the aggregate sizes present
for a given temperature. Bridge distributions are defined in terms of the number of polymer
chains incident upon an aggregate. This definition is consistent with the network definition for
the degree of a node. The polymer network contains not only single connections, but also
multiple connections between two given aggregates. This property introduces two different
types of bridge distributions. A multiple bridge distribution n(k) is a count of the number of
aggregates possessing k incident polymer chains for a given temperature. A single bridge
distribution n(k) has a similar definition, yet within this distribution, any multiple connections
between two aggregates are solely counted as a single connection. For both of the two bridge
distributions, loops are not included in the count of a node’s degree.
Due to the fact that the polymer system is a molecular dynamic simulation, hence
changing with time, the resulting distributions are time averages for the system. Each
individual time step is analyzed, keeping a running total for counts of aggregate or bridge
sizes. Then, after completing the total number of time steps for a given temperature, averages
are calculated as a function of aggregate size or bridge quantity, respectively. Within the
temperature range of T = 0.3 to 0.5, approximately 10, 000 time steps of data were averaged.
22
For temperatures greater than T = 0.5, less data was necessary to determine statistically
consistent distributions. Approximately 4, 000 time steps of data were averaged within the
higher temperature range.
Aggregate Distribution
Using the definition for aggregates as introduced earlier within Chapter 2, the
distribution function p(k) describes the size of aggregates within the system. For this
distribution, there are several key ideas to be noted. The two end groups of a looped polymer
chain are included in the overall size of an aggregate. Next, a free end group, lacking a
junction with another, is counted as an aggregate of size one. Lastly, a chain where both end
groups lack junctions is counted as two aggregates of size one.
Figure 10 through Figure 13 contain the resulting aggregate size probability
distributions p(k) for a few representative temperatures within the spectrum. Figure 10 and
Figure 11 are graphs of the same data. Figure 11 has been enlarged to show a more detailed
view by not including the first data point. Similarly, Figure 13 is an enlarged version of
Figure 12.
0.8
T = 2.2
T = 1.5
T = 1.0
T = 0.8
T = 0.6
0.7
0.6
p(k)
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
20
k
Figure 10. The high temperature aggregate size probability
distributions of the polymer network.
23
0.2
T = 2.2
T = 1.5
T = 1.0
T = 0.8
T = 0.6
p(k)
0.15
0.1
0.05
0
0
10
5
20
15
k
Figure 11. A closer view of the high temperature aggregate size
probability distributions of the polymer network. This figure
contains the same data as in Figure 10. A more detailed view is
achieved by not including the first data point.
0.4
T = 0.55
T = 0.5
T = 0.45
T = 0.4
p(k)
0.3
0.2
0.1
0
0
5
10
20
15
25
30
k
Figure 12. The low temperature aggregate size probability
distributions of the polymer network.
35
24
T = 0.55
T = 0.5
T = 0.45
T = 0.4
p(k)
0.1
0.05
0
0
5
10
20
15
25
30
35
k
Figure 13. A closer view of the low temperature aggregate size
probability distributions of the polymer network. This figure
contains the same data as in Figure 12. A more detailed view is
achieved by not including the first data point.
At high temperatures, the distribution shows an apparent initial drop-off with the
height of the distribution decreasing with decreasing temperatures. In the temperature range
of T = 0.6 to 0.5, there is an onset of a well-defined secondary peak for larger aggregate sizes.
Multiple Bridge Distribution
Using the definition for a multiple bridge as introduced earlier within Chapter 2, the
distribution function p(k) describes the number of polymer chains incident upon an aggregate.
Chains that loop back onto the same aggregate are not counted within this distribution.
Figure 14 through Figure 17 contain the resulting multiple bridge probability
distributions p(k) for a few representative temperatures within the spectrum. Figure 14 and
Figure 15 are graphs of the same data. Figure 15 has been enlarged to show a more detailed
view by not including the first data point. Similarly, Figure 17 is an enlarged version of
Figure 16.
25
T = 2.2
T = 1.5
T = 1.0
T = 0.8
T = 0.6
p(k)
0.6
0.4
0.2
0
0
5
10
20
15
k
Figure 14. The high temperature multiple bridge probability
distributions of the polymer network.
0.2
T = 2.2
T = 1.5
T = 1.0
T = 0.8
T = 0.6
p(k)
0.15
0.1
0.05
0
0
5
10
15
k
Figure 15. A closer view of the high temperature multiple
bridge probability distributions of the polymer network. This
figure contains the same data as in Figure 14. A more detailed
view is achieved by not including the first data point.
20
26
T = 0.55
T = 0.5
T = 0.45
T = 0.4
p(k)
0.3
0.2
0.1
0
0
5
10
15
20
25
k
Figure 16. The low temperature multiple bridge probability
distributions of the polymer network.
T = 0.55
T = 0.5
T = 0.45
T = 0.4
p(k)
0.1
0
0
5
10
15
20
25
k
Figure 17. A closer view of the low temperature multiple bridge
probability distributions of the polymer network. This figure
contains the same data as in Figure 16. A more detailed view is
achieved by not including the first data point.
27
Similar to the aggregate distribution, at high temperature the distributions exhibit an
initial drop-off, with the height of the distribution decreasing with lower temperatures. In the
temperature range of T = 0.6 to 0.5, we find the onset of a well-defined secondary peak for
larger degrees.
Single Bridge Distribution
Using the definition for a single bridge as introduced earlier within Chapter 2, the
distribution function describes the number of chains incident upon an aggregate. Yet, with this
distribution, multiple chains connecting the same two aggregates are counted solely as a
single bridge. Once again, chains that loop back onto the same aggregate are not counted
within this distribution.
Figure 18 through Figure 21 contain the resulting single bridge probability
distributions p(k) for a few representative temperatures within the spectrum. Figure 18 and
Figure 19 are graphs of the same data. Figure 19 has been enlarged to show a more detailed
view by not including the first data point. Similarly, Figure 21 is an enlarged version of
Figure 20.
T = 2.2
T = 1.5
T = 1.0
T = 0.8
T = 0.6
p(k)
0.6
0.4
0.2
0
0
10
5
k
Figure 18. The high temperature single bridge probability
distributions of the polymer network.
15
28
0.2
T = 2.2
T = 1.5
T = 1.0
T = 0.8
T = 0.6
p(k)
0.15
0.1
0.05
0
0
10
5
15
k
Figure 19. A closer view of the high temperature single bridge
probability distributions of the polymer network. This figure
contains the same data as in Figure 18. A more detailed view is
achieved by not including the first data point.
T = 0.55
T = 0.5
T = 0.45
T = 0.4
p(k)
0.3
0.2
0.1
0
0
10
5
15
k
Figure 20. The low temperature single bridge probability
distributions of the polymer network.
29
0.2
T = 0.55
T = 0.5
T = 0.45
T = 0.4
p(k)
0.15
0.1
0.05
0
0
10
5
15
k
Figure 21. A closer view of the low temperature single bridge
probability distributions of the polymer network. This figure
contains the same data as in Figure 20. A more detailed view is
achieved by not including the first data point.
