APS/123-QED Cascading Edge Failures: A Dynamic Network Process∗ June Zhang and José M.F. Moura Department of Electrical and Computer Engineering Carnegie Mellon University Pittsburgh, PA, USA (Dated: January 13, 2016) Thispaperconsidersthedynamicsofedgesinanetwork. TheDynamicBondPercolation(DBP) processmodels,throughstochastic localrules,thedependenceofanedge(a,b)inanetworkonthe 6 states of its neighboring edges. Unlike previous models, DBP does not assume statistical indepen- 1 dencebetweendifferentedges. Inapplications, thismeansforexamplethatfailures oftransmission 0 lines in a power grid are not statistically independent, or alternatively, relationships between in- 2 dividuals (dyads) can lead to changes in other dyads in a social network. We consider the time n evolution of the probability distribution of the network state, the collective states of all the edges a (bonds),and show that it converges to a stationary distribution. We use this distribution to study J the emergence of global behaviors like consensus (i.e., catastrophic failure or full recovery of the 2 entire grid) or coexistence (i.e., some failed and some operating substructures in the grid). In par- 1 ticular, weshow that,dependingon thelocal dynamicalrule, differentnetwork substructures,such as hubor triangle subgraphs, are more proneto failure. ] PACS numbers: h p PACSnumbers: 64.60.aq,89.75.Fb,02.50.-r,89.65.-s - c o s I. INTRODUCTION according to some dynamics. . s Dynamicedgemodelshavebeenusedincomputational c In 2003, the failure of a single power line triggered a sociology to study the time evolution of dyads [7–10]. i s series of cascading failures throughout the Northeastern However, some models assume that the dynamics of dif- y United States and Southeastern Canada [1]. Connectiv- ferent dyads are statistically independent. These mod- h p ityofthepowergridledtoanuncontrollablepropagation els can not capture effects such as triadic closure, where [ offailuresthatresultedinover50millionscustomerslos- friendships are more likely to form between individuals ingpower. Similarly,therapidspreadofinfectionduring with common neighbors [11]. Alternatively, models try 1 epidemicsisduetocontactsbetweenindividualsofapop- tocaptureinterdependenciesbyincorporatingknowledge v ulation. Thesephenomenaarerareanddifficulttostudy of network structure into the dynamics of a dyad. This 3 2 via experiments. Due to their scale,they arealsoexpen- assumesthatindividualdyadshavefullknowledgeofthe 9 sive to simulate. It is therefore advantageous to be able overall network structure, which may not be feasible in 2 to study such phenomena analytically. practice. 0 Cascading failures and epidemics can be modeled as The bond percolation model from statistical mechan- . 1 network processes —dynamical processes whose sub- ics has been used to study the robustness and resilience 0 strateisaparticularnetworkstructure. Itisofinterestto of network structures to stochastic bond (i.e., edge) re- 6 understand the dynamics of such processes, particularly movals [3, 12, 13]. However, the standard percolation 1 the impact of the network structure [2, 3]. In some ap- : model is not a network process, as it does not model v plications such as epidemiology, where nodes represent dynamical evolution of bond states over time [14]. Ad- i individuals, it is more intuitive to study network pro- X ditionally, the bond percolation model assumes that the cesses on nodes [4–6]. This means that the state of each size of the network is infinite, and accountfor the topol- r nodechangesaccordingtodeterministicorstochasticdy- a ogy only through the degree distribution and not higher namicsoftendependingonthestatesofthenodes’neigh- orderdegreecorrelations. Lastly,percolationis amacro- bors. This inclusion couples the evolution of each node scopic model; it characterizes global properties such as with thatofallthe other nodes inthe network,resulting the percentageofremovedcomponents,but cannotpro- in a complex dynamical system. For modeling phenom- videinformationonmicroscopicpropertiessuchasprob- enasuchascascadingfailuresoftransmissionlinesinthe ability that a set of bonds is removed. powergridortheevolutionofrelationships,calleddyads, The Dynamic Bond Percolation (DBP) model we in social networks, it is more appropriate to study edge- presentin this paper differsfrompreviousmodels by the centric network processes where the edges change state inclusion of local coupling dependencies in time between edges as represented by a heterogenous network structure. Eachedgeinanetworkcanbeinoneoftwostates: ∗ ThisworkispartiallysupportedbyNSFgrantsCCF1011903and