Random intersection graphs with communities

TL;DR

The paper introduces the random intersection graph with communities ($\\mathrm{RIGC}$), a flexible model that allows overlapping, non-complete community graphs, constructed via a bipartite configuration model between individuals and communities. It establishes local weak convergence of the RIGC to a community-projected limit $\\mathrm{CP}$, derives the asymptotic degree distribution and local clustering, and analyzes the overlapping structure of communities, including both typical overlaps and a global perspective under a second-moment condition. The analysis hinges on a detailed study of the underlying BCM, whose local weak limit is a mixture of branching processes, and on a projection framework that maps BCM neighborhoods to RIGC neighborhoods. The results show that the RIGC can reproduce tunable degree and clustering properties while providing precise descriptions of how communities overlap, with implications for modeling real networks and for tasks like community detection. A companion paper further investigates giant components and percolation phenomena in this framework, highlighting the practical relevance of these local-structure results.

Abstract

Random intersection graphs model networks with communities, assuming an underlying bipartite structure of groups and individuals, where these groups may overlap. Group memberships are generated through the bipartite configuration model. Conditionally on the group memberships, the classical random intersection graph is obtained by connecting individuals when they are together in at least one group. We generalize this definition, allowing for arbitrary community structures within the groups. In our new model, groups might overlap and they have their own internal structure described by a graph, the classical setting corresponding to groups being complete graphs. Our model turns out to be tractable. We analyze the overlapping structure of the communities, derive the asymptotic degree distribution and the local clustering coefficient. These proofs rely on local weak convergence, which also implies that subgraph counts converge. We further exploit the connection to the bipartite configuration model, for which we also prove local weak convergence, and which is interesting in its own right.

Figures (4)

• Figure 1: Two models for overlapping communities: RIG and RIGC
• Figure 2: An example of the parameters. Individuals form the lhs partition $\mathscr{V}^\mathscr{l}$, and their $\mathscr{l}$-degree, i.e., the number of group memberships, is represented by outgoing half-edges. Communities form the rhs partition $\mathscr{V}^\mathscr{r}$, and each is assigned an arbitrary connected community graph. As before, we represent the $\mathscr{r}$-degree, i.e., the number of community members, by outgoing half-edges. In fact, each half-edge represents a specific vertex (role) in the community graph, thus they are labeled the same way. In the next step, we assign group memberships (community roles) through a (bipartite) matching of the half-edges.
• Figure 3: The community projection
• Figure 4: A neighborhood and a possible pre-image

Theorems & Definitions (21)

• Remark 2.1: Algorithmic pairing
• Definition 2.2: The "underlying BCM"
• Remark 2.4: Consequences of \ref{['asmp:convergence']}
• Remark 2.5: Random parameters
• Definition 2.6: Rooted graph, rooted isomorphism and $r$-neighborhood
• Definition 2.7: Local weak convergence in probability
• Theorem 2.8: Local weak convergence of the $\mathrm{RIGC}$
• Corollary 2.9: Degrees in the $\mathrm{RIGC}$
• Corollary 2.10: Local clustering in the $\mathrm{RIGC}$
• Corollary 2.11: Condition for positive asymptotic clustering