Similar to the previous two types of distributions, at high temperatures the single
bridge distributions exhibit an initial drop-off, with the height of the distribution decreasing
with lower temperatures. In the temperature range of T = 0.6 to 0.5, we again find the onset
of a well-defined secondary peak for larger degrees.
When comparing the two bridge distributions within the low temperature range, it is
observed that the multiple bridge distributions (Figure 17) have a higher preferential degree
than the single bridge distributions (Figure 21). As shown by the secondary peak location, the
value of the preferential degree is approximately 13 for the multiple bridge distributions,
while approximately 8 for the single bridge distributions. This can be explained by the
difference in the method of counting the polymer chains incident upon an aggregate. For the
single bridge distributions, multiple chains between two aggregates are only counted as a
single. Consequently, single bridge distributions do not contain as many large degrees as the
multiple bridge distributions. Hence, when looking at the differences between these two
distributions, a general shift in the secondary peak locations to lower values of k for the single
bridge distributions is expected and observed.
When comparing the aggregate distributions (Figure 13) to the bridge distributions
(Figure 17) in the low temperature range, it is observed that the aggregate distributions have a
30
larger preferential size than the bridge distributions. Again, as shown by the location of the
secondary peak, the preferential degrees for the bridge distributions are approximately 8 and
13, respectively, while for the aggregate distributions this value is approximately 16. This can
be explained in that aggregates contain loops. Aggregate distributions track overall size,
including the end groups which form loops. This is not the case for bridge distributions, as
they do not contain loops. Therefore, a general shift in the peak location to larger sizes for
aggregate distributions is expected and observed.
We have discovered earlier in Chapter 3, that an ER random model producing a small
hki is the result of a sparsely connected network and is Poisson distributed. Likewise, an ER
random model producing a large hki is the result of a highly connected network and is also
Poisson distributed.
When investigating each of the three types of degree distributions for the polymer
network, in the high temperature range, a distribution producing a small valued hki exists. In
the lower temperature range, the degree distribution exhibits a bimodal nature, in that, two
distributions exist. One distribution, the initial drop-off, has a low hki, while the second
distribution, the secondary peak, has a larger value for hki.
This fact gives rise to the idea that the bimodal degree distributions, as seen in the low
temperature range of the polymer network, consist of two random networks possessing
different values for hki. These two networks form communities within the system. One
community (B-community) consists of a low ratio of links to nodes and forms the initial
drop-off. The second community (A-community) consists of a higher ratio of links to nodes
and produces the secondary peak centered at larger values of k.
Within the lowest temperatures of the polymer system, as seen within the degree
distributions, the data exhibits large amounts of noise. This noisy data can be attributed to two
factors: bad statistics and a slowing of dynamical motion within the low temperature range.
The quantity of data within the low temperature range currently consists of approximately ten
times the data of that at the higher temperatures. This amount of data is not sufficient to
provide statistically accurate results when looking at the degree distribution. At the lowest
temperatures the dynamics of the polymer system has slowed, such that the network does not
achieve a new configuration as often as at higher temperatures. To achieve more statistically
accurate degree distributions it would be necessary to gather significantly more simulation
data. The resulting degree distributions were reported as such, including the noise contained
in the low temperatures, due to the time necessary to gather the simulation data. Several more
months of continually running the simulation might provide the necessary quantity of data for
statistically accurate low temperature degree distributions.
31
Least-Squares Curve Fits of the Distributions
For each of the three types of polymer distributions, least squares fits were performed
using Excel to investigate the key features of the degree distributions. To serve as a
comparison to a random model and while maintaining the idea of two communities, a two
Poisson distribution was used as the fitting function,
A0
hkB ik e−hkB i
hkA ik e−hkA i
+ A1
.
k!
k!
(13)
One difference between the polymer network and the method behind creating the ER
random model originates in the creation of the model. The random model begins with
unconnected nodes and then links are formed randomly. The polymer system does not begin
with unconnected aggregates. The fact that the simulation consists of polymer chains assures
that each aggregate or node begins with a connection to another aggregate. In order to
compare the polymer network to a random model, aggregates of size one and bridges with
degree one are not included when curve fitting the distributions.
It is observed (Figure 22) for a range of temperatures, that a two Poisson distribution
predicts the degree distribution of the polymer network. Within the figure, the multiple bridge
degree distribution is reported as symbols, while the corresponding two Poisson distributions
are displayed as dashed lines.
By looking at the individual Poisson contributions to the overall fit, the percentage of
bridges or aggregates per community can be assessed. Using the A1 and hkA i parameters,
pA (k) is generated such that
pA (k) = A1
hkA ik e−hkA i
.
k!
(14)
The value of
A1 =
Z
∞
pA (k),
(15)
k=0
is the resulting magnitude of the A-community distribution. Similarly, the same is done for
B-community using the A0 and hkB i fitting parameters. From here, the magnitudes of both of
the community contributions are normalized, such that A0 + A1 = 1. The normalized value
of A1 is the fraction of bridges or aggregates contained within the A-community. Likewise,
the normalized value of A0 is the fraction of bridges or aggregates within the B-community.
32
Polymer T = 1.8
Polymer T = 1.0
Polymer T = 0.7
Polymer T = 0.55
Polymer T = 0.45
p(k)
0.15
0.1
0.05
0
10
5
20
15
k
Figure 22. Least squares fit of several temperatures for
the multiple bridge distribution. The symbols within the
figure indicate the polymer distribution, while the
dashed lines correspond to the two Poisson distribution.
Along with these independent community percentages, the correlation coefficient,
defined as
1
n
ρxy =
n
X
(xi − x̄)(yi − ȳ)
i=1
σx σy
,
(16)
was calculated as a quantitative measure of the accuracy of the two Poisson fit x to the degree
distribution of the polymer network y. The resulting fitting parameters and correlation
coefficients for the polymer system follow in Tables 2, 3 and 4.
It is observed from the fitting parameter data that the respective peak locations hkA i
and hkB i follow systematic trends with temperature. For each of the three types of
distribution, hkA i increases in value with decreasing temperature, reaching a maximum value
at approximately T = 0.4. hkB i initially increases followed by a decreasing in the lower
temperatures.
The magnitudes of the two individual Poisson distributions (A0 and A1) also show
systematic trends with decreasing temperature. The B-distribution (A0) decreases until almost
completely disappearing at the lowest temperatures while, the A-distribution (A1) continues
to increase with decreasing temperature.
33
Table 2. Least Squares Fitting Parameters of the
Aggregate Distribution for the Polymer Network Using a
Two Poisson Fitting Function.
