Elastic properties of small-world spring networks

TL;DR

This paper investigates the elastic behavior of small-world spring networks, revealing how their stiffness and connectivity influence their response to external forces, with implications for understanding complex network mechanics.

Contribution

It introduces a model for small-world spring networks with variable shortcut extensions and derives scaling relations and statistical properties of their elastic response.

Findings

Effective stiffness scales with network parameters when $k=k'$

Shortcut extension distribution follows a scale-free law with exponent -2

CEED changes can be abrupt or continuous, with power-law distributions in the latter case

Abstract

We construct small-world spring networks based on a one dimensional chain and study its static and quasistatic behavior with respect to external forces. Regular bonds and shortcuts are assigned linear springs of constant $k$ and $k'$, respectively. In our models, shortcuts can only stand extensions less than $\delta_c$ beyond which they are removed from the network. First we consider the simple cases of a hierarchical small-world network and a complete network. In the main part of this paper we study random small-world networks (RSWN) in which each pair of nodes is connected by a shortcut with probability $p$. We obtain a scaling relation for the effective stiffness of RSWN when $k=k'$. In this case the extension distribution of shortcuts is scale free with the exponent -2. There is a strong positive correlation between the extension of shortcuts and their betweenness. We find that the chemical end-to-end distance (CEED) could change either abruptly or continuously with respect to the external force. In the former case, the critical force is determined by the average number of shortcuts emanating from a node. In the latter case, the distribution of changes in CEED obeys power laws of the exponent $-\alpha$ with $\alpha \le 3/2$.