Abstract
We introduce the random intersection graph with communities, a new model for networks with overlapping communities with arbitrary internal structure. We construct the model from a list of arbitrary community graphs that are the building blocks, and a separate list of individuals, each with a prescribed number of community membership tokens. Randomness is introduced by matching these tokens uniformly at random to vertices of the community graphs. We then identify the community members assigned to the same individual, thus overlaps arise due to individuals having several tokens. This gives a highly flexible model for networks with community structure.We are able to derive a wide range of analytic results on this model. We derive an asymptotic description of the local structure of the graph, which further yields the asymptotic degree distribution, local clustering coefficient, and results on the overlapping structure of the communities. For the global connectivity structure, we identify a phase transition in the size of the largest component. When the largest component constitutes a positive proportion of the graph, we can further characterize its asymptotic local structure. Finally, we study how the connectivity structure changes under a randomized attack, where we remove edges randomly, according to independent coin flips.
Highlights
Network science is an active and quickly developing field, due to joint efforts from practitioners and theoretical researchers
To the best knowledge of the authors, the first model to combine the above two key features is the random intersection graph with communities (RIGC) that we introduce here
Results we introduce our analytic results on the RIGC model
Summary
Network science is an active and quickly developing field, due to joint efforts from practitioners and theoretical researchers. Theoretical studies allow us to make predictions or approximations when data analysis is not feasible, and refine our understanding of the causality relations between properties of the network. As such, both sides provide their invaluable contributions, facilitating the progress of one another. Each community is represented by a ‘small’ graph, in the sense described above Each vertex of these small graphs represents a unique community role, and connections between community members are represented by edges. These graphs may be arbitrary, as long as they are connected, allowing for different applications. We note that the given list may contain several communities with the same structure, for example, each community that is a three-member family would be represented by a triangle
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