Infinite paths and cliques in random graphs

Counting cliques and cycles in scale-free inhomogeneous random graphs

Scale-free networks contain many small cliques and cycles. We model such networks as inhomogeneous random graphs with regularly varying infinite-variance weights. For these models, the number of cliques and cycles have exact integral expressions amenable to asymptotic analysis. We obtain various asymptotic descriptions for how the average number of cliques and cycles, of any size, grow with the network size. For the cycle asymptotics we invoke the theory of circulant matrices.

次临界随机时间图中时间团的大小

On the size of temporal cliques in subcritical random temporal graphs

本文改进了次临界随机时间图中最大时间团大小的上界，证明了当 $p=c\log n/n$ 且 $c<1$ 时，高概率最大时间团的大小为常数。

Epidemics on random intersection graphs

In this paper we consider a model for the spread of a stochastic SIR (Susceptible $\to$ Infectious $\to$ Recovered) epidemic on a network of individuals described by a random intersection graph. Individuals belong to a random number of cliques, each of random size, and infection can be transmitted between two individuals if and only if there is a clique they both belong to. Both the clique sizes and the number of cliques an individual belongs to follow mixed Poisson distributions. An infinite-type branching process approximation (with type being given by the length of an individual's infectiou…