General optimization technique for high-quality community detection in complex networks.

Recent years have witnessed the development of a large body of algorithms for community detection in complex networks. Most of them are based upon the optimization of objective functions, among which modularity is the most common, though a number of alternatives have been suggested in the scientific literature. We present here an effective general search strategy for the optimization of various objective functions for community detection purposes.

Delineating geographical regions with networks of human interactions in an extensive set of countries.

Large-scale networks of human interaction, in particular country-wide telephone call networks, can be used to redraw geographical maps by applying algorithms of topological community detection. The geographic projections of the emerging areas in a few recent studies on single regions have been suggested to share two distinct properties: first, they are cohesive, and second, they tend to closely follow socio-economic boundaries and are similar to existing political regions in size and number. Here we use an extended set of countries and clustering indices to quantify overlaps, providing ample additional evidence for these observations using phone data from countries of various scales across Europe, Asia, and Africa: France, the UK, Italy, Belgium, Portugal, Saudi Arabia, and Ivory Coast.

Random collisions on branched networks: how simultaneous diffusion prevents encounters in inhomogeneous structures.

A huge variety of natural phenomena, including prey-predator interaction, chemical reaction kinetics, foraging, and pharmacokinetics, are mathematically described as encounters between entities performing a random motion on an appropriate structure. On homogeneous structures, two random walkers meet with certainty if and only if the structure is recurrent, i.e.

Statistics of reciprocal distances for random walks of three particles in one dimension.

We investigate the collective properties of three particles performing both independent and interacting random walks on an infinite line by studying the problem on the corresponding distance graph. In the large times limit, we obtain the asymptotically exact behavior of several probability functions regarding maximum and minimum mutual distances among particles. Finally, we suggest possible applications of our results.

Generalization of the Peierls-Griffiths theorem for the Ising model on graphs.

We present a sufficient condition for the presence of spontaneous magnetization for the Ising model on a general graph, related to its long-range topology. Applying this condition we are able to prove the existence of a phase transition at temperature T>0 on a wide class of general networks. The possibility of further extensions of our results is discussed.