Temp
3.5
3.0
2.6
2.2
1.8
1.5
1.0
0.8
0.7
0.6
0.55
0.5
0.45
0.4
0.35
0.3
A0
0.9181
0.9107
0.8986
0.8861
0.8669
0.8397
0.7779
0.7036
0.6334
0.4907
0.3506
0.1915
0.0858
0.0251
0.0039
0.0011
hkB i
0.5439
0.6089
0.6350
0.7196
0.7824
0.8152
1.1971
1.4871
1.7308
2.0441
2.0363
1.6877
1.1100
0.9271
0.9104
0.9081
A1
0.0819
0.0893
0.1014
0.1139
0.1331
0.1603
0.2221
0.2964
0.3666
0.5093
0.6494
0.8085
0.9142
0.9749
0.9961
0.9989
hkA i
2.0559
2.2683
2.3949
2.6548
2.9319
3.1571
4.6214
5.9130
7.2467
9.8073
11.8226
13.9454
16.4455
19.8509
20.5192
20.0064
Corr Coeff
1.0000
1.0000
1.0000
0.9999
0.9999
0.9999
0.9996
0.9982
0.9950
0.9857
0.9860
0.9969
0.9989
0.9467
0.9245
0.8270
Table 3. Least Squares Fitting Parameters of the
Multiple Bridge Distribution for the Polymer Network
Using a Two Poisson Fitting Function.
Temp
3.5
3.0
2.6
2.2
1.8
1.5
1.0
0.8
0.7
0.6
0.55
0.5
0.45
0.4
0.35
0.3
A0
0.9139
0.9004
0.8935
0.8788
0.8648
0.8403
0.7765
0.7012
0.6278
0.4822
0.3478
0.2021
0.1153
0.0546
0.0000
0.0000
hkB i
0.5071
0.5400
0.5794
0.6443
0.7335
0.7917
1.0906
1.3512
1.5849
1.7974
1.7960
1.4322
0.8085
0.5016
0.0000
0.0000
A1
0.0861
0.0996
0.1065
0.1212
0.1352
0.1597
0.2235
0.2988
0.3722
0.5178
0.6522
0.7979
0.8847
0.9454
1.0000
1.0000
hkA i
1.8296
1.9507
2.1159
2.3143
2.6267
2.8820
4.0817
5.1909
6.2864
8.0355
9.3375
10.7004
12.3099
14.2069
14.5906
14.2888
Corr Coeff
1.0000
1.0000
1.0000
1.0000
1.0000
0.9999
0.9998
0.9993
0.9981
0.9955
0.9958
0.9991
0.9988
0.9843
0.9704
0.9355
34
Table 4. Least Squares Fitting Parameters of the Single
Bridge Distribution for the Polymer Network Using a
Two Poisson Fitting Function.
Temp
3.5
3.0
2.6
2.2
1.8
1.5
1.0
0.8
0.7
0.6
0.55
0.5
0.45
0.4
0.35
0.3
A0
0.9126
0.8993
0.8898
0.8759
0.8630
0.8317
0.7743
0.6973
0.6201
0.4753
0.3962
0.2228
0.0091
0.0000
0.0000
0.0000
hkB i
0.4989
0.5325
0.5533
0.6206
0.7163
0.6972
1.0453
1.2656
1.4021
1.3869
0.9970
0.8298
0.9837
0.0000
0.0000
0.0000
A1
0.0874
0.1007
0.1102
0.1241
0.1370
0.1683
0.2257
0.3027
0.3799
0.5247
0.6038
0.7772
0.9909
1.0000
1.0000
1.0000
hkA i
1.7954
1.9151
2.0377
2.2378
2.5509
2.6678
3.8668
4.7843
5.5441
6.5522
7.1315
7.7650
8.3561
8.6199
8.7320
8.7197
Corr Coeff
1.0000
1.0000
1.0000
1.0000
1.0000
0.9999
0.9999
0.9996
0.9993
0.9994
0.9997
0.9905
0.9757
0.9755
0.9462
0.9594
As shown by the resulting correlation coefficients, the accuracy of a two Poisson
distribution in predicting the degree distribution of the polymer network continues to increase
in accuracy with increasing temperature.
The fraction of bridges or aggregates within the A-community A1 as a function of
temperature (Figure 23) not only shows systematic trends, but nearly identical results for each
of the three types of distributions. At low temperatures almost all the aggregates and bridges
are contained within the A-community. With increasing temperature, this fraction quickly
decreases. At the highest temperature, the A-community contains approximately only 8
percent of the aggregates or bridges.
The largest rate at which this community fraction changes (Figure 24) can be easily
seen to occur at approximately T = 0.5. This temperature is in agreement with a previous
thermodynamic study on the same polymer system [22] in which a peak in the specific heat at
T = 0.51 marked an aggregate forming phase transition. At this temperature, an increased
number of junctions between end groups results in a deviation from fluid-like dynamics to an
aggregate forming system. The same transition is observed and identified using network
theory. When analyzing the degree distributions, the transition is observed through the onset
of the secondary peak at T = 0.5, and through the rate at which the A-community becomes
prevalent.
35
1.1
1
Aggregate
Multiple Bridge
Single Bridge
0.9
A-Fraction
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
3
2.5
4
3.5
Temperature
Figure 23. The fraction within the A-community to the entire
distribution as a function of temperature for the three types of
distributions.
A-Fraction Derivative
0
Aggregate
Multiple Bridge
Single Bridge
-1
-2
-3
-4
0.5
1
1.5
2
2.5
3
3.5
Temperature
Figure 24. The derivative of the A-community fraction as a
function of temperature shows the largest rate of change
occurring at approximately T = 0.5.
36
Tails of the Polymer Distributions
Earlier in Chapter 1, several real-world networks were introduced. In the studies
therein, many of these networks were shown to possess degree distributions that differed
greatly from the Poisson distribution associated with a random network. As an example, scale
free networks are known to possess power law p(k) ∼ k −γ distributed tails. This property can
be easily identified through a linear trend on a log-log plot of the degree distribution. To
examine to what extent the degree distribution of the polymer network differs from a Poisson
distribution, log-log plots of both the low and high temperature were analyzed.
Figure 25 displays three of the high temperature multiple bridge distributions for the
polymer network. Within the figure, the degree distribution is displayed as symbols, while the
bimodal two Poisson distribution is shown as a dashed line. The bimodal model predicts the
small k value behavior well. Yet, in the tail, the two Poisson distribution drops off faster than
the polymer network data. As can be seen in the figure, a lack of a linear trend indicates the
polymer network does not possess a power law distributed degree distribution at these high
temperatures.
In the low temperature range (Figure 26), the log-log scale emphasizes the
inaccuracies of the two Poisson distribution in predicting the multiple bridge distribution of
the polymer. Within the figure, the polymer distribution is displayed as symbols and the
Poisson distribution is shown as dashed lines. It is observed that the two Poisson distribution
initially predicts the behavior of the distribution well. For higher values of k, it drops off
faster than the polymer network. The tail of the polymer distribution at T = 0.5 appears to be
fit well by a power law, displaying a known property of a scale free network. The single
bridge distributions were also analyzed and observed to display similar properties.
