Effective Resistance in Simplicial Complexes as Bilinear Forms: Generalizations and Properties

2025-11-13

AI Summary

This paper addresses the challenge of generalizing the graph-theoretic concept of effective resistance to higher-order topological structures represented by simplicial complexes. We introduce a basis-independent bilinear form defined uniformly on both chain and cochain spaces, extending circuit-theoretic resistance modeling to simplices of arbitrary dimension. This form induces a pseudometric on the chain space and a genuine metric on the cycle space, rigorously satisfying symmetry, positive definiteness (on cycles), and the triangle inequality. Furthermore, we generalize Foster’s classical theorem from graph theory to simplicial complexes, establishing an explicit relationship between effective resistance and higher-order topological invariants—namely, Betti numbers and the dimension of the cycle space. Our approach integrates tools from algebraic topology, matrix analysis, and electrical network theory, yielding a unified metric framework for higher-order network analysis, topological data analysis, and discrete geometric modeling.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Social Networks and Social Media: Computational social science
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Extends effective resistance from graphs to simplicial complexes using bilinear forms
• Generalizes effective resistance to simplices, chains, and cochains in complexes
• Establishes higher-order analogues of graph properties like metrics and Foster's Theorem

Innovation

• Bilinear form extends effective resistance from graphs
• Generalizes resistance to simplices chains cochains complexes
• Unifies existing formulations and proves higher-order properties