open and closed. The dynamics of edges are no longer CCF1513936. independent as an edge (a,b) changes state according to 2 astochasticruledepending onlyonthe statesofits local 1 2 4 neighboringedges. Forexample,thismeansthatthefail- ure rate of a transmission line is affected by the failures of other transmission lines. Alternatively, the formation a b 3 of friendship links between individuals a and b depends on the other relationships of a and b. FIG. 1: Maximal network represented by Amax. Dashed The state of DBP at time t is the collection of the edges (i.e., bonds) are the only possible edges in the states of all the edges at time t. We show that the prob- network. ability distribution over the states of DBP reaches an equilibrium distribution. For certain local rules, we can 1 2 4 compute this distribution explicitly for any finite-size, unweighted, undirected network structure, unlike previous models that approximate the underlying network a b 3 by simpler structures (e.g., complete graphs) or infinite- size networks (i.e., mean-field approximation) [15, 16]. FIG. 2: Invalid network state. Edge (2,4) is not in the Analysisoftheequilibriumdistributioninformsusofthe maximal network and can not change state. Solid edges emergence of global behavior. For example, in certain are closed. Dashed edges are open. parameter regimes, the most probable network state at equilibriumisthatofconsensus,denotingeithercomplete failure or complete recovery of all transmission lines. In can fail, social ties between individuals can weaken or other parameter regimes, it provides information about strengthen with time. We are particularly interested in the subset of transmission lines more prone to cascading scenarioswherethereisacontagionaspecttothedynam- failures or more likely to form relationships. We will see icalprocess: asingletransmissionlinefailurecanleadto that these effects depend on the local dynamical rules. other line failures such as in blackouts. When the imbalance between node a and node b does In this paper, we present the Dynamic Bond Percola- not matter,the criticaledgestend to belong to star sub- tion (DBP) process, a dynamical extension of the bond graphs(i.e.,hubstructures). Whereascriticaledgestend percolation model [12]. Edges in the maximal network tobelongtotrianglesubgraphsandothernetworkstruc- open or close according to stochastic dynamical rules tures (e.g., P or P , see Section IIIA) when imbalance specifiedbyDBP.Weassumethatthecloseofanedgede- 3 4 or mutual relationships matter. We illustrate our analy- pendsonthestateoftheneighboringedges,therebycou- sis within two real-world networks: 118-node IEEE test pling the underlying network topology with the process bus power grid [17] and 198-node social network [18]. dynamics. In applications, DBP can be used to model SectionIIexplainstheDynamicBondPercolationpro- cascadingfailures of transmissionlines in the power grid cess in detail. Section III derives the closed-form equi- or the formation and dissolution of ties in a social net- librium distribution for 3 different dynamical rules. Sec- work. Figure 3 shows a possible realization of the DBP tion IV describes the Most-Probable Network Problem. process on the maximal network shown in Figure 1. Section V through Section VII analyzes the solution The state of the DBP process at time t ≥ 0, A(t), is spaceoftheMost-ProbableNetworkProblemforthedif- represented by the N ×N adjacency matrix A, where ferentparameterregimes. SectionVIIIconcludesthepa- A =1 if edge (a,b) is close per. i,j =0 if edge (a,b) is open. We will call A the network state. The set of nodes in II. DYNAMIC BOND PERCOLATION PROCESS the network state V(A)=V . The set of edges in the max networkstatecorrespondstothesetofclosededgesinthe ConsiderapopulationofN individualsorcomponents maximal network; therefore, E(A) ⊆ E . The space max represented as nodes in a network. The adjacency ma- of all possible network states is A. Since each edge in trix, A , describes the largest set of potentially open the maximal network can be either open or closed, then max or closed bonds between the nodes. We will refer to the |A|=2|Emax|. DBP makes the following assumptions: network represented by A the maximal network. For Assumption 1: Network states that contain edges max example, the maximal network may be a representation not in E are invalid. For example, with respect to max ofthe underlyingpowergridorasocialnetworkbetween the maximal network in Figure 1, the network state in N individuals. We assume that the maximal network Figure2isnotvalidsinceedge(2,4)isnotinthemaximal is a