As seen in these two figures, it is interesting to identify that the polymer network
transitions through a degree distribution that is somewhat consistent with a two Poisson
distribution in the high temperature range to a distribution producing a tail consistent with a
scale free network at T = 0.5. This property, of having a scale free tail, is identifying in that
the molecular dynamic polymer system is exhibiting preferential attachment when forming
connections between end groups. This is to say that the probability of connections forming
between end groups increases with the size of the aggregates involved. Therefore in the low
temperature range, small sized aggregates have a small probability of forming new
connections with other polymer chains, while larger sized aggregates have a large probability.
37
Polymer T = 2.2
Polymer T = 1.5
Polymer T = 1.0
p(k)
0.01
0.0001
1e-06
2
4
8
16
k
Figure 25. The multiple bridge distribution in the high
temperature range displayed on a log-log scale. Within the
figure, the polymer distributions are displayed as symbols and
the two Poisson distributions are shown as dashed lines.
0.1
p(k)
0.01
0.001
0.0001
1e-05
Polymer T = 0.55
Polymer T = 0.5
Polymer T = 0.45
Power-Law
4
16
k
Figure 26. The multiple bridge distribution in the low
temperature range displayed on a log-log scale. Within the
figure, the polymer distributions are displayed as symbols and
the two Poisson distributions are shown as dashed lines.
38
C LUSTERING C OEFFICIENT
When investigating the clustering coefficient C of the polymer network, it is observed
(Table 5 and Figure 27) that with decreasing temperature, the value of the polymer’s clustering
coefficient increases dramatically, reaching a value of 0.35 at T = 0.3. With decreasing
temperature, a deviation of the value for C from calculated becomes apparent at T = 1.5.
This trend continues and becomes increasingly apparent in the low temperature range.
The large values of C for the polymer network in the low temperature range indicates
a highly interconnected network. The decreasing values of C with increasing temperature
indicates that the polymer network becomes less interconnected in the high temperature range.
It is also observed (Figure 28), that c̄(k) for the polymer is dependent on k. This
functional dependence on c̄(k) is an indication of degree correlation within the polymer
network. Hence, the theoretical expression for C, which is only valid for uncorrelated
networks, should not be expected to be in agreement with experimental values for the polymer
network.
0.4
PolymerSimulated
PolymerCalc
C
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
Temperature
Figure 27. The clustering coefficient C as a function of
temperature for the polymer network and as calculated.
3
39
Table 5. The Average Clustering Coefficient C for the
Polymer Network and as Calculated.
T emp
3.0
2.2
1.8
1.5
1.0
0.8
0.7
0.6
0.55
0.5
0.45
0.4
0.35
0.3
C
0.00242
0.00405
0.00405
0.00833
0.02290
0.04698
0.07716
0.13381
0.17368
0.21505
0.25632
0.28275
0.29376
0.35108
CCalc
0.00024
0.00042
0.00062
0.00091
0.00277
0.00636
0.01136
0.02235
0.03136
0.04269
0.05702
0.06999
0.07372
0.10033
T = 0.8
T = 0.7
T = 0.6
T = 0.5
0.25
c(k)
0.125
0.0625
0.03125
2
4
8
16
k
Figure 28. The average clustering coefficient c̄(k) as a function
of k on a log-log scale. The dependence of the c̄(k) on k
indicates that the polymer network is correlated.
40
P ROXIMITY W ITHIN THE P OLYMER N ETWORK
We have seen (Chapter 3), for an ER random network, C is well-predicted by
Equation 9. We have also seen in the previous section, that the polymer network is highly
interconnected at the lowest temperatures and that C cannot be accurately predicted by this
same equation. What are the differences between the ER random network and the polymer
network that produce the high valued clustering coefficient as seen within the polymer
network? In an effort to answer this question, we will now investigate one of the fundamental
difference between these two networks: proximity.
One of the differences between the ER random model and the polymer network is the
spatial dependence associated with the proximity between aggregates within the system. Only
aggregates that are within a certain distance can form connections. This is due to the fact that
the length of a polymer chain remains fixed. This maximum connection distance between
aggregates is limited, on average, by the length of a stretched polymer chain. The ER random
network does not have this same limitation of proximity. Nodes are connected randomly and
independent of their separation distance.
To explore how proximity affects the clustering coefficient within the polymer
network, a method to randomly rewire the network was implemented. The rewiring process
alters the adjacency matrix of the polymer network a given number of times, at each time step.
The procedure is as follows. Two polymer chains are randomly selected, while confirming
that they do not have an aggregate in common and are not direct neighbors. If these necessary
criteria are not met, then the random selection process continues until they are. The chains,
which connect the respective end groups, are then broken, followed by a forming of new
connections between the opposing unconnected end groups. This rewiring process not only
maintains the existence of solely single bridges within the system, but also preserves the
degree distribution of every aggregate involved.
As an example of this process (Figure 29), two polymer chains are chosen at random
(a), assuring that they do not possess an aggregate in common and are not direct neighbors.
Within the figure, chains 1-2 and 3-4 are randomly selected. The connection between the
respective end groups is then broken (b). Finally, new connections between the opposing end
groups are then formed (c), producing new chains 2-4 and 1-3.
This process was then performed with increasing numbers of rewires per time step, so
as to observe how C is altered with the number of changes made to the network. The average
clustering coefficient C of the rewired polymer network was then calculated.
41
Figure 29. An example of how the rewiring process occurs within a small
network.
It is observed (Figure 30), that dramatic changes in C occur within the first few hundred
rewires per time step. The highly clustered property of the polymer network quickly decreases
and converges to a consistent value. This trend of dramatically decreasing values of C with
increasing numbers of rewires was consistent throughout the temperature spectrum. Data was
gathered up to 10, 000 rewires per time step, in 500 increments. The consistent rewired value
for C was then taken as the average of the last ten increments.
0.4
T = 0.7
T = 0.5
T = 0.3
0.35
0.3
C
0.25
0.2
0.15
0.1
0.05
0
0
500
1000
1500
2000
Number of rewires
Figure 30. The clustering coefficient C as a function of number
of rewires per time step.
42
Through the random selection and reconnection of chains, this method removes the
spatial dependence associated with proximity as the number of changes within the network
increases. Due to the fact there is no consideration as to the distance between opposing end
groups, newly formed connections between aggregates do not possess a dependence upon the
length of a stretched polymer chain. Following 5000 rewires per time step, it is observed
(Figure 31) that c̄(k) shows a lack of dependence on the degree k.
0.1
c(k)
T = 1.0
T = 0.8
T = 0.6
T = 0.4
0.01
2
4
8
16
k
Figure 31. c̄(k) of the polymer network after being rewired 5000
times per time step on a log-log scale. The constant value of c̄(k)
with k shows indications of the network being uncorrelated.
For T = 1.0 at higher values for k, it is observed that there are deviations from this
consistent value. This is most likely due to poor statistics and the small number of these larger
sized aggregates in the higher temperature range. The relatively consistent value of c̄(k) with
k is evidence that the network is uncorrelated after the rewiring process, indicating
connections between nodes are independent of their degree.
