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ERGM et al

MGT 780 SPRING 2022

STEVE BORGATTI

25-Apr-22

26 APR

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Agenda

Problem statement
ERGM
Relation to QAP
Overview of other stochastic models

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Problem statement

Model structure of networks

Presumably related to micro mechanisms governing tie formation

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Tie dependencies

Some of these mechanisms can be expressed as tie dependencies

Presence/absence of one tie affects probability of another tie

Reciprocity effect means that when i🡪j, there is an increased chance that i🡨j

In short, prob(xji|xij > prob(xji)
There are more mutual dyads than expect by chance

Transitivity effect means that �presence of i🡪j and j🡪k �increases chance of i🡪k

More triangles than expectedc

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Node attributes & dyadic covariates

Node attributes can also be incorporated
Probability of a tie involving node j is higher to the extent that node j is wealthy
Probability of a tie between two nodes is higher if they have the same gender
Probability of a tie between two nodes is higher if they are physically closer together

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Model assumptions

We assume the observed network is the result of a set of social processes that favor the formation of certain ties and hinders the formation of others.

The process maximized an unseen objective function by forming / not forming certain ties

The objective function consists of a set of parameters that indicate strength and direction of certain tendencies, e.g.,

A mild tendency toward mutual (reciprocated) dyads
A strong tendency toward closed triangles

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Evolutionary process

Suppose the objective function has

A strong preference for closed triangles
A general dislike for forming ties (keep density low)

Consider a possible tie between holly & john�vs holly & bill

Holly-John creates no (closed) triangles
Holly-Bill creates 3 triangles
The selection process will prefer�the Holly-Bill tie because it maxes�objectives

If objective function has negative�parameter for triangles, the holly-john tie becomes the more probable tie

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Formal model

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What is ERGM?

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Converting problem to a logistic regression

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Thanks to Filip Agneessens

Can cross out the Ks

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Change in graph statistics that result from tie present vs absent

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Odds <- change statistics

Take the natural log of this equation to remove the e^power , giving logit:

Which results in a “simple” logistic regression equation

(or would be if the cases were independent – i.e. not “conditional on the rest of the graph, except for the parts included in the model”)
where��is the difference between graph statistics ( “change statistics” ) caused by adding a single tie

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The “logit” of a probability is the log of the associated odds ratio

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Logits to probabilities

The logit of a probability is the natural log of the associated odds
The odds of an event = the probability of it happening divided by probability it doesn’t happen

Odds = p/(1 – p)

Can convert odds to probabilities:

Multiply both sides by 1-p

odds*1 – odds*p = p

Add adds*p to both sides

odds = p + odds*p = p(1 + odds),

Divide both sides by 1 + odds

odds/(1+odds) = p

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So, a model written in terms of odds can ultimately be converted back to probabilities

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Interpreting parameters

Simplest possible ERGM:

Logit(x_ij = 1) = b0*(Δ edges)

Edges is number of edges in network
Δ edges is change in number of edges when tie is present vs absent

Obviously, Δ edges is always 1

So, model says log odds of a tie = b0*1

Odds = exp(b0)
Probability of a tie = exp(b0)/(1 + exp(b0) = density of the network

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Simplest ERGM

B0 = -1.609*1, which is log odds of a tie,
Odds of a tie = exp(-1.609) = 0.200088
Prob of a tie is odds/(1+odds) = .2/(1+.2) = 0.1667

which is the density of padgm

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Estimate Std. Error MCMC % p-value

edges -1.609 0.245 NA <1e-04 ***

Using PAGM data

Logit(Xij =1) = b0(Δedges)

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Adding closure parameter

Logit(Xij =1) = b0(Δedges) + b1(Δtriangles)
b0 is like an intercept

Not normally interpreted
~ density of network when no other parameters present
Typically negative

Parameter b1 is the increase in the log odds of a tie brought about by an increase of 1 triangle in the network due to adding a tie

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Padgm results

For a tie that will create

0 triangles, such as a tie between a and b

the conditional log-odds is: -1.76, and probability of a tie is 0.146

1 triangle, such as a tie between b and d

the logit is: -1.76 + 0.091 = -1.67, and probability of a tie is 0.158

2 triangles, such as a tie between b and c

the logit is -1.76 +0.091*2= -1.58 (prob of tie is 0.17)

