Research : Complex networks
Our work in network theory is mainly articulated around 3 different topics:
We have obtained a formalism that models the discrete-time evolution of a Susceptible-Infected-Recovered (SIR) epidemic propagation on random networks for a given degree distribution. Each time step corresponds to a generation of infections: the initial case(s) is(are) of generation 0, those infected by generation 0 are of generation 1 ans so forth. Unlike many approaches where only the expected (mean) behaviour is studied, our method provides a probability for each infection size at each time step. Although our results are exact for infinite networks ( i.e. simulations converge to the analytical results for increasing networks size), the method needs modifications for finite networks. We have succeeded in incorporating the major finite-size effects in an extended formalism such that, even for populations as small as 1 000, excellent agreement is achieved between simulations and the results of the analysis. Details of this work can be found in: Noel et al. 2009 (see Publications section for further details).
We are now working on extending our formalism to a continuous time description of the dynamics.
Real-world networks possess complex topologies that can significantly influence its static and/or dynamical properties. However tackling this structural complexity in an analytical framework is not an easy task. Therefore, one must rely on simplifying assumptions -- although realistic in some context -- for the analytical models to be solvable or use extensive and time-consuming computer simulations.
We have developed a percolation model using a multitype approach, i.e. where nodes in the network belong to a specic type, to di fferentiate nodes into categories (e.g. gender, age, sociocultural group). This allows to include correlations in the way nodes are connected in the network based on the type they belong to. Moreover, knowing who is who allows to know who infects whom and therefore to use an heterogeneous probability of transmission between node types. Thus this model considers a more realistic propagation dynamics and can give more detailed information on how the disease could propagate in the population (e.g. which groups could be more a ffected than others). (This model is published as Phys. Rev. E 79, 036113 2009.)
Adaptive networks are a class of networks in which there exists a mutual interaction between the dynamics on the networks and the dynamics of the networks. In epidemiological settings, the status of an individual's health influences the underlying social network topology, which in turn a ffects the way the disease will spread further. While it is clear that adaptive networks are ubiquitous in nature, they have not received until recently the attention they deserve.
Until now, adaptive networks, in epidemiology and elsewhere, have been analyzed using low-complexity models revealing nevertheless some novel dynamical features. However, the current approaches have failed to reproduce with accuracy the simultaneous time evolution of a process and the network topology on networks with di erent degrees of heterogeneity. We are currently working on an improved compartmental formalism capable of handling this task successfully. This will provide a first tool able to make accurate quantitative predictions on the complex behaviour of processes occuring on adaptive networks.
Funding for this research is provided by the Canadian Institutes for Health Research (CIHR), the Natural Sciences and Engineering Research Council (NSERC) of Canada, and the Fonds Québécois de la Recherche sur la Nature et les Technologies (FQRNT).