Abstract
In dense Erdős–Renyi random graphs, we are interested in the events where large numbers of a given subgraphs occur. The mean behaviour of subgraph counts is known, and only recently were the related large deviations results discovered. Consequently, it is natural to ask, what is the probability of an Erdős–Renyi graph containing an excessively large number of a given subgraph? Using the large deviation principle, we study an importance sampling scheme as a method to numerically compute the small probabilities of large triangle counts occurring within Erdős–Renyi graphs. The exponential tilt used in the importance sampling scheme comes from a generalized class of exponential random graphs. Asymptotic optimality, a measure of the efficiency of the importance sampling scheme, is achieved by the special choice of exponential random graph that is indistinguishable from the Erdős–Renyi graph conditioned to have many triangles. We show how this choice can be made for the conditioned Erdős–Renyi graphs both in the replica symmetric phase and also in parts of the replica breaking phase. Equally interestingly, we also show that the exponential tilt suggested directly by the large deviation principle does not always yield an optimal scheme.
Highlights
In this paper we study the use of importance sampling schemes to numerically estimate the probability that an Erdos-Rényi random graph contains an unusually large number of triangles
The presence of a large deviation principle for the random graphs Gn,p as n → ∞, leads to a way to quantify the efficiency of the importance scheme in an asymptotic sense, as is done in other contexts [4]
Which is consistent with the intuition that a good choice of Qn should put Xk ∈ Wn,t with high probability. To understand in this context the tilts that could be relevant, let us describe in a little more detail, properties of the rate function 1.6 as well as structural results of the Erdos-Rényi model conditioned on rare events and their connections to a sub-family of the famous exponential random graph models
Summary
In this paper we study the use of importance sampling schemes to numerically estimate the probability that an Erdos-Rényi random graph contains an unusually large number of triangles. Importance Sampling for rare events in Erdos-Rényi graphs of the space Ωn = {0, 1}(n2). One first needs to understand the structure of such random graphs, conditioned on this rare event, more precisely the large deviation rate function for such events. In the dense graph case, where the edge probability p stays fixed as n → ∞, [6] derived a large deviation principle (LDP) for the rare event. More recently [8] showed a general large deviation principle for dense Erdos-Rényi graphs, using the theory of limits of dense random graph sequences developed recently by Lovasz et al [14, 16, 15, 3]. Before describing the relevant tilts we formally define our aims
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