The Fate of Articulation Points and Bredges in Percolation
Haggai Bonneau, Ido Tishby, Ofer Biham, Eytan Katzav, Reimer Kuehn
aa r X i v : . [ c ond - m a t . d i s - nn ] J a n The Fate of Articulation Points and Bredges in Percolation
Haggai Bonneau , Ido Tishby , Ofer Biham , Eytan Katzav , and Reimer K¨uhn Racah Institute of Physics, The Hebrew University, Jerusalem, 9190401, Israel Mathematics Department, King’s College London, Strand, London WC2R 2LS, United Kingdom (Dated: January 26, 2021)We investigate the statistics of articulation points and bredges (bridge-edges) in complex networksin which bonds are randomly removed in a percolation process. Articulation points are nodes ina network which, if removed, would split the network component on which they are located intotwo or more separate components, while bredges are edges whose removal would split the networkcomponent on which they are located into two separate components. Both articulation points andbredges play an important role in processes of network dismantling and it is therefore useful to knowthe evolution of the probability of nodes or edges to be articulation points and bredges, respectively,when a fraction of edges is randomly removed from the network in a percolation process. Dueto the heterogeneity of the network, the probability of a node to be an articulation point, or theprobability of an edge to be a bredge will not be homogeneous across the network. We thereforeanalyze full distributions of articulation point probabilities as well as bredge probabilities, using amessage-passing or cavity approach to the problem, as well as a deconvolution of these distributionsaccording to degrees of the node or the degrees of both adjacent nodes in the case of bredges. Ourmethods allow us to obtain these distributions both for large single instances of networks as well asfor ensembles of networks in the configuration model class in the thermodynamic limit of infinitesystem size. We also derive closed form expressions for the large mean degree limit of Erd˝os-R´enyinetworks.
PACS numbers: 64.60.aq,64.60.ah
I. INTRODUCTION
Networks affect many dimensions of human existence.They manifest themselves in everyday life, they underpinadvanced information and communication technologies,and they provide a powerful paradigm to analyse complexproblems in the natural sciences, in engineering, and ineconomics and the social sciences [1–5]. Key functionalityof a network often depends of the fact that pairs of nodesare mutually connected through one or several paths ofcontiguous edges, or whether on the contrary they residein different disconnected components of a net. One ofthe key questions of network science has therefore beenthe identification of conditions under which networks —natural or artificial — exhibit a so-called giant connectedcomponent (GCC), which occupies a finite fraction of asystem in the limit of large system size [6]. A naturalquestion then is how random or intentional removal ofnodes or edges will affect the functionality of a network,in particular whether a GCC would survive such a processof node or edge removals; this question has been exten-sively studied in the past two decades using percolationtheory [1–5, 7–11].Of crucial importance for the functionality of a networksare articulation points (APs) and bredges (or bridge-edges) [12]. Articulation points are nodes whose removalwould break the network component on which they arelocated into two or more disconnected components [13–16], while bredges are edges whose removal would break the network component on which they are located intotwo components [17, 18]. APs and bredges thus play acentral role in network attack strategies where networkdismantling is achieved by systematic removal of cycles,a process known as decycling [19–21], which in turn gen-erates further APs and bredges. It is clear from theirdefinition that all nodes on trees that are not be leaf-nodes of a network are APs, and conversely that nodesthat belong to any cycle cannot be APs, unless they arealso root nodes of a tree. In a similar vein, any edge lo-cated on a subtree of a network will always be a bredge.Figure 1 provides an illustrative example.It is worth mentioning, however, that there is also a con-structive role for APs and bredges, as these are preciselythe nodes or edges in a network one needs to remove inefficient strategies for