simple, connected, undirected graph. Let E de- network. max note the set of edges and V denote the set of nodes Assumption 2: Multiple edgeopeningorclosurecan max in the maximal network respectively. Figure 1 shows an not occur simultaneously. Only one edge changes state example of a maximal network. per transition. Edges in the maximal network may change state over Assumption 3: The time it takes for an edge (a,b) time. For instance, transmission lines in the power grid to go from close to open (e.g., t −t as in Figure 3) is 4 3 3 1 2 4 1 2 4 1 2 4 1 2 4 a b 3 a b 3 a b 3 a b 3 (a) X(t1) (b) X(t2) (c) X(t3) (d) X(t4) FIG. 3: Example evolution of A(t). Solid edges are closed. Dashed edges are open. exponentially distributed with rate SUD-DBP,theclosurerateofedge(a,b)isλ′γ′6 forboth ScenariosAandB.Under POD-DBP,the closurerateof q(A,Ta−,bA)=µ′. (1) edge (a,b) is λ′γ′9 for Scenario A and λ′γ′5 for Scenario B. This means that the probability that edge (a,b) TRI-DBPassumesthatthe closurerateofanedgede- switches from close to open ∆t time units in the future pendsonthenumberofclosedneighboringedgessharing is a common agent. This comes from the concept of tri- adicclosure,whichstatesthat, for 3agentsa,b,andc, if P(A(t+∆t)=T− A|A(t)=A)≈µ′∆t+o(∆t), a,b there is a connection between (a,b) and (a,c), then it is more likely that there is a connectionbetween(b,c) [19]. where o(·) is the little-o notation. We call µ′ > 0 the Under TRI-DBP, the closure rate of edge (a,b) is λ′ for edge opening rate. DBP assumes that µ′ is the same for both Scenario A and λ′γ′ for Scenario B. all closed edges. For applications where edge opening is When f(N ,N ) = 0, the transition rate rare, µ′ can be arbitrarily small as long as it is not 0. a b q(A,T+ A) = λ′. This means that it it is possible Assumption 4: The time it takes for an edge (a,b) a,b to go from open to close (e.g., t −t as in Figure 3) is for an edge (a,b) to close independent of the state of 2 1 neighboring edges. Therefore, we consider λ′ to be the exponentially distributed with rate spontaneousedgeclosurerateandγ′ tobe the cascading q(A,T+ A)=λ′γ′f(Na,Nb), (2) edge closure rate. a,b Sincethetransitionratesarenottimedependent,DBP where Na and Nb denotes the set of closed edges con- is a time-homogenous process. Since the model assumes nected to node a and the set of closed edges connected both spontaneous edge opening and close, there is no to node b in network state A, respectively. We call the absorbing state. Under the stated assumptions, the Dy- functionf(Na,Nb)thecascade function. Itcaptureshow namic Bond Percolation model, {A(t),t ≥0}, is an irre- theedgeclosurerateof(a,b)dependsonthelocalneigh- ducible,continuous-timeMarkovprocesswithfinitestate borhood of edge (a,b). In this paper, we consider the space, A = {A}; each state in the Markov process cor- following cascade functions: responds to a potential network state A. The dimension of the configuration space is |A|=2|Emax|. 1. SUD-DBP(Sum-Dependent Dynamic BondPerco- lation): f(N ,N )=|N |+|N | (3) III. TIME-ASYMPTOTIC BEHAVIOR a b a b 2. TRI-DBP (Triangle-Dependent Dynamic Bond While it is difficult to completely characterize the Percolation): dynamics of the Dynamic Bond Percolation process, its time-asymptotic behavior (i.e., lim A(t)) can be t→∞ f(Na,Nb)=|Na∩Nb|. (4) studiedusingitsequilibriumdistribution,π. Theequilib- rium distribution of DBP is a probability mass function 3. POD-DBP (Product-Dependent Dynamic Bond (PMF) over A. Since DBP is a finite-state, continuous- Percolation): time Markov process, the equilibrium distribution π is unique[20]. Itcanbefoundbysolvingthelefteigenvalue- f(Na,Nb)=|Na||Nb|. (5) eigenvector problem SUD-DBP assumes that the closure rate of an edge πQ=0, depends on the sum ofthe number of closedneighboring edges. POD-DBP assumesthat the closurerate depends where Q is the transition rate matrix, also knownas the ontheproductofthenumberofclosedneighboringedges. infinitesimal matrix [21]. These model different dynamics because POD-DBP im- The matrix Q characterizes the transition rates be- plicitly accounts for imbalance between the number of tweenallthestatesinAusing (1),(2). ForDBP,itisan closed edges on node a, |Na|, and node b, |Nb|. For ex- asymmetric 2|Emax| ×2|Emax| matrix. Element Qij cor- ample, consider the two scenarios in Figure 4. Under responds to the transition rate between 2 states i,j∈ A 4 c a b a b (a) Scenario A: |Na|=3,|Nb|=3. (b) Scenario B: |Na|=5,|Nb|=1. FIG. 4: Different edge removal scenarios. Solid edges are closed. Dashed edges are open. where i and j are decimal scalar representations of the 1 2 3 network states