Scaling of random spreading in small world networks

Using both numerical simulations and scaling arguments, we study the behavior of a random walker on a one-dimensional small-world network. For the properties we study, we find that the random walk obeys a characteristic scaling form. These properties include the average number of distinct sites visited by the random walker, the mean-square displacement of the walker, and the distribution of first-return times. The scaling form has three characteristic time regimes. At short t...

We study the simple random walk dynamics on an annealed version of a Small-World Network (SWN) consisting of $N$ nodes. This is done by calculating the mean number of distinct sites visited S(n) and the return probability $P_{00}(t)$ as a function of the time $t$. $S(t)$ is a key quantity both from the statistical physics point of view and especially for characterizing the efficiency of the network connectedness. Our results for this quantity shows features similar to the SWN...

We present the analytical and numerical results of a random walk on the family of small-world graphs. The average access time shows a crossover from the regular to random behavior with increasing distance from the starting point of the random walk. We introduce an {\em independent step approximation}, which enables us to obtain analytic results for the average access time. We observe a scaling relation for the average access time in the degree of the nodes. The behavior of av...

We give exact relations which are valid for small-world networks (SWN's) with a general `degree distribution', i.e the distribution of nearest-neighbor connections. For the original SWN model, we illustrate how these exact relations can be used to obtain approximations for the corresponding basic probability distribution. In the limit of large system sizes and small disorder, we use numerical studies to obtain a functional fit for this distribution. Finally, we obtain the sca...

Numerous studies show that most known real-world complex networks share similar properties in their connectivity and degree distribution. They are called small worlds. This article gives a method to turn random graphs into Small World graphs by the dint of random walks.

A dynamical model of small-world network, with directed links which describe various correlations in social and natural phenomena, is presented. Random responses of every site to the imput message are introduced to simulate real systems. The interplay of these ingredients results in collective dynamical evolution of a spin-like variable S(t) of the whole network. In the present model, global average spreading length \langel L >_s and average spreading time <T >_s are found to...

We study the distribution function for minimal paths in small-world networks. Using properties of this distribution function, we derive analytic results which greatly simplify the numerical calculation of the average minimal distance, $\bar{\ell}$, and its variance, $\sigma^2$. We also discuss the scaling properties of the distribution function. Finally, we study the limit of large system sizes and obtain some analytic results.

In this paper we study the small-world network model of Watts and Strogatz, which mimics some aspects of the structure of networks of social interactions. We argue that there is one non-trivial length-scale in the model, analogous to the correlation length in other systems, which is well-defined in the limit of infinite system size and which diverges continuously as the randomness in the network tends to zero, giving a normal critical point in this limit. This length-scale go...

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.

Watts and Strogatz [Nature 393, 440 (1998)] have recently introduced a model for disordered networks and reported that, even for very small values of the disorder $p$ in the links, the network behaves as a small-world. Here, we test the hypothesis that the appearance of small-world behavior is not a phase-transition but a crossover phenomenon which depends both on the network size $n$ and on the degree of disorder $p$. We propose that the average distance $\ell$ between any t...