containment (islanding) of black-outs in power grids, for containment of financial shocksor vaccination in the context of epidemics to name buta few relevant cases. From a different perspective, APsand bredges are fundamental quantities in describing thegeometry of networks.The statistical properties of APs and bredges were re-cently studied in detail for models in the configurationmodel class in [22] and [23], respectively. Closed form ex-pressions were obtained for the average fraction of nodesthat are APs, both in the entire network, and on the GCCof configuration model networks. Further details concern-ing the degree distribution of APs as well as the distri-bution of articulation ranks which specify the number of
FIG. 1: (Color online) Articulation points and bredges in anER network of mean degree c = 2 and N = 30 nodes. ar-ticulation points are indicated by circles, with a radius pro-portional to their articulation rank, i.e. proportional to thenumber of new network components that would be created bytheir removal, while vertices that are not articulation pointsare indicated as black dots. Bredges are indicated by brokenlines, with long dashed lines indicating so-called root bredgeswhich are directly linked to the 2-core of the network, andshort dashes indicating the remaining bredges located on treebranches of the GCC or on finite isolated clusters. Full linesindicate edges which are part of at least one cycle, and hencetheir removal would not break the network into two compo-nents. These non-bredge edges form the so-called 2-core of anetwork. additional network components created by the removal ofa node were also evaluated in closed form; for further de-tails see [22]. In [23], analogous results were obtained forbredges, including the probability for edges to be bredges,both in the entire network, and separately for the GCC,as well as joint degree distributions of nodes connectedby a bredge, once more both in the entire network, andrestricted to the GCC. Both sets of results heavily rely onan earlier analysis of the micro-structure of the GCC andthe finite network components of configuration models in[24]. Figure 1 shows an example of an Erd˝os-R´enyi (ER)network exhibiting APs and bredges.In [22] and [23] the authors looked at average AP andbredge probabilities and their deconvolution according todegree for ensembles of random networks in the configu-ration model class. In the present paper we look at theevolution of these probabilities in a percolation processwhere a certain fraction of edges is randomly removedfrom the network with probability 1 − p (and hence re-tained with probability p ). Moreover, we go beyond av- a i (cid:1853) (cid:2874) (cid:1853) (cid:2869) , (cid:1853) (cid:2877) , (cid:1853) (cid:2870)(cid:2877) , (cid:1853) (cid:2871)(cid:2868) b ij (cid:1854) (cid:2869) , (cid:2871) , (cid:1854) (cid:2872) , (cid:2873) , (cid:1854) (cid:2874) , (cid:2869)(cid:2868) , (cid:1854) (cid:2870)(cid:2873) , (cid:2870)(cid:2874) (b)(a) FIG. 2: (Color online) (a) : Articulation point probabilities a i of a selected set of nodes i , and (b) : bredge probabilities b ij of a selected set of edges ( i, j ) from the network of Fig. 1 asfunctions of the bond retention probability p in a percolationprocess. erage probabilities and their deconvolution according todegree in recognition of the fact that the probability ofa node to be an AP or of an edge to be a bredge willdepend on higher order coordination shells around thenode or edge in question, rather than just on the degreeof a node or the two degrees of the end-nodes of an edge.This type of heterogeneity was first properly highlightedand analyzed in detail for percolation probabilities andcluster-size distributions in [25], and we will use a vari-ant of that analysis for the study of APs and bredges.Figure 2 illustrates the heterogeneity of AP and bredgeprobabilities that is to be expected in this problem. Itshows AP probabilities for a selected set of of nodes andbredge probabilities for a selected set of edges for theexample network of Fig. 1 as functions of the bond re-tention probability p of a percolation process. Resultsare obtained using the theory outlined in the present pa-per, as predicted by Eq. (9) for APs and Eq. (15) forbredges. They demonstrate, in particular, that nodes onthe 2-core of the network at p = 1 cannot be APs, butmay become APs as p is decreased