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Overview of Social Network Models
George G. Vega Yon<br><a href="https://ggvy.cl" target="_blank" style="color: black;">ggvy.cl</a><br><br>Center for Applied Network Analysis (CANA)<br>Department of Preventive Medicine
September 12, 2018
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knitr::opts_chunkset( echo = FALSE, fig.width=6, fig.align = "center" )  ## 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 $$Y_i = f\left(\mathbf{X}, Y_{-i}, \mathbf{G}; \theta\right)$$ whereY_{-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] ## {style="font-size:35px"} 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.

## Summary

library(magrittr)