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# Jagers-Nerman stable age distribution theory, change point detection and power of two choices in evolving networks

## March 24 @ 11:00 am

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Abstract: The last few years have seen an explosion in the amount of data on real world networks, including networks that evolve over time. A number of mathematical models have been proposed to understand the evolution of such networks and explain the emergence of a wide array of structural features such as heavy tailed degree distribution and small world connectivity of real networks. One sophisticated mathematical tool in the arsenal of a modern probabilist is the so-called Jagers and Nerman stable age distribution theory. In this talk we will describe two such settings:

(i) Change point detection for networks: We consider the preferential attachment model. We formulate and study the regime where the network transitions from one evolutionary scheme to another. In the large network limit we derive asymptotics for various functionals of the network including degree distribution and maximal degree. We study functional central limit theorems for the evolution of the degree distribution which feed into proving consistency of a proposed estimator of the change point.

(ii) Power of choice and network evolution: Recently, Raissa D’Souza et al proposed a new model of random tree growth with choice, where each new vertex first picks a set of vertices uniformly at random from the existing network and then chooses to attach itself according to some functional of the degree e.g. to either the vertex with the maximal or minimal degree. We prove that these random trees converge locally to limiting infinite random sin-trees. These limiting objects are described through a continuous time branching process whose offspring distribution is obtained via a recursive construction from the limiting degree distribution. The local weak convergence of these trees implies convergence of global functionals such including the spectral distribution of the adjacency matrix.