and cycles are areeliminated through bond removal. Conversely nodes ontree-components of the network that are not endpoints at p = 1 are APs with probability 1, but may lose that prop-erty by becoming endpoints or isolated nodes throughbond removal as p is decreased. For bredges, the fig-ure illustrates in a similar vein that edges that belong totree components of the network, whether attached to theGCC or not, are bredges with probability 1, while edgeslocated on cycles on the original network, i.e. at p = 1,are bredges with probability 0 but may become bredges as cycles are broken with increasing probability as p isdecreased. Our paper explains how this fairly intricatephenomenology can be captured analytically.Our main results are the following. We demonstrate thatthe message passing approach to percolation probabilitiescan be used to evaluate node dependent probabilities ofvertices in a complex networks to be APs as well as edgedependent probabilities of pairs of neighboring verticesin a network to be connected by a bredge. We derive aformulation for single instances of large networks and usethat to obtain a formulation for ensembles of networksin the configuration model class in the thermodynamiclimit. We obtain closed form approximations for the largemean degree limit of Erd˝os-R´enyi (ER) networks whichwe find to efficiently capture probability density functionsof AP and bredge probabilities already for relatively mod-erate mean degrees. Distributions of AP probabilitiesand bredge probabilities are evaluated for ER networksas well as scale free networks in the thermodynamic limit,and we also obtain deconvolutions of these distributionsaccording to degree(s) of the node(s) involved. We alsoapply the single instance theory to obtain distributionsof AP and bredge probabilities for a real-world network.Finally, we use the message passing approach to identifyAPs and bredges in a given network, and for any givenrealization of a percolation process, and find that it per-forms far better than might be expected, given that themethod is known to be exact only on trees.The remainder of this paper is organized as follows. InSect. II we introduce the version of the message passingapproach to percolation that we are using for the analy-sis of APs and bredges. We formulate the approach bothfor the analysis of finite instances of large networks, andfor the thermodynamic limit of networks in the config-uration model class. In Sect. III we use these results toanalyze the probabilities of edges to be bredges and theprobabilities of nodes to be APs as a function of the edgeretention probability p . Once more we do this for largesingle network instances and for networks in the config-uration model class, when the thermodynamic limit ofinfinite system size is taken. In Sect. IV we derive closedform approximations for the probability density functions(pdfs) of AP and bredge probabilities for the large meandegree limit of Erd˝os-R´enyi (ER) networks. Results arepresented in Sect. V. Section VI finally concludes witha summary and a discussion that also considers variouspossible applications of our main results. II. BOND PERCOLATION
The analysis of AP and bredge probabilities is closely re-lated to the analysis of percolation in complex networks,a process by which bonds (or vertices) of a network arerandomly and independently either kept with probabil-ity p or deleted with probability 1 − p . The present in-vestigation is based on a message passing approach topercolation in complex networks [26, 27]. We will, how-ever, complement the results of these papers by evaluat-ing the results of the message passing approach beyond average percolation probabilities or, for that matter, av-erage AP and bredge probabilities, using ideas proposedin [25]. The approach taken in [25] evaluates distributionsof percolation probabilities in terms of size distributionsof finite clusters. Here we use a more direct approachcloser to [26] which is formulated directly in terms ofnode dependent percolation probabilities.We will formulate the message passing approach for sin-gle instances of large complex networks, and then use theresults to obtain a description for networks in the configu-ration model class in the thermodynamic limit. Networksin the configuration model class are maximally randomsubject to a prescribed degree distribution [28, 29]. Us-ing k i to denote the degree of node i in a network wethus assume that a network is characterized by a degreedistribution Prob( k i = k ) = p k for k ∈ N , and that thereare no degree-degree correlations. A. Single-Instance Theory for PercolationProbabilities
