Analytical topological bounds and favorable elastic paths in discrete molecular networks
4 affiliations
- 1Faculty of Sciences
- 2Universitat Autònoma de Barcelona (UAB
- 3ESIT – School of Engineering and Technology
- 4International University of La Rioja (UNIR
Abstract
The mechanical response of discrete networks is fundamentally governed by their underlying connectivity. While traditional matrix-based approaches successfully capture global mechanical responses, their utility is delimited by an 𝒪(𝑁3)computational cost that limits direct access to specific topological pathways and localized load transmission. Here, we formulate a graph-theoretical framework that determines the effective elastic constant of molecular networks through analytical topological rules, bypassing global matrix diagonalization. Central to this formulation is the concept of favorable elastic paths—the specific topological highways that actively transmit mechanical force between terminal vertices. For series-parallel architectures, we compute effective stiffness via recursive edge reductions characterized by excluded minors. To quantify localized mechanical contributions, we introduce a normalized Relevance Ratio and a hierarchical (𝑘, 𝑑)-subgraph classification based on graph distance and vertex connectivity. For complex, non-reducible topologies, we leverage Menger's theorem to derive rigorous topological upper and lower bounds for the effective stiffness using purely combinatorial opera tions. By mapping atoms and bonds to vertices and edges, we demonstrate how discrete connectivity dictates continuum mechanical response. This topology-driven approach provides a computationally efficient (𝒪(∣ 𝐸 ∣)) and analytically transparent baseline, opening new avenues for the algorithmic screening and targeted chemical functionalization of complex biopolymer networks and mechanical metamaterials.
Related
- Phase-delays shape multistability and basin sizes in Kuramoto networks: analytical estimates from network structure
- Topological Characteristics for the Analysis of Vector Fields. New stable characteristics for discrete and continuous dynamical systems
- Discrete Diffusion Models via Evolving Variational Autoregressive Networks
- Discrete p-Fold Directional Models for Binned Orientation Data with Applications to Sand-Grain Orientations and Urban Street Networks
- Morse-Smale dynamics in discrete Huygens networks
BibTeX
@misc{pons2026analytical,
title = {{Analytical topological bounds and favorable elastic paths in discrete molecular networks}},
author = {G. Pons and Y. García and Yamila Garcia-Martinez},
howpublished = {ChemRxiv},
year = {2026},
month = sep,
doi = {10.26434/chemrxiv.15009036/v1},
url = {https://chemrxiv.org/doi/full/10.26434/chemrxiv.15009036/v1},
note = {CC BY},
}
RIS
TY - UNPB
TI - Analytical topological bounds and favorable elastic paths in discrete molecular networks
AU - G. Pons
AU - Y. García
AU - Yamila Garcia-Martinez
PY - 2026
DA - 2026/09/18
PB - ChemRxiv
DO - 10.26434/chemrxiv.15009036/v1
UR - https://chemrxiv.org/doi/full/10.26434/chemrxiv.15009036/v1
AB - The mechanical response of discrete networks is fundamentally governed by their underlying connectivity. While traditional matrix-based approaches successfully capture global mechanical responses, their utility is delimited by an 𝒪(𝑁3)computational cost that limits direct access to specific topological pathways and localized load transmission. Here, we formulate a graph-theoretical framework that determines the effective elastic constant of molecular networks through analytical topological rules, bypassing global matrix diagonalization. Central to this formulation is the concept of favorable elastic paths—the specific topological highways that actively transmit mechanical force between terminal vertices. For series-parallel architectures, we compute effective stiffness via recursive edge reductions characterized by excluded minors. To quantify localized mechanical contributions, we introduce a normalized Relevance Ratio and a hierarchical (𝑘, 𝑑)-subgraph classification based on graph distance and vertex connectivity. For complex, non-reducible topologies, we leverage Menger's theorem to derive rigorous topological upper and lower bounds for the effective stiffness using purely combinatorial opera tions. By mapping atoms and bonds to vertices and edges, we demonstrate how discrete connectivity dictates continuum mechanical response. This topology-driven approach provides a computationally efficient (𝒪(∣ 𝐸 ∣)) and analytically transparent baseline, opening new avenues for the algorithmic screening and targeted chemical functionalization of complex biopolymer networks and mechanical metamaterials.
ER -