Mixing Time of Exponential Random Graphs

Abstract

A variety of random graph models have been developed in recent years to study a range of problems on networks, driven by the wide availability of data from many social, telecommunication, biochemical and other networks. A key model, extensively used in the sociology literature, is the exponential random graph model. This model seeks to incorporate in random graphs the notion of reciprocity, that is, the larger than expected number of triangles and other small subgraphs. Sampling from these distributions is crucial for parameter estimation hypothesis testing, and more generally for understanding basic features of the network model itself. In practice sampling is typically carried out using Markov chain Monte Carlo, in particular either the Glauber dynamics or the Metropolis-Hasting procedure.In this paper we characterize the high and low temperature regimes of the exponential random graph model. We establish that in the high temperature regime the mixing time of the Glauber dynamics is Θ(n 2 log n), where n is the number of vertices in the graph; in contrast, we show that in the low temperature regime the mixing is exponentially slow for any local Markov chain. Our results, moreover, give a rigorous basis for criticisms made of such models. In the high temperature regime, where sampling with MCMC is possible, we show that any finite collection of edges are asymptotically independent; thus, the model does not possess the desired reciprocity property, and is not appreciably different from the Erdős-Rényi random graph.

Econometrics of network models

“…The procedure can be very slow to converge to an invariant distribution. This is highlighted, for instance, in the discussion by Chandrasekhar and Jackson (2014) and Mele (2015) and formally demonstrated in Bhamidi, Bresler, and Sly (2011). In particular, for certain regions of the parameter space (defined as "low temperature" regions, in analogy to spin systems in physics), where the distribution (7) is multimodal, the mixing time for the MCMC procedure, i.e., the time it takes for the MCMC procedure to be within e −1 in total variation distance from the desired distribution, is exponential on the number of nodes (Theorem 6 in that paper).…”

The Economic Consequences of Social Network Structure

“…Unfortunately, standard sampling methods based on MCMC techniques do not work except in extreme cases, as shown by Bhamidi, Bresler, and Sly (2008); and yet those models are being widely applied even though the estimation of parameters as well as standard errors may be inaccurate.…”

Stochastic Network Formation and Homophily

“…All available approximation techniques (see Bhamidi et al 2008 andChatterjee andDiaconis 2013) rely on the assumption of independent links, which is against the spirit of the model. Chandrasekhar and Jackson (2014) show that these techniques prove themselves to be inaccurate when tested against simulations.…”