We begin by briefly describing the message passing ap-proach to percolation, concentrating for specificity on bond percolation . The case of site percolation wherenodes are randomly removed with probability 1 − p andretained with probability p can be analyzed using thesame ideas and methods.We consider networks consisting of N vertices, labeled i =1 , , . . . , N which are connected by a set of non-directededges ( ij ). We introduce indicator variables n i to denotewhether vertex i is in the giant connected component(GCC) of the network ( n i = 1) or not ( n i = 0), andindicator variables x ij which denote whether the edge ( ij )is kept in a single realization of the percolation process( x ij = 1) or not ( x ij = 0). In terms of these we have n i = 1 − Y j ∈ ∂i (cid:16) − x ij n ( i ) j (cid:17) (1)in which ∂i denotes the set of nodes connected to i in theoriginal graph, and n ( i ) j is an indicator variable denotingwhether the vertex j adjacent to i is ( n ( i ) j = 1) or is not( n ( i ) j = 0) on the GCC on the cavity graph from whichvertex i and the edges emanating from it are removed.Equation (1) states the fact that a site belongs to theGCC if it is connected to it through at least one of itsneighbors, which in turn requires that both that neigh-bor is in the GCC, and that the link connecting to thatneighbor is actually kept in the given instance of a perco-lation experiment. For the cavity indicator variables wehave, by the same line of reasoning, that n ( i ) j = 1 − Y ℓ ∈ ∂j \ i (cid:16) − x jℓ n ( j ) ℓ (cid:17) . (2)Equations (1) and (2) can be averaged over all possiblerealizations of the percolation process on the given net-work. This gives g i = 1 − Y j ∈ ∂i (cid:16) − pg ( i ) j (cid:17) (3)for the probability that vertex i will be part of the GCCin a realization of the percolation process, while g ( i ) j = 1 − Y ℓ ∈ ∂j \ i (cid:16) − pg ( j ) ℓ (cid:17) (4)is the probability that vertex j neighboring on i on theoriginal graph will be part of the GCC on the cavity graphfrom which i and edges emanating from it are removed.In Eqs. (3) we exploited the fact that that x ij and n ( i ) j are independent, and the same clearly holds for x jℓ and n ( j ) ℓ in Eq. (4). As usual, a factorization of averages thatassumes independence of random variables along differentedges incident on a given node. This assumption is exactonly on trees, but is known to be an excellent approxima-tion on locally tree-like graphs, which becomes exact forfinitely coordinated systems in the thermodynamic limit N → ∞ of infinite system size.Eqs. (4) can be solved through forward iteration — start-ing from random initial conditions — on a single instanceof a graph, and from the solution site dependent perco-lation probabilities g i can be computed using Eqs. (3).Alternatively, for certain random network ensembles, dis-tributions of percolation probabilities can be evaluated inthe thermodynamic limit. B. Thermodynamic Limit
We will evaluate distributions of percolation probabili-ties, and subsequently distributions of articulation pointprobabilities and distributions of bredge probabilities in the thermodynamic limit for networks in the configura-tion model class.In the thermodynamic limit Eqs. (4) constitute an infinitesystem of coupled self-consistency equations for the cavityprobabilities g ( i ) j . Assuming that a statistical law or aprobability density ˜ π (˜ g ) of the g ( i ) j exists, it can be found,following meanwhile standard arguments [25, 30–32] bydemanding probabilistic self-consistency. The value of˜ π (˜ g ) is obtained by summing probabilities of all instancesof of the r.h.s. of Eqs. (4) for which g ( i ) j ∈ (˜ g, ˜ g + d˜ g ].Using this procedure, Eqs. (4) result in˜ π (˜ g ) = X k kc p k Z h k − Y ν =1 d˜ π (˜ g ν ) i δ (cid:16) ˜ g − h − k − Y ν =1 (1 − p ˜ g ν ) i(cid:17) , (5)in which kc p k is the probability that a randomly chosenneighbour of a node has degree k , and we have adoptedthe shorthand d˜ π (˜ g ν ) = ˜ π (˜ g ν ) d˜ g ν . Although this equa-tion is a highly non-linear integral equation, it can besolved efficiently and to any desired degree of precision(limited only by computational power) using a popula-tion dynamics algorithm [30]. In terms of the solution ofEq. (5), the distribution π ( g ) of node dependent percola-tion probabilities is found from Eq. (3) as π ( g ) = X k p k Z h k Y ν =1 d˜ π (˜ g ν ) i δ (cid:16) g − h − k Y ν =1 (1 − p ˜ g ν ) i(cid:17) . (6) III. STATISTICS OF ARTICULATION POINTSAND BREDGES IN PERCOLATIONA. Articulation Points