i and j, respectively. The equilibrium distribution of a continuous-time (a) P3 graph Markov process is the left eigenvector of the transition 2 rate matrix corresponding to the zero eigenvalue. How- ever, the challenge of finding π is that the dimensions of Q scales exponentially with the total number of edges 1 3 in the maximal network, |E |. This makes computing max the equilibrium distribution prohibitively expensive for (b) C3 graph large networks. In the next section, we will show that 1 2 3 4 wecanavoidthis computationfor SUD-DBP,TRI-DBP, and POD-DBP by finding the equilibrium distribution, (c) P4 graph π, up to a constant, in closed form. 1 A. Review of Graph Theoretic Concepts 6 4 2 First, we review graph theory terms necessary for the rest of the paper [2, 22]. 5 3 Definition III.1. Awalk is alist v ,e ,v ,e ,...,e ,v 0 1 1 2 k k (d) S6(4) star graph of vertices and edges. The length of the walk is the number of edges in the list. The number of walks in an undi- FIG. 5: P , C , P and S (4) subgraphs 3 3 4 6 rected graph from node i to node j of length k is (Ak) , i,j where Ak is the adjacency matrix of an undirected, un- Definition III.5. [22] A matching, M, of the graph weighted graph raised to the kth power. G(V,E), alsocalledtheIndependentEdgeSet,isasubset of edges E such that no vertex in V is incident to more Definition III.2. A path is a walk where all the ver- than one edge in M; see Figure 6a. Maximum Matching tices aredistinct (although some literaturedoes not make is a matching with the maximum number of edges; see this distinction between paths and walks). A graph that Figure 6b. is a path is called a path graph and written as P , where n n is the number of vertices (not edges) in the path. By convention, the path graph P is equivalent to a path of The number of edges in the maximum matching is n length n−1. Figure 5a shows the P subgraph and Fig- known as the matching number, ν(G). 3 ure 5c shows the P subgraph. 4 In this paper, we introduce a generalization to the Definition III.3. A cycle is a path where the endpoints matching set. v = v . A graph that is a cycle is called a cycle graph 0 k andwrittenas C , wheren is thenumberof vertices (not n Definition III.6. A star matching, S, of the graph edges) in the cycle. By convention, the cycle graph C is n G(V,E), is a subset of edges E such that these edges equivalent to a cycle of length n. Figure 5b shows the C 3 form a collection of disconnected star graphs; see Fig- subgraph. C subgraphs are also called triangles. 3 ure 6c. Maximum star matching is a star matching with the maximum number of edges; see Figure 6d. Note that Definition III.4. A star graph, S (i), has n vertices n M⊂S. that are only connected to the center vertex i. Figure 5d shows the S (4) subgraph. 6 5 2 4 6 2 4 6 1 3 5 7 1 3 5 7 (a) Matching: {(2,4),(3,5)} (b) Max Matching: {(1,2),(3,4),(5,6)} 2 4 6 2 4 6 1 3 5 7 1 3 5 7 (d) Max StarMatching: (c) StarMatching: {(1,2),(2,3),(4,5),(5,6)} {(1,2),(3,5),(4,5),(5,6),(6,7)} FIG. 6: Matching and Star Matching B. Reversibility and Equilibrium Distribution The equilibrium distribution is the product of three terms: the partition function Z, a structure-free term, Some Markov processes possess the property that the andastructure-dependent termthatdependsonthemax- process forward in time is statistically equivalent to the imalnetwork. LetE(A)denotethesetofclosededgesin |E(A)| process backward in time. These Markov processes are network state A. The term λ′ is structure-free called reversible Markov processes. The following theo- (cid:16)µ′(cid:17) because it depends only on the number of close edges in rem is important in deriving the stationary distribution networkstatewhereasthetermγ′g(E(A)) dependsonthe of a reversible Markov process: underlying maximal network topology. Theorem III.7 (From [20]). A stationary Markov process is reversible if and only if there exists a collection of positive number π(j),j ∈ L, summing to unity that C. (SUD-DBP) Sum-Dependent Dynamic Bond Percolation Model satisfy the detailed balance conditions π(j)q(j,k)=π(k)q(k,j), j,k,∈L, Theorem III.8. The Sum-Dependent Dynamic Bond Percolation model, {A(t),t ≥ 0}, is a reversible Markov where q(·,·) is the transition rate of the Markov process. process and the equilibrium distribution is When there exists such a collection π(j),j ∈ L, it is the equilibrium distribution of the process. 1 λ′ |E(A)| π(A)= γ′g(E(A)), A∈A, (8) Z (cid:18)µ′(cid:19) Using this theorem, we will show that the equilibrium distributions for SUD-DBP, POD-DBP, and TRI-DBP where g(E(A)) it the number of P subgraphs induced by 3 have the form: the set of closed edges E(A). 