# Percolation in Complex Networks

Reuven Cohen, Shlomo Havlin

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

## Abstract

ComponentThe set of nodes reachable from a given node. The nodes of a component are all reachable from each other.DegreeNumber of edges emanating from a node.Giant componentThe component of a graph with size (number of nodes) of order of the number of nodes in the graph.GraphA set of nodes (sites) and edges (links or bonds) connecting them.LoopA path that start and ends at the same node.Minimum spanning treeIn a weighted graph -- the tree subgraph of the graph with the minimum total weight.Optimal pathIn a weighted graph -- the path with minimum total weight connecting two nodes.Percolation theoryThe theory studying the connectivity behavior of networks when a fraction of the nodes or edges are removed. Site (or node) percolation involves occupying a fraction, p, of the nodes of the graph, or alternatively, removing a fraction q = 1 -- p. In bond (or edge) percolation edges are occupied, or removed, with some probability. A combined site-bond percolation, where both processes occur simultaneously, is also considered.Percolation thresholdThe fraction, pc of occupied nodes or edges, under the graph is fragmented into small components, and above which a giant component emerges.Random graphA graph selected from an ensemble (probability space) of graphs.Scale free networkA network whose nodes' degrees are distributed according to a power law.Shortest pathThe path with minimum number of edges connecting two nodes.TreeA connected graph (a graph consisting of a single component) with no loops.Weighted graphA graph where each edge is assigned a (usually non-negative) weight.
Original language English Complex Media and Percolation Theory Muhammad Sahimi, Allen G. Hunt New York, NY Springer US 419-431 13 978-1-0716-1457-0 https://doi.org/10.1007/978-1-0716-1457-0_383 Published - 2009

### Publication series

Name Complex Media and Percolation Theory

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