In order for a node i of the system not to be an articula-tion point, all its neighbors must reside on the giant com-ponent of the reduced network from which i is removed[22]. Introducing ˆ n i ∈ { , } as an indicator variablewhich denotes whether i is an articulation point (ˆ n i = 1)or not (ˆ n i = 0), and noting that only the links that arestill present , for which thus x ij = 1, should contribute tothe logic as to whether or not a node is an articulationpoint, we getˆ n i = h − Y j ∈ ∂i (cid:0) n ( i ) j (cid:1) x ij i × | x ∂i |≥ = h − Y j ∈ ∂i (cid:0) − x ij + x ij n ( i ) j (cid:1)i × | x ∂i |≥ . (7)Here we have introduced the vector x ∂i = ( x ij ) j ∈ ∂i andthe norm | x ∂i | = P j ∈ ∂i x ij , and we have invested thefact that nodes connected to fewer than 2 other nodescannot be articulation points. Upon averaging this overrealizations of a percolation process, this gives the prob-ability a i = h ˆ n i i (8)that node i is an articulation point, where angled bracketsdenote an average over bond configurations in an ensem-ble of percolation processes in which bonds (of a givennetwork) are randomly and independently removed withprobability 1 − p and kept with probability p . Performingthe average over bond configurations in Eq. (7) we get a i = (cid:28)h − Y j ∈ ∂i (cid:0) − x ij + x ij n ( i ) j (cid:1)i × | x ∂i |≥ (cid:29) = 1 − p (1 − p ) k i − X j ∈ ∂i (cid:0) − g ( i ) j (cid:1) − Y j ∈ ∂i (cid:0) − p + pg ( i ) j (cid:1) . (9)The heterogeneity of the original network entails that the a i depend in a highly non-trivial way on the location ofthe nodes i in the original network. Following the reason-ing used above to obtain the distribution of percolationprobabilities, one obtains the probability density function π ( a ) of the node dependent articulation point probabil-ities in the thermodynamic limit of infinite system sizeas π ( a ) = (cid:0) p + p (cid:1) δ ( a ) + X k ≥ p k π ( a | k ) (10)with π ( a | k ) = Z h k Y ν =1 d˜ π (˜ g ν ) i δ (cid:18) a − (cid:20) − p (1 − p ) k − × k X ν =1 (cid:0) − ˜ g ν (cid:1) − k Y ν =1 (cid:0) − p + p ˜ g ν (cid:1)(cid:21)(cid:19) (11)giving the pdfs of articulation point probabilities condi-tioned on degrees for which k ≥ ≤ g ( i ) j ≤ g ( i ) j appearingin Eq. (9), it is straightforward to obtain p -dependent ex-pressions for articulation point probabilities of and lim-iting probabilities for some subclasses of vertices. Forexample, for any node i residing on a finite cluster of anetwork — examples are nodes labeled 25, 28, 29 an 30in Fig. 1 — we have g ( i ) j = 0 for all j ∈ ∂i . For thesenodes Eq. (9) then entails that a i (cid:12)(cid:12) k i = k = a FC k ( p ) = 1 − kp (1 − p ) k − − (1 − p ) k . (12)These form a family of continuous curves for which a FC k (0) = 0 and a FC k (1) = 1, and the a FC k ( p ) would mark p -dependent locations of δ -peaks in pdfs of AP proba-bilities for any network in which finite isolated clustersexist. Curves labeled a , a , a and a in Fig. 2 pro-vide examples belonging to this family. Given that iso-lated clusters are generated through percolation whenever p < p = 1. Below the percolationthreshold, they completely describe the support of thedistribution of AP probabilities.Next, suppose that i is a node on the GCC of a networkwith k i = k , and suppose that k t < k − i belong to a tree rooted in i , whereasthe remaining k ℓ = k − k t ≥ i are part of one or several loops on the GCC. Examples ofsuch nodes are nodes 10, 16, and 18 in Fig. 1. Their cor-responding p dependent AP probabilities a i ( p ) as givenby Eq. (9) are marked in Fig. 2. For nodes of this type wehave g ( i ) j t = 0 for all j t ∈ ∂i which are located on the tree,whereas for the the remaining k ℓ neighbors of i , have thethe inequality (1 − g ( i ) j ℓ ) ≤ (1 − p ) k jℓ − which follows byusing the upper bound 1 in g ( j ℓ ) ℓ ≤ ℓ ∈ ∂j ℓ \ i inEq. (4). Note that g ( j ℓ ) ℓ ’s close to (the upper bound ) 1are only likely to be found sufficiently far above any per-colation transition, thus for p .
