Social Network Analysis and
Complex Systems Science
Pip Pattison
University of Melbourne
CSIRO Complex Systems Symposium, Pelican Beach, 10-12 Aug 2004
In collaboration with:
Garry Robins, University of Melbourne
Tom Snijders, University of Groningen
Henry Wong, University of Melbourne
Jodie Woolcock, University of Melbourne
Emmanuel Lazega, University of Lille I
Kim Albert, University of Melbourne
Anne Mische, Rutgers University
John Padgett, University of Chicago
Peng Wang, University of Melbourne
1. Why are social networks important?
For understanding action in relation to its social context
network ties link actors to each other as well as to groups, cultural resources,
neighbourhoods, communities
networks structure opportunities and constraints
For understanding social dynamics
social action is interactive: one person’s action changes the context for those
to whom they are connected
To understand the cumulation of local processes into population level
outcomes
The structure of networks and the dynamics of local processes are critical to
understanding how locally interactive, context-dependent actions
cumulate into outcomes at higher levels (eg communities, populations)
A simplified multi-layered and relational
framework for the social world
Social units
individuals
groups
...
For example:
Interactions between social units depend
on proximity through ties
Ties among social units
person-to-person
person-to-group
...
Interactions between ties depend on
proximity through settings
Settings
geographical
sociocultural
...
There are interactions within and
between levels
Social structure: regularities in
interactions
2: Typical data structures
Network observations give rise to relational data
structures, e.g.:
People  groups, people  attributes, groups  attributes, people  settings, groups
 settings, people  people
people  people  types of tie, people  people  settings, …
Some important design issues:
Network boundaries?
Complete: which “nodes” to include?
Which network ties?
What are the relevant network links?
How do we best “measure” them?
Example 1: Management consulting firm
node colour codes workgroup membership
node size codes extent of cohesive beliefs
ties: “Who do you ask when you want to find out what is going on..?”
Example 2:
Network
of Mutual
Collaboration
Ties (Lazega,
1999)
Example 3: Change in interorganizational
networks (Goldman et al, 1994)
Data are from an evaluation of the Robert Wood Johnson Program on
Chronic Mental Illness in 6 US cities (one of which was a “control”
site)
Organisations
Mental health agencies in the “control” site (n =37)
Networks at time 1 and time 2 (x1, x2)
Client referrals
Information-sharing
Fund-sharing
Data are from key informants and were gathered two years apart
Client referrals: time 1
Client referrals: time 2
3: Modelling networks and other relational
structures
Guiding principles:
1. Network ties (and other observations) are the outcome of unobserved
processes that tend to be local and interactive
2. There are both regularities and irregularities in these local interactive
processes
Hence we aim for a stochastic model formulation in which:
– local interactions are permitted and assumptions about “locality” are explicit
– regularities are represented by model parameters and estimated from data
– consequences of local regularities for global network properties can be understood
and can also provide an exacting approach to model evaluation
Building models for social networks
We model tie variables: X = [Xij]
Xij = 1 if i has a tie to j
0 otherwise
realisation of X is denoted by x = [xij]
Two modelling steps:
methodological: define two network tie variables to be neighbours if they
are conditionally dependent, given the values of all other tie variables
Substantive: what are appropriate assumptions about the neighbourhood
relation (ie about the network topology)?
Network topologies:
which tie variables are neighbours?
Two tie variables are neighbours if:
they share a dyad
dyad-independent model
they share an actor
Markov model
they share a connection
with the same tie
realisation-dependent model
They share a connection
with two ties
k-triangle model
etc...
Models for interactive systems of variables
(Besag, 1974)
Hammersley-Clifford theorem: A model for X has a form determined by its
neighbourhoods, where a neighbourhood is a set of mutually neighbouring variables
This general approach leads to:
P(X = x) = (1/c) exp{Q QzQ(x)}
normalizing quantity
parameter
network statistic
the summation is over all neighbourhoods Q
zQ(x) = XijQxij signifies whether
all ties in Q are observed in x
c = xexp{Q

QzQ(x)}
Neighbourhoods depend on
proximity assumptions
Assumptions: two ties are neighbours:
Configurations for neighbourhoods
if they share a dyad
dyad-independence
if they share an actor
Markov
edge
+
2-star 3-star 4-star ...
if they share a connection with the
same tie
realisation-dependent
+
triangle
...