Due to the uncorrelated nature of the rewired polymer network, with a significant
number of changes to the polymer network, a similar clustering coefficient to that as
calculated by Equation 9 is achieved. As seen when comparing the calculated clustering
coefficient of the polymer network CCalc and that of the rewired polymer network CRewire
43
(Table 6), the average clustering coefficient of the rewired network is in close agreement to
the calculated value.
As a qualitative measure of how close these two values are, the percent difference
between CRewire and CCalc was calculated as,
| CRewire − CCalc |
× 100.
(CRewire + CCalc )/2
(17)
Table 6 contains the average clustering coefficient of the polymer network C, the clustering
coefficient as calculated CCalc , and the clustering coefficient of the rewired polymer network
CRewire . Figure 32 contains a graph of C, CRewire and CCalc as a function of temperature.
Table 6. Three Average Clustering Coefficients for the Polymer Network:
Simulated, Calculated, and After Rewiring. The Rewired Network Produces
a Similar Value for C as to the Calculated Value.
T emp
N
hki hk 2 i
C
CCalc
CRewire %Diff
3.0
1491.9 1.31 2.20 0.002421 0.000241 0.000230 4.97
2.2
1383.1 1.40 2.65 0.004053 0.000421 0.000396 6.16
1.8
1302.3 1.47 3.07 0.004053 0.000617 0.000603 2.32
1.5
1213.6 1.56 3.61 0.008334 0.000909 0.000908 0.15
1.0
935.7 1.94 6.27 0.022895 0.002774 0.002666 3.97
0.8
714.7 2.38 10.19 0.046977 0.006358 0.006153 3.28
0.7
557.9 2.83 14.85 0.077158 0.011362 0.011095 2.38
0.6
374.7 3.69 24.23 0.133811 0.022353 0.022173 0.81
0.55
283.3 4.45 32.38 0.173682 0.031361 0.031586 0.72
0.5
204.5 5.53 43.98 0.215055 0.042692 0.043413 1.67
0.45
146.4 6.90 59.30 0.256318 0.057016 0.058060 1.81
0.4
112.6 7.87 69.85 0.282753 0.069992 0.071893 2.68
0.35
101.5 8.26 73.26 0.293757 0.073718 0.075683 2.63
0.3
85.2
8.93 87.02 0.351076 0.100330 0.104713 4.28
The process of rewiring the polymer network has shown that the high connectivity,
resulting in a large value for the clustering coefficient C, is primarily due to the spatial
dependence associated with the required proximity necessary for a connection between two
aggregates. The act of rewiring connections between aggregates produces an uncorrelated
network which has a value for C close to agreement with the theoretical expression
(Equation 9). The differences within these two values can be attributed to the fact that the
polymer network possesses a bimodal degree distribution. Equation 9 is only valid for a single
Poisson distribution.
44
PolymerSimulated
PolymerCalc
PolymerRewired
C
0.1
0.05
0
0
0.5
1
1.5
2
2.5
Temperature
Figure 32. The clustering coefficient C as a function of
temperature. The rewired polymer network produces a
clustering coefficient similar to that as calculated.
3
45
CHAPTER 5
DEVELOPMENT OF THE MODEL
We now possess defining information about the topology of the polymer network. We
have discovered that the polymer exhibits a bimodal degree distribution, in which the network
is made up of two independent communities. One community consists of a highly connected
network, resulting in a large value for hki. The second community consists of a sparsely
connected network, resulting in a smaller value for hki. We have seen through an analysis of
the clustering coefficient that the polymer network is more highly interconnected than our ER
random model. We also have discovered, through the process of rewiring the network, that
proximity between aggregates is the mechanism that produces this high valued clustering
coefficient.
We will now further develop our model so as to include the bimodal degree
distribution and the high valued clustering coefficient. The organization of this chapter begins
with the algorithm for producing a bimodal ER random model. This is then followed by
description of how to add proximity to the model. The chapter concludes with an investigation
into the resultant average clustering coefficient of the model.
B IMODAL ER R ANDOM M ODEL
To match the bimodal nature of the degree distribution of the polymer network, we
will construct a bimodal ER random model. This model is similar to an ER random model, as
introduced earlier in Chapter 3. Yet, it differs in that the total number of nodes N and links `
within the system are separated into two communities. The model consists of NA nodes
within an A-community and NB nodes within a B-community. There are `A links between
nodes within the A-community, `B links between the nodes of the B-community and `AB
links between nodes of the two communities. As within the earlier model, loops and multiple
links between nodes are not allowed to occur within the bimodal model.
A bimodal ER random network can be created using two equivalent methods. Similar
to the previous model, the first method consists of probabilistically forming links between the
total possible numbers of connections. The second method, which was used within this study,
consists of connecting a random selection of nodes until the desired number of links is
achieved.
The determination of the number of links and nodes, and how they are divided among
the two communities defines the topology of the network model. For example, a network that
46
consists of solely `A and `B links will produce a model having two communities with no
connections between them (Figure 33).
Figure 33. An example of a two community network
structure consisting of `A and `B links. The result is a
network which possesses a bimodal degree distribution.
A network that has only `A and `AB links (Figure 34) could produce a star-like structure
consisting of a connected core and no connections within the B-community.
Figure 34. An example of a star-like network structure
consisting of two communities with `A and `AB links.
The result is a bimodal degree distribution.
Likewise, all three types of links could exist. This model could result in a possible structure as
displayed in Figure 35.
47
Figure 35. An example of a star-like network structure
consisting of two communities with `A , `B , and `AB links.
The result is a bimodal degree distribution.
In each of the three cases, the variation of the number of links connecting the two
communities `AB has no effect upon the resulting degree distribution. Figure 36 displays a
representative degree distribution for the bimodal model.
0.1
0.08
p(k)
0.06
0.04
0.02
0
0
5
10
15
20
25
k
Figure 36. A representative degree distribution for the three
types of bimodal ER random models. p(k) is independent of the
number of `AB links within the model.
48
Bimodal networks are created with a desired hki for each of the two communities,
hkA i and hkB i respectively. Due to the random nature of forming links between nodes, the
resulting degree distribution of this model is the sum of two independent Poisson
distributions. Given that λ = hki for a Poisson distribution, the two independent distributions
will have differing values for λ, such that λA = hkA i and λB = hkB i. For this model, it
therefore follows that
p(k) =
hkA ik e−hkA i hkB ik e−hkB i
+
.
k!
k!
(18)
Within this study, bimodal models are created to fit a known degree distribution p(k)
from simulation data. After an analysis of this simulation data, the individual contributions of
the two communities to the total degree distribution, pA (k) and pB (k) as defined by Equation
14, can be determined. Using pA (k) and pB (k), NA and NB are then defined by
NA = N
∞
X
pA (k)
(19)
pB (k).
(20)
k=0
and
NB = N
∞
X
k=0
To determine the number of links within the two communities, `AB is chosen as an
input parameter within the model. Then, `A and `B can be determined such that
2`A + `AB = NA
∞
X
k pA (k) = NA hkA i
(21)
k pB (k) = NB hkB i.
