We study a simple model of how social behaviors, like trends and opinions, propagate in networks where individuals adopt the trend when they are informed by threshold T neighbors who are adopters. Using a dynamic message-passing algorithm, we develop a tractable and computationally efficient method that provides complete time evolution of each individual's probability of adopting the trend or of the frequency of adopters and nonadopters in any arbitrary networks. We validate the method by comparing it with Monte Carlo-based agent simulation in real and synthetic networks and provide an exact analytic scheme for large random networks, where simulation results match well. Our approach is general enough to incorporate non-Markovian processes and to include heterogeneous thresholds and thus can be applied to explore rich sets of complex heterogeneous agent-based models.
A new percolation problem is posed which can exhibit a first-order transition. In bootstrap percolation, sites on an empty lattice are first randomly occupied, and then all occupied sites with less than a given number m of occupied neighbours are successively removed until a stable configuration is reached. On any lattice for sufficiently large m, the ensuing clusters can only be infinite. On a Bethe lattice for m>or=3, the fraction of the lattice occupied by infinite clusters discontinuously jumps from zero at the percolation threshold. From an analysis of stable and metastable ground states of the dilute Blume-Capel model (1966), it is concluded that effects like bootstrap percolation may occur in some real magnets.  This work extends to dimension d≥3 the main result of Dehghanpour and Schonmann. We consider the stochastic Ising model on Zd evolving with the Metropolis dynamics under a fixed small positive magnetic field h starting from the minus phase. When the inverse temperature β goes to ∞, the relaxation time of the system, defined as the time when the plus phase has invaded the origin, behaves like exp(βκd). The value κd is equal to κd=1d+1(Γ1+⋯+Γd), where Γi is the energy of the i-dimensional critical droplet of the Ising model at zero temperature and magnetic field h.   A theory is constructed to describe the zero-temperature jamming transition of repulsive soft spheres as the density is increased. Local mechanical stability imposes a constraint on the minimum number of bonds per particle; we argue that this constraint suggests an analogy to k-core percolation. The latter model can be solved exactly on the Bethe lattice, and the resulting transition has a mixed first-order continuous character reminiscent of the jamming transition. In particular, the exponents characterizing the continuous parts of both transitions appear to be the same. Finally, numerical simulations suggest that in finite dimensions the k-core transition can be discontinuous with a nontrivial diverging correlation length.   Thek-core of a graph is the largest subgraph with minimum degree at leastk. For the Erdo?s?R?nyi random graphG(n,?m) onnvertives, withmedges, it is known that a giant 2-core grows simultaneously with a giant component, that is, whenmis close ton 2. We show that fork?3, with high probability, a giantk-core appears suddenly whenmreachesckn 2; hereck=min?>0? ?k(?) and?k(?)=P Poisson(?)?k?1 . In particular,c3?3.35. We also demonstrate that, unlike the 2-core, when ak-core appears for the first time it is very likely to be giant, of size ?pk(?k)n. Here?kis the minimum point of? ?k(?) andpk(?k)=P Poisson(?k)?k . Fork=3, for instance, the newborn 3-core contains about 0.27nvertices. Our proofs are based on the probabilistic analysis of an edge deletion algorithm that always find ak-core if the graph has one. Bootstrap percolation models, or equivalently certain types of cellular automata, exhibit interesting finite-volume effects. These are studied at a rigorous level. The authors find that for an initial configuration obtained by placing particles independently with probability p or=2), the density of the 'bootstrapped' (final) configurations in the sequence of cubes (-L 2, L 2)d typically undergoes an abrupt transition, as L is increased, from being close to 0 to the value 1. With L fixed at a large value, the mean final density as a function of p changes from 0 to 1 around a value which varies only slowly with L-the pertinent parameter being lambda =p1(d-1) ln L. The driving mechanism is the capture of a 'critical droplet'. This behaviour is analogous to the decay of a metastable state near a first-order phase transition, for which the analysis offers some suggestive ideas.