The critical effect of dependency groups on the function of networks
aa r X i v : . [ phy s i c s . d a t a - a n ] O c t The critical effect of dependency groups on the function of networks
Roni Parshani, Sergey V. Buldyrev,
2, 3 and Shlomo Havlin Minerva Center & Department of Physics, Bar-Ilan University, Ramat Gan, Israel Center for Polymer Studies and Dept. of Physics, Boston Univ., Boston, MA 02215 USA Department of Physics, Yeshiva University, 500 West 185th Street, New York, New York 10033, USA (Dated: May 20, 2018)Current network models assume one type of links to define the relations between the networkentities. However, many real networks can only be correctly described using two different types ofrelations. Connectivity links that enable the nodes to function cooperatively as a network and de-pendency links that bind the failure of one network element to the failure of other network elements.Here we present for the first time an analytical framework for studying the robustness of networksthat include both connectivity and dependency links. We show that the synergy between the twotypes of failures leads to an iterative process of cascading failures that has a devastating effect onthe network stability and completely alters the known assumptions regarding the robustness of net-works. We present exact analytical results for the dramatic change in the network behavior whenintroducing dependency links. For a high density of dependency links the network disintegrates in aform of a first order phase transition while for a low density of dependency links the network disin-tegrates in a second order transition. Moreover, opposed to networks containing only connectivitylinks where a broader degree distribution results in a more robust network, when both types of linksare present a broad degree distribution leads to higher vulnerability.
Many friendships between individuals in a social net-work, numerous business connections in a financial net-work or multiple cables between Internet routers, are allexamples of networks with a high density of connectivitylinks [1–10]. Such networks are regarded as very stableto attacks since even after a failure of many nodes thenetwork still remains connected. In contrast, dependen-cies between the network nodes endanger the networkstability since the failure of several nodes may lead tothe immediate failure of many others. As an exampleconsider a financial network: Each company has tradingand sales connections with other companies (connectivitylinks). These connections enable the companies to inter-act with each other and function together as a global fi-nancial market. But there are also dependencies relationsbetween companies, several companies that belong to thesame owner depend on one another. If one company failsthe owner might not be able to finance the other compa-nies that will fail too. Such dependencies jeopardize thenetwork stability and are the possible cause of many ma-jor financial crises. Another example is an online socialnetwork (Facebook or Twitter): Each individual commu-nicates with his friends (connectivity links), thus forminga social network through which information and rumorscan spread. However, many individuals will only partici-pate in a social network if other individuals with commoninterests also participate (dependency links) in that so-cial network.The effect of failing nodes on the network stability hasbeen studied separately for networks containing only con-nectivity links [11–16] and for networks containing onlydependency links [17–21]. The fundamental differencebetween connectivity and dependency links is that fordependency links the failure of a direct neighbor of anode leads to the direct failure (with some probability) of that node, but for connectivity links a node fails onlywhen it (or the cluster it is in) becomes completely dis-connected from the network. Percolation theory is a ma-jor tool for studying network stability when the networkis connected only with connectivity links. In a percola-tion process on a network of size N , a fraction 1 − p ofthe network nodes are removed. If the remaining frac-tion of nodes, p , is larger then a critical value ( p > p c ),a spanning cluster connecting order N nodes exists, ifhowever, p < p c , the network collapses into small clus-ters. At p = p c the network undergoes a second orderphase transition [11–16].Previous studies of networks containing dependenciescan be divided into two categories: (i) Overload failuresin networks containing a flow of a physical quantity. Forexample, disturbances in power transmission systems orcongestion instabilities in transportation networks andInternet traffic [17–20]. These models show that whenone node is overloaded and the traffic cannot be routedthrough it, choosing alternative paths will cause othernodes to also become overloaded. This process may de-velop into a series of cascading failures that can disablethe entire network. (ii) Models based on local dependen-cies, such as decision making of interacting agents [21].In these models the state of a node depends on the stateof its neighbors and therefore a failing node will causeit’s neighbors to also fail and so on. RESULTS
Here we present an analytical framework for studyingthe robustness of networks that include both connectiv-ity and dependency links. When nodes fail in a networkcontaining both types of links, two different processes
