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Joint social selection and social influence models for networks:
The interplay of ties and attributes.
Garry Robins
Michael JohnstonUniversity of Melbourne,
Australia
Symposium on the dynamics of networks and behavior
Slovenia, May 10-11, 2004
Thanks to Pip Pattison, Tom Snijders, Henry Wong, Yuval Kalish, Antonietta Pane
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A thought experiment:
Most models that purport to explain important global network properties are homogeneous across nodes.
Might a simple model of interactions between node-level and tie-level effects be sufficient to explain global properties?
1. Develop a model that incorporates both social selection and social influence processes.
2. Which global properties of networks are important?
3. Simulate the model to see whether the these properties can be reproduced in a substantial proportion of graphs?
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1. Develop a model incorporating both social selection and social influence effects
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Simple random graph models
For a fixed n nodes, edges are added between pairs of nodes independently and with fixed probability p
(Erdös & Renyi, 1959)
Pr( ) exp ijx
1X x
Bernoulli random graph distribution:
X is a set of random binary network variables [Xij]; Xij = 1 when an edge is observed, = 0 otherwise;x is a graph realization;θ is an edge parameter.
an exponential random graph (p*) model.
1
ep
e
a homogeneous model – (node homogeneity)
p and θ are independent of node labels
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A Bernoulli random graph model will not fit this network well
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In this example, actor attributes are important to tie formation Social selection
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Yellow: Jewish
Blue: Arab
(Kalish, 2003)Exogenous attributes affect network ties
In this example, actor attributes are important to tie formation Social selection
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Binary variables:Xij network tiesZi actor attributes
Exogenous attributes affect network ties
Zi
Zj
Xij
1 2Pr( ) exp ij ij i ij i jx x z x z z
1X x
Robins, Elliott & Pattison, 2001
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1 2Pr( ) exp ij ij i ij i jx x z x z z
1X x
Effects in the model
Baseline edge effectirrespective of attributes
Propensity for actors with attribute z=1 to have more partners
Propensity for ties to form between actors who both have attribute z=1
Equivalent (blockmodel) parameterization:
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Social influence: Are actor attributes influenced by fixed network structure?
Robins, Pattison & Elliott, 2001
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Robins, Pattison & Elliott, 2001
Social influence: Are actor attributes influenced by fixed network structure?
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A cutpoint
Social influence: Are actor attributes influenced by fixed network structure?
Exogenous network ties affect attributes
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Binary variables:Xij network tiesZi actor attributes
Exogenous network ties affect attributes
Zi
Zj
Xij
1 2Pr( ) exp i ij iz x z
1X x
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Effects in the model
Baseline effect for number of attributed nodes (z=1)
Propensity for attributed nodes to have more partners
1 2Pr( ) exp i ij iz x z
1X x
No effect for an actor being influenced by a network partner
need to introduce dependencies among attribute variables
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Assume attribute variables are dependent if the actors are tied
partial conditional dependence (Pattison & Robins, 2002)
Zi
Zj
Xij
1 2 3Pr( ) exp i ij i ij i jz x z x z z
1X x
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Effects in the model
Baseline effect for number of attributed nodes (z=1)
Propensity for attributed nodes to have more partners
1 2 3Pr( ) exp i ij i ij i jz x z x z z
1X x
Propensity for attributed nodes to be connected
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Friendship network for training squad in 12th week of training (Pane, 2003)Green: detachedYellow: team orientedRed: positive
Why should attributes or ties be exogenous?
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Models for joint social selection/social influence
Zi
Zk
Xik
1 2 1 2Pr( ) exp
where [ , ]
i i j ij ij i ij i jz z z x x z x z z
1Y y
Y X Z
Xij Zj
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1 2 1 2Pr( ) exp i i j ij ij i ij i jz z z x x z x z z
1Y y
Effects in the modelQuadratic effect in no. of attributed nodes
Propensity for attributed nodes to have more partners
Propensity for attributed nodes to be connected
Baseline effect for no. of edges
Equivalent (blockmodel) parameterization:
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1 2 1 2Pr( ) exp i i j ij ij i ij i jz z z x x z x z z
1Y y
Change statistics
1 2
Pr( 1 , )log ( )
Pr( 0 , )
Cij ij
i j i jCij ij
Xz z z z
X
z x
z x
2 21 1
Pr( 1 , )log
2 2Pr( 0 , )
Cr r
s rs rs sCs r s r s rr r
Zz x x z
Z
x z
x z
Conditional log-odds for a tie to be observed:
Conditional log-odds for an attribute to be observed:
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2. Which global network properties are important ?
– Small worlds• Short average geodesics
• High clustering
– Skewed degree distributions– Regions of higher density among nodes
• cohesive subsets, “community structures”
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Confiding (trust) network (Pane, 2003)
An example network (without attributes)
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Many observed networks have short average geodesics – small worlds
The confiding network has a median geodesic (G50) of 2: not extreme compared to a distribution of Bernoulli graphs
The confiding network has a third quartile geodesic (G75) of 2:
also not extreme compared to a distribution of Bernoulli graphs.
