School of Information Sciences
University of Pittsburgh
Network Science: A Short Introduction
i3 Workshop
Konstantinos Pelechrinis
Summer 2014
Figures are taken from:
M.E.J. Newman, “Networks: An Introduction”
Network models

We want to have formal processes which can give rise to
networks with specific properties
 E.g., degree distribution, transitivity, diameters etc.

These models and their features can help us understand
how the properties of a network (network structure) arise

By growing networks according to a variety of different
rules/models and comparing the results with real
networks, we can get a feel for which growth processes
are plausible and which can be ruled out
 Random graphs represent the “simplest” model
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Erdos-Renyi random network model

Simplest model
 Basic idea: nodes are connected completely at random

Given: number of nodes n
 Number of edge: m
 In
this case for each edge m we pick uniformly at random a pair of
nodes
 Probability that any two nodes in the network are connected: p
 In
this case, we go over all possible pairs of the n nodes and
connect each one of them with a probability p
 Both processes/models are equivalent
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Properties of random networks

The random network model can generate networks with:
 Short paths
 Giant components

It cannot generate networks with:
 High clustering
 Skewed degree distribution
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Small-world model



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Random graphs exhibit small paths but not clustering
Figure 15.1: A triangular lattice. Any vertex in a triangular lattice, such as the one highlighted
here, has six neighbors and hence
pairs of neighbors, of which six are connected by edges,
giving a clustering coefficient of = 0.4 for the whole network, regardless of size.
If we consider an ordered network (lattice) exhibits high
clustering but large paths
To calculate the number of triangles in such a network, we observe that a trip around any
triangle must consist of two steps in the same direction around the circle—say clockwise—
followed by one step back to close the triangle. The number of triangles per vertex in the whole
network is then equal to the number of such triangles that start from any given point.
Why not combine both these models ?
Small-world models

The small-world model (Watts and Strogatz 1998) tries to
do exactly this
 We start with a circle model of n vertices in which every vertex
The small-world model, in its original form, in
has a degree of c
random graph by moving or rewiring edges from
structure
the model
is shown
in Fig. 15.3a.
 We go through each of the edges
andofwith
some
probability
p Start
every vertex has degree c, we go through each of th
we rewire it
remove that edge and replace it with one that join
 Remove
randomlyuniformly
placed edgesatarerandom
commonlyand
referred
this edge and pick twoThe
vertices
connect them with a new edge they create shortcuts from one part of the circle to an
o Shortcut edge
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Small-world models

The parameter p controls the interpolation between the
circle model and the random graph
 p=0  ordered situation/circle model
 p=1  random graph
 Intermediate values of p give networks somewhere in between

The crucial and interesting point is that small paths appear
even for small values of p as we increase from p=0, while
the high clustering remains until fairly large values of p
 Hence, there is a regime for values of p where both small paths
as well as high clustering exists!
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Small-world models
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Small world models

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For c=6 and n=600
Small-world
regime
Cannot generate:
Skewed degree distribution
Preferential attachment

Both previous models cannot generate skewed degree
distributions
 How can we have networks where there are a few nodes with a
large number of edges, while the majority of them has few
edges only?

A simple growth process can provide insights!
 Until now we have fixed topology models
 Given
number of nodes and edges from the beginning
o In other words, nodes do not appear one-by-one in time
 A growth process refers to the evolution of the network by the
addition of nodes (and edges for these nodes)
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Preferential attachment

Nodes prefer to attach to existing nodes that have high
degree!

At every point of time a new node is created and this node
generates b edges
 Each of this edges is connected to the existing nodes randomly
 NOT
UNIFORMLY AT RANDOM BUT WITH A PROBABILITY
PROPORTIONAL TO THE NUMBER OF EDGES AN EXISTING
NODE ALREADY HAS!

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Rich-gets-richer, cumulative advantage, Matthew effect
etc.