3-path
4-cycle
“coathanger”
Neighbourhoods, continued
k-triangle model
configurations include:
k nodes
2 ties are neighbours if they create
a 4-cycle
k-independent
2-path
k-triangle
useful for higher-order clustering
effects
Homogeneous Markov random graphs
(Frank & Strauss, 1986)
P(X = x) = (1/c) exp{L(x) + 2S2(x) + … + kSk(x) + … + T(x)}
where: L(x)
no of edges in x
S2(x) no of 2-stars in x
…
Sk(x) no. of k-stars in x
…
T(x)
no of triangles in x
…
Simulating from homogeneous Markov random graph
distributions on 36 nodes: a typical graph
Parameter values:
 = -3
2 = 2
=0
3 = -2
Average statistics: edges 57.0 2-stars 133.8 triangles 2.3 3-stars 68.4
Typical graphs for  = 0, 2, 5, 6
A typical graph for  = 10
Parameter values:
 = -3
2 = 2
 = 10
3 = -2
Average statistics: edges 92.0 2-stars 390.0 triangles 130.0 3-stars 440.0
These models can represent very different network
structures: eg small worlds: =-4, 2=0.1, 3=-0.05, =1
[Robins, Pattison & Woolcock, in press]
No of edges
L=126
path length distribution
Q1 = 4 (5)
Q2 = 5 (7)
Q3 =  ()
clustering coefficient
Cluster = 0.09 (0.02)
figures for Bernoulli distribution in red
Longer path worlds: =-1.2, 2=0.05, 3=-1, =1
but levels of clustering are still high
No of edges=118
Q1 = 5 (5)
Q2 = 7 (7)
Q3 = 9 ()
Cluster = 0.08 (0.02)
Very long path worlds: =-2.2, 2=0.05, 3=-2, =1
(no clustering)
L=82
Q1 =  (11)
Q2 =  ()
Q3 =  ()
Cluster = 0.00 (0.02)
Simulations of two-star models (n=30)
(a)  = 0, 2 =[0.00, 0.01,…0.10]
(see also Handcock, 2004; Park & Newman, 2004; Snijders, 2002)
30
average
degree
14000
no of
2-stars
28
26
24
12000
10000
22
8000
number of 2-stars
average degree
20
18
16
14
-.02
0.00
.02
.04
.06
.08
.10
6000
4000
2000
.12
-.02
2-star parameter
0.00
.02
.04
.06
.08
.10
.12
2-star parameter
no of successful proposals in 500,000 steps
600000
no of successful moves
Metropolis algorithm
complete graph has
high probability for
high values of 2
500000
400000
300000
200000
100000
0
-100000
-.02
0.00
.02
2-star parameter
.04
.06
.08
.10
.12
multiple random starts
(b)  = -2.5, 2 =[-0.50, -0.45,…,0.25]
14000
30
12000
average
degree
no of
2-stars
average degree
10
8000
6000
4000
number of 2-stars
20
10000
2000
0
-2000
0
-.6
-.6
-.4
-.2
-.0
.2
-.4
-.2
-.0
.2
.4
.4
2-star parameter
2-star parameter
sharp transition
from low to high
density graphs
around 2 = -/(n-2)
no of successful proposals in 500,000 steps
120000
no of successful moves
100000
80000
60000
40000
20000
0
-20000
-.6
-.4
2-star parameter
-.2
-.0
.2
.4
“Freezing” at 2 = -/(n-2):
(,2) = (-14,0.5)/t for t = 0,1,…
Average degree
Successful moves
600000
no of successful proposals in 500,000 steps
30
20
10
0
-10
-.1
0.0
2-star parameter
.1
.2
500000
400000
300000
200000
100000
0
-100000
.3
.4
.5
.6
-.1
0.0
.1
.2
.3
.4
2-star parameter
See Park and Newman (2004) for an analytical solution
(including phase diagram)
.5
.6
4: Applications: Estimation of model
parameters and model evaluation
A. Estimation of model parameters from data:
MLE via MCMC approaches (Snijders, 2002; Handcock et al, 2004)
B. Model evaluation: do substantively important global properties of the
observed data resemble simulated data?
For example:
Degree distribution
Path length distribution
Presence of clustering, cycles
The overall aim is to identify regularities in local relational structures,
and at the same time build models that reproduce global network
structure from empirically-grounded local regularities
The alternating k-star, k-independent 2-path and k-triangle
hypotheses (Snijders, Pattison, Robins & Handcock, 2004)
Suppose that:
k = -k-1/
where   1 is a (fixed) constant
alternating k-star hypothesis
Then kSk(x)k = S[](x) 2 where:
S[](x) = 2 i{(1 - 1/)d(i) + d(i)/ - 1}
and d(i) denote the degree of node i
alternating k-star statistic
Likewise:
If Uk(x) = no of k-independent 2-paths in x, with corresponding parameter k
and Tk(x) = no of k-triangles in x, with corresponding parameter k
We can suppose that:
k+1 = - k/
k+1 = - k/
alternating independent 2-path hypothesis
alternating k-triangle hypothesis
Network
of Collaboration
Ties
Realisation-dependent model for colaboration ties
among lawyers (Pattison & Robins, 2002)
neighbourhood
estimate
_________________________________________
edge
-3.669 (.474)
2-star
0.307 (.053)
3-star
-0.001 (.002)
triangle
0.173 (.047)
3-path
-0.019 (.002)
4-cycle
0.086 (.009)
_________________________________________
MCMCML parameter estimates for collaboration
network (SIENA, conditioning on total ties, partners only)
Model 1
Parameter
est
s.e.
alternating k-stars (=3)
-0.083 0.316
Alternating ind. 2-paths (=3) -0.042 0.154
Alternating k-triangles (=3)
0.572 0.190
No pairs connected by a 2–path -0.025 0.188
No pairs lying on a triangle
0.486 0.513
Seniority main effect
0.023 0.006
Practice (corp. law) main effect 0.391 0.116
Same practice
0.390 0.100
Same gender
0.343 0.124
Same office
0.577 0.110
Model 2
est
s.e.