Observed networks: Path lengths
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Many observed networks have high clustering – small worlds
Observed networks: Clustering
Global Clustering coefficient:
3 × (no. of triangles in graph) / (no. of 2-paths in graph) = 3T / S2
The confiding network has a global clustering coefficient of 0.41:
a comparable Bernoulli graph sample has a mean clustering coefficient of 0.25 (sd=0.03)
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Many observed networks have high clustering – small worlds
Observed networks: Clustering
Local Clustering coefficient: For each node i, compute density among nodes adjacent to i.
Average across the entire graph.
The confiding network has a local clustering coefficient of 0.58:
a comparable Bernoulli graph sample has a mean clustering coefficient of 0.25 (sd=0.04)
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Many observed networks have high clustering – small worlds
The confiding network has a global clustering coefficient of 0.41
The confiding network has a local clustering coefficient of 0.58
Observed networks: Clustering
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Many observed networks have skewed degree distributions
as is the case for the confiding network
Observed networks: Degree distribution
DEGREE
20181614121086420
fre
qu
en
cy
10
8
6
4
2
0
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Observed networks: Higher order clusteringk-triangles
(Snijders, Pattison, Robins & Handcock, 2004)
Alternating k-triangles
23 221 2 3
( ) ... ( 1)n nn
T TTu T
x
1-triangle
2-triangle
3-triangle
Permits modeling of (semi) cohesive subsets of nodes (cf community structures)
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Observed networks: Higher order clustering
Observed networks often exhibit regions (subsets of nodes) with higher density
In which case, we will see an alternating k-triangle statistic higher than for Bernoulli graphs
Alternating k-triangle statistic
4003002001000
Glo
bal c
lust
erin
g
.7
.6
.5
.4
.3
.2
.1
0.0
The k-triangle statistic is not simply equivalent to global clustering
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• Short median geodesics (G50)
• Short third quartile geodesics (G75) – perhaps?
• High clustering
• High k-triangle statistics
• Skewed degree distributions
Bernoulli distributions tend to have short median geodesics, low clustering and low k-triangles
Hence a basis for comparison
SummarySome global features not uncommon in observed networks
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3. Simulate the model to see whether global properties can be reproduced
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Use the Metropolis algorithm –procedure similar to Robins, Pattison & Woolcock (in press)
Typically 300,000 iterationsreject initial simulations for burnin
Sample every 1000th graph
Inspect degree distributions across sample
Compare each graph in sample with a Bernoulli graph distribution with same expected density
Hence can determine if graph- has short G50, G75
- highly clustered; high k-triangles
Define highly clustered and short G50 as SW50 (small world)Similarly define SW75
Simulation of the model
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1 2 1 2Pr( ) exp i i j ij ij i ij i jz z z x x z x z z
1X x
Quadratic effect in no. of attributed nodes
Propensity for attributed nodes to have more partners
Propensity for attributed nodes to be connected
Baseline effect for no. of edges
First simulation series: 30 node graphs
1 2
1 2 2 2 1
fix: 7; 0.5; 1.0;
vary: , s.t. 0, 2
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Change statistics
1 2
Pr( 1 , )log ( )
Pr( 0 , )
Cij ij
i j i jCij ij
Xz z z z
X
z x
z x
Conditional log-odds for a tie to be observed:
Expect density to be same among:• non-attributed nodes (zi = zj = 0)• attributed nodes (zi = zj = 1)
1 2 2 1vary: , s.t. 2
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Numbers of edges and attributed nodes
Beta1
-1.00
-1.50
-1.80
-2.00
-2.20
-2.50
-2.70
-3.00
-3.50
Mea
n nu
mbe
r of
edg
es a
nd a
ttrib
utes
100
80
60
40
20
0
EDGE
Attributed nodes
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Assortative and dissasortative mixing
Beta1
-1.00
-1.50
-1.80
-2.00
-2.20
-2.50
-2.70
-3.00
-3.50
.4
.3
.2
.1
0.0
-.1
Mean density
non-attributed nodes
Mean density betw een
att & non-att nodes
Mean density
attributed nodes
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Acceptance rates
Beta1
-1.00
-1.50
-1.80
-2.00
-2.20
-2.50
-2.70
-3.00
-3.50
.5
.4
.3
.2
.1
0.0
-.1
Acceptance rate
attributes
Acceptance rate
edges
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Clustering
Beta1
-1.00-1.50-1.80-2.00-2.20-2.50-2.70-3.00-3.50
.28
.26
.24
.22
.20
.18
Global clustering
Local clustering
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k- triangles
Beta1
-1.00-1.50-1.80-2.00-2.20-2.50-2.70-3.00-3.50
t-st
atis
tic f
or k
-tria
ngle
s
1.6
1.4
1.2
1.0
.8
.6
.4
.2
0.0
-.2
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Geodesics and clustering
Beta1
-1.00-1.50-1.80-2.00-2.20-2.50-2.70-3.00-3.50
Per
cent
of
sam
ple
120
100
80
60
40
20
0
Short median
geodesic
Short third quartile
geodesic
High clustering
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Small worlds
Beta1
-1.00-1.50-1.80-2.00-2.20-2.50-2.70-3.00-3.50
Per
cent
of
sam
ple
40
30
20
10
0
SW50
SW75
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1= 3.0 Degree distributions
D20D18D16D14D12D10D8D6D4D2D0
12
10
8
6
4
2
0
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1= 3.0 Graph is SW50 (but not SW75)
t-statistic for k-triangles (relative to Bernoulli) = 2.02
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1= 3.0 The graph also has a skewed degree distribution:
Although unusual for graphs in this distribution
Degree
D12D10D8D6D4D2D0
Fre
quen
cy
14
12
10
8
6
4
2
0
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Conclusions for this series of simulations
• The parameter estimates results in approximately equal numbers of attributed and non-attributed nodes– Density within the two sets of nodes are similar and high.
