A functional limit theorem for random graphs with applications to subgraph count statistics.

*(English)*Zbl 0708.05052Summary: 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 |

##### Keywords:

central limit theorems; functional limit theorems; Skorokhad topology; subgraph counts; random graph; statistics; martingale convergence theorem; asymptotic distribution
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\textit{S. Janson}, Random Struct. Algorithms 1, No. 1, 15--37 (1990; Zbl 0708.05052)

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##### References:

[1] | Convergence of Probability Measures, Wiley, New York, 1968. · Zbl 0172.21201 |

[2] | and , The asymptotic distribution of generalized U-statistics with applications to random graphs Tech. Report, Univ. Lund (1988). |

[3] | Ruciński, Prob. Th. Rel. Fields 78 pp 1– (1988) |

[4] | and , A central limit theorem for decomposable random variables with applications to random graphs, J. Comb. Theory B. (1988). |

[5] | and , Limit Theorems for Stochastic Processes, Springer, Berlin, 1987. · Zbl 0635.60021 |

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