  Discontinuous percolation transitions and the associated tricritical points
are manifest in a wide range of both equilibrium and non-equilibrium
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 (GEP) 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
GEP 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", or formally equilibrium) random graph models and includes the
Strauss and the 2-star model, where 'chemical potentials' control the densities
of links, triangles or 2-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.
