Extreme events on complex networks

arXiv:1102.1789v1 [cond-mat.stat-mech] 9 Feb 2011 Extreme events on complex networks Vimal Kishore,1, ∗M. S. Santhanam,2, † and R. E. Amritkar1, ‡ 1Physical Research Laboratory, Navrangpura, Ahmedabad 380 009, India. 2Indian Institute of Science Education and Research, Pashan Road, Pune 411 021, India. (Dated: October 30, 2018) We study the extreme events taking place on complex networks. The transport on networks is modelled using random walks and we compute the probability for the occurance and recurrence of extreme events on the network. We show that the nodes with smaller number of links are more prone to extreme events than the ones with larger number of links. We obtain analytical estimates and verify them with numerical simulations. They are shown to be robust even when random walkers follow shortest path on the network. The results suggest a revision of design principles and can be used as an input for designing the nodes of a network so as to smoothly handle an extreme event. PACS numbers: 05.45.-a, 03.67.Mn, 05.45.Mt Extreme events(EE) taking place on the networks is a fairly common place experience. Traffic jams in roads and other transportation networks, web servers not re- sponding due to heavy load of web requests, floods in the network of rivers, power black outs due to tripping of power grids are some of the common examples of EE on networks. Such events can be thought of as an emergent phenomena due to the transport on the networks. As EE lead to losses ranging from financial and productivity to even of life and property [1], it is important to estimate probabilities for the occurance of EE and, if possible, in- corporate them to design networks that can handle such EE. Transport phenomena on the networks have been stud- ied vigorously in the last several years [2, 3] though they were not focussed on the analysis of EE. However, one kind of extreme event in the form of congestion has been widely investigated [4]. For instance, a typical approach is to define rules for (a) generation and transport of traf- fic on the network and (b) capacity of the nodes to service them. Thus, a node will experience congestion when its capacity to service the incoming ’packets’ has been ex- ceeded [5]. In this framework, several results on the sta- bility of networks, cascading failures to congestion tran- sition etc. have been obtained. Extreme event, on the other hand, is defined as exceedences above a prescribed quantile and is not necessarily related to the handling capacity of the node in question. It arises from natu- ral fluctuations in the traffic passing through a node and not due to constraints imposed by capacity. Thus, in rest of this paper, we discuss transport on the networks and analyse the probabilities for the occurance of EE arising in them without having to model the dynamical processes or prescribe capacity at each of the nodes. The transport model we adopt in this work is the ran- dom walk on complex networks [3]. Random walk is of fundamental importance in statistical physics though in real network settings many variants of random walk could be at work [6]. For instance, in the case of road traffic, the flow typically follows a fixed, often shortest, path from node A to B and can be loosely termed deterministic. As we show in this paper, thresholds and corresponding probabilities for the EE depend on such details as the operating principle of the network. Thus, given the oper- ational principle of network dynamics, i.e., deterministic or probabilistic or a combination of both, can the nodes of the network be designed to have sufficient capacity to smoothly handle EE of certain magnitude? We show that we can obtain apriori estimates for the volume of transport on the nodes given the static parameters and operating principle of the network. Currently, for uni- variate time series, there is a widespread interest on the extreme value statistics and their properties, in particu- lar in systems that display long memory [7]. Thus, we place our results in the context of both the random walks and EE in a network setting. We consider a fully connected, undirected, finite net- work with N nodes with E edges. The links are described by an adjacency matrix A with whose elements Aij are either 1 or 0 depending on whether i and j are connected by a link or not respectively. On this network, we have W non-interacting walkers performing the standard random walk. A random walker at time t sitting on ith node with Ki links can choose to hop to any of the neigh- bouring nodes with equal probability. Thus, transition probability for going from ith to jth node is Aij/K. We can write down a master equation for the n−step transi- tion probability of a walker starting from node i at time n = 0 to node j at time n as, Pij(n + 1) = X k Akj Kk Pik(n) (1) It can be shown that the n−step time-evolution operator corresponding to this transition, acting on an initial dis- tribution, leads to stationary distribution with eigenvalue u

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