Network Discovery by Generalized Random Walks

We investigate network exploration by random walks defined via stationary and adaptive transition probabilities on large graphs. We derive an exact formula valid for arbitrary graphs and arbitrary walks with stationary transition probabilities (STP), for the average number of discovered edges as function of time. We show that for STP walks site and edge exploration obey the same scaling $\sim n^{\lambda}$ as function of time $n$. Therefore, edge exploration on graphs with many loops is always lagging compared to site exploration, the revealed graph being sparse until almost all nodes have been discovered. We then introduce the Edge Explorer Model, which presents a novel class of adaptive walks, that perform faithful network discovery even on dense networks.Comment: 23 pages, 7 figure

Brownian-Vacancy Mediated Disordering Dynamics

The disordering of an initially phase segregated system of finite size, induced by the presence of highly mobile vacancies, is shown to exhibit dynamic scaling in its late stages. A set of characteristic exponents is introduced and computed analytically, in excellent agreement with Monte Carlo data. In particular, the characteristic time scale, controlling the crossover between increasing disorder and saturation, is found to depend on the exponent scaling the number of vacancies in the sample.Comment: 6 pages, typeset using Euro-LaTex, 6 figures, compresse

Optimization in Networks

The recent surge in the network modeling of complex systems has set the stage for a new era in the study of fundamental and applied aspects of optimization in collective behavior. This Focus Issue presents an extended view of the state of the art in this field and includes articles from a large variety of domains where optimization manifests itself, including physical, biological, social, and technological networked systems.Comment: Opening article of the CHAOS Focus Issue "Optimization in Networks", available at http://link.aip.org/link/?CHA/17/2/htmlto