Self-avoiding walks on scale-free networks

We present a novel algorithm that generates scale free small world graphs such as those found in the World Wide Web,social and metabolic networks. We use the generated graphs to study the dynamics of a realistic search strategy on the graphs, and find that they can be navigated in a very short number of steps.

We present an algorithm to grow a graph with scale-free structure of {\it in-} and {\it out-links} and variable wiring diagram in the class of the world-wide Web. We then explore the graph by intentional random walks using local next-near-neighbor search algorithm to navigate through the graph. The topological properties such as betweenness are determined by an ensemble of independent walkers and efficiency of the search is compared on three different graph topologies. In add...

Although the ``scale-free'' literature is large and growing, it gives neither a precise definition of scale-free graphs nor rigorous proofs of many of their claimed properties. In fact, it is easily shown that the existing theory has many inherent contradictions and verifiably false claims. In this paper, we propose a new, mathematically precise, and structural definition of the extent to which a graph is scale-free, and prove a series of results that recover many of the clai...

We present a simple mechanism for generating undirected scale-free networks using random walkers, where the network growth is determined by choosing parent vertices by sequential random walks. We show that this mechanism produces scale-free networks with degree exponent gamma=3 and clustering coefficients depending on random walk length. The mechanism can be interpreted in terms of preferential attachment without explicit knowledge of node degrees.

Folklore has, that the universal scaling properties of linear polymers in disordered media are well described by the statistics of self-avoiding walks Folklore has, that the universal scaling properties of linear polymers in disordered media are well described by the statistics of self-avoiding walks (SAWs) on percolation clusters and their critical exponent $\nu_{\text{SAW}}$, with SAW implicitly referring to \emph{average} SAW. Hitherto, static averaging has been commonly u...

We calculate the mean neighboring degree function $\bar k_{\rm{nn}}(k)$ and the mean clustering function $C(k)$ of vertices with degree $k$ as a function of $k$ in finite scale-free random networks through the static model. While both are independent of $k$ when the degree exponent $\gamma \geq 3$, they show the crossover behavior for $2 < \gamma < 3$ from $k$-independent behavior for small $k$ to $k$-dependent behavior for large $k$. The $k$-dependent behavior is analyticall...

Very often, when studying topological or dynamical properties of random scale-free networks, it is tacitly assumed that degree-degree correlations are not present. However, simple constraints, such as the absence of multiple edges and self-loops, can give rise to intrinsic correlations in these structures. In the same way that Fermionic correlations in thermodynamic systems are relevant only in the limit of low temperature, the intrinsic correlations in scale-free networks ar...

Extensive empirical investigation has shown that a plethora of real networks synchronously exhibit scale-free and modular structure, and it is thus of great importance to uncover the effects of these two striking properties on various dynamical processes occurring on such networks. In this paper, we examine two cases of random walks performed on a class of modular scale-free networks with multiple traps located at several given nodes. We first derive a formula of the mean fir...

In this paper we describe the emergence of scale-free degree distributions from statistical mechanics principles. We define an energy associated to a degree sequence as the logarithm of the number of indistinguishable simple networks it is possible to draw given the degree sequence. Keeping fixed the total number of nodes and links, we show that the energy of scale-free distribution is much higher than the energy associated to the degree sequence of regular random graphs. Thi...

This paper presents an algorithm for generating scale-free networks with adjustable clustering coefficient. The algorithm is based on a random walk procedure combined with a triangle generation scheme which takes into account genetic factors; this way, preferential attachment and clustering control are implemented using only local information. Simulations are presented which support the validity of the scheme, characterizing its tuning capabilities.