Small-world networks and RNA secondary structures

TL;DR

This paper analyzes the properties of RNA secondary structure networks, showing they are not small-world due to their asymptotic average degree and clustering coefficient, using combinatorial and analytical methods.

Contribution

It introduces a novel analysis of RNA secondary structure networks, establishing their non-small-world nature through asymptotic behavior.

Findings

Average degree grows linearly with n

Clustering coefficient decreases as 1/n

Networks are not small-world for large n

Abstract

Let Sn denote the network of all RNA secondary structures of length n, in which undirected edges exist between structures s, t such that t is obtained from s by the addition, removal or shift of a single base pair. Using context-free grammars, generating functions and complex analysis, we show that the asymptotic average degree is O(n) and that the asymptotic clustering coeffcient is O(1/n), from which it follows that the family Sn, n = 1,2,3,... of secondary structure networks is not small-world.