ATTRIBUTE NETWORK MODELS, STOCHASTIC APPROXIMATION AND NETWORK SAMPLING

Motivated by the central role of social networks in the diffusion of information, the study of network valued data where nodes and/or edges have attributes, which modulate the dynamics of both network evolution, and information flow on the network itself, has witnessed significant research interest across multiple disciplines. A key ingredient of this general area comprises probabilistic network models that incorporate (a) heterogeneity in edge creation across different attribute groups; (b) temporal network evolution and (c) popularity bias. Such models are then used to understand a host of domain specific questions, including bias in network sampling, PageRank and degree centrality scores and their impact in network ranking and recommendation algorithms. Despite significant interest, for these network models, the main network functional amenable to analysis has so far been degree distribution asymptotics. In this paper, we analyze dynamic random network models where younger vertices connect to older ones with probabilities proportional to their degrees as well as a propensity kernel governed by their attribute types. Using stochastic approximation techniques we show that, in the large network limit, such networks converge in the local weak sense to limiting infinite random trees with an explicit description in terms of randomly stopped multi-type branching processes. This allows for the derivation of asymptotics for a wide class of network functionals implying, for example, that while degree distribution tail exponents depend on the attribute type (already derived by (Elec-tron. J. Probab. 18 (2013) 8)), PageRank centrality scores have the same tail exponent across attributes. The limit results also give explicit formulae for the performance of various network sampling mechanisms. One surprising consequence is the efficacy of PageRank and walk based network sampling schemes for directed networks in the setting of rare minorities.