Compound Poisson approximations of subgraph counts in random graphs

Abstract

The upper tail problem in a sparse Erdős-Rényi graph asks for the probability that the number of copies of some fixed subgraph exceeds its expected value by a constant factor. We study the analogous problem for oriented subgraphs in directed random graphs. By adapting the proof of Cook, Dembo, and Pham [11, Theorem 1.1], we reduce this upper tail problem to the asymptotic of a certain variational problem over edge weighted directed graphs. We give upper and lower bounds for the solution to the corresponding variational problem, which differ by a constant factor of at most 2. We provide a host of subgraphs where the upper and lower bounds coincide, giving the solution to the upper tail problem. Examples of such digraphs include triangles, stars, directed k-cycles, and balanced digraphs.

Compound Poisson approximation of subgraph counts in stochastic block models with multiple edges

“…The Stein-Chen method for Poisson approximation was introduced by Chen [11], and was extended to compound Poisson approximation by Roos [27], [28]. A comprehensive account of the application of the Stein-Chen method for Poisson approximation in random graph theory can be found in [10], and the method was used to derive compound Poisson approximation of subgraph counts in Erdős-Rényi random graphs in [30]. Here, we present the compound Poisson framework given in [27], [28], and [30].…”

“…A comprehensive account of the application of the Stein-Chen method for Poisson approximation in random graph theory is given in [7], and the method is used to derive compound Poisson approximation of subgraph counts in Erdős-Rényi random graphs in [31]. Here, we present the compound Poisson framework that is given in [27,28] and [31].…”

Compound Poisson approximation of subgraph counts in stochastic block models with multiple edges

“…The Stein-Chen method for Poisson approximation was introduced by Chen [11], and was extended to compound Poisson approximation by Roos [27], [28]. A comprehensive account of the application of the Stein-Chen method for Poisson approximation in random graph theory can be found in [10], and the method was used to derive compound Poisson approximation of subgraph counts in Erdős-Rényi random graphs in [30]. Here, we present the compound Poisson framework given in [27], [28], and [30].…”

“…A comprehensive account of the application of the Stein-Chen method for Poisson approximation in random graph theory is given in [7], and the method is used to derive compound Poisson approximation of subgraph counts in Erdős-Rényi random graphs in [31]. Here, we present the compound Poisson framework that is given in [27,28] and [31].…”

Stein's method in network analysis

“…Such counts can often be well approximated by a compound Poisson distribution. Stein's method for compound Poisson distributions is summarised for example in [5]; results specialised to Bernoulli random graphs can be found for example in [38], and [14] covers subgraph counts in stochastic blockmodels.…”

U -Statistics on bipartite exchangeable networks

“…The frequencies of occurences of motifs (sometimes called network moments) are widely studied in network theory. Motifs frequencies have known asymptotic distribution under many generative models, so they can be used to analyze binary networks (Stark, 2001;Picard et al, 2008;Reinert and Röllin, 2010;Bickel, Chen and Levina, 2011;Bhattacharyya and Bickel, 2015;Levin and Levina, 2019;Maugis et al, 2020;Naulet et al, 2021;Ouadah, Latouche and Robin, 2022). Many probabilistic graph models also rely on motif frequencies, such as the Exponential Random Graph Model (Frank and Strauss, 1986) or the dk -random graphs (Orsini et al, 2015).…”

Compound Poisson approximation of subgraph counts in stochastic block models with multiple edges

“…The Stein-Chen method for Poisson approximation was introduced by Chen [11], and was extended to compound Poisson approximation by Roos [27], [28]. A comprehensive account of the application of the Stein-Chen method for Poisson approximation in random graph theory can be found in [10], and the method was used to derive compound Poisson approximation of subgraph counts in Erdős-Rényi random graphs in [30]. Here, we present the compound Poisson framework given in [27], [28], and [30].…”

“…A comprehensive account of the application of the Stein-Chen method for Poisson approximation in random graph theory is given in [7], and the method is used to derive compound Poisson approximation of subgraph counts in Erdős-Rényi random graphs in [31]. Here, we present the compound Poisson framework that is given in [27,28] and [31].…”

Stein's method in network analysis

“…Such counts can often be well approximated by a compound Poisson distribution. Stein's method for compound Poisson distributions is summarised for example in [5]; results specialised to Bernoulli random graphs can be found for example in [38], and [14] covers subgraph counts in stochastic blockmodels.…”

U -Statistics on bipartite exchangeable networks

“…The frequencies of occurences of motifs (sometimes called network moments) are widely studied in network theory. Motifs frequencies have known asymptotic distribution under many generative models, so they can be used to analyze binary networks (Stark, 2001;Picard et al, 2008;Reinert and Röllin, 2010;Bickel, Chen and Levina, 2011;Bhattacharyya and Bickel, 2015;Levin and Levina, 2019;Maugis et al, 2020;Naulet et al, 2021;Ouadah, Latouche and Robin, 2022). Many probabilistic graph models also rely on motif frequencies, such as the Exponential Random Graph Model (Frank and Strauss, 1986) or the dk -random graphs (Orsini et al, 2015).…”

Compound Poisson approximation of subgraph counts in stochastic block models with multiple edges

“…The Stein-Chen method for Poisson approximation was introduced by Chen [11], and was extended to compound Poisson approximation by Roos [27], [28]. A comprehensive account of the application of the Stein-Chen method for Poisson approximation in random graph theory can be found in [10], and the method was used to derive compound Poisson approximation of subgraph counts in Erdős-Rényi random graphs in [30]. Here, we present the compound Poisson framework given in [27], [28], and [30].…”

“…A comprehensive account of the application of the Stein-Chen method for Poisson approximation in random graph theory is given in [7], and the method is used to derive compound Poisson approximation of subgraph counts in Erdős-Rényi random graphs in [31]. Here, we present the compound Poisson framework that is given in [27,28] and [31].…”

Stein's method in network analysis

“…Such counts can often be well approximated by a compound Poisson distribution. Stein's method for compound Poisson distributions is summarised for example in [5]; results specialised to Bernoulli random graphs can be found for example in [38], and [14] covers subgraph counts in stochastic blockmodels.…”

U -Statistics on bipartite exchangeable networks

“…The frequencies of occurences of motifs (sometimes called network moments) are widely studied in network theory. Motifs frequencies have known asymptotic distribution under many generative models, so they can be used to analyze binary networks (Stark, 2001;Picard et al, 2008;Reinert and Röllin, 2010;Bickel, Chen and Levina, 2011;Bhattacharyya and Bickel, 2015;Levin and Levina, 2019;Maugis et al, 2020;Naulet et al, 2021;Ouadah, Latouche and Robin, 2022). Many probabilistic graph models also rely on motif frequencies, such as the Exponential Random Graph Model (Frank and Strauss, 1986) or the dk -random graphs (Orsini et al, 2015).…”