Exponential random graph (p*) models for social networks

Workshop

Harvard University

February 2002

Philippa Pattison

Garry Robins

Department of Psychology

University of Melbourne

Australia

Plan for the workshop

Model construction: dependence graphs

Dyadic independence and Bernoulli graph models– Example: The network structure of a US law firm

Markov random graphs– Example: Mutual work ties among the lawyers

Estimation– Pseudo-likelihood estimation

– Monte Carlo maximum likelihood

Incorporating individual level variables– Directed dependence graphs

– Social selection models

– Example: Advice ties among the lawyers

Further steps

General framework for model construction

1. Regard each network tie as a random variable (often binary)

Xij = 1 if there is a network tie from person i to person j

= 0 if there is no tie

for i, j members of some set of actors N.

A directed network: Xij and Xji are distinct.

A non-directed network: Xij = Xji

X … matrix of all variables

x … matrix of observed ties (the network)

General framework for model construction

2. Formulate a hypothesis about interdependencies and construct a dependence graph

3. The Hammersley-Clifford theorem produces a general model : – each parameter corresponds to a configuration

in the network

4. Consider homogeneity constraints: should some parameters be equal?

5. Estimate model parameters

Dependence graphs

Spatial statistics (Besag, 1974)

Spread of a contagious disease in a field of plants.– Whether or not a plant has the disease in part

depends on whether nearby plants have the disease.

The probability of a plant having the disease is conditionally dependent on whether neighbouring plants have the disease.

What counts as neighbouring plants?

How to represent neighbours?

For example: a 2-dimensional lattice

1 2 3 4

65

9 10

7 8

11 12

More abstractly … a dependence graph!

Random variables

Zi = 1 if plant i has the disease

Zi = 0 if plant i does not have the disease

Z1 Z2Z3 Z4

Z6

Z5

Z9 Z10

Z7Z8

Z11Z12

The lattice then represents the contingencies among thevariables.

We can have neighbouring variables, just as we haveneighbouring plants.

And then ...

Besag (1974) used the Hammersley-Clifford theorem to derive a probabilistic description of the entire system

• in terms of the hypothesised local dependencies.

Cliques of the dependence graph.

• The sufficient statistics of the model are all the combinations of variables that are all neighbours of each other.

Cliques of the dependence graph

Z7, Z8, Z12 are all neighbours of each other.

Whether plant 7 has the disease depends not only on whether plant 8 or plant 12 has the disease, but also on whether both 8 and 12 have the disease simultaneously.

Z1 Z2Z3 Z4

Z6

Z5

Z9 Z10

Z7Z8

Z11Z12

For social networks

The dependence graph represents the contingencies among network variables Xij.

The cliques of the dependence graph represent local configurations in the network.

(a configuration is a small subgraph of the network)

There is one parameter in the model for each clique.

(ie one parameter for each configuration.)

The model then is an expression of the probability of the network expressed in terms of local configurations.

Example: Dyadic independence models

Hypothesis about a possible local process: • Whether person i considers person j a

friend is contingent on whether person j considers person i a friend.

Cliques of the dependence graph

{Xij}

{Xji}

{Xij , Xji}

Dependence graph:

Xij Xji

i j

i j

i j

The Hammersley-Clifford theorem

Pr(X = x) = p*(x) = (1/c) exp{all cliques AzA}

where:the summation is over all cliques A;

zA = xijA xij is the network statistic corresponding to the clique A;

A is the parameter corresponding to clique A;

c = X exp{AAzA(x)} is a normalising constant

(Besag, 1974)

For dyadic independence

Cliques: {Xij}, {Xji}, {Xij, Xij}

So: Pr(X = x) = (1/c) exp{all cliques AzA}

– with zA = A xij

becomes:

Pr(X = x) = (1/c) exp{i,j ij xij + i,j ij,ji xij xji }

Homogeneity: ij= ; ij,ji =

Pr(X = x) = (1/c) exp{ i,j xij + i,j xij xji }

= (1/c) exp{ L + M }where L is no. of ties, M is no. of mutual ties

HomogeneityThe model cannot be estimated (unidentifiable) unless

some parameters are equated:– ie, assume that certain effects are the same for all (or

at least large parts) of the network

– eg, a single tendency for mutuality across the entire network.

Homogeneity of isomorphic network configurations– parameters equated for the same types of

configuration ignoring the numbering on the nodes

– statistics become the counts of the configurations in the network

– parameter interpretation: tendency for the configuration to be present in the network, given the other configurations.

Isomorphic configurations within blocks

Bernoulli graphs: the simplest dependence structure

Dependence assumption: all ties are independent.