0.608 0.089
0.024
0.375
0.385
0.359
0.572
0.006
0.109
0.101
0.120
0.100
Modelling group cohesion (Albert, 2002)
Network ties are important in understanding social processes, but so are:
cultural and psychological resources and aspirations (beliefs, values,
attitudes, knowledge)
settings (geographical locations, physical and organisational constraints)
Lindenberg (1997) on groups:
Three overlapping forms of interdependence:
functional (common goals and tasks)
cognitive (psychological representations)
structural (patterning of interpersonal ties)
workgroup membership
beliefs
network ties
Albert (2002) on group cohesion:
An illustrative analysis of interdependent functional, cognitive and structural
aspects of group cohesion using generalised relational structures
Management consulting firm
node colour codes group membership
node size codes extent of cohesive beliefs
ties: “Who do you ask when you want to find out what is going on..?”
Functional, structural and cognitive
interdependence
Evidence for separable tendencies:
structural logic of information seeking: hierarchical
with differentiation in information seeking
structural interdependence
information ties within groups
shared beliefs within groups
structural & functional interdependence
cognitive and functional interdependence
shared beliefs within groups among those linked by an information tie
cognitive, structural and functional interdependence
5: A dynamic perspective
co-evolution of action, networks, settings
Dynamic models
Suppose that Xij(t) are time-dependent relational variables
At any moment t, suppose that there is a possible change in status for
some randomly chosen Xij with a transition rate
logistic(Q Q(zQ(x*ij(t)) - zQ(x(t))))
where:
x(t) denotes the state of the network at time t;
x*ij(t) equals x(t) but with the value of Xij(t) changed from xij(t) to 1-xij(t);
 is a rate parameter;
logistic(z)=exp(z)/(1+exp(z))
Then this continuous-time Markov process converges to the distribution
Pr (X = x) = (1/c) exp{QQ zQ(x)}
parameters can be estimated from longitudinal data (using approach adapted
from that developed by Snijders, 2001, 2002)
Client referrals: time 1
Client referrals: time 2
Modelling client referrals
Time 2
PLE
Edge
-3.02
-3.20
2-in-star
0.01
0.05
0.06 (.03)
0.04 (0.03)
2-path
-0.08
-0.07
-0.05 (.02)
-0.05 (0.02)
2-out-star
0.09
0.10
0.08 (.02)
0.09 (0.02)
mutual tie
2.54
1.73
1.72 (.29)
1.39 (0.28)
3-cycle
-0.20
-0.14
-0.15 (.09)
-0.14 (0.09)
transitive triad
0.21
0.19
0.16 (.03)
0.14 (0.03)
*using SIENA, conditioning on number of ties
Time 2
cond MCMCMLE*
Time1Time2
cond estimate
Time 1
PLE
-
-2.74 (0.35)
Early 1990s in Brazil:
student, civic, political and business groups
Key :
time 1
time 2
-3.222(.44)
-2.223(1.1)
-4.405(.98)
0.099(.02)
0.123(.17)
0.198(.02)
0.204(.04)
0.745(.10)
0.320(.06)
-0.177(.04)
-0.461(.06)
-0.146(.07)
0.808(.08)
-3.805( .44)
-6.665(1.8)
-6.333(1.5)
0.116(.02)
0.734(.17)
0.207(.03)
0.309(.06)
0.886(.14)
0.443(.09)
-0.123(.05)
-0.307(.06)
-0.041(.05)
0.472(.07)
organisation
project
time 3
-4.678( .46)
-10.71(1.5)
-9.322(1.8)
0.170(.02)
1.051(.15)
0.202(.02)
0.459(.14)
0.906(.12)
0.444(.06)
-0.022(.04)
0.000(.06)
-0.024(.03)
0.139(.06)
event
6. Concluding comments
Models can display complex behaviour (e.g. nonlinearities, phase
transitions)
creating some statistical difficulties!
Nonetheless, a statistical approach allows us to stay close to empirical
data, and model parameters can be estimated from data. For a wellspecified model
We can test hypotheses about local contextual effects
We can predict the evolution of the system (and its variability)
We can understand the aggregate-level consequences of local contextual
effects (and their variability)
Realisation-dependent models appear to be necessary, and reflect a
“capacity for actors to transform as well as reproduce long-standing
structures, frameworks and networks of interaction” (Emirbayer &
Goodwin, 1994)
Some modelling challenges
Scaling up: the role of space
Spatial random graph models (Henry Wong)
Co-evolution
Dynamic interactions across levels
Evolution of multiple networks
Social “innovation” and transformation
Multiple networks are implicated theoretically
e.g Padgett et al on the evolution of markets in Florence
“Emergent” phenomena?
Eg emergence of social institutions such as groups
Technical issues
Sampling, estimation, missing data…