c11_netw_res_handout - Topological perturbation of complex...

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1 Topological perturbation of complex networks Effects on the global topology: • increase of path lengths, Perturbations in complex systems can deactivate some of the edges or nodes. Edge loss: the edge is deleted Node loss: the node and all its edges are deleted • separation into isolated clusters. More connected network - less effect of an edge removal But bridges are definite points of vulnerability! The effect of a node removal depends on the number and characteristics of its edges. Resilience to perturbations Topological resilience studied in the literature: the remaining nodes are still connected. the average distance does not increase. Ex. Propose other measures of resilience. Testing resilience to incremental damage: remove edges/nodes one by one, and look at the size of the giant connected component the average distance between nodes in the giant connected component Ex. What factors affect the topological resilience of a network? Review: components in a random graph Erd ő s-Rényi (uniform) random graph: •If the graph contains only isolated trees. • If the graph has isolated trees and cycles. •At a giant connected component appears. • The size of the giant connected component approaches N rapidly as c increases 1 c cN p 1 < = with 0 lim = pN N 1 c cN p 1 = = with rapidly as c increases. • The graph is connected if Random graph with degree distribution P(k): • A giant connected component exists if Ex. How is this related to topological resilience? N / N ln p > 2 2 k k Edge removal in random graphs Start with a connected ER random graph with conn. prob. p. Remove a random fraction f of the edges. Expected result: an ER graph with conn. prob. p(1-f) N / N ln p > Connected if For a broad class of starting graphs, there exists a threshold edge removal probability such that if a smaller fraction of edges is removed the graph is still connected. B. Bollobas, Random Graphs, 1985 N / N ln ) f 1 ( p >
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2 Node removal Removing a node deactivates all its edges. We can expect that the effect of the node removal will depend on the number of edges it had. The size of the connected component will decrease at least by one. Assume we have two networks with the same number of nodes and edges, and remove a randomly chosen fraction f of the nodes. Can the two networks’ resilience be different? Breakdown transition in general random graphs Consider a random graph with arbitrary P(k 0 ) A giant cluster exists if each node is connected to at least two other nodes. 2 2 k k After the random removal of a fraction f of the nodes, ) f 1 ( f k ) f 1 ( k k ), f 1 ( k k 0 2 2 0 2 0 + = = 1 k k 1 1 f 0 2 0 c = Breakdown threshold: Cohen et al., Phys. Rev. Lett. 85, 4626 (2000). Application: ER random graphs Consider a random graph with connection probability p such that at least a giant connected component is present in the graph.
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c11_netw_res_handout - Topological perturbation of complex...

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