Effective Resistance in Simplicial Complexes as Bilinear Forms: Generalizations and Properties Article Swipe
García-Redondo, Inés
,
Landi, Claudia
,
Percival, Sarah
,
Skeja, Anda
,
Wang Bei
,
Zhou Ling
·
YOU?
·
· 2025
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2511.10749
YOU?
·
· 2025
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2511.10749
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.
Related Topics
Concepts
Mathematics
Simplicial complex
Algebraic number
Vertex (graph theory)
Bilinear interpolation
Discrete mathematics
Toric variety
Bilinear form
Topology (electrical circuits)
Betti number
Homotopy
Algebraic properties
Multiplicative function
Combinatorics
Graph
Regular polygon
Simplicial set
Generalization
Pure mathematics
Parametric statistics
Random walk
Metric space
Invariant (physics)
Hypergraph
Resistance distance
Vector space
Space (punctuation)
Formalism (music)
Metadata
- Type
- preprint
- Landing Page
- https://doi.org/10.48550/arxiv.2511.10749
- OA Status
- green
- OpenAlex ID
- https://openalex.org/W7105883441
All OpenAlex metadata
Raw OpenAlex JSON
- OpenAlex ID
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https://openalex.org/W7105883441Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.48550/arxiv.2511.10749Digital Object Identifier
- Title
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Effective Resistance in Simplicial Complexes as Bilinear Forms: Generalizations and PropertiesWork title
- Type
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preprintOpenAlex work type
- Publication year
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2025Year of publication
- Publication date
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2025-11-13Full publication date if available
- Authors
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García-Redondo, Inés, Landi, Claudia, Percival, Sarah, Skeja, Anda, Wang Bei, Zhou LingList of authors in order
- Landing page
-
https://doi.org/10.48550/arxiv.2511.10749Publisher landing page
- Open access
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YesWhether a free full text is available
- OA status
-
greenOpen access status per OpenAlex
- OA URL
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https://doi.org/10.48550/arxiv.2511.10749Direct OA link when available
- Concepts
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Mathematics, Simplicial complex, Algebraic number, Vertex (graph theory), Bilinear interpolation, Discrete mathematics, Toric variety, Bilinear form, Topology (electrical circuits), Betti number, Homotopy, Algebraic properties, Multiplicative function, Combinatorics, Graph, Regular polygon, Simplicial set, Generalization, Pure mathematics, Parametric statistics, Random walk, Metric space, Invariant (physics), Hypergraph, Resistance distance, Vector space, Space (punctuation), Formalism (music)Top concepts (fields/topics) attached by OpenAlex
- Cited by
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0Total citation count in OpenAlex
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| abstract_inverted_index.injected | 47 |
| abstract_inverted_index.physical | 58 |
| abstract_inverted_index.products | 121 |
| abstract_inverted_index.proposed | 113 |
| abstract_inverted_index.spanning | 102 |
| abstract_inverted_index.standard | 123 |
| abstract_inverted_index.subsumes | 181 |
| abstract_inverted_index.theorem, | 90 |
| abstract_inverted_index.topology | 84 |
| abstract_inverted_index.vertices | 34 |
| abstract_inverted_index.Moreover, | 191 |
| abstract_inverted_index.algebraic | 152 |
| abstract_inverted_index.analogues | 195 |
| abstract_inverted_index.appearing | 98 |
| abstract_inverted_index.classical | 160 |
| abstract_inverted_index.complexes | 110 |
| abstract_inverted_index.effective | 3, 30, 61, 106, 139, 161, 189, 209 |
| abstract_inverted_index.establish | 193 |
| abstract_inverted_index.extracted | 52 |
| abstract_inverted_index.framework | 180 |
| abstract_inverted_index.geometric | 67 |
| abstract_inverted_index.introduce | 143 |
| abstract_inverted_index.operators | 124 |
| abstract_inverted_index.resistor. | 25 |
| abstract_inverted_index.associated | 125 |
| abstract_inverted_index.complexes. | 178, 238 |
| abstract_inverted_index.determines | 92 |
| abstract_inverted_index.electrical | 8 |
| abstract_inverted_index.formulated | 118 |
| abstract_inverted_index.introduced | 6 |
| abstract_inverted_index.opposition | 38 |
| abstract_inverted_index.originally | 5 |
| abstract_inverted_index.properties | 199 |
| abstract_inverted_index.quantifies | 35 |
| abstract_inverted_index.resistance | 31, 62, 107, 162, 210 |
| abstract_inverted_index.simplices, | 172 |
| abstract_inverted_index.simplicial | 109, 177, 237 |
| abstract_inverted_index.structural | 65 |
| abstract_inverted_index.underlying | 71 |
| abstract_inverted_index.fundamental | 198 |
| abstract_inverted_index.information | 68 |
| abstract_inverted_index.probability | 94 |
| abstract_inverted_index.resistance, | 4 |
| abstract_inverted_index.resistance. | 140, 190 |
| abstract_inverted_index.formulations | 187 |
| abstract_inverted_index.higher-order | 194 |
| abstract_inverted_index.interpreting | 20 |
| abstract_inverted_index.matrix-based | 186 |
| abstract_inverted_index.pseudometric | 213 |
| abstract_inverted_index.generalization | 136, 232 |
| abstract_inverted_index.Generalizations | 104 |
| abstract_inverted_index.interpretation, | 59 |
| abstract_inverted_index.reinterpretation | 153 |
| abstract_inverted_index.basis-independent | 146 |
| cited_by_percentile_year | |
| countries_distinct_count | 0 |
| institutions_distinct_count | 6 |
| citation_normalized_percentile |