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A functional limit theorem for random graphs with applications to subgraph count statistics. (English) Zbl 0708.05052
Summary: We consider a random graph that evolves in time by adding new edges at random times (different edges being added at independent and identically distributed times). A functional limit theorem is proved for a class of statistics of the random graph, considered as stochastic processes. The proof is based on a martingale convergence theorem. The evolving random graph allows us to study both the random graph model \(K_{n,p}\), by fixing attention to a fixed time, and the model \(K_{n,N}\), by studying it at the random time it contains exactly n edges. In particular, we obtain the asymptotic distribution as \(n\to \infty\) of the number of subgraphs isomorphic to a given graph G, both for \(K_{n,p}\) (p fixed) and \(K_{n,N}\left(N/(\begin{matrix} n\\ 2\end{matrix} \right)\to p)\). The results are strikingly different; both models yield asymptotically normal distributions, but the variances grow as different powers of n (the variance grows slower for \(K_{n,N}\); the powers of n usually differ by 1, but sometimes by 3). We also study the number of induced subgraphs of a given type and obtain similar, but more complicated, results. In some exceptional cases, the limit distribution is not normal.

MSC:
05C80 Random graphs (graph-theoretic aspects)
60F17 Functional limit theorems; invariance principles
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