Joint social selection and social influence models for networks: The interplay of ties and attributes.

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Joint social selection and social influence
models for networks The interplay of ties and
attributes.

• Garry Robins
• Michael Johnston
• University 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?
• Develop a model that incorporates both social
  selection and social influence processes.
• Which global properties of networks are
  important?
• 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)

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 ? is an edge
parameter. an exponential random graph (p) model.
a homogeneous model (node homogeneity) p and ?
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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In this example, actor attributes are important
to tie formation
Social selection
Yellow Jewish Blue Arab (Kalish, 2003)
Exogenous attributes affect network ties
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Binary variables Xij network ties Zi
actor attributes
Exogenous attributes affect network ties
Zi
Xij
Zj
Robins, Elliott Pattison, 2001
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Effects in the model
Baseline edge effect irrespective of attributes
Propensity for actors with attribute z1 to have
more partners
Propensity for ties to form between actors who
both have attribute z1
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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Social influence Are actor attributes
influenced by fixed network structure?
Robins, Pattison Elliott, 2001
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Social influence Are actor attributes
influenced by fixed network structure?
A cutpoint
Exogenous network ties affect attributes
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Binary variables Xij network ties Zi
actor attributes
Exogenous network ties affect attributes
Zi
Xij
Zj
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Effects in the model
Baseline effect for number of attributed nodes
(z1)
Propensity for attributed nodes to have more
partners
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)
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Effects in the model
Baseline effect for number of attributed nodes
(z1)
Propensity for attributed nodes to have more
partners
Propensity for attributed nodes to be connected
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Why should attributes or ties be exogenous?
Friendship network for training squad in 12th
week of training (Pane, 2003) Green
detached Yellow team oriented Red positive
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Models for joint social selection/social influence
Xij
Zj
Zi
Xik
Zk
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Effects in the model
Quadratic effect in no. of attributed nodes
Baseline effect for no. of edges
Propensity for attributed nodes to have more
partners
Propensity for attributed nodes to be connected
Equivalent (blockmodel) parameterization
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Change statistics
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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An example network (without attributes)
Confiding (trust) network (Pane, 2003)
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Observed networks Path lengths
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.
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Observed networks Clustering
Many observed networks have high clustering
small worlds
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
(sd0.03)
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Observed networks Clustering
Many observed networks have high clustering
small worlds
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
(sd0.04)
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Observed networks Clustering
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
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Observed networks Degree distribution
Many observed networks have skewed degree
distributions as is the case for the confiding
network
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Observed networks Higher order
clustering k-triangles (Snijders, Pattison,
Robins Handcock, 2004)
1-triangle
2-triangle
3-triangle
Alternating k-triangles
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
The k-triangle statistic is not simply equivalent
to global clustering
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Summary Some global features not uncommon in
observed networks

• 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

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• 3. Simulate the model to see whether global
  properties can be reproduced

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Simulation of the model

• Use the Metropolis algorithm
• procedure similar to Robins, Pattison Woolcock
  (in press)
• Typically 300,000 iterations
• reject 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

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Quadratic effect in no. of attributed nodes
Baseline effect for no. of edges
Propensity for attributed nodes to have more
partners
Propensity for attributed nodes to be connected
First simulation series 30 node graphs
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Change statistics
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)

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Numbers of edges and attributed nodes
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Assortative and dissasortative mixing
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Acceptance rates
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Clustering
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k- triangles
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Geodesics and clustering
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Small worlds
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Degree distributions
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Graph is SW50 (but not SW75) t-statistic for
k-triangles (relative to Bernoulli) 2.02
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The graph also has a skewed degree
distribution Although unusual for graphs in this
distribution
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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 (ß1) parameter
  becomes more negative, and the attribute
  connection (ß2) 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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Quadratic effect in no. of attributed nodes
Baseline effect for no. of edges
Propensity for attributed nodes to have more
partners
Propensity for attributed nodes to be connected
Second simulation series 30 node graphs
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Numbers of edges and attributed nodes
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Assortative and dissasortative mixing
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Acceptance rates
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Clustering
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k- triangles
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Geodesics and clustering
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Small worlds
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Degree distributions
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Degree distributions
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Degree distributions
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Graph is SW50 (but not SW75) t-statistic for
k-triangles (relative to Bernoulli) 3.98
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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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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 (ß2) 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.