Anomalous diffusion on complex networks

CSSS Santa Fe 2010 Home Surveys/Evaluations U.S. Visa Requirements Travel Logistics Faculty Participants Schedule Readings Background Readings Projects & Working Groups Tutorials Blog Student Notes After Hours Photos TShirts Project Presentations Final Papers

Next meeting Friday (when?)

Members/interested:

• Lucas Antiqueira
• Damian Blasi
• Borys Wrobel

A standard random walk over a n-dimensional lattice is a stochastic process where the average square of the distance from the origin is proportional to the number of steps of the process -namely <r^2>=c<t> where c is a constant. Among other names of this process in different contexts -brownian motion, Wiener process- is mostly know as diffusion. A possible generalization of this process is to define an anomalous case as <r^2>=c(<t>^h). The exponent h fixes the behavior of the walker. If 0<h<1 we have an antipersistent behavior: the walker prefers to return to its original location; by the other hand, if we choose 1<h<2 we will have a persistent behavior: the walker tends to apart from its origin. This kind of processes are easily mapped to complex networks. Now the distance is the chemical distance between the original birth node and the node reached at time t. The interesting fact is that even if the exponent of the walker is 1 (this is a random walker) if we put it over a complex network the resulting average behavior will not be necessary that of a diffusive process. So, there is a topology-induced anomalous diffusion.

Why this matters? Well, many communication processes in networks are well described as local random walks in the sense that the transmission its realized from one node to another chosen randomly between its neighbors. The idea is to run this walkers over different interesting networks -random, Small World, preferential attachment and real ones- to see wha