1 λ′ ′|E(A)| The number of close edges in a network state is π(A)= γ′g(E(A)), A∈A, (6) Z (cid:18)µ′(cid:19) 1TA1 |E(A)|= , 2 where the partition function, Z, is where 1=[1,1,...,1]T. λ′ |E(A)| ThenumberofP subgraphsinducedbytheedgesE(A) Z = γ′g(E(A)). (7) 3 (cid:18)µ′(cid:19) in a network state is AX∈A N N k The exponent|E(A)| is the totalnumber ofclosededges g(E(A))= (A2) = i , (9) i,j (cid:18)2(cid:19) inanetworkstate,andg(E(A))isthe numberofspecial Xi=1Xj>i Xi=1 network structures induced by the set of closed edges E(A). These special network structures are derived for where k = N A and is the number of closed edges i j=1 ij SUD-DBP,POD-DBP,TRI-DBPinsectionsIIIC, IIIE, at node i. TPhe derivation of the closed-from equation of and IIID, respectively. g(E(A)) is in Appendix A 6 In the SUD-DBP model, the sufficient statistics are the total number of close edges and the total number π(T+ A)q(T+ A,A) of paths of length 2 (i.e., the number of P subgraphs) a,b a,b 3 iinndduucceeddbbyytthheecclolosseeddededggese.sTishreenlautmedbearlsooftPo3tshuebgdreagprhees = Z1 (cid:18)µλ′′(cid:19)|E(Ta+,bA)|γ′g(E(Ta+,bA))µ′ (13) of the network state A. Surprisingly, this means that a sufficient statistic of a process where edges change state = 1 λ′|E(A)|+1 γ′g(E(A))+ka+kb. in time is a nodal characteristic. Z (cid:18) µ′|E(A)| (cid:19) Proof. We prove that the equilibrium distribution π(A) The LHSandRHSof (10)areequivalent. Similar rea- of the SUD-DBP is given by (8). soning holds for (11). Since the detailed balance equa- By Theorem III.7, π(A) satisfies the detailed balance tions are satisfied by (8), Theorem III.7 proves Theo- equations: rem III.8. π(A)q(A,T+ A)=π(T+ A)q(T+ A,A) (10) a,b a,b a,b and D. (TRI-DBP) Triangle-Dependent Dynamic Bond Percolation Model π(A)q(A,T− A)=π(T− A)q(T− A,A). (11) a,b a,b a,b Theorem III.9. The Triangle-Dependent Dynamic We prove (10) first. Bond Percolation model, {A(t),t ≥ 0}, is a reversible Let T+ A denote the network state that is the same a,b Markov process and the equilibrium distribution is asnetworkstateAexceptedge(a,b)switchesfromopen toclose. Accordingtothetransitionratein(2)forSUD- 1 λ′ |E(A)| DBP (3), π(A)= γ′g(E(A)), A∈A, (14) Z (cid:18)µ′(cid:19) q(A,T+ A)=λ′γ′ka+kb, a,b where g(E(A)) it the number of C subgraphs induced by 3 the set of closed edges E(A). where ka = |Na| = Ni=1Aai and kb =|Nb| = Ni=1Abi The number of close edges in a network state is are the number of cPlosed edges incident at nodePa and b, respectively. 1TA1 |E(A)|= , AccordingtotheequilibriumdistributionofSUD-DBP 2 (8), the equilibrium probability of network state A is where 1=[1,1,...,1]T. 1 λ′ |E(A)| ThenumberofC3 subgraphsinducedbytheedgesE(A) π(A)= γ′g(E(A)), in a network state is [22] Z (cid:18)µ′(cid:19) N (A3) where g(E(A)) is the number of P3 subgraphs induced g(E(A))= i,i. (15) by the set of closed edges E(A). The LHS of (10) is Xi=1 6 In the TRI-DBP model, the sufficient statistics are π(A)q(A,T+ A) the total number of close edges and the total number a,b 1 λ′ |E(A)| of triangles (i.e., the number of C3 subgraphs) induced = γ′g(E(A)) λ′γ′kakb by the closed edges. The proof for Theorem III.9 fol- Z (cid:18)µ′(cid:19) (12) lows the same steps as the proof for Theorem III.8 ex- (cid:0) (cid:1) 1 λ′|E(A)|+1 cept 1) the transition rate q(A,T+ A) for TRI-DBP = γ′g(E(A))+ka+kb. a,b Z (cid:18) µ′|E(A)| (cid:19) is given by (4), and 2) the number of C3 subgraphs g(E(T+ A))=|N ∩N |+g(E(A)). a,b a b According to the transition rate in (1), q(Ta+,bA,A)=µ′. E. (POD-DBP) Product-Dependent Dynamic Bond Percolation Model Recognize that by definition, |E(T+ A)| = 1+|E(A)|. a,b Consider the closure of edge (a,b). This means that the Theorem III.10. The Product-Dependent Dynamic number of paths of length 2 from any node in Na to Bond Percolation model, {A(t),t ≥ 0}, is a reversible the node b is ka and the number of paths of length 2 Markov process and the equilibrium distribution is from the node a to any node in N in k . Therefore, b b g(E(Ta+,bA))=g(E(A))+ka+kb. π(A)= 1 λ′ |E(A)|γ′g(E(A)), A∈A, (16) The RHS of (10) is Z (cid:18)µ′(cid:19) 7 where A is the adjacency matrix, g(E(A)) is the number on the dynamic parameters, λ′,µ′,γ′ and the maximal of triangles C , and the paths of length 3, P formed by network topology. 3 4 the set of closed edges, E(A). Further,theMost-ProbableNetworkProblemdoesnot The number of close edges, |E(A)|, is depend on finding the partition function, Z, (7). This meanswedonothavetosumover2|Emax| configurations. 