Featured Graphons with Applications to SIR Models

The complexity of a dense graph increases combinatorically as its size increases. One approach to alleviate this complexity is to use graphon analysis to find an approximation of a very large graph’s adjacency matrix. Standard graphons are defined as functions on the unit square, but mapping nodes of a graph onto the unit interval may entail the loss of information. To account for this, a type of random graph is introduced called a featured graph which is a graph where each vertex has meaningful attributes determining connectivity. Featured graphons also provide an approach to the problems arising with graphs embedded in higher dimensional spaces. It is shown that in an appropriate norm the adjacency matrix operator converges to the associated featured graphon. Convergence is illustrated numerically with an SIR epidemic model generalized to multiple communities.