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title author date output bibliography editor_options
Overview of Social Network Models
George G. Vega Yon<br><a href="" target="_blank" style="color: black;"></a><br><br>Center for Applied Network Analysis (CANA)<br>Department of Preventive Medicine
September 12, 2018
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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 {.smaller}

  • 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" [@Shalizi2011].

  • 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) {.smaller}

  • 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.) {.smaller}

  • 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 @LeSage2008]

  • 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! [@Hunter2008]


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 @Hunter2008].

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 @hoff2002

Estimation of ERGMs

In statnet, the default estimation method is based on a method proposed by @Geyer1992, 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) {style="font-size:25px"}

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:

    a. 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 $\theta^{(t)}$

    b. 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): $\theta^{(t)}\to\theta^{(t+1)}$

    c. 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 @lusher2012;@admiraal2006;@Snijders2002;@Wang2009 provides details on the algorithm used by PNet (which is the same as the one used in RSiena). @lusher2012 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

![](awful-chains.png){style="width:500px"} ![](ergm-fit.png){style="width:500px"}

Example of problems encountered in the estimation process of ERGMs: No convergence (left), and model degeneracy (right).

Stochastic Actor Oriented Models (SOAMs) {style="font-size:30px"}

  • Also known as Siena: Simulation Investigation for Empirical Network Analysis.

  • Models both, structure and behavior as a time-continuous Markov process where changes happen one at a time (as a poisson process).

  • In other words, individuals choose between states $x$ and $x'$ in which either a tie changes, or their behavior changes.

  • Ultimately, we maximize the following function:

$$ \frac{\exp{f_i^Z(\beta^z,x, z)}}{\sum_{Z'\in\mathcal{C}}\exp{f_i^{Z}(\beta, x, z')}} $$

  • Like ERGMs, the denominator is what makes estimating this models hard.

See @Snijders2010;@lazega2015;@Ripley2011.

Network Matching: Aral et al. (2009) {style="font-size:30px"}

  • Built on top of the Rubin Causal Model (RCM). Uses matching (non-parametric method) to estimate the effect that changes on exposure has over behavior

  • As a difference from RCM, we don't have one but multiple treatments

  • In the dynamic case, for each time $t$, we can build multiple levels of treatments, in particular, given that individual $i$ had an exposure $E_{t-1}=j-1$ at $t-1$, we write:

    $$ T_{itj} = \left{\begin{array}{ll} 1 &\mbox{if }E_t = j \ 0 &\mbox{otherwise.} \end{array}\right. $$

  • Based on the previous equation, we can use some matching algorithm to build counter factuals and estimate a simil to Average Treatment Effect on the Treated (ATT).

  • For more on matching methods see @Imbens2009;@sekhon2008neyman;@king2016propensity (special attention to the last one).

Other Models

  • GERGM: Generalized Exponential Random Graph Models (using weighted graphs, see @Desmarais2012).

  • SERGMs: Statistical Exponential Random Graph Models, suitable for large graphs, uses sufficient statistics. [see @Chandrasekhar2012]

  • DyNAM: dynamic network actor models [see @Stadtfeld2017].

  • REM: Relational Event Models [see @Butts2008], which are very similar to DyNAMs.

  • ALAAM: Autologistic actor attribute models [see @Daraganova2013;@Kashima2013]

Some other models can be found in @Snijders2011.


knitr::kable(read.csv("models_summary.csv")) %>%
  kableExtra::column_spec(1, bold=TRUE)

References {style="font-size:9px"}