1TA1 Wewillproveinlatersectionthatthemost-probablenet- |E(A)|= , 2 work configuration for SUD-DBP, TRI-DBP, and POD- DBP can be found using polynomial-time algorithms. where 1=[1,1,...,1]T. We can partition the parameter space of DBP into four The number of C and P subgraphs is 3 4 regimesanddetermine ineachregimethe most-probable network. N (A3) g(E(A))= i,i+ Regime I): RecoveryDominant: 0< λ′ ≤1,0<γ′ ≤1 6 µ′ Xi=1 N N Regime II): Cascading Failure: 0< λ′ ≤1,γ′ >1 µ′ (A3)i,j −(Ai,j)(A2)i,i+ Ak,j. Xi=1Xj>i k=1X,k6=i,j Regime III): CascadingPrevention: λµ′′ >1,0<γ′ ≤1    (17) Regime IV): Removal Dominant: λ′ >1,γ′ >1. µ′ The derivation of the closed-from equation of g(E(A)) is We are especially interested in contrasting the most- in Appendix B. probable networks of SUD-DBP, TRI-DBP, and POD- In the POD model, the sufficient statistics are the to- DBP models in the different parameter regimes. This tal number of closed edges and the total number of C3 willgiveusinsightinhowthecascadefunctionf(Na,Nb) andP subgraphsinducedbytheclosededges. ThePOD affects the dynamic process. 4 model and the SUD model do not have the same suffi- cientstatistics. The proofforTheoremIII.10 followsthe same steps as the proof for Theorem III.8 except 1) the V. REGIME I) RECOVERY DOMINANT AND transitionrateq(A,T+ A)forPOD-DBPisgivenby(5), REGIME IV) REMOVAL DOMINANT a,b and 2) the number of paths of length 3 from any node in N to any node in N through edge (a,b) is |N ||N |. In Regime I) Recovery Dominant, the structure-free a b a b Therefore g(E(T+ A))=k k +g(E(A)). and the structure dependent terms are both decreas- a,b a b ing exponential functions of the number of closed edges. Therefore, the most-probable network for SUD-DBP, TRI-DBP, and POD-DBP is the network with no edges, IV. CRITICAL STRUCTURES AND THE MOST-PROBABLE NETWORK PROBLEM A0 = {A ∈ A : |E(A)| = 0}; for this regime, the opening rate is high enough that the most-probable network is a consensus state such that none of the edges in the In the second part of the paper, we will use DBP to maximal network are considered at-risk. study critical structures in the maximal networks. As- InRegime IV),the topologydependentandthe topol- suming that DBP models cascading failures of transmis- ogy independent terms are both increasing exponential sion lines in the power grid, we are interested in under- functions. Therefore, the most-probable network for standing the interaction between the underlying power SUD-DBP, TRI-DBP, and POD-DBP is A = {A ∈ grid structure and the closure (i.e., failure) and open- max A : |E(A)| = E }; for this regime, the closure rate ing (i.e., recovery)rates. Alternatively, DBP may model max is high enough that the most-probable network is a con- formation of social ties between individuals; the critical sensus state such that all of the edges in the maximal structures are then relationships that are more likely to network are considered at-risk. be formed and maintained. The most-probable network for SUD-DBP, TRI-DBP, The most-probable network state is the network state and POD-DBP are the same in Regime I) and Regime in A with the highest equilibrium probability: IV). We will show in the next section that it may be different for these models in Regime II) and Regime III) λ′ |E(A)| as in these regimes, there is competition between the A∗ =argmaxπ(A)=argmax γ′g(E(A)). structure-free term and the structure-dependent term . A∈A A∈A(cid:18)µ′(cid:19) (18) Once the DBP process has reached equilibrium, it is VI. REGIME II) CASCADING FAILURE thenetworkstatemostlikelytobeobserved. Wecall(18) the Most-Probable Network Problem and A∗ the most- probable network. By evaluating A∗, we see that some In Regime II) Cascading Failure: 0 < λ′ ≤ 1,γ′ > 1, µ′ edges are more prone to closure than others depending the structure-free term is driven by edge opening and 8 the structure-dependent term is driven by edge closure. Proof. 1. When E(A )⊂E(A ), edges inE(A ) can 2 1 2 ForSUD-DBP,TRI-DBP,andPOD-DBP,weexpectthe not induce more subgraphs than edges in E(A ). 1 solution space of the Most-Probable Network Problem When E(A ) = E(A ), then the edges in E(A ) 2 1 1 (18) to exhibit phase transition depending on if edge and E(A ) will induce the same number of sub- 2 opening or edge closure dominates. From the analysis graphs. Hence, g(E(A ))≥g(E(A ))≥0. 1 2 of Regime I) and Regime IV), we expect that when the 2. Every edge in E(A ) is also an edge in E(A ). closureprocessdominates,the most-probablenetworkof 2 1 Therefore, adding edge i to E(A ) will generate SUD-DBP, TRI-DBP, and POD-DBP will be driven to- 2 the same or less number of subgraphs as adding ward A ; whereas if the opening process dominates, max edge i to E(A ). Hence, m ≥m ≥0. themost-probablenetworkforbothmodelswillbedriven 1 1 2 toward A . 