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

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 ${\mu }_N\sim N^{(3-\gamma )/2}$.

arXiv comment13 pages, 9 figures

References in corpus (57)

• Statistical mechanics of complex networks
• The structure and function of complex networks
• Power laws, Pareto distributions and Zipf's law
• Lethality and centrality in protein networks
• The structure of scientific collaboration networks
• The diameter of the world wide web
• Evolution of networks
• Characterization of complex networks: A survey of measurements
• Critical phenomena in complex networks
• Search in Power-Law Networks
• Growing Scale-Free Networks with Tunable Clustering
• Epidemics and percolation in small-world networks
• Dynamics of Rumor Spreading in Complex Networks
• Generation of uncorrelated random scale-free networks
• Small world effect in an epidemiological model
• Random graphs with clustering
• Highly clustered scale-free networks
• Ising Model on Networks with an Arbitrary Distribution of Connections
• Classification of scale-free networks
• Sandpile on Scale-Free Networks
• Accurate estimate of the critical exponent $\nu$ for self-avoiding walks via a fast implementation of the pivot algorithm
• Explosive percolation in scale-free networks
• Tuning clustering in random networks with arbitrary degree distributions
• The robustness of interdependent clustered networks
• How clustering affects the bond percolation threshold in complex networks
• Identifying influential spreaders and efficiently estimating infection numbers in epidemic models: a walk counting approach
• Bond percolation on a class of clustered random networks
• Exploring Complex Networks through Random Walks
• First- and second-order phase transitions in scale-free networks
• Ising model in scale-free networks: A Monte Carlo simulation
• Random spread on the family of small-world networks
• Conformation of Circular DNA in 2 Dimensions
• Propagation dynamics on networks featuring complex topologies
• Self-avoiding walks on scale-free networks
• Scaling Properties of Random Walks on Small-World Networks
• Belief-propagation algorithm and the Ising model on networks with arbitrary distributions of motifs
• Beyond the locally tree-like approximation for percolation on real networks
• Ising models on power-law random graphs
• Explosive spreading on complex networks: the role of synergy
• Two-dimensional self-avoiding walks and polymer adsorption: Critical fugacity estimates
• Self-avoiding walks and connective constants in small-world networks
• Ising model in clustered scale-free networks
• Pulling adsorbed self-avoiding walks from a surface
• The distribution of path lengths of self avoiding walks on Erdős-Rényi networks
• Spin glass behavior of the antiferromagnetic Ising model on a scale-free network
• Irreversible Opinion Spreading on Scale-Free Networks
• Kinetic growth walks on complex networks
• Antiferromagnetic Ising model in scale-free networks
• Critical behavior and correlations on scale-free small-world networks. Application to network design
• Kinetic-growth self-avoiding walks on small-world networks
• Asymptotic scaling behavior of self-avoiding walks on critical percolation clusters
• Fractal transit networks: self-avoiding walks and Lévy flights
• Exactly Solvable Random Graph Ensemble with Extensively Many Short Cycles
• Self-Avoiding Walk on Fractal Complex Networks: Exactly Solvable Cases
• A series test of the scaling limit of self-avoiding walks
• Structural characterization of ice polymorphs from self-avoiding walks
• Relaxation dynamics of maximally clustered networks

Cited by in corpus (3)

• A comparative analysis of knowledge acquisition performance in complex networks
• Imaginary replica analysis of loopy regular random graphs
• Efficient network exploration by means of resetting self-avoiding random walkers