Network Capacity Bound for Personalized PageRank in Multimodal Networks
M. A. Kłopotek, S. T. Wierzchoń, R. A. Kłopotek
TL;DR
This work generalizes PageRank to multimodal networks represented as $M$-uniform multimodal hypergraphs by introducing MuMoRank, a random-walk based ranking that treats each modality separately with its own supernode and damping factor. It formalizes a generalized graph structure and derive update equations for node and hyperedge ranks, then proves upper bounds on the outflow of authority from a node subset under both identical and heterogeneous damping across modalities. The key contributions include the MuMoRank framework, modality-aware random walks, and provable bounds that depend only on boundary and modality-degree statistics, with a concrete example illustrating the concepts. These results enable validation of cluster quality in multimodal hypergraphs and guide design choices for damping and personalization in such networks, with future work pointing to tighter bounds and convergence analysis.
Abstract
In a former paper the concept of Bipartite PageRank was introduced and a theorem on the limit of authority flowing between nodes for personalized PageRank has been generalized. In this paper we want to extend those results to multimodal networks. In particular we deal with a hypergraph type that may be used for describing multimodal network where a hyperlink connects nodes from each of the modalities. We introduce a generalisation of PageRank for such graphs and define the respective random walk model that can be used for computations. We state and prove theorems on the limit of outflow of authority for cases where individual modalities have identical and distinct damping factors.