FIG. 1: Demonstration of the synergy between the percola-tion process and the failures caused by dependency links (de-pendency process) that lead to an iterative process of cascad-ing failures. The network contains two types of links: connec-tivity links (solid lines) and dependency links (dashed lines).(a) The process starts with the initial failure of two nodes(marked in red). The connectivity links connected to themalso fail (marked in red). (b) Percolation process - in thisstage all the nodes and the connectivity links that are con-nected to them, that are not connected to the giant cluster(largest cluster) by connectivity links also fail (marked in red).(c) Dependency process - the nodes that depend (connectedby dependency links) on the failing nodes also fail (marked inred). (d) The next step of connectivity failure in which twomore nodes fail because they are not connected to the largestcluster (currently containing only two nodes). occur. (i) Connectivity links are disconnected, causingother nodes to disconnect from the network (percolationprocess). (ii) Failing nodes cause other nodes that de-pend on them to also fail even though they are still con-nected via connectivity links (dependency process). Weshow that the synergy between the percolation processand the dependency process leads to a cascade of failuresthat can fragment the entire network (Fig.1). We findthat the density of dependency links, q , plays a key rolein determining the robustness of such networks. For net-works containing connectivity links and a high density ofdependency links, an initial failure of even a small frac-tion of the network nodes disintegrates the network in aform of a first order phase transition.If however, the fraction of dependency links is reducedbelow a certain threshold, q c , the network disintegratesin a form of a second order phase transition. The cas-cading process leading to a first order transition existsfor a wide range of topologies including lattices, ER and p α SF q=1ER q=1Lattice q=1SF q=0ER q=0Lattice q=0 (a) q p I , p II ER networkSF network second order first order (b) transitiontransition
FIG. 2: (a) Simulation results showing the first and secondorder phase transitions in lattice, ER and SF networks. Thefraction of nodes in the giant component at the end the cas-cade process, α ∞ , is shown as a function of p for q = 1 (filledsymbols) and for q = 0 (open symbols), where q is the fractionof dependent nodes. For q = 1, α ∞ abruptly drops to zero atthe transition point characterizing a first order transition. For q = 0, α ∞ gradually approaches zero as expected in a secondorder transition. The SF (circle) and ER (square) networkspresented both have the same average degree of h k i = 3 . q = 0) becomemost vulnerable when dependency links are added (very hightransition point for q = 1). (b) The transition points, p I forthe first order region (solid line) and p II for the second or-der region (dashed line) are plotted as a function of q (thefraction of dependent nodes) for ER (squares) and SF (cir-cles) networks with the same average degree h k i = 4. For ERnetworks theoretical results (confirmed by simulation results)are obtained according to Eq.(3) and Eq.(4) presented in thepaper. For SF networks simulation results of a network with λ = 2 . SF networks, indicating it is a general property of manynetworks (Fig.2(a)). Comparing networks with both con-nectivity and dependency links but with different topolo-gies, reveals a new relation between topology and therobustness to random failure: Networks with a broaderdegree distribution of connectivity links are more vul-nerable to random failure in the presence of dependencylinks. This is opposed to the known result for networkscontaining only connectivity links, where networks with abroader degree distribution are significantly more robustto random failures. Fig.2(a) and Fig.2(b) show that whencomparing ER and SF networks with the same averagedegree, SF networks with a high density of dependentnodes are more vulnerable to random failures then ERnetworks.
FORMALISM
Next we present an analytical approach for studyingthe robustness to random failure of networks containingthe two types of links. Without loss of generality wedefine a model in which only pairs of nodes depend onone another, forming dependency groups of size 2. Whenthe dependency group contain more then two nodes thecascade effect is even more extreme and the transitionfrom the regular second order percolation transition to afirst order transition occurs even for more stable networks(see SI). Therefore, the new properties we present for thecase of dependency groups of size 2 are also valid in thegeneral case of larger dependency groups (see Fig.1 in SI).The model is defined as follows: A network containing N nodes is randomly connected by connectivity links witha degree distribution P ( k ) and an average degree h k i .In addition, pairs of nodes are connected by dependencylinks as follows: a) A node can only have one dependencylink. b) If node i depends on node j then node j dependson node i . For this model we denote by q the fraction ofnodes that have dependencies.We start by presenting the formalism describing theiterative process of cascading failures for the simple caseof q = 1 (see Fig.1 in SI). Each iteration (step) includesfailures that are the result of the percolation process andfailures that are the result of the dependency process.The goal of the formalism is to describe the accumulatedprocess up to step n as an equivalent single randomremoval, r n , from the original