1. Denoting by k = ( k j ℓ )the set of degrees of the k ℓ terminal nodes that link i toloops, we can conclude that for vertices of this type wehave a i (cid:12)(cid:12) k i = k ; k t , k → a k ; k t , k ( p )with a k ; k t , k ( p ) = 1 − p (1 − p ) k − h k t + k ℓ X ℓ =1 (1 − p ) k jℓ − i − (1 − p ) k t k ℓ Y ℓ =1 (cid:2) − p (1 − p ) k jℓ − (cid:3) . (13)These form families of curves for which a k ; k t , k (0) = 0,just as in the family of curves that describe the situationon finite clusters. If k t > i ∈ GCC), i.e. if i is aroot-node of a tree attached to the GCC, (examples arenodes 10, 16, and 18 in Fig. 1) , then a k ; k t , k ( p ) → p →
1, just as in the case of finite clusters. However, if k t = 0 (and i ∈ GCC), then i is not the root node ofa tree attached to the GCC (examples are nodes 12, 13,14, 15, and 17 in Fig. 1), and we have a k ; k t , k ( p ) → p → unlike in the finite cluster case. Close to p = 1 itis expected that the probability of g ( j ℓ ) ℓ ’s saturating theirupper bound is expected to be reasonably high, so thisfamily of curves is expected to be reasonably well visi-ble, as they would correspond to locations of pronouncedmaxima in p -dependent pdfs of AP probabilities, at leastin networks which are reasonably densely connected at p = 1. We shall find that this is clearly borne out by theresults presented below.In order to rationalize further structures in p -dependentpdfs of AP probabilities, one would have to include in-formation about the configuration of higher coordinationshells around a chosen vertex i , and use iterated versionsof the self-consistency equation (4) to express the g ( i ) j for j ∈ ∂i in terms of cavity percolation probabilities onedges further removed from i . Following that strategy,one would in principle be able to characterize the p de-pendence of such structures in terms of sums of powers of p and (1 − p ). In that context, the small example providedin Figs. 1 and 2 can be instructive. B. Bredges
Moving on to bredges, we can follow the same line ofreasoning. For a randomly chosen edge ( ij ) in a network not to be a bredge [16, 23], both of its end-nodes mustbelong to the GCC in a network from which the edge( ij ) is removed. Introducing n ij as an indicator variablethat denotes whether the edge ( ij ) is a bredge ( n ij = 1)or not ( n ij = 0), this can be expressed in terms of thecavity indicator variables n ( i ) j and n ( j ) i introduced aboveas n ij = 1 − n ( j ) i n ( i ) j (14)Averaging this equation over all realizations of the per-colation process gives the probability b ij = h n ij i = 1 − g ( j ) i g ( i ) j , (15) for a link ( ij ) to be a bredge in an ensemble of percola-tion processes where links (on a given network) are ran-domly and independently removed with probability 1 − p and kept with probability p . Equation (15) allows oneto obtain link-dependent bredge-probabilities b ij in largesingle network instances from the solutions of Eqs. (4).Once again, the heterogeneity of the original networkentails that the b ij depend in a non-trivial way on thelocation of the edge ( ij ) in the original network. Theprobability density function π ( b ) of the link dependentbredge-probabilities in the thermodynamic limit is ob-tained following the reasoning used to find the distribu-tion of percolation probabilities as π ( b ) = Z d˜ π (˜ g )d˜ π (˜ g ′ ) δ (cid:0) b − (1 − ˜ g ˜ g ′ ) (cid:1) . (16)To access the dependence of bredge probabilities on thedegrees of the terminal nodes of an edge, one can use theself-consistency equations (4) in Eq. (15) giving b ij = 1 − h − Y ℓ ∈ ∂i \ j (cid:0) − pg ( i ) ℓ (cid:1)ih − Y ℓ ′ ∈ ∂j \ i (cid:0) − pg ( j ) ℓ ′ (cid:1)i (17)In the thermodynamic limit this then translates into π ( b ) = X k,k ′ kc p k k ′ c p k ′ π ( b | k, k ′ ) , (18)with the π ( b | k, k ′ ) = Z h k − Y ν =1 d˜ π (˜ g ν ) ih k ′ − Y ν ′ =1 d˜ π (˜ g ν ′ ) i δ b − (cid:26) − h − k − Y ν =1 (1 − p ˜ g ν ) ih − k ′ − Y ν ′ =1 (1 − p ˜ g ν ′ ) i(cid:27)! (19)as pdfs of the bredge probabilities, conditioned on thedegrees k and k ′ of the terminal nodes of an edge.For bredge probabilities described by Eq. (15) one canobtain p -dependent families of curves that depend on thelocal environment of the terminal nodes i and j definingthe edge. If i or j (or both) are nodes residing on atree of the original network then the product g ( j ) i g ( i ) j isidentically zero, hence for edges of this type we have b ij = b tij ( p ) ≡ . (20) Examples are edges (1,3), (4,5), (6,10) and (25,26) inFig. 