Research
Cascades on random networks
Previous results for a wide class of dynamical problems on random undirected networks, including Watts’ model of threshold dynamics, site and bond percolation, and k-core size calculations can be shown to be special cases of a general method developed in our group [1], which is based on the approximation of a network by a tree with the same degree distribution of nodes. We would like to know whether it is necessary and/or sufficient to use additional information about degree-degree correlations, and degree-dependent clustering [2] to reproduce dynamical results from real-world information, technological, social, and biological networks.
Social networks online
Online social networks (Facebook and similar) became very popular and widely spread over the past few years and give a fair approximation to real social networks. Overcoming the difficulty of collecting sufficient and reliable information, these networks are a valued source of empirical data about human relationships that researchers never had before. We have recently calculated k-core and percolation cluster sizes on Facebook datasets provided by our collaborators [3]. The opportunity of extending theories for the spread of cascades through social networks, investigating statistical properties of such networks, as well as the issues of connectivity, centrality, and answering such questions as how small our world actually is, appears to be really intriguing.
Bifurcation analysis
Epidemic and other related models with spatial correlations [4] can exhibit complex dynamical behavior, which is only beginning to be understood. We are interested in performing a thorough bifurcation analysis of these models, which should yield a deeper understanding of already studied phenomena, and perhaps the discovery of new dynamical effects.
Parallel computation
Large network sizes and dynamical complexity demand large scale computations, developing efficient algorithms and high performance parallel computing.
Dynamics on networks
[1] J.P. Gleeson, “Cascades on correlated and modular random networks”, Phys. Rev. E 77, 046117 (2008).
[2] J.P. Gleeson and S. Melnik, “Analytical results for bond percolation and k-core sizes on clustered networks”, arXiv:0811.4511v1 [cond-mat.stat-mech], submitted to Phys. Rev. E.
[3] A.L. Traud, E.D. Kelsic, P.J. Mucha, and M.A. Porter, “Community Structure in Online Collegiate Social Networks”, arXiv:0809.0690v1 [physics.soc-ph] (2008).
[4] R. M. Anderson and R. M. May, “Infectious Diseases of Humans”, Oxford Univ. Press, New York (1991); M.E.J. Newman, “Spread of epidemic disease on networks”, Phys. Rev. E 66, 016128 (2002); J. Joo and J.L. Lebowitz, ”Pair approximation of the stochastic susceptible-infected-recovered-susceptible epidemic model on the hypercubic lattice”. Phys. Rev. E 70, 036114 (2004); A. Vazquez, “Epidemic outbreaks on structured populations”, J. Theor. Bio. 245, 125 (2007).
