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Competing types in preferential attachment graphs with community structure

John HaslegraveJonathan JordanMark Yarrow
Dec 2022
摘要
We extend the two-type preferential attachment model of Antunovi\'c, Mosseland R\'acz to networks with community structure. We show that different typesof limiting behaviour can be found depending on the choice of communitystructure and type assignment rule. In particular, we show that, for essentially all type assignment rules wheremore than one limit has positive probability in the unstructured model,communities may simultaneously converge to different limits if the communityconnections are sufficiently weak. If only one limit is possible in theunstructured model, this behaviour still occurs for some choices of typeassignment rule and community structure. However, we give natural conditions onthe assignment rule and, for two communities, on the structure, either of whichwill imply convergence to this limit, and each of which is essentially bestpossible. Although in the unstructured two-type model convergence almost surely occurs,we give an example with community structure which almost surely does notconverge.
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