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Logit(Xij =1) = b0(Δedges) + b1(Δtriangles)

The conditional log-odds of two actors having a tie is:

-1.764*(change in the number of ties) + 0.091*(change in number of triangles)

Estimate Std. Error MCMC % z value Pr(>|z|)

edges -1.76367 0.33136 0 -5.323 <0.0001 ***

triangle 0.09115 0.15760 0 0.578 0.563 NS

---

Note: your results will vary

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Degree variance (centralization)

2-stars

Nodes with high degree create many 2-stars

A network with low degree variance will have negative 2-star parameter

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Parameter	Name	Interpretation	Image
Arc	Edges	This is a baseline propensity for tie formation
Reciprocity	Mutual	This is often positive, suggesting that reciprocated ties are very likely to be observed for positive affect networks
Simple connectivity	twopath	This measures the extent to which actors who send ties also receive them. It controls for the correlation between in and out degree. It is often negative.
Popularity spread	gwidegree	popularity spread. indicates a network with high in-degree nodes (i.e., centralized)—gwidegree is opposite
Activity spread	gwodegree	popularity spread. indicates a network with high out-degree nodes (i.e., centralized)—gwiegree is opposite
Triangulation	gwesp	A positive effect indicates there is a high degree of closure, or multiple clusters of triangles in the data
Cyclic closure	ctriple	A negative effect indicates tendencies against cyclic triads (sometimes this is interpreted as a tendency against generalized exchange or generalized reciprocity)
Multiple connectivity	gwdsp	2-path in the networks. A negative estimate in conjunction with positive triangulation indicates that 2-paths tend to be closed (i.e., triangles)

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Directed effects

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Exogenous effects

Here, exogenous means ‘not calculated from the network itself’
Node attributes

Wealthier nodes might have more ties
Women may have more ties

Similarities

Nodes with similar wealth (smaller diff) more likely to have tie
If nodes are the same gender, they may be more likely to have a tie

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Adding a node characteristic (exogenous var)

We have the wealth of each node, and believe wealthier nodes are more likely to connect
Logit(Xij =1) = b0(Δedges) + b1(wealth of pair)

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Estimate Std. Error MCMC % p-value

edges -2.59493 0.53606 NA <1e-04 ***

nodecov.wealth 0.01055 0.00467 NA 0.026 *

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Adding a node characteristic (exogenous var)

Log odds of a tie is -2.59*{change in the number of ties} + 0.01*{wealth of node i} + 0.01*{wealth of node j}
for a tie between two nodes with minimum wealth

the conditional log-odds is: -2.59 + 0.01*(3+3) = -2.53. Prob of a tie is 0.07

for a tie between two nodes with maximum wealth: -2.59 + 0.01*(146+146) = 0.33. prob of tie .58
for a tie between the node with maximum wealth and the node with minimum wealth:�-2.59 + 0.01*(146+3) = -1.1 prob of tie is 0.25

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Estimate Std. Error MCMC % p-value

edges -2.59493 0.53606 NA <1e-04 ***

nodecov.wealth 0.01055 0.00467 NA 0.026 *

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ERGM Goodness of fit

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ERGM Goodness of fit – cont.

We use two sets of statistics
First, all statistics actually used in the model

# of triangles, # of reciprocated ties, # of cycles, # of 2-paths etc
The simulated networks had better reproduce these statistics very well

Second, additional statistics not explicitly included in the model

Number of nodes with degree 1, 2, 3 … etc
Number of pairs of nodes with distance 1, 2, 3 … etc
Simulated networks should reproduce these fairly well

Note that with only endogenous parameters, ERGMs do not predict specific ties at all well

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ERGM in R

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Download this script:

https://tinyurl.com/ergmscript

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Padgm example

> padgm = Padgett_FlorentineFamilies$Marriage
>netpadgm = as.network(padgm)
> res = ergm(formula = netpadgm ~ edges)
> summary(res)
> odds = exp(-1.609)
> odds/(1 + odds)
> xDensity(padgm)

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Padgm triangles

> res = ergm(formula = netpadgm ~ edges + triangle)
> summary(res)�

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Goodness of fit

res = ergm(formula = netpadgm ~ edges + triangle)
summary(res)
gof(res)

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Reflections on ERGM

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General comments

Much more complicated than I indicated today
It’s not really a dyadic model

Meant to characterize network, not predict specific ties
More so, though, when node covariates are used

Model estimation sometimes (often?) fails

It’s not that some parameter turns out not to predict – you can’t run model

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Endogenous effects & exogenous covariates

Endogenous variables are all variables that can be calculated from the network itself
Exogenous variables are things that require outside information

Node attributes, such as gender
Other tie types, such as friendship ties when modeling advice ties

Endogenous does not mean that the effects reflect some process that is inherent in networks

It’s people (organizations, whatever) forming the ties

Beware of claims that these effects are “self-organizing processes”

This is a misconstrual of the idea that larger patterns in a network may emerge from local tendencies such as transitivity

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Emergence & self-organization

Note clumpiness of graph
Networks with many �transitive triples tend to be clumpy
So clumpiness can be seen as an �emergent or self-�organizing property
But it is not �transitivity that is �self-organizing

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Parameters and social processes

We assume that the observed network is the result of a long-term sequence of tie changes governed by social processes or mechanisms
We use the ERGM parameters to infer the social processes
There isn’t a 1-to-1 correspondence between social processes and ERGM parameters

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The triangle parameter

Positive significant triangle parameter indicates a tendency toward transitivity (i🡪j, j🡪k, and i🡪k)
Transitivity is not a social process. It is the outcome of a social process

And not just one

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Triangle parameter – cont.

Mechanism 1 - Balance theory (Heider; Festinger)

If A likes B, and B likes C, it induces cognitive dissonance in A if A doesn’t also appreciate C

May also cause conflict

So A will tend to like C, or drop B. Either way, we should see many triples in which A--B, B--C, and A--C

Mechanism 2 – Opportunity

If A makes friends with B, there is a good chance that B will introduce A to B’s friends, closing the gap

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Other models

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How does ergm relate to qap?

Qap is a dyadic model that predicts presence/absence/strength of ties between each pair of nodes as a function of other dyadic variables

Friends 🡨 distance + samegender

Ergm is fundamentally a whole network model that models the overall pattern as the result of a combination of micro-processes

Not ideal for predicting specific ties

They intersect if you use ergm with a dyadic covariate

P(Y = y) = exp(b0*Edges + b1*Triangles + b2*nodematch(gender) + b3*dyadxcov(distance))

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When using a dyadic covariate in ergm …

When using a dyadic covariate in ergm, then:

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QAP	ERGM
Conceptually simple Easy to implement Easily customizable – any standard statistic can be recast in QAP terms Parameters identical to standard regression Handles all sources of interdependence, even unknown ones Difficult/impossible to quantify these sources of interdependence Dependent variable can be anything -- e.g., correlation coefficients Goodness of fit measured at the tie level�-- how well did we predict each tie?	Statistically complex Difficult to estimate – often fails to converge Effects to be tested must be pre-programmed by an expert Parameters can be hard to interpret An unknown source of interdependence can invalidate the model Provides rich description of sources of inter-dependence, interpreted as tendencies toward certain micro-configurations Dependent variable must be a network and, in practice, binary Goodness of fit measured at the level of whole network statistics like degree dist.

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Longitudinal extensions to ERGM

TERGM (temporal ergm)

Essentially an ERGM in which the network at time t-1 is included as a dyadic covariate
Assumes tie changes are independent of each other

STERGM (separable tergm)

Separately models tie formation and tie dissolution

LERGM (longitudinal ergm)

Similar to SAOM (covered next)
Continuous time model in which change is made one dyad at a time

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SAOMs

Stochastic actor-oriented models

Implemented as rSiena package. So, often called Siena models

Designed to model process of change in ties

For directed networks, it is assumed that each actor only has control over their own outgoing ties
For undirected networks, actor proposes a tie to an alter, and the alter may or may not agree
The ties have inertia: they are states rather than events

Models have similar parameters to ERGMs

Endogenous effects - # of 2-paths, triangles, etc
Exogenous effects – node attributes, other relations, etc.

