Nonlinear Sciences – Cellular Automata and Lattice Gases
Scientific paper
2006-05-25
B. Samuelsson and J. E. S. Socolar, Phys. Rev. E 74, 036113 (2006)
Nonlinear Sciences
Cellular Automata and Lattice Gases
19 pages, 7 figures; corrected errors in eq. (73) and some typos
Scientific paper
10.1103/PhysRevE.74.036113
We consider propagation models that describe the spreading of an attribute, called "damage", through the nodes of a random network. In some systems, the average fraction of nodes that remain undamaged vanishes in the large system limit, a phenomenon we refer to as exhaustive percolation. We derive scaling law exponents and exact results for the distribution of the number of undamaged nodes, valid for a broad class of random networks at the exhaustive percolation transition and in the exhaustive percolation regime. This class includes processes that determine the set of frozen nodes in random Boolean networks with indegree distributions that decay sufficiently rapidly with the number of inputs. Connections between our calculational methods and previous studies of percolation beginning from a single initial node are also pointed out. Central to our approach is the observation that key aspects of damage spreading on a random network are fully characterized by a single function specifying the probability that a given node will be damaged as a function of the fraction of damaged nodes. In addition to our analytical investigations of random networks, we present a numerical example of exhaustive percolation on a directed lattice.
Samuelsson Björn
Socolar Joshua E. S.
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