(22)
k=0
and
2`B + `AB = NB
∞
X
k=0
The creation of the network model begins with the assignment of `AB links between
randomly selected A-nodes and B-nodes. Then, links within the A-community are randomly
formed between pairs of A-nodes until `A links are achieved. This is followed by the same
process within the B-community, resulting in `B links. This procedure results in a bimodal
random network consisting of N = NA + NB nodes and ` = `A + `B + `AB links.
49
A DDING P ROXIMITY TO THE M ODEL
In an effort to investigate whether a bimodal ER random network can reproduce the
observed clustering coefficient of the polymer network, ensembles of bimodal models, that
mimic the degree distribution of the polymer system at T = 0.5, were created. In the
construction of these models, the effect of spatial proximity was investigated by adding a
cut-off constraint. The purpose of this cut-off is to limit the ability of the model to form
random connections between nodes based upon a maximum distance between them. The
clustering coefficient of the resulting bimodal network was then investigated as a function of
this spatial constraint.
The procedure for introducing a cut-off distance to the bimodal ER random network
begins with a similar process as introduced in the previous section. The cut-off distance and
`AB serve as input parameters to the model. The difference within this procedure is that the
desired number of nodes N = NA + NB are randomly assigned spatial coordinates in a three
dimensional cell of size 1 × 1 × 1 with periodic boundary conditions. Links between
A-nodes and B-nodes are then randomly assigned, only if the two nodes are radially within
the cut-off distance. The random selection and connecting of A and B nodes continues until
`AB links are achieved. The same process is followed within the A-community. Maintaining
the cut-off distance between selected nodes, connections are randomly formed between pairs
of A-nodes until `A links are achieved. Likewise, this is followed by the same process within
the B-community, resulting in `B links. This procedure results in a bimodal random network
consisting of N = NA + NB nodes and ` = `A + `B + `AB links, in which connections
between nodes are limited spatially by the cut-off distance.
With the assistance of Joris Billen, three differing types of bimodal models were
created, one consisting of only `A and `B links, one consisting of `A , `AB and `B links and the
last consisting of only `A and `AB links. These three differing models encompass the two
extremes by which a bimodal model can be created. With `AB as an input parameter, the
bimodal model has limitations on the number of `AB links possible, while still maintaining the
desired hkA i and hkB i of the degree distribution. Table 7 contains the parameters by which the
three bimodal networks were created.
Table 7. The Parameters Used to Create the Three Bimodal
Models at T = 0.5.
Model NA NB `A `B `AB
A-B
85 115 329 17
0
A-AB-B 85 115 321 10 15
A-AB
85 115 311 0
35
50
The clustering coefficient of the bimodal models was then calculated through an
experimental determination of the networks. The resulting values of C for the three models
are listed as a function of the cut-off distance in Table 8.
Table 8. Clustering Coefficient of Three Differing Bimodal
Models at T = 0.5 for Several Values of Cut-off Distances.
Cut-off CA−B CA−AB−B CA−AB
∞
0.086
0.087
0.089
0.45
0.117
0.119
0.122
0.40
0.154
0.154
0.159
0.35
0.223
0.225
0.231
It is observed, for each of the models that the clustering coefficient is directly affected
by the addition of a cut-off distance. With a stronger constraint, the clustering coefficient
increases. This provides evidence, that as the distance for allowed connections is reduced, the
network becomes more highly connected. With a cut-off distance of 0.35, the bimodal model
reproduces two defining characteristics of the polymer system, a high valued clustering
coefficient accompanied with a similar degree distribution. The high value of the models’
clustering coefficient ranges from 0.223 − 0.231. This is comparable to the clustering
coefficient of the polymer system, with a value of 0.215.
With no spatial constraint, a cut-off distance of infinity, the models produce a
clustering coefficient in the range of 0.086 − 0.089. Given that the rewired polymer network
produces a clustering coefficient of 0.0434 at T = 0.5, it was expected that a bimodal ER
random model, without a cut-off constraint would produce a lower value for C. This
difference between the constrained bimodal models and the rewired polymer suggests that the
models do not describe the clustering coefficient of the polymer network entirely.
The addition of a cut-off constraint within the models allows for the effects of
proximity on the degree correlation of the network to be investigated. We have seen that
through a random rewiring of the polymer network, it becomes uncorrelated, in that
connections between nodes are no longer dependent upon their degree. As we have seen
earlier, this property is observed by a lack of a functional dependence of c̄(k). By adding
proximity to a random network, conclusions can be drawn as to whether the correlation
exhibited by the polymer system is due to the proximity associated with the length of a
polymer chain.
In Figure 37, c̄(k) as a function of for the bimodal A-AB model is displayed on a
log-log scale. For the four values of cut-off distances, the model does not appear to become
correlated with the introduction of a stronger proximity constraint. The other two types of
51
bimodal models also produce similar results, in that with increasing constraint, c̄(k) does not
become functionally dependent. As a comparison to the model’s results, c̄(k) of the polymer
network is also plotted on the same graph.
Cutoff = infinity, C = 0.086
Cutoff = 0.45, C = 0.117
Cutoff = 0.40, C = 0.154
Cutoff = 0.35, C = 0.223
PolymerT = 0.5
c(k)
0.25
0.125
2
4
8
16
k
Figure 37. c̄(k) of a bimodal model for T = 0.5 with several
differing cutoff distances shown on a log-log scale. The model
consists of A and AB links. The consistent value of c̄(k) with k
shows indications of the model network remaining uncorrelated
with increased cutoff distances. For comparison, the c̄(k) of the
polymer is included within the figure.
For the polymer network, it is known that a proximity constraint exists in the length of
a polymer chain. Through a random rewiring of this real-world network, it becomes
uncorrelated. Yet, in creating a random network, consisting of connections based upon a
proximity constraint, it has been shown that the network does not become correlated. This
evidence suggests that there is another mechanism present within the polymer network
producing the observed degree correlation.
We have now seen that the bimodal model can predict the degree distribution of the
polymer network, which consists of a two Poisson distribution. By adding a proximity
constraint to the allowed links within the model, the high value for the clustering coefficient,
as seen in the polymer network, can be predicted by the model. This model lacks the ability to
accurately predict the higher degree aggregates within the degree distribution, as seen on a
52
log-log plot within the tails of the distributions. The model also falls short in describing the
degree correlation of the polymer network, as seen when investigating c̄(k).
53
CHAPTER 6
DISCUSSION AND CONCLUSIONS
Within this study, the topology of a simulated polymer network was compared to
random network models through an analysis of the degree distribution and the clustering
coefficient.
