Title: Cascades on correlated and modular networks Speaker: Prof. J.P. Gleeson, University of Limerick, Ireland Abstract: An analytical approach to determining the mean avalanche size in a broad class of dynamical models on random networks is introduced. Previous results on percolation transitions and epidemic sizes are shown to be special cases of the method. The time-dependence of cascades and extensions to networks with community structure or degree-degree correlations are discussed. Analytical results for the rate of spread of innovations (Watts, 2002) in a modular network and for the size of k-cores (Dorogotsev et al, 2006) in networks with degree-degree correlations are confirmed with numerical simulations. The dynamics of cascades are strongly dependent upon the topological structure of the underlying network and on the details of how the cascade spreads among the nodes of the network. In the class of examples considered here, each node of the network can be in one of two states, either active (also termed damaged or infected) or inactive (undamaged or susceptible), with nodes updating their states depending on the number and state of the node's immediate neighbors in the (undirected) network. Networks are chosen from an ensemble of graphs with specified degree distribution (i.e. using the configuration model (Newman et al, 2001)), and both synchronous and asynchronous updating may be considered. We show that for a class of such models the average cascade size may be determined analytically (averages being taken over an ensemble of realizations) (Gleeson and Cahalane, 2007). This basic model is also extended to networks with strong community structure or with degree-degree correlations. Previous results on percolation and k-core sizes are shown to be special cases of our general approach. References: S.N. Dorogovtsev, A.V. Goltsev, and J.F.F. Mendes (2006): k-Core organization of complex networks. Phys. Rev. Lett., 96, 040601. J.P. Gleeson and D.J. Cahalane (2007): Seed size strongly affects cascades on random networks. Phys. Rev. E., 75, 056103. M.E.J. Newman, S.H. Strogatz, and D.J. Watts (2001): Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E, 64, 026118. D.J. Watts (2002): A simple mode of global cascades on random networks. Proc. Nat. Acad. Sci. 99, 5766.