# Self-avoiding walks and connective constants in clustered scale-free networks

## TL;DR

This paper investigates self-avoiding walks in clustered scale-free networks, revealing how clustering and degree distribution influence the asymptotic behavior of these walks and their connective constants.

## Contribution

It provides a combined enumeration and analytical approach to understand SAWs in clustered scale-free networks, highlighting the impact of clustering and degree exponent on connective constants.

## Key findings

- Average number of SAWs is higher in clustered networks.
- Connective constant converges for γ > 3, diverges logarithmically at γ=3, and scales as a power law for γ < 3.
- Results agree between enumeration and analytical methods.

## Abstract

Various types of walks on complex networks have been used in recent years to model search and navigation in several kinds of systems, with particular emphasis on random walks. This gives valuable information on network properties, but self-avoiding walks (SAWs) may be more suitable than unrestricted random walks to study long-distance characteristics of complex systems. Here we study SAWs in clustered scale-free networks, characterized by a degree distribution of the form $P(k) \sim k^{-\gamma}$ for large $k$. Clustering is introduced in these networks by inserting three-node loops (triangles). The long-distance behavior of SAWs gives us information on asymptotic characteristics of such networks. The number of self-avoiding walks, $a_n$, has been obtained by direct enumeration, allowing us to determine the {\em connective constant} $\mu$ of these networks as the large-$n$ limit of the ratio $a_n / a_{n-1}$. An analytical approach is presented to account for the results derived from walk enumeration, and both methods give results agreeing with each other. In general, the average number of SAWs $a_n$ is larger for clustered networks than for unclustered ones with the same degree distribution. The asymptotic limit of the connective constant for large system size $N$ depends on the exponent $\gamma$ of the degree distribution: For $\gamma > 3$, $\mu$ converges to a finite value as $N \to \infty$; for $\gamma = 3$, the size-dependent $\mu_N$ diverges as $\ln N$, and for $\gamma < 3$ we have $\mu_N \sim N^{(3 - \gamma) / 2}$.