title: Discontinuous percolation transitions in epidemic processes, surface depinning in random media, and Hamiltonian random graphs.
abstract: Discontinuous percolation transitions and the associated tricritical points are manifest in a wide range of both equilibrium and nonequilibrium cooperative phenomena. To demonstrate this, we present and relate the continuous and first-order behaviors in two different classes of models: The first are generalized epidemic processes that describe in their spatially embedded version--either on or off a regular lattice--compact or fractal cluster growth in random media at zero temperature. A random graph version of these processes is mapped onto a model previously proposed for complex social contagion. We compute detailed phase diagrams and compare our numerical results at the tricritical point in d = 3 with field theory predictions of Janssen et al. [Phys. Rev. E 70, 026114 (2004)]. The second class consists of exponential ("Hamiltonian," i.e., formally equilibrium) random graph models and includes the Strauss and the two-star model, where "chemical potentials" control the densities of links, triangles, or two-stars. When the chemical potentials in either graph model are O(logN), the percolation transition can coincide with a first-order phase transition in the density of links, making the former also discontinuous. Hysteresis loops can then be of mixed order, with second-order behavior for decreasing link fugacity, and a jump (first order) when it increases.
authors.affiliation: Complexity Science Group, University of Calgary, Calgary T2N 1N4, Canada.
authors.forename: Golnoosh
authors.initials: G
authors.lastname: Bizhani
authors.forename: Maya
authors.initials: M
authors.lastname: Paczuski
authors.forename: Peter
authors.initials: P
authors.lastname: Grassberger
mesh_terms: D000465:Algorithms; D000818:Animals; D003198:Computer Simulation; D003627:Data Interpretation, Statistical; D004196:Disease Outbreaks / Q000706:statistics & numerical data*; D058872:Epidemics / Q000706:statistics & numerical data*; D006801:Humans; D015233:Models, Statistical; D012307:Risk Factors