Dependence graph:

Xij Xkl

Cliques: {Xij} i j

Hammersley-CliffordPr(X = x) = (1/c) exp{i,j

ij xij }

Homogeneity: ij= Pr(X = x) = (1/c) exp i,j xij

Pr(X = x) = (1/c) exp ( L )where L is no. of ties.

Bernoulli block models

Suppose actors are either in block 1 or 2

Hammersley-Clifford: Pr(X = x) = (1/c) exp{i,j ij xij }

Block homogeneity:ij = 11 if i and j both in block 1

ij = 12 if i in block 1 and j in block 2, etc.

Pr(X = x) = (1/c) exp{11 L11+12 L12+21 L21+22 L22}

where L rs is the number of edges from block r to block s.

Example: The network structure of a US law firm(Lazega & Pattison, 1999)

Respondents: All 71 lawyers Blocks: 36 partners (block 1), 35 associates (block

2)

Coworker and Advice ties:

The general question: – How is collective participation organised?

– Small, flexible, and temporary work-teams must be able to form quickly and cooperate efficiently in order to react to complex and non-standard problems.

– Regularities in local patterns of exchange provide one possible solution.

Bernoulli and dyad-independent models for advice ties

a. Homogeneous Bernoulli:

Pr(X = x) = (1/c) exp ( L )

b. Bernoulli blockmodel:

Pr(X = x) = (1/c) exp{11 L11+12 L12+21 L21+22 L22}

c. Homogeneous Dyad-independent:

Pr(X = x) = (1/c) exp{ L + M }

d. Dyad-independent block

Pr(X = x) = (1/c) exp{r,s=1,2 rs Lrs+ M }

e. Dyad-independent block with block reciprocity parameters:

Pr(X = x) = (1/c) exp{r,s=1,2 (rs Lrs+ rs Mrs )}

Results:Using pseudolikelihood estimation (to come)

PL deviance is a measure of fitMAR = Mean absolute residual

PL deviance MAR Parameters

a. Bernoulli model 4677.8 .294 1

b. Bernoulli block 4341.1 .276 5

c. Homogeneous dyad-independent model 4391.1 .275

2

d. Dyad-independent block model 4072.6 .258

5

e. Dyad-indpt block with block reciprocity

4071.2 .257 8

Parameter estimates for model c

Advice ties: (P=Partner, A=Associate)

Parameter MPLE

11 -1.34 Density of ties in P

12 -3.51 Ties from P to A

21 -1.53 Ties from A to P

22 -1.98 Density of ties in A

1.52 Reciprocity

Markov Random Graphs

Dyadic independence is an unrealistic assumption– Both theoretically and empirically.

Markov Dependencies (Frank & Strauss, 1986)

– Assume that a tie from i to j is contingent only on other ties involving person i and person j.

• network ties assumed conditionally dependent if and only if they share a common actor

– There is an edge (Xij, Xkl) in the dependence graph if and only if {i,j} {k,l}

Markov dependence graph for a directed network

Cliques of size 1 or 2 :

{ Xij }

{ Xij , Xji }

{ Xij , Xik }

{ Xij , Xjk }

{ Xij , Xkj }

Xjk

Xij

Xik

Xji

Xkj

Xki

Markov dependence graph for a directed network

Cliques of size 3 :

1. Stars

{ Xij , Xik , Xil }

etc2. TriadsTransitive triads

{ Xij , Xjk , Xik }

3-Cycles{ Xij , Xjk , Xki }

etc

Xjk

Xij

Xik

Xji

Xkj

Xki

Parameters for a homogeneous Markov random directed graph model

Density (15)

Two-in-stars (14)

Two-mixed-stars (13)

Two-out-stars (12)

Reciprocity (11)

Three-Cycles (10)

Transitive triads (9)

8

7

plus higher order stars

NB: Bernoulli models- Density only; Dyad independent - Density plus reciprocity

For non-directed graphs, parameters are simpler

Density or edge ()

Two-stars (2)

Triangle ()

Three-stars (3)

and higher order stars…

Interpretation of Markov random graph models

Broadly, positive parameter indicates a high occurrence of the associated configuration.

But the effects are marginal to each other.– A positive parameter indicates a greater number

of the configuration than expected, given the observation of other configurations

– Interpret a higher order parameter in relation to its lower order constituents.

– e.g. interpret cyclic triads in relation to 2-mixed-stars (2-paths): Positive cyclic parameter suggests presence of more cycles than expected from the number of 2-paths.

