# Local graph stability in exponential family random graph models

## Publication Date

1-1-2021

## Document Type

Article

## Publication Title

SIAM Journal on Applied Mathematics

## Volume

81

## Issue

4

## DOI

10.1137/19M1286864

## First Page

1389

## Last Page

1415

## Abstract

Exponential family random graph models (ERGMs) are widely used to model networks by parameterizing graph probability in terms of a set of user-selected sufficient statistics. Equivalently, ERGMs can be viewed as expressing a probability distribution on graphs arising from the action of competing social forces that make ties more or less likely, depending on the state of the rest of the graph. Such forces often lead to a complex pattern of dependence among edges, with nontrivial large-scale structures emerging from relatively simple local mechanisms. While this provides a powerful tool for probing macro-micro connections, much remains to be understood about how local forces shape global outcomes. One very simple question of this type is that of the conditions needed for social forces to stabilize a particular structure: that is, given a specific structure and a set of alternatives (e.g., arising from small perturbations), under what conditions will said structure remain more probable than the alternatives? We refer to this property as local stability and seek a general means of identifying the set of parameters under which a target graph is locally stable with respect to a set of alternatives. Here, we provide a complete characterization of the region of the parameter space inducing local stability, showing it to be the interior of a convex cone whose faces can be derived from the change scores of the sufficient statistics vis-`a-vis the alternative structures. As we show, local stability is a necessary but not sufficient condition for more general notions of stability, the latter of which can be explored more efficiently by using the ``stable cone"" within the parameter space as a starting point. In addition to facilitating the understanding of model behavior, we show how local stability can be used to determine whether a fitted model implies that an observed structure would be expected to arise primarily from the action of social forces, versus by merit of the model permitting a large number of high probability structures, of which the observed structure is one (i.e., entropic effects). We also use our approach to identify the dyads within a given structure that are the least stable, and hence predicted to have the highest probability of changing under the current social forces. The utility of the ``stable cone"" for ERGM parameter optimization is then demonstrated on a physical model of amyloid fibril formation. This demonstration features a visualization of the stable region, whereby it is shown that the majority of the region of the model's parameter space where ERGM simulations produce the highest fibril yield lies within the ``stable cone.""

## Funding Number

DMS-1361425

## Funding Sponsor

National Science Foundation

## Keywords

Computational biophysics, Computational sociology, Graph theory, Mathematical modeling of graphs, Network data, Network Hamiltonian, Social network analysis

## Department

Chemistry

## Recommended Citation

Yue Yu, Gianmarc Grazioli, Nolan E. Phillips, and Carter T. Butts. "Local graph stability in exponential family random graph models" *SIAM Journal on Applied Mathematics* (2021): 1389-1415. https://doi.org/10.1137/19M1286864