1, with the corresponding b ij ( p ) curves marked inFig. 2 (b). This results in the appearance of a δ -peak at b = 1 in the pdf of the bredge probabilities at all p (pro-vided the network does contain trees (be they attachedto the GCC or not). If networks are constructed withouttree components at p = 1, i.e., prior to random bond dilu-tion, then this δ -peak at b = 1 will appear with increasingweight as p decreases.Assuming that an edge ( ij ) connects nodes with k i = k and k j = k ′ , and that both are indeed on the GCC ofa network from which the edge in question is removed,then one can use the bounds g ( j ) i ≤ − (1 − p ) k i − andsimilarly g ( i ) j ≤ − (1 − p ) k j − to conclude b ij (cid:12)(cid:12) k i = k,k j = k ′ ≥ b k,k ′ ( p ) (21)with b k,k ′ ( p ) = 1 − (cid:2) (1 − (1 − p ) k − (cid:3)(cid:2) (1 − (1 − p ) k ′ − (cid:3) (22)As discussed above in the case of AP probabilities oneexpects that close to p = 1 the probability of g ( j ) i ’s and g ( i ) j ’s saturating their upper bound should be reasonablyhigh so this family of curves is expected to be reason-ably well visible in representations of p -dependent pdfsof bredge probabilities, as long as the networks are fairlydensely connected in the p → IV. LARGE MEAN DEGREE APPROXIMATION
For networks exhibiting ‘narrow’ degree distributions inthe sense that the the standard deviation of the degreesis negligibly small in comparison to the mean degree, itis possible to derive closed form approximations of theresults above [25, 33]. The obvious candidate to consideris the Poisson degree distribution of ER graphs with largemean degree h k i = c , for which the standard deviation σ k = √ c is small compared to the mean for c ≫ δ -distribution˜ π (˜ g ) = δ (˜ g − ˜ g ∗ ) . (23)The value of g ∗ is obtained by inserting this ansatz intoEq. (5), and deriving a self-consistency equation for g ∗ .Assuming a Poisson distribution for the degrees, we getthe equation g ∗ = 1 − e − pcg ∗ . (24)as the self-consistency equation for g ∗ . In order to obtaina non-trivial solution in the large c limit, one has to adoptthe scaling p = ρ/c at fixed ρ , so that Eq. (24) becomes g ∗ = 1 − e − ρg ∗ , (25)which can be solved in closed form, giving g ∗ = 1 + W ( − ρ e − ρ ) ρ , (26) where W ( · ) is the Lambert W -function; see Sect. 4.13 in[34].In order to obtain the large mean degree limit of thedistribution π ( a ) of articulation point probabilities, weinsert the ansatz of Eq. (23) into Eq. (11), which impliesthat the conditional probability a ( k ) for a node of degree k to be an articulation point is at large k ( ≥
2) given by a ( k ) = 1 − kp (1 − p ) k − (1 − g ∗ ) − (1 − p + pg ∗ ) k (27)For a Poisson distribution of mean degree c , the distribu-tion of scaled degrees x = k/c is well approximated by anormal distribution of mean 1 and variance 1 /c for c ≫ x ∼ N (1 , /c ) in this limit. From Eq. (27), we canobtain an expression of the scaled degree x = x ( a ) as afunction of the AP probability a as the solution x = x ( a )of a = 1 − ρx (1 − p ) cx − (1 − g ∗ ) − (1 − p + pg ∗ ) cx ≃ − ρx e − ρx (1 − g ∗ ) − e − ρx (1 − g ∗ ) , (28)where we have used the large c limit in the second line.This then allows us to obtain a closed form expression forthe pdf π ( a ) using the fact that the distribution of scaleddegrees x is a normal distribution π ( x ) = r c π exp h − c x − i (29)which, via a standard identity about transformations ofpdfs under a change of variable, transforms into π ( a ) = π ( x ) (cid:12)(cid:12)(cid:12) d x d a (cid:12)(cid:12)(cid:12) = p c π exp h − c ( x − i(cid:12)(cid:12) ρ (1 − g ∗ )e − ρx (cid:2) ( ρx −
1) + e ρxg ∗ (cid:3)(cid:12)(cid:12) , (30)with x = x ( a ) given in terms of the solution of Eq. (28).Note that the most efficient way to evalutate this den-sity, however, is to avoid solving Eq. (28), but to simplytreat the pair of equations (28) and (30) as a parametricrepresentation of the pdf π ( a ) in terms of the parameter x ