# Overview of Social Network Models

## Models for Social Networks

Today, we will take a brief (very brief) look at the following models:

• Spatial Auto-correlation Models

• Exponential Random Graph Models

• Stochastic Actor Oriented Models

• Network Matching

First, a brief introduction

## Context: Identifying Contagion in Networks

• Ideally we would like to estimate a model in the form of $Y_i = f\left(\mathbf{X}, Y_{-i}, \mathbf{G}; \theta\right)$

where $$Y_{-i}$$ is the behavior of individuals other than $$i$$ and $$\mathbf{G}$$ is a graph.

• The problem: So far, traditional statistical models won’t work, why? because most of them rely on having IID observations.

• On top of that, when it comes to explain behavior, “Homophily and Contagion Are Generically Confounded” (Shalizi and Thomas 2011).

• This has lead to the development of an important collection of statistical models for social networks.

## Spatial Autocorrelation Models (SAR a.k.a. Network Auto-correlation Models)

• Spatial Auto-correlation Models are mostly applied in the context of spatial statistics and econometrics.

• A wide family of models, you can find SA equivalents to Probit, Logit, MLogit, etc.

• The SAR model has interdependence built-in using a Multivariate Normal Distribution:

\begin{align} Y = & \alpha + \rho W Y + X\beta + \varepsilon,\quad\varepsilon\sim MVN(0,\sigma^2I_n) \\ \implies & Y = \left(I_n -\rho W\right)^{-1}(\alpha + X\beta + \varepsilon) \end{align}

Where $$\rho\in[-1,1]$$ and $$W=\{w_{ij}\}$$, with $$\sum_j w_{ij} = 1$$

## Spatial Autocorrelation Models (SAR) (cont.)

• This is more close than we might think, since the $$i$$-th element of $$Wy$$ can be expressed as $$\frac{\sum_j a_{ij} y_j}{\sum_j a_{ij}}$$, what we usually define as exposure in networks, where $$a_{ij}$$ is an element of the $$\{0,1\}$$-adjacency matrix .

• Notice that $$(I_n - \rho W)^{-1} = I_n + \rho W + \rho^2 W^2 + \dots$$, hence there autocorrelation does consider effects over neighbors farther than 1 step away, which makes the specification of $$W$$ no critical. (see LeSage 2008)

• These models assume that $$W$$ is exogenous, in other words, if there’s homophily you won’t be able to use it!

• But there are solutions to this problem (using instrumental variables).

## Exponential Random Graph Models (ERGMs)

The distribution of $$\mathbf{Y}$$ can be parameterized in the form

$\Pr\left(\mathbf{Y}=\mathbf{y}|\theta, \mathcal{Y}\right) = \frac{\exp{\theta^{\mbox{T}}\mathbf{g}(\mathbf{y})}}{\kappa\left(\theta, \mathcal{Y}\right)},\quad\mathbf{y}\in\mathcal{Y} \tag{1}$

Where $$\theta\in\Omega\subset\mathbb{R}^q$$ is the vector of model coefficients and $$\mathbf{g}(\mathbf{y})$$ is a q-vector of statistics based on the adjacency matrix $$\mathbf{y}$$.

• Model (1) may be expanded by replacing $$\mathbf{g}(\mathbf{y})$$ with $$\mathbf{g}(\mathbf{y}, \mathbf{X})$$ to allow for additional covariate information $$\mathbf{X}$$ about the network. The denominator,

$\kappa\left(\theta,\mathcal{Y}\right) = \sum_{\mathbf{z}\in\mathcal{Y}}\exp{\theta^{\mbox{T}}\mathbf{g}(\mathbf{z})}$ 0
• Is the normalizing factor that ensures that equation (1) is a legitimate probability distribution.

