Scaling laws for properties of random graphs that grow via successive combination

Abstract

We consider undirected graphs that grow through the successive combination of component sub-graphs. For any well-behaved functions defined for such graphs, taking values in a Banach space, we show that there must exist a scaling law applicable when successive copies of the same component graph are combined. Crucially, we extend the approach introduced in previous work to the successive combination of component random sub-graphs. We illustrate this by generalizing the preferential attachment operation for the combination of stochastic block models. We discuss a further wide range of random graph combination operators to which this theory now applies, indicating the ubiquity of growth scaling laws (and asymptotic decay scaling laws) within applications, where the modules are quite distinct, yet may be considered as instances drawn from the same random graph. This is a type of statistically self-similar growth process, as opposed to a deterministic growth process incorporating exact copies of the same motif, and it represents a natural, partially random, growth processes for graphs observed in the analysis of social and technology contexts.