Scaling of random spreading in small world networks

Random networks with complex topology are common in Nature, describing systems as diverse as the world wide web or social and business networks. Recently, it has been demonstrated that most large networks for which topological information is available display scale-free features. Here we study the scaling properties of the recently introduced scale-free model, that can account for the observed power-law distribution of the connectivities. We develop a mean-field method to pre...

The cover time of random walks on a graph has found wide practical applications in different fields of computer science, such as crawling and searching on the World Wide Web and query processing in sensor networks, with the application effects dependent on the behavior of cover time: the smaller the cover time, the better the application performance. It was proved that over all graphs with $N$ nodes, complete graphs have the minimum cover time $N\log N$. However, complete gra...

Supplementing a lattice with long-range connections effectively models small-world networks characterized by a high local and global interconnectedness observed in systems ranging from society to the brain. If the links have a wiring cost associated to their length l, the corresponding distribution q(l) plays a crucial role. Uniform length distributions have received most attention despite indications that q(l) ~ l^{-\alpha} exist, e.g. for integrated circuits, the Internet a...

Small-world networks by Watts and Strogatz are a class of networks that are highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. These characteristics result in networks with unique properties of regional specialization with efficient information transfer. Social networks are intuitive examples of this organization with cliques or clusters of friends being interconnected, but each person is really only 5-6 people away from a...

In this work we investigate the dynamics of random walk processes on scale-free networks in a short to moderate time scale. We perform extensive simulations for the calculation of the mean squared displacement, the network coverage and the survival probability on a network with a concentration $c$ of static traps. We show that the random walkers remain close to their origin, but cover a large part of the network at the same time. This behavior is markedly different than usual...

We use scaling results to identify the crossover to mean-field behavior of equilibrium statistical mechanics models on a variant of the small world network. The results are generalizable to a wide-range of equilibrium systems. Anomalous scaling is found in the width of the mean-field region, as well as in the mean-field amplitudes. Finally, we consider non-equilibrium processes.

Systems as diverse as genetic networks or the world wide web are best described as networks with complex topology. A common property of many large networks is that the vertex connectivities follow a scale-free power-law distribution. This feature is found to be a consequence of the two generic mechanisms that networks expand continuously by the addition of new vertices, and new vertices attach preferentially to already well connected sites. A model based on these two ingredie...

We explore a new variant of Small-World Networks (SWNs), in which an additional parameter ($r$) sets the length scale over which shortcuts are uniformly distributed. When $r=0$ we have an ordered network, whereas $r=1$ corresponds to the original SWN model. These short-range SWNs have a similar degree distribution and scaling properties as the original SWN model. We observe the small-world phenomenon for $r \ll 1$ indicating that global shortcuts are not necessary for the sma...

Small-world architectures may be implicated in a range of phenomena from disease propagation to networks of neurons in the cerebral cortex. While most of the recent attention on small-world networks has focussed on the effect of introducing disorder/randomness into a regular network, we show that that the fundamental mechanism behind the small-world phenomenon is not disorder/randomness, but the presence of connections of many different length scales. Consequently, in order t...

Small world models are networks consisting of many local links and fewer long range `shortcuts'. In this paper, we consider some particular instances, and rigorously investigate the distribution of their inter--point network distances. Our results are framed in terms of approximations, whose accuracy increases with the size of the network. We also give some insight into how the reduction in typical inter--point distances occasioned by the presence of shortcuts is related to t...