During the last decade it has become clear that the topology in many systems, ranging from technological to social to biological, is not well described by regular lattices nor by random graphs. Complex networks, characterized by small-world effects, large connectivity fluctuations, clustering, correlations and other nontrivial features are often a better description of many natural and man-made systems. Since many of such networks describe the topological patterns that mediate various sorts of interactions among nodes, it is natural and interesting to wonder what is the effect of complex topologies on dynamical processes taking place on them.

A paradigmatic example of how nontrivial phenomena can emerge due to the complex interaction patterns is provided by the study of standard epidemic models [as the Susceptible-Infected-Susceptible (SIS) model] on scale-free networks. When the interaction pattern is given by a regular lattice or a random graph, an epidemic transition occurs, depending on the transmission rate. For high values of such rate the epidemics propagates to a finite fraction of the whole population. For low transmission rates instead, it dies out rapidly, affecting only a vanishingly small fraction of the system. When the dynamical process takes place on a scale-free network, i.e. on a network where the degree distribution decays as P(k) ~ k^(-gamma) with gamma<= 3, the large connectivity fluctuations have the consequence that the epidemic threshold becomes zero: even a very small transition rate leads to a finite prevalence for a sufficiently large system.

Our activity focuses on very simple models similar to SIS, like the Contact Process (CP). The kind of questions we are interested in have to do with the formulation of (heterogeneous) mean-field approaches, the validity of such approaches to quantitatively describe nonequilibrium transitions, the possibility to formulate criteria

a la Ginzburg for them. Other interesting issues are the applicability of Finite-Size-Scaling theory to processes on scale-free topologies and the problem of how to characterize dynamical correlations on network.

a la Ginzburg for them. Other interesting issues are the applicability of Finite-Size-Scaling theory to processes on scale-free topologies and the problem of how to characterize dynamical correlations on network.

**People**: Claudio Castellano

**References**:

- C. Castellano and R. Pastor-Satorras

Non mean-field behavior of the contact process on scale-free networks

Phys. Rev. Lett.96, 038701 (2006). [cond-mat/0506605]

- C. Castellano and R. Pastor-Satorras

- Zero temperature Glauber dynamics on complex networks

J. Stat. Mech. P05001 (2006). [cond-mat/0602315]

- C. Castellano and R. Pastor-Satorras

Reply to the Comment on the paper "Non-mean-field behavior of the contact process on scale-free networks"

Phys. Rev. Lett.

98, 029802 (2007). [cond-mat/0701275]

- C. Castellano and R. Pastor-Satorras

- Routes to thermodynamic limit on scale-free networks

Phys. Rev. Lett.**100**, 148701 (2008). [arXiv:0710.2784]

- M. Boguna', C. Castellano and R. Pastor-Satorras

- Langevin approach for the dynamics of the contact process on annealed scale-free networks
- [arXiv:0810.3000]