network. The remainingfraction of nodes after such a removal is β n = 1 − r n .The new network after the removal of a fraction r n of the nodes, has a giant component consisting of afraction g ( β n ) of the remaining nodes which is a fraction α n +1 = β n g ( β n ) from the original network.The iterative process is initiated by the removal of afraction r = 1 − p of the network nodes. The remainingpart of the network is β = p . This initial removal willcause additional nodes to disconnect from the giantcluster due to the percolation process. The fractionof nodes that remain functional after the percolationprocess is α = β g ( β ). Each node from the nonfunctional part (1 − α ) will cause the node thatdepends on it to also fail (dependency process). Theprobability that a node depending on a non functionalnodes has survived until now is α . Therefore thefraction of new nodes that will fail due to dependenciesis δ = (1 − α ) α . The accumulated failure includ-ing the initial failure of 1 − β and δ is equivalentto a random removal of r = (1 − β ) + (1 − α ) β from the original network (see SI). The remainingfraction of nodes after the new removal is therefore β = 1 − r = β α = β g ( β ). The remaining functionalpart of the giant component is now α = β g ( β ). Tocalculate the fraction δ of nodes that are disconnecteddue to dependencies at the second stage, recall that atthe previous stage a fraction δ failed from α . Theremaining part of α was therefore α − δ = α . Thus δ = ( α /α )( α − α ) = [1 − ( α /α )] α . This is equiva-lent to a random removal of r = (1 − β )+[1 − ( α /α )] β from the original network. The remaining fraction ofnodes is β = 1 − r = α α /β = β g ( β ). Following this approach we can construct the sequence, β n , of theremaining fraction of nodes in the network after eachiteration. β = p . β = p g ( β ). β = p g ( β )... β n = p g ( β n − ).Following a similar approach for the general case of 0 ≤ q ≤ β n = qp g ( β n − ) + p (1 − q ). Given, β n , the fraction of nodes in the giantcluster is α n +1 = β n g ( β n ) = p (1 − q (1 − pg ( β n ))) g ( β n ).Fig. 3(a) compares theory and simulations of α n , for thecase of an ER network. n α n (a) p NO I p10 -3 -3 -3 -2 s ec ond c l u s t e r (b) FIG. 3: (a) Comparison between simulations and theoreticalresults for the fraction of nodes in the giant cluster on everystep n of the iterative process of failing nodes. The resultsare shown for an ER network with q = 0 . p = 0 . p ≃ p I ). The theoretical results (line) are calculated accord-ing to Eq.(1) (the explicit form of g ( x ) is presented in thetext) and are compared to several realizations of computersimulations on networks of size N = 200 K . (b) The numberof iterative failures (NOI) are shown for a scale free networkwith λ = 2 . q = 1. At the first order transition point,the number of iterative failures that the network undergoesbefore disintegrating scales as N / (see SI). This numbersharply drops as the distance from the transition is increased.Thus, plotting the number of iterations as a function of p pro-vides a useful method for identifying the transition point, p I ,at the first order region. The inset shows that the size of thesecond largest cluster reaches its maximum value at the sec-ond order transition point, p II , therefore providing a usefulmethod for identifying p II at the second order region. To determine the state of the system at the end of thecascade process we analyze β n at the limit of n → ∞ .This limit must satisfy the equation β n = β n +1 since at theend of the process the cluster is not further fragmented.Denoting β n = β n − = x we arrive to the equation: x = p qg ( x ) + p (1 − q ); (1)This equation can be solved graphically as the intersec-tion of a straight line y = x and a curve y = p qg ( x ) + p (1 − q ). When p is small enough the curve increasesvery slowly and does not intersect with the straight line(except at the origin which corresponds to the trivial so-lution). The critical case for which the nontrivial solutionemerges, corresponds to the case when the line touchesthe curve at a single point x and in this point we havethe condition 1 = p q dgdx ( x ), which together with Eq.(1)gives the solution for the critical fraction of failing nodesthat will fragment the network and the critical size of thegiant component. ANALYTICAL SOLUTION
An exact analytical solution can be obtained using theapparatus of generating functions. As in Refs. [22–24]we introduce the generating function of the degree distri-bution G ( ξ ) = P k P ( k ) ξ k . Analogously, we also intro-duce the generating function of the underlying branchingprocess, G ( ξ ) = G ′ ( ξ ) /G ′ (1). A random removal of afraction 1 − p of nodes will change the degree distribu-tion of the remaining nodes, so the generating function ofthe new distribution is equal to the generating functionof the original distribution with the argument ξ replacedby 1 − p (1 − ξ ) [22]. The fraction of nodes that belongto the giant component after the removal of 1 − p nodesis g ( p ) = 1 − G [1 − p (1 − f )], where f = f ( p ) satisfies atranscendental equation f = G [1 − p (1 − f )] [24].In the case of an ER network with a Poisson degreedistribution [11–13], the problem can be solved explicitlysince G ( ξ ) = G ( ξ ) = exp( h k i ( ξ − g ( x ) = 1 − f and f = exp[ h k i x ( f − x is definedin Eq.(1). The fraction of nodes in the giant componentat the end of the cascade process is then given by α ∞ = β ∞ g ( β ∞ ) = p (1 − q (1 − p (1 − f )))(1 − f ). The equation f = f ( q, p, k ) has a trivial solution at f = 1. The non-trivial solutions of f can be presented by the crossingpoints of the two curves in a system of equations thatare given with respect to x and f : (cid:26) x = p q (1 − f ) + p (1 − q ) x = ln f h k i ( f − . , ≤ f < f = 1 the size of the giantcomponent is zero ( α ∞ = 0). For the solutions thatare the crossing points of the two curves, f <
