The reasonable effectiveness of continuous time branching processes in understanding evolving network models

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A wide array of network growth models have been proposed across various domains as test beds to understand questions such as the effect of network change point (when a shock to the network changes the probabilistic rules of its evolution) or the role of attributes in driving the emergence of network structure and subsequent centrality measures in real world systems. 

The goal of this talk will be to describe two specific settings where continuous time branching processes give mathematical insight into asymptotic properties of such models. In the first setting, a natural network change point model can be directly embedded into continuous time thus leading to an understanding of long range dependence of the initial network system on subsequent properties imply the difficulty in understanding and estimating network change point. In the second application, we will describe a notion of resolvability where convergence of a simple macroscopic functional in a model of networks with vertex attributes, coupled with stochastic approximation techniques implies local weak convergence of a standard model of nodal attribute driven network evolution to a limit infinite random structure driven by a multitype continuous time branching process. In the second setting, continuous time branching processes naturally emerge in the limit. If time permits, we will describe a final setting of network evolution with delay where once again such processes arise only in the limit. 

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