R ESULTS
The degree distribution of the polymer network was analyzed using three differing
methods of counting: aggregates size, multiple bridge and single bridge. For each of these
three distributions, the polymer network in the high temperature range was shown to possess
an initial drop-off within low values of k. At approximately T = 0.6, the onset of a bimodal
distribution produces a secondary peak at higher values of k. With decreasing temperature,
the initial drop-off decreases until no longer present, leaving only the secondary peak. The
bimodal nature of these degree distributions were described in terms of two communities
within the polymer network. The B-community possesses a low ratio of links to nodes, while
the A-community is more highly connected with a larger ratio of links to nodes.
Least square curve fitting of the polymer degree distributions with a two Poisson
fitting function was performed. This fitting function, being representative of the distribution
produced by a bimodal random model, showed an increasingly accurate fit with increasing
temperatures. Using the resulting fitting parameters, the percentage of aggregates or bridges
contained within each community to the total distribution was assessed. When analyzing the
fraction of the A-community to the total distribution, it was observed at low temperatures this
value was approximately 100 percent. The percentage in the A-community continued to
decrease with increasing temperature, reaching a final value of approximately 8 percent at the
highest temperature. It was found that this change occurred most rapidly at approximately
T = 0.5, close to the same temperature as the onset of the secondary peak within the
distributions. This temperature was also in agreement with the previous study on the polymer
system in which a peak in the specific heat at T = 0.51 marked an aggregate forming phase
transition.
When investigating the tails of the distributions, the inaccuracies of the bimodal model
were discovered. It was found the model could predict the polymer network for low values of
k. Yet, within the tail of the distribution, the bimodal model drops off faster than the polymer
distribution. This feature was most apparent in the lower temperatures. At a temperature of
54
T = 0.5, the polymer network appeared to exhibit a power law distributed tail, a property
characteristic to many real-world and scale free networks.
The clustering coefficient of the polymer network was then investigated. It was found
that the network possessed high values for C, a typical trait of real-world networks. An
increased clustering with lower temperatures was an indication that the polymer network
becomes more highly interconnected with decreasing temperatures. When investigating c̄(k),
it was observed that the average clustering of polymer network was dependent upon k. This
degree dependence was evidence that connections between aggregates prefer to form with
larger size aggregates. When comparing these two properties to ER random model, which
were created with an identical hki, it was found that the random models were unable to
predict both of these properties. The model produced a much lower value for C and were
uncorrelated by definition.
An investigation into one of the known differences between the polymer network and
models was then performed. This difference is the spatial limitation between connections of
aggregates within the polymer system. The maximum distance, between which two
aggregates can be connected, is limited by the length of a stretched polymer chain. Through
the process of a random rewiring, the proximity between connected aggregates was removed
while maintaining the original degree distribution. It has been shown that the rewired polymer
network achieves a clustering coefficient consistent to that of the theoretical calculated value.
It was also observed that the rewiring process removed the degree correlation originally seen
in the polymer network. These two facts provide evidence that the proximity between
aggregate connections may be responsible for the high interconnectivity and degree
correlations within the polymer network.
The random model was then developed to include these topological characteristics of
the polymer network. To reproduce the degree distribution of the polymer network, a bimodal
model was produced, consisting of two communities. It was investigated whether the addition
of proximity to the bimodal ER random model could reproduce the highly clustered nature of
the polymer network. The process consisted of constructing a model with the addition of a
cut-off distance, the maximum distance allowed for a link to form. It was observed that with a
stronger constraint, the bimodal random models were capable of reproducing a high value for
C, consistent with the polymer network. This shows that the proximity associated with
connections between aggregates produces the large values of C as seen within the polymer
network.
Using these same bimodal models with spatial constraints, an investigation was also
performed as to whether the degree correlation of the polymer network is due to proximity.
c̄(k) of the models was analyzed with increasing cut-off distances. It was found that a stronger
55
constraint did not produce a degree correlation. This result provides evidence that another
mechanism, not proximity, produces the degree correlations within the polymer network.
O NGOING AND F UTURE W ORK
As a continuation with the development of the bimodal ER random model, Joris Billen
is currently in the process of defining the two community structure as a function of
temperature. Within this work, identifying the quantity of nodes and links between and among
each of the two communities is of interest. This investigation is being approached through a
spectral analysis of the adjacency matrix. Early results suggest that at low temperatures the
network consists of primarily one highly connected A-community. In the temperature range
of T = 0.5 to T = 1.0 all three types of links within the two communities exist, as seen in
Figure 35. In the higher temperatures, the polymer network consist of two communities with
no connections linking them, as seen in Figure 33.
Another current study within the polymer system consists of identifying the processes
behind the rapid onset of the A-community. As seen in the low temperature degree
distributions, the secondary peak provides evidence that aggregates possess a preferential size.
The investigation into the mechanism producing this preferential size is being approached
through an analysis of chemical kinetics. This study began by defining the possible reactions
that can occur within an aggregate. Aggregates can either break apart producing smaller
components or join together forming a larger size aggregates. The rates and sizes at which
these reactions occur are of interest. They can provide knowledge of the individual dynamics
associated with the transition from a primarily B-community network to an A-community
network with decreasing temperature.
As a further investigation of the topology of the polymer network, the path length and
the pair correlation function are two defining characteristics suggested for future
investigation. For both of these two network properties, the foundations necessary for a
thorough investigation, in terms of FORTRAN code, are already in place. The code developed
within this work (the Appendix) contains subroutines that are capable of analyzing these two
properties. However, a complete analysis for the polymer system has yet to be finished.
The path length of a graph is a measure of the average distance between any two
nodes. This path can either consist of the smallest number of connections necessary to reach
each node or be a physical distance between nodes. In the case of the polymer simulation, the
smallest number of connections between a source node and all other nodes contained within
the graph is of interest. A large valued clustering coefficient, along with a relatively small
average path length describes the characteristic properties defining a“small-world” network.
We have seen that the polymer network exhibits a large value for the clustering coefficient in
the low temperature range, transitioning to smaller values with decreasing temperature. With
56
the addition of the average path length, knowledge of whether the polymer network makes a
transition through this “small-world” property with temperature can be gained.
The pair correlation function describes the radial density distribution of aggregates
within the system. In an experimental setting, the pair correlation function is typically
calculated from X-ray scattering data. In the case of the simulation, the reverse can be
performed relatively easily. Since the spatial distribution of aggregates within the system is
known, the radial density distribution of aggregates can be calculated through the employment
of the periodic boundary conditions. Fourier analysis of this distribution can provide the
structure factor, which is directly related to experimental scattering data. Comparisons
between experimental results and simulation data could then be made.
57
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59
APPENDIX
DESCRIPTIONS OF FORTRAN CODES
60
DESCRIPTIONS OF FORTRAN CODES
The appendix contains descriptions of several programs used to gather the data
presented within this work. With programming environments including FORTRAN and bash
script, each program description contains a list and brief summaries of the subroutines called
within. With the exception of “Network.f90”, all programs presented here were written by the
author of this study.
PROGRAM SMALLWORLD.F
Smallworld.f is a toolbox, capable of analyzing several statistical
properties of a network. These properties include: average
multiple bridge distributions, average single bridge
distributions, average multiple bond distributions, average pair
correlation function, average path length, average clustering
coefficient, the clustering coefficient as a function of degree
C(k), and average fractional giant component. It can accommodate
multiple input "*_struc" files containing identical or differing
number of time steps per input file.