Example: Markov random graph model for mutual work ties among the

lawyers

Model parameters -2LPL MAR

_________________________________________

1. Edge (density) 2119.0 .258

2. edge, 2-star, 3-star, triangle 1760.8 .213

_________________________________________

parameter configuration parameter estimate

– 2.785 (.369)

2 – 0.019 (.030)

3 0.002 (.002)

0.482 (.035)

Lazega AdviceWork.ppt

Pseudo-likelihood estimation:an approximate technique

Conditional form of the Hammersley-Clifford theorem:

where:

(1) The sum is over all cliques A that contain Xij;

(2) A is the parameter corresponding to clique A;

(3) dA(ij) is the change statistic

• the change in the value of the network statistic zA(x) when xij changes from 1 to 0

Note: this version of the theorem takes the form of a conditional logistic regression.

( )

( 1 " ")log ( )

( 0 " ")ij

ijA A

A Xij

P X restd ij

P X rest

Calculating the change statistic

Eg: Markov random graph model for a nondirected network.– Graph statistics: edges, 2-stars, 3-stars, triangles.

Set up standard logistic regression file:– “Cases” are each possible tie (i, j).

– Variables: Xij , edges, 2-stars, 3-stars, triangles.

Eg, to calculate 2-stars value for each (i, j):– count 2-stars in the network when Xij =1

– count 2-stars in the network when Xij =0

– take the difference.

Predict Xij =1in a standard logistic regression from the other variables

Some warnings

This looks like a logistic regression: It is not!!– The standard errors are at best approximate

(probably too small).

– Do not rely on the Wald statistic.

– You cannot assume the pseudo-likelihood deviance is asymptotically chi-squared

But the PL deviance still indicates how well the model fits the observed network. – If a variable does not contribute “substantially”

to the PL deviance, maybe exclude it?

– But maybe keep lower order variables in the model if higher order variables are important.

– Report PL deviance and mean absolute residual.

Monte Carlo Maximum Likelihood Estimation

Some recent developments in this area, eg Snijders 2002– Degenerate regions in the parameter space

– So far, studies only involve simple non-directed Markov random graph models

– Possible importance of model specification

Implications for Pseudolikelihood estimation– Do not treat PL as hypothesis testing; it is more

exploratory.

– More work is needed to determine types of models, and regions of parameter space, for which we can be confident about PL estimation.

– Nevertheless, PL is convenient, and for complex models is still our only available estimation technique

Incorporating individual level variables

Attribute variables:e.g. sex, rank, attitudes– Yi = 1 if actor i possesses the attribute (e.g.

actor i is female)

– Yi = 0 otherwise

Example: Social selection models (Robins, Elliott & Pattison, 2001).

Similarity or homophily hypothesis: There is a propensity for a tie to develop between actors with similar attributes.

Directed dependence graphs

One type of variable predictive of another– Two block chain graphs

Attribute variables Network variables

(possibly some non-directed dependencies among network variables)

Example: Dyadic independence Markov attribute models

Yi

Yj

Xij

Xji

Configurations for Bernoulli graph attribute

models:

Additional configurations for dyadic

independence models

Markov graph - Markov attribute models

Xij

Xji

Xki

Xik

Xjk

Xkj

Yi

Yj

Additional configurations

Markov graph - partial dependent attribute models

A possible tie between two people is contingent on a third person’s attribute if the third person is tied to at least one of the first two.

E.g. k’s attribute possibly shapes whether there is a tie from i to j

i j

k

Lawyers’ advice network with status attribute

Model - 2 LPL Parameters MAR

Bernoulli graph models

1.No attribute 4678 1 0.295

2.Markov attribute 4341 4 0.276

Dyadic independence models

3.No attribute 4391 2 0.275

4.Markov attribute 4071 7 0.257

Markov graph models

5.No attribute 2855 7 0.175

6.Markov attribute 2819 15 0.173

7.Partial dpt. attribute 2693 39 0.166

8.Reduced part. dpt. att. 2749 15 0.169

____________________________________________________

Configurations and estimates for reduced partial dependence model

-2.13 0.77

-2.06 2.44

1.60

-0.03

-0.15 -0.22

0.10 0.13

0.05 0.09

- 0.09

-0.14 0.29

Other applications of directed dependencies

• Social influence models– Robins, Pattison & Elliott (2001)

• Discrete time models– Robins & Pattison (2001)

• Tripartite graphs– Mische & Robins (2002)

Further stepsPattison & Robins (2002)

Partial dependence models:

observed ties create new neighbourhoods – longer range dependencies

– longer cycles (eg generalized exchange)

Further stepsPattison & Robins (2002)

Setting structures:A setting: a subset of possible ties

potential "site of social action" e.g., focus, Feld (1981)network-domain (White, 1995)

Limits the scope of the dependencies in the dependence graph:

Xjk

Xij

Xik

Xji

Xkj

Xki