On local weak limit and subgraph counts for sparse random graphs

Abstract

We use an inequality of Sidorenko to show a general relation between local and global subgraph counts and degree moments for locally weakly convergent sequences of sparse random graphs. This yields an optimal criterion to check when the asymptotic behaviour of graph statistics, such as the clustering coefficient and assortativity, is determined by the local weak limit. As an application we obtain new facts for several common models of sparse random intersection graphs where the local weak limit, as we see here, is a simple random clique tree corresponding to a certain two-type Galton–Watson branching process.

Normal and stable approximation to subgraph counts in superpositions of Bernoulli random graphs

“…Another line of research pursued in [11, 16] addresses the concentration of subgraph counts in . We also mention related work on local weak limits and subgraph counts: the results of [18, 26] imply the linear growth in n of the numbers of small dense subgraphs for a large class of sparse affiliation network models. Establishing the distributional asymptotics here is an interesting problem for future research.…”

“…The remaining case, ρj−1 − ρj = −1, is realised by the configuration where the vertex sets of H j and Hj−1 have no common elements. In this case (18) follows from the identity vj = vj−1 + v j . By summing the inequalities in (18), we obtain, using…”

“…In this case (18) follows from the identity vj = vj−1 + v j . By summing the inequalities in (18), we obtain, using…”

Estimating Global Subgraph Counts by Sampling

“…A similar problem for sequences of graphs with a weak local limit has been studied in [8]; there a uniform integrability condition on the (h − 1)th power of the random vertex degree was used. Uniform integrability of graph degrees or their powers is natural for sequences of graphs, and has been used both in theoretical work and in applications, see, e.g., [1,2].…”

“…Finally, Theorem 2 allows us to generalize (the difficult part of) Theorem 2.1 of [8], with a simpler proof that does not require the local weak convergence assumption.…”