• Even after fixing $$\mathcal{Y}$$ to be all the networks that have size $$n$$, the size of $$\mathcal{Y}$$ makes this type of models hard to estimate as there are $$N = 2^{n(n-1)}$$ possible networks! (Hunter et al. 2008)

How does ERGMs look like (in R at least)

network ~ edges + nodematch("hispanic") + nodematch("female") +
mutual +  esp(0:3) +  idegree(0:10)

Here we are controlling for:

• edges: Edge count,
• nodematch(hispanic): number of homophilic edges on race,
• nodematch(female): number of homophilic edges on gender,
• mutual: number of reciprocal edges,
• esp(0:3): number of shared parterns (0 to 3), and
• indegree(0:10): indegree distribution (fixed effects for values 0 to 10)

(See Hunter et al. 2008).

## Separable Exponential Random Graph Models (a.k.a. TERGMs)

• A discrete time model.

• Estimates a set of parameters $$\theta = \{\theta^-, \theta^+\}$$ that capture the transition dynamics from $$\mathbf{Y}^{t-1}$$ to $$\mathbf{Y}^{t}$$.

• Assuming that $$(\mathbf{Y}^+\perp\mathbf{Y}^-) | \mathbf{Y}^{t-1}$$ (the model dynamic model is separable), we estimate two models: \begin{align} \Pr\left(\mathbf{Y}^+ = \mathbf{y}^+|\mathbf{Y}^{t-1} = \mathbf{y}^{t-1};\theta^+\right),\quad \mathbf{y}^+\in\mathcal{Y}^+(\mathbf{y}^{t-1})\\ \Pr\left(\mathbf{Y}^- = \mathbf{y}^-|\mathbf{Y}^{t-1} = \mathbf{y}^{t-1};\theta^-\right),\quad \mathbf{y}^-\in\mathcal{Y}^-(\mathbf{y}^{t-1}) \end{align}

• So we end up estimating two ERGMs.

## Latent Network Models

• Social networks are a function of a latent space (literally, xyz for example) $$\mathbf{Z}$$.

• Individuals who are closer to each other within $$\mathbf{Z}$$ have a higher chance of been connected.

• Besides of estimating the typical set of parameters $$\theta$$, a key part of this model is find $$\mathbf{Z}$$.

• Similar to TERGMs, under the conditional independence assumption we can estimate:

$\Pr\left(\mathbf{Y} =\mathbf{y}|\mathbf{X} = \mathbf{x}, \mathbf{Z}, \theta\right) = \prod_{i\neq j}\Pr\left(y_{ij}|z_i, z_j, x_{ij},\theta\right)$

See Hoff, Raftery, and Handcock (2002)

## Estimation of ERGMs

In statnet, the default estimation method is based on a method proposed by Geyer and Thompson (1992), Markov-Chain MLE, which uses Markov-Chain Monte Carlo for simulating networks and a modified version of the Newton-Raphson algorithm to do the parameter estimation part.

## Estimation of ERGMs (cont’ d)

In general terms, the MC-MLE algorithm can be described as follows:

1. Initialize the algorithm with an initial guess of $$\theta$$, call it $$\theta^{(t)}$$

2. While (no convergence) do:

1. Using $$\theta^{(t)}$$, simulate $$M$$ networks by means of small changes in the $$\mathbf{Y}_{obs}$$ (the observed network). This part is done by using an importance-sampling method which weights each proposed network by it’s likelihood conditional on $$\theta^{(t)}$$

2. With the networks simulated, we can do the Newton step to update the parameter $$\theta^{(t)}$$ (this is the iteration part in the ergm package): $$\theta^{(t)}\to\theta^{(t+1)}$$

3. If convergence has been reach (which usually means that $$\theta^{(t)}$$ and $$\theta^{(t + 1)}$$ are not very different), then stop, otherwise, go to step a.

For more details see Lusher, Koskinen, and Robins (2012);Admiraal and Handcock (2006);T. A. Snijders (2002);Wang et al. (2009) provides details on the algorithm used by PNet (which is the same as the one used in RSiena). Lusher, Koskinen, and Robins (2012) provides a short discussion on differences between ergm and PNet.

The main problems with ERGMs are:

1. Computational Time: As the complexity of the model increases, it gets harder to achieve convergence, thus, more time is needed.

2. Model degeneracy: Even if convergence is achieved, model fitness can be very bad