0 Unlike Regime I) and IV), there may be solutions to Using Lemma VI.1, we can prove that theMost-ProbableNetworkProblemthatareneitherA 0 nor A . We call these solutions the non-degenerate Theorem VI.2. In Regime II), for SUD-DBP, TRI- max most-probablenetworks. Theexistenceofthesesolutions DBP, and POD-DBP, the function means that subset of edges in the maximal network are λ′ moreat-riskofclosure(i.e.,failure)thanotheredgesdur- −log(Zπ(A))=−|E(A)|log −g(E(A))log(γ′). (cid:18)µ′(cid:19) ing cascading failures. To find these non-degenerate solutions, we have to is submodular. The most-probable network, solvetheMost-ProbableNetworkProblem(18),acombi- natorial optimization problem. In general, such compu- A∗ =argmaxπ(A)=argmin−log(Zπ(A)), tation is NP-hard [23]. For SUD-BDP, POD-BDP, and TRI-BDPhowever,theMost-ProbableNetworkProblem is the minimum of a submodular function and can there- can be solved exactly using polynomial-time algorithm fore be computed in polynomial-time [24] . using submodularity [24]. Proof. By the additive closure property of submodu- Recall the definition of a submodular function: lar functions, the function −log(Zπ(A)) is submodular when Definition VI.1 ([25]). A set function, h:P(E)→R, is submodular if and only if for any E(A1),E(A2) ⊆ E λ′ with E(A2)⊆E(A1),i6∈E\E(A1): f1(E(A))=−|E(A)|log(cid:18)µ′(cid:19) h(E(A1)∪{i})−h(E(A1))≤h(E(A2)∪{i})−h(E(A2)). and (19) f (E(A))=−g(E(A))log(γ′) 2 Submodular functions are closed under nonnegative linear combination [26]. Consider the function are submodular functions in regime II). We need to show that f (A) and f (A) satisfy Defini- 1 2 M tion VI.1 when 0< λ′ ≤1 and γ′ >1. h(E)= α f (E). µ′ i i Consider E(A ),E(A ) ⊆ E with E(A ) ⊆ Xi=1 1 2 2 E(A ),i6∈E\E(A ). 1 1 If α ≥0∀i=1,...,M and the functions f (E) are sub- i i modular, then h(E) is also submodular. f (E(A )∪{i})−f (E(A )) First, we need the following lemma: 1 1 1 1 λ′ λ′ =−(|E(A )|+1)log +|E(A )|log Lemma VI.1. Consider two sets of closed edges, 1 (cid:18)µ′(cid:19) 1 (cid:18)µ′(cid:19) (20) E(A ),E(A ) ⊆ E and an additional closed edge 1 2 max λ′ i∈Emax\{E(A1)∪E(A2)}. For a given maximal net- =−log . work, the number of subgraphs induced by the edges in (cid:18)µ′(cid:19) E(A ) and E(A ) are g(E(A )) ≥ 0 and g(E(A )≥ 0, 1 2 1 1 respectively. Let the number of subgraphs induced by f (E(A )∪{i})−f (E(A )) 1 2 1 2 the edges E(A )∪{i} = g(E(A ))+m and the edges 1 1 1 λ′ λ′ nEu(mAb2e)r∪o{fia}d=ditgio(Ena(lAi2n)d)u+cedms2u;btghrearpehfosrecrmea1te≥d w0itihs tthhee =−(|E(A2)|+1)log(cid:18)µ′(cid:19)+|E(A2)|log(cid:18)µ′(cid:19) (21) inclusion of edge i in E(A ) and m ≥ 0 is the number λ′ 1 2 =−log . ofadditional inducedsubgraphs createdwiththeinclusion (cid:18)µ′(cid:19) of edge i in E(A ). If E(A )⊆E(A ), then: 2 2 1 Equations(20)and(21)satisfythe submodularcondi- 1. g(E(A1))≥g(E(A2))≥0, tion (19) when 0< λµ′′ ≤1. In fact, function f1(E(A)) is 2. m ≥m ≥0. always submodular regardless of λ′. 1 2 µ′ 9 structures aremore vulnerableunder SUD-DBP dynamics. f (E(A )∪{i})−f (E(A )) 2 1 2 1 The dynamics of TRI-DBP (4) put high probability =−(g(E(A ))+m )log(γ′)+g(E(A )log(γ′) (22) on network states that minimize the number of closed 1 1 1 =−m1log(γ′). edges, |E(A)| but maximize the number of induced C3 subgraphs. From Figure 7f, we can see the the most- probablenetworkconsistsoftriangles. Since the cascade functionofTRI-DBPdependsonthenumberofcommon f (E(A )∪{i})−f (E(A )) 2 2 2 2 neighbors, networks with few triangles have the lowest =−(g(E(A ))+m )log(γ′)+g(E(A )log(γ′) (23) 2 2 2 rate of cascading failures. =−m2log(γ′). The dynamics of POD-DBP (5) put high probability on network states that minimize the number of closed According to Lemma VI.1, m1 ≥ m2 ≥ 0. The func- edges, |E(A)| but maximize the number of induced C3 tion f2(E(A) satisfies the definition of submodularity and P4 subgraphs. We can see from Figure 7h that the when γ′ >1. most-probable network consists of more triangles (i.e., Furthermore, since Lemma VI.1 is true for all sub- C s) and longer paths than the most-probable network 3 graphs induced by E(A). It is applicable for SUD-DBP, ofSUD-DBP.POD-DBPhashigherrateofcascadethan TR-DBP, and POD-DBP. SUD-DBP and TRI-DBP and therefore more lines are vulnerable to cascading failures for the same parameter values. A. Power Grid Example B. Social Network Example Inthissection,weuseDBPtomodelcascadingfailures oftransmissionlinesonthe IEEE118BusTestCase[17]. We also used DBP to model evolving social ties on a The network is a portion of the Midwestern US power 