Effective Resistance in Simplicial Complexes as Bilinear Forms: Generalizations and Properties
Inés García-Redondo, Claudia Landi, Sarah Percival, Anda Skeja, Bei Wang, Ling Zhou
TL;DR
The paper develops a basis-free, operator-theoretic generalization of effective resistance from graphs to simplicial complexes by defining a pair of dual bilinear forms on p-chains and p-cochains using up-Laplacians and Moore–Penrose inverses. This framework unifies existing matrix formulations (Osting–Palande–Wang, Kook–Lee, Black–Maxwell) as specific basis choices, and yields new higher-order metric properties: a pseudometric on p-chains and a true metric on p-cycles. A higher-dimensional Foster-type theorem is established, linking the spectrum of the effective-resistance operator to the rank of boundary maps. Collectively these results deepen the connection between network structure and higher-dimensional diffusion-like processes, with potential applications in topology-based data analysis and higher-order machine learning on complexes.
Abstract
The concept of effective resistance, originally introduced in electrical circuit theory, has been extended to the setting of graphs by interpreting each edge as a resistor. In this context, the effective resistance between two vertices quantifies the total opposition to current flow when a unit current is injected at one vertex and extracted at the other. Beyond its physical interpretation, the effective resistance encodes rich structural and geometric information about the underlying graph: it defines a metric on the vertex set, relates to the topology of the graph through Foster's theorem, and determines the probability of an edge appearing in a random spanning tree. Generalizations of effective resistance to simplicial complexes have been proposed in several forms, often formulated as matrix products of standard operators associated with the complex. In this paper, we present a twofold generalization of the effective resistance. First, we introduce a novel, basis-independent bilinear form, derived from an algebraic reinterpretation of circuit theory, that extends the classical effective resistance from graphs. Second, we extend this bilinear form to simplices, chains, and cochains within simplicial complexes. This framework subsumes and unifies all existing matrix-based formulations of effective resistance. Moreover, we establish higher-order analogues of several fundamental properties known in the graph case: (i) we prove that effective resistance induces a pseudometric on the space of chains and a metric on the space of cycles, and (ii) we provide a generalization of Foster's Theorem to simplicial complexes.