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Discontinuous percolation transitions and the associated tricritical pointsare manifest in a wide range of both equilibrium and non-equilibriumcooperative phenomena. To demonstrate this, we present and relate thecontinuous and first order behaviors in two different classes of models: Thefirst are generalized epidemic processes (GEP) that describe in their spatiallyembedded version - either on or off a regular lattice - compact or fractalcluster growth in random media at zero temperature. A random graph version ofGEP is mapped onto a model previously proposed for complex social contagion. Wecompute detailed phase diagrams and compare our numerical results at thetricritical 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", or formally equilibrium) random graph models and includes theStrauss and the 2-star model, where 'chemical potentials' control the densitiesof links, triangles or 2-stars. When the chemical potentials in either graphmodel are O(logN), the percolation transition can coincide with a first orderphase transition in the density of links, making the former also discontinuous.Hysteresis loops can then be of mixed order, with second order behavior fordecreasing link fugacity, and a jump (first order) when it increases.

Discontinuous Percolation Transitions in Epidemic Processes, Surface Depinning in Random Media and Hamiltonian Random Graphs