1, i.e., α ∞ >
0. Thus, the case where the curves tangentially in-tersect corresponds to a first order phase transition point( p = p I ) where α ∞ abruptly jumps from a finite sizeabove p I to zero below p I [25]. The condition for thefirst order transition is that the derivatives of the equa-tions of system (2) with respect to f are equal. Togetherwith system (2) this yields:( p I ) h k i q = − / [( f − f ] + ln f / ( f − (3)However, for a solution of system (2) where f → α ∞ = 0) there is no jump in the size of the gi-ant cluster and thus the transition is a second order transition ( p = p II ). Solving system (2) for f → p II h k i (1 − q ) = 1 (4)The analysis of Eq.(3) and Eq.(4) shows that the first or-der transition at p = p I occurs for networks with a highdensity of dependency links ( q > q c ), while the second or-der transition at p = p II , occurs for networks with a lowdensity of dependency links ( q < q c ). This is confirmedby Fig. 4(a) that compares theory and simulations for p II ( q ) and p I (q). The critical value of q c (and p c ) forwhich the phase transition changes from first order to asecond order is obtained when the conditions for boththe first and second order transitions are satisfied simul-taneously. Applying both conditions we obtain (cid:26) q c = ( h k i + 1 − p h k i + 1) / h k i p c = 1 / ( p h k i + 1 − . (5) SIMULATIONS
Next, we support our analytical results by simulations.Finding the transition point via simulations is always adifficult task that requires high precision. In the caseof the first order transition we are able to calculate thetransition point with good precision by identifying thespecial behavior characterizing the number of iterations(NOI) in the cascading process. At the first order tran-sition point, the NOI scales as N / (see SI) which isalso demonstrated by the long plateau in Fig. 3(a). Thisnumber sharply drops as the distance from the transitionpoint is increased, since away from the transition point, p I , the NOI scales as log N/ ( p − p I ) (see SI). Thus, plot-ting the NOI as a function of p , provides a useful and pre-cise method for identifying the transition point p I at thefirst order region. For the second order region a similarbehavior exists for the size of the second largest clusterwhich also reaches its maximum at the transition point[16]. Fig. 3(b) presents simulation results of the NOI.The transition point, p I , can easily be identified by thesharp peek characterizing the transition point. The insetof Fig. 3(b) presents a similar behavior for the size of thesecond largest cluster near the second order transitionpoint, p II . Fig. 4(a) compares simulation results andtheory for the transition points p I ( q ) at the first orderregion (solid line) and p II ( q ) at the second order region(dashed line). The transition points were obtained usingthe NOI and the second cluster size techniques respec-tively. The theoretical results for different values of q and h k i were calculated by solving system (2) togetherwith Eq.(3) or Eq.(4) respectively. Fig. 2(b) comparesthe values of the transition points p I ( q ) and p II ( q ) re-spectively between SF and ER networks with the same q p I , p II first ordertransitionsecond ordertransition q c =(
Here we show that in order to properly model real net-works two different type of links are needed: connectivitylinks and dependency links. We present an analytical for-malism for a general network model including both con-nectivity and dependency links. According to our model,networks with high density of dependency links are ex-tremely vulnerable to random failure and when a criticalfraction of nodes fail the network disintegrates in a formof a first order phase transition. Networks with a lowdensity of dependency links are significantly more robustand disintegrate in a form of a second order phase tran-sition. In the limit of zero fraction of dependency linksour general solution yields the known results for networkswith only one type of links. Our framework also provides an analytical solution for the critical density of depen-dency links for which the phase transition changes froma first order to a second order percolation transition. Wedevelop a powerful simulation method for accurately es-timate the transition point, based on the unique behaviorof the NOI (number of iterations in the iterative processof cascading failures) that diverges at the first order tran-sition point. Using this method we are able to providevery accurate simulation results supporting our analyti-cal results.We thanks the European EPIWORK project, theIsrael Science Foundation, the ONR and the DTRAfor financial support. S.V.B. thanks the Office of theAcademic Affairs of Yeshiva University for funding theYeshiva University high-performance computer clusterand acknowledges the partial support of this researchthrough the Dr. Bernard W. Gamson ComputationalScience Center at Yeshiva College. [1] Watts D-J, Strogatz S-H (1998) Collective dynamics of’small-world’ networks.
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