Network heterogeneity: a focus on links and types

Network heterogeneity is a fundamental concept in defining the structural complexity of interacting systems, yet traditional definitions based on degree distribution often overlook the diversity of connections between distinct entity types. In this paper, we propose a novel theoretical framework centered on "link heterogeneity" within the context of bipartite and layered networks. By employing a matrix-based approach to transform bipartite structures into layered representations, we demonstrate that heterogeneity is driven exclusively by "external links"—connections between different node layers. We introduce the HE index to rigorously quantify this property, establishing that networks lacking cross-layer connections are mathematically equivalent to homogeneous systems. Our findings indicate that a single bipartite network generates two distinct weighted one-mode projections, each possessing a unique heterogeneity profile. However, this structural diversity presents a paradox: while high heterogeneity, characterized by pronounced modularity, facilitates information diffusion, it simultaneously introduces systemic risks by increasing vulnerability to the removal of key hubs. While currently limited to undirected projections and requiring further empirical validation on large-scale datasets, this study provides a mathematically grounded tool for analyzing bibliographic coupling, co-citation, and affiliation networks. The findings suggest that effective R&D policy management should balance the efficiency of connectivity with the protection of critical nodes to mitigate structural fragility.