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SAOM process

At each time point, randomly chosen node given opportunity to make a tie, keep a tie, or dissolve a tie with some other node
The actor makes choices that maximize an objective function based on the state of the network that would be true after the change
Actors act without coordination

But they change each other’s environment, so they are interdependent

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	MR-QAP	ERGM	LERGM	SAOM	REM
Summary description	Multiple regression in which cases are dyads and dependent variable is presence/absence or strength of tie. Significance assessed via permutation test	Odds of a tie modeled as a function of the contribution of that tie to graph statistics reflecting prevalence of micro-configurations	Odds of a tie modeled as a function of the contribution of that tie to graph statistics reflecting prevalence of micro-configurations	Actors make choices to add/drop ties to maximize utility function
What is modeled	Presence/absence or strength of ties	- Probability of observed network given set of network statistics (e.g., counts of micro-configurations)�- Presence/absence of ties as equilibrium outcome of tie-wise change process	Presence/absence of ties	- Presence/absence of ties�- (optional) Change in node-level attributes (e.g., behaviors)	Occurrence of relational events
Kinds of effects	Any variables that can be expressed in dyadic form, including both endogenous and exogeneous variables, as well as attribute-based variables�2. Endogeneous variables (e.g., whether a tie completes a transitive triple)	1. Exogenous covariates (e.g., similarity of interests)�2. Endogenous variables (e.g., # of transitive triples that would be added if tie were added)	1. Exogenous covariates (e.g., similarity of interests)�2. Endogenous variables (e.g., # of transitive triples that would be added if tie were added)	1. Exogenous covariates (e.g., similarity of interests)�2. Endogenous variables (e.g., # of transitive triples that would be added if tie were added)
Meaning of parameters	Positive parameter for X indicates that larger values of X are associated with greater probability or strength of tie	1. Exogenous variables: change in odds of tie given unit increase in X�2. Endogenous variables: Positive parameter for X (e.g., transitivity) indicates that odds of a tie is higher to the extent its presence contributes to change in X (e.g., change in # of transitive triples)	1. Exogenous variables: change in odds of tie given unit increase in X�2. Endogenous variables: Positive parameter for X (e.g., transitivity) indicates that odds of a tie is higher to the extent its presence contributes to change in X (e.g., change in # of transitive triples)	1. Exogenous variables: change in odds of tie given unit increase in X�2. Endogenous variables: Positive parameter for X (e.g., transitivity) indicates that odds of a tie is higher to the extent its presence contributes to change in X (e.g., change in # of transitive triples)
Approach to change	Implicit. Model gives effect of X on Y. Hence, we predict change in Y given a change in X. Variants can model Y(t)-Y(t-1) or control for Y(t-1) when modeling Y(t)	Implicit. Model gives effect of X on Y. Hence, we predict change in Y given a change in X. TERGM variant can control for Y at t-1	Explicit. Continuous time framework updates dyadic dependencies as each dyad changes. Ordering of tie changes makes a difference	Explicit. Continuous time framework updates dyadic dependencies as each dyad changes. Ordering of tie changes makes a difference
Approach to non-independence of observations in dyadic data	Control for sources of dependence via permutation method	Explicitly model sources of dependence between dyads	Explicitly model sources of dependence between dyads	Explicitly model sources of dependence between dyads; dependencies can be asymmetric (dependency of i-->j on u-->v does not imply the reverse)
Goodness of fit	Dyadic. Difference, across all dyads, between observed Yij and predicted Yij	Whole network. Do networks simulated from the model have the same overall network statistics as the observed network?	Whole network. Do networks simulated from the model have the same overall network statistics as the observed network?	Whole network. Do networks simulated from the model have the same overall network statistics as the observed network?
Temporal variants	- Can include lagged versions of both Xs and Y as controls�- Can explicitly model Y(t) - Y(t-1)	- Include lagged versions of both Xs and Y as controls (aka TERGM)�- Model tie formation and dissolution independently (aka STERGM)	Not applicable	Not applicable	Not applicable