Elastic properties of small-world spring networks

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 $k^{\prime }$, 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 $k=k^{\prime }$. 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 $\alpha \le 3/2$.

16 pages, 14 figures, 1 table, published version

References in corpus (13)

• Statistical mechanics of complex networks
• The structure and function of complex networks
• Classes of behavior of small-world networks
• Incomplete ordering of the voter model on small-world networks
• Nonequilibrium Phase Transitions in Directed Small-World Networks
• Small-World Networks: Links with long-tailed distributions
• Associative memory on a small-world neural network
• Relaxation Properties of Small-World Networks
• Scaling Properties of Random Walks on Small-World Networks
• Dynamic critical behavior of the XY model in small-world networks
• Self-organized Criticality on Small World Networks
• Small-world phenomena and the statistics of linear polymer networks
• Conformations of Randomly Linked Polymers