• As the “attribute expansiveness” (β1) parameter becomes more negative, and the “attribute connection” (β2) parameter more positive:– acceptance rate for attributes decreases, – clustering and community structure increases, 3rd quartile geodesics
decrease, but median geodesic remain relatively short
• Graphs with “small world” features, but not with skewed degree distributions, are common within a medium range of the “attribute expansiveness” parameter.
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1 2 1 2Pr( ) exp i i j ij ij i ij i jz z z x x z x z z
1X x
Quadratic effect in no. of attributed nodes
Propensity for attributed nodes to have more partners
Propensity for attributed nodes to be connected
Baseline effect for no. of edges
Second simulation series: 30 node graphs
1 2 1
2
fix: 6; 1; 2.5; 1
vary: to be increasingly positive
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Beta2
4.504.304.204.104.003.903.803.503.00
95%
Con
fiden
ce in
terv
als
80
60
40
20
0
Mean no of edges
Mean no of nodes
Numbers of edges and attributed nodes
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Beta2
4.504.304.204.104.003.903.803.503.00
.6
.5
.4
.3
.2
.1
0.0
-.1
Density
Non-attributed nodes
Density
att and non-att
Density
Attributed nodes
Assortative and dissasortative mixing
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Beta2
4.504.304.204.104.003.903.803.503.00
.4
.3
.2
.1
0.0
-.1
Acceptance rate
attributes
Acceptance rate
edges
Acceptance rates
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Beta2
4.504.304.204.104.003.903.803.503.00
.5
.4
.3
.2
.1
0.0
Global clustering
Local Clustering
Clustering
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Beta2
4.504.304.204.104.003.903.803.503.00
t-st
atis
tic f
or k
-tria
ngle
s
4
3
2
1
0
-1
k- triangles
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Beta2
4.504.304.204.104.003.903.803.503.00
Per
cent
age
of s
ampl
e
120
100
80
60
40
20
0
Short median
geodesic
Short third quartile
geodesic
High clustering
Geodesics and clustering
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Beta2
4.504.304.204.104.003.903.803.503.00
Per
cent
of
dist
ribut
ion
40
30
20
10
0
SW50
SW75
Small worlds
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D20D18D16D14D12D10D8D6D4D2D0
16
14
12
10
8
6
4
2
0
2 =3.0 Degree distributions
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D20D18D16D14D12D10D8D6D4D2D0
12
10
8
6
4
2
0
2 =4.0 Degree distributions
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D20D18D16D14D12D10D8D6D4D2D0
12
10
8
6
4
2
0
2 =4.5 Degree distributions
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2 =4.0Graph is SW50 (but not SW75)
t-statistic for k-triangles (relative to Bernoulli) = 3.98
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Degree
D12D10D8D6D4D2D0
Fre
quen
cy
8
6
4
2
0
Graph is SW50 (but not SW75)
t-statistic for k-triangles (relative to Bernoulli) = 3.98
And with skewed degree distribution
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2 =4.0
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2 =4.0
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Conclusions for the second series of simulations
• The parameter estimates result in a minority of attributed nodes with high internal density, and a majority of non-attributed nodes with lower density.
• As the “attribute connection” (β2) parameter increases, no of edges and attributes increase somewhat, and acceptance rate for attributes decreases, – clustering and community structure increases, 3rd quartile and median geodesic
become longer.
– Degree distributions become skewed, and then bimodal
• Graphs with “small world” features, and with skewed degree distributions, make up a sizeable proportion of distributions with large “attribute similarity” parameter.
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Some final comments
• This “thought experiment” demonstrates that several important global features of social networks may be emergent from attribute-based processes of mutually interacting social influence and social selection:– Short average paths
– High clustering
– Small world properties
– Community structures
– Skewed degree distribution
• Moreover, the models do not presume fixed attributes– although the structural properties begin to emerge as attributes become
“sticky” (changing more slowly)
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Some final comments
• Network models typically assume homogeneity across graphs.– This assumption may not be appropriate to the actual processes that are
generating the network.
– One way that homogeneity may break down is through attribute-based processes.
– Other possibilities include: social settings; geographic proximity
• Network studies may require a careful conceptualisation of “process” to ensure that models are properly specified.– Because process is (usually) local, with global implications, the
possibility of node-level effects should not be excluded.