Resilience of the Internet to Random Breakdowns

Abstract

A common property of many large networks, including the Internet, is that the connectivity of the various nodes follows a scale-free power-law distribution, P (k) = ck −α . We study the stability of such networks with respect to crashes, such as random removal of sites. Our approach, based on percolation theory, leads to a general condition for the critical fraction of nodes, pc, that need to be removed before the network disintegrates. We show analytically and numerically that for α ≤ 3 the transition never takes place, unless the network is finite. In the special case of the physical structure of the Internet (α ≈ 2.5), we find that it is impressively robust, with pc > 0. 99. 02.50.Cw, 05.40.a, 05.50.+q, 64.60.Ak

Wave Localization Transitions in Complex Systems

“…For this purpose we have analyzed the size N 2 of the second largest cluster in the system which should increase with C 0 if the giant component exists (C 0 < C 0,c ) and decrease for higher values of C 0 if it broke down (C 0 > C 0,c ) [26,40]. We found no indications of a classical transition for λ < 4.5, i.e.…”

“…The upper cut-off value K = mN 1/(λ−1) depends on the system size N [26]. Such networks are called scale-free, because the second moment, k 2 , diverges in the limit of infinite system size (N → ∞).…”

“…The critical occupation probability p c of a network depends on its degree distribution. A randomly connected network with arbitrary degree distribution has an infinite cluster if [26,39,40] …”

“…Scale-free networks are particularly interesting, since it is nearly impossible to break them, i.e. p c > 0.99 is found for exponents 2 < λ < 3 [26,40,41]. Equation (5.7) is only true for uncorrelated networks with zero clustering.…”

Semantic Modelling of Virtual Organizations with Service Network Schemata

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