198-node social network of drug users in Hartford, CT gridfrom1962. Itremainsoneofthestandardtestcases [18]. The nodesinthe networkrepresentindividualdrug today. We only use the network topology provided by users. Edge closure correspondsto formationorreestab- the test case. The nodes in the network represent the lishment of social ties, and edge opening corresponds to busesandtheedgesrepresenttransmissionlinesbetween dissipation of social ties. In this case, edges that are the buses. We model edge closures as line failures and closed in the most-probable networks are the social ties edge opening as recovery. During cascading failures like that are often established. They are the stronger social blackouts, transmission line (a,b) is more likely to fail if ties between individuals. there are already many failed transmission lines at bus Assuming that γ′ >1 means that a relationship (a,b) (i.e., node) a and bus b. Therefore, it is intuitive to ismorelikelytoformifagentaandagentbalreadyhave assume that γ′ > 1. On the other hand, spontaneous many other relationships. In particular, TRI-DBP as- failures are rare events and since the power grid is well sumes that a relationship (a,b) is more likely to form if maintained, the recoveryrate offailedtransmissionlines agent a and agent b already have many friends in com- isrelativelyhigh;itisnaturaltoassumethat0< µλ′′ ≤1. mon. This is based on the theory of triadic closure from Figure 7 shows the most-probable network with dif- social networks [19]. On the other hand, spontaneous ferent dynamic parameters λ′,γ′,µ′ for SUD-DBP, TRI- friendships are possible and relationships can dissipate DBP,andPOD-DBP.Whentheclosure(i.e.,failure)rate over time. Therefore, it is natural to assume nonzero λ′ ofanedgeislowcomparedwiththeopeningrate(i.e.,re- and µ′ and that 0< λ′ ≤1. covery),A∗ =A asinRegimeI).Whentheclosure(i.e., µ′ 0 Figure 8 shows the most-probable network with dif- failure) rate of an edge is high compared with the open- ferent dynamic parameters λ′,γ′,µ′ for SUD-DBP, TRI- ing rate (i.e., recovery), A∗ = A as in Regime IV). max DBP,andPOD-DBP.WecanseefromFigure8fandFig- However, for a subset of parameters, the most-probable ure 8h that, for some range of parameter values, social networks are non-degenerate. This means that a subset ties in triadic closures are stronger assuming TRI-DBP of edges are more vulnerable to cascading failures than and POD-DBP cascade functions. On the other hand, others. We can also see that the contagion dynamic, socialtiesofhighlyconnectedindividualsarestrongeras captured by the cascade function f(N ,N ), has a large a b in SUD-DBP. impact on the susceptibility of the network to failure. The dynamics of SUD-DBP (3) put higherprobability on network states that minimize the number of closed VII. REGIME III) CASCADING PREVENTION edges, |E(A)| and maximize the number of induced P 3 subgraphs. We can see from Figures 7a, that the most- probable network of SUD-DBP consists of closed edge In Regime III) Cascading Prevention: λ′ > 1 and µ′ structuresresemblingmanystarsubgraphs. Central,hub 0<γ′ ≤1, the structure-free term is driven by edge clo- 10 (a) SUD-DBP:|E(A∗)|=9 (b) SUD-DBP:|E(A∗)|=0 (c) SUD-DBP:|E(A∗)|=179 λ′ =0.0028,γ′=4.4222 λ′ =0.0167,γ′ =1.844 λ′ =0.27542,γ′=28.889 µ′ µ′ µ′ (d) TRI-DBP: |E(A∗)|=0 (e) TRI-DBP:|E(A∗)|=0 (f) TRI-DBP: |E(A∗)|=28 λ′ =0.0028,γ′ =4.4222 λ′ =0.0167,γ′=1.844 λ′ =0.27542,γ′ =28.889 µ′ µ′ µ′ (g) POD-DBP: |E(A∗)|=178 (h) POD-DBP: |E(A∗)|=98 (i) SUD-DBP:|E(A∗)|=179 λ′ =0.0028,γ′=4.4222 λ′ =0.0167,γ′ =1.844 λ′ =0.27542,γ′ =28.889 µ′ µ′ µ′ FIG. 7: 118-bus Test Case Most-Probable Network (solid edges = close, dashed edges = open) sure and the structure-dependent term is driven by edge DBP,we expect the solutionspace of the Most-Probable opening. The dynamics of Regime III) Cascading Pre- Network Problem (18) to exhibit phase transition de- ventionisthe opposite ofRegime II)CascadingFailures. pending on if edge opening or edge closure dominates. The averagetime anedge is open increaseswith increas- However, since 0 < γ′ ≤ 1 in Regime III), the Most- ingnumber ofclosededgesonits endnodes; diffusionef- Probable Network Problem can not be solvedusing sub- fects, instead of driving cascading failures, preventsedge modularity according to Theorem VI.2. However, we closures. Therefore,thisregimeiscalledRegimeIII)Cas- will prove in the next section that we can still solve cadingPrevention. ForSUD-DBP,TRI-DBP,andPOD- for the most-probable network in polynomial-time for a

Cascading Edge Failures: A Dynamic Network Process

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