# Autologistic actor attribute models

File:Social-Network-Sensors-for-Early-Detection-of-Contagious-Outbreaks-pone.0012948.s001.ogv Autologistic actor attribute models (ALAAMs) are a family of statistical models used to model the occurrence of node attributes (individual-level outcomes) in network data. They are frequently used with social network data to model social influence, the process by which connections in a social network influence the outcomes experienced by nodes. However, they may be applied to any type of network data that incorporates binary node attributes.

## Background

Autologistic actor attributes models (ALAAMs) are a method for social network analysis. They were originally proposed as alteration of Exponential Random Graph Models (ERGMs) to allow for the study of social influence. ERGMs are a family of statistical models for modeling social selection, how ties within a network form on the basis of node attributes and other ties in the network. ALAAMs adapt the structure of ERGM models, but rather than predicting tie formation on the based on fixed node attributes, they predict node attributes based on fixed ties. This allows for the modeling of social influence processes, for instance how friendship among adolescents (network ties) may influence whether they smoke (node attributes), influences of networks on other health-related practices, and how attitudes or perceived attitudes may change.

ALAAMs are distinct from other models of social influence on networks, such as epidemic/SIR models, because ALAAMs are used for the analysis of cross-sectional data, observed at only a single point in time.

## Definition

ALAAMs, like ERGMs, are part of the Exponential family of probability models. ALAAMs are exponential models that describe, for a network, a joint probability distribution for whether or not each node in the network exhibits a certain node-level attribute.

$\displaystyle{ P(Y = y | \theta , X) = \frac{\exp(\theta^{T} s(y,X))}{c(\theta)} }$

where $\displaystyle{ \theta }$ is a vector of weights, associated with $\displaystyle{ s(y,X) }$,the vector of model parameters, and $\displaystyle{ c(\theta) }$ is a normalization constant to ensure that the probabilities of all possible combination of node attributes sum to one.

## Estimation

Estimation of model parameters, and evaluation of standard errors (for the purposes of hypothesis testing), is conducted using Markov chain Monte Carlo maximum likelihood estimation (MCMC-MLE), building on approaches such as the Metropolis–Hastings algorithm. Such approaches are required to estimate the model's parameters across an intractable sample space for moderately-size networks. After model estimation, good-of-fit testing, through the sampling of random networks from the fitted model, should be performed to ensure that the model adequately fits the observed data.

ALAAM estimation, while not perfect, has been demonstrated to be relatively robust to partially missing data, due to random sampling or snowball sampling data collection techniques.

Currently, these algorithms for estimating ALAAMs are implemented in the PNet and MPNet software, published by Melnet, a research group at the University of Melbourne