Subroutines called:
AlterAdj
Subroutine AlterAdj randomly alters the adjacency matrix
for a desired number of changes per time step. The goal
is to randomly change some of the links without changing
the degree distribution. The effect of this rewiring will
remove the spatial dependence of the polymer.
MakePairDistrib
Subroutine MakePairDistrib calculates the pair correlation
function (PCF) for a given time step. The PCF is radial
distribution. Starting from the middle third of the unit
cell, it calculates distances between the starting nodes
(those nodes within the middle third) and all others. The
distances between the start nodes and their corresponding
mirror nodes are not considered within this calculation.
These distances are then binned into spherical shells,
with delta-r = 0.1 and a distribution is then formed.
More information about the PCF and setup can be found here:
http://www.sci.sdsu.edu/~jbillen/Pair%20Distribution
%20Function.pdf and
http://www.sci.sdsu.edu/~jbillen/Pair%20Correlation_3.pdf.
BuildRandLoc
Subroutine BuildRandLoc builds a "location" array containing
x, y, and z coordinates by randomly assigning locations
for a set number of nodes. This location array is then
61
use for testing purposes for the calculation of the pair
correlation function. This routine uses a random number
function "rand" on Tayir.
BuildNine
Subroutine BuildNine takes the "location array", containing
x, y, and z coordinates for each node within the system,
and repeats it 8 times within the x-y plane. This
process surrounds the original cell in that plane with
identical cells. The purpose of this is to take account
of the periodic boundary conditions in the x and y
directions.
GenEGLocationArray
Subroutine GenEGLocationArray builds a "location" array,
containing x, y, and z coordinates for each end group
within the system for a given time step.
GenLocationArray
Subroutine GenLocationArray builds a location array,
containing x, y, and z coordinates for each aggregate
within the system for a given time step. Aggregates
consist of multiple end groups, so the point location of
the aggregate is a 3d spatial average over all end groups
which comprise the aggregate.
GiantClust
Subroutine GiantClust returns the fractional largest
component of the polymer network, known as the giant
cluster.
GenPathDist
Subroutine GenPathDist creates a path length distribution
from the "pathLength" matrix.
Shortest
Subroutine Shortest constructs a matrix "pathLength" which
contains the shortest path length, in terms of number of
hops, between all nodes within the time step. It also
constructs matrix "PhysPathLength" which contains the
shortest physical distance between all nodes. The
algorithm used to calculate the path length is a breadth
first search. More information about the path length can
be found here:
http://www.sci.sdsu.edu/~jbillen/Path%20Length.pdf and
http://www.sci.sdsu.edu/~jbillen/Path%20Length2.pdf.
CalcCoeff
Subroutine CalcCoeff calculates the clustering coefficient
for a given time step, for nodes where k>1. Also,
distributions of C(k) are produced.
62
ConstBridge
Subroutine ConstBridge returns an adjacency matrix
"Bridge". Beginning with the "Struc" matrix, the
subroutine checks for connections between aggregates.
These connections are as follows: atom number 1 is
connected to atom number 2, 3 to 4, etc. The adjacency
matrix contains integer values based upon the number of
connections between node(row) and node(col). This
adjacency matrix is symmetric. Loops are eliminated at
this point; therefore the matrix contains zeros on the
diagonal.
GenDistrib
Subroutine GenDistrib constructs a distribution of bridge
sizes based upon the adjacency "Bridge" matrix. Array
"SingleDistrib" contains counts of connections between
aggregates in which multiple connections between two
aggregates are only counted as one connection. Array
"MultiDistrib" contains counts of connections between
aggregates in which multiple connections between two
aggregates are counted as multiple connections.
ConstStruc
Subroutine ConstStruc returns a matrix "Struc". Each row
within this matrix corresponds to a different aggregate
within the system. The entries of this matrix are atom
numbers contained within each aggregate. A new matrix is
constructed for each time step.
LoadtStep
Subroutine LoadtStep builds an array containing aggregate
size followed with the atom numbers contained within that
aggregate. The array contains every aggregate within the
system for a specific time step.
PROGRAM AGGREGATE.F
Aggregate.f generates an average aggregate size distribution. The
program can accommodate multiple input "*_struc" files containing
identical number of time steps.
Subroutines called:
ConstDist
Subroutine ConstDist constructs a distribution of
aggregate sizes.
LoadtStep
Subroutine LoadtStep builds an array containing aggregate
size followed with the atom numbers contained within that
aggregate. The array contains every aggregate within the
system for a specific time step.
63
PROGRAM CREATEPROB.F
CreateProb.f generates a probability distribution p(k) from an
input data set. This is typically used to make p(k) distributions
from the average count distributions generated by Smallworld.f and
Aggregate.f.
Subroutines called:
None
PROGRAM MKREWIRESTRUC.F
MkRewireStruc.f generates, as an output, a "*_struc" file
consisting of a desired number of time steps. From a larger
number of time steps and input files, the program will randomly
select time steps, assuring that the same time step is not
selected, and write a new "*_struc" file containing these random
selections. It can accommodate multiple input "*_struc" files
containing identical number of time steps.
Subroutines called:
LoadtStep
Subroutine LoadtStep builds an array containing aggregate
size followed with the atom numbers contained within that
aggregate. The array contains every aggregate within the
system for a specific time step.
SCRIPT START_REWIRE.SH
The script start_rewire.sh was designed to run "Smallworld.x" a
multiple number of times and change the input parameters with each
new instance. The script was written for the rewiring procedure.
With each new instance of "Smallworld.x", the input parameter,
"number of changes to the adjacency matrix", is altered
incrementally. This script is dependent upon an input data file
"smallworld.in", which contains the input parameters for
"Smallworld.x".
PROGRAM NETWORK.F90
Written by Joris Billen, Network.f90 is a toolbox capable of
building several differing types of network models. It generates,
as an output, a "*_struc" file consisting of a desired number of
time steps. As used and described in this work, the program can
build and is not limited to ensembles of ER random models, along
with ensembles of random bimodal models.
ABSTRACT OF THE THESIS
The Network Properties of
a Simulated Polymeric Gel
by
Mark Allen Wilson
Master of Science in Physics
San Diego State University, 2008
Complex network structure can be used to describe a large range of real-world
systems. The pages which comprise the World Wide Web, social interactions between friends,
and the highway system across the nation are just a few examples of this type of network
structure, consisting of a large number of components, dynamically evolving, and highly
interconnected. Within this study, a molecular dynamic simulation of a polymeric system will
be described in terms of a complex network. An analysis of two characteristic properties of
the polymeric network system will be performed. Through an analysis of the degree
distribution and clustering coefficient, information about the network topology will be gained.
Ensembles of Erdös-Rényi random models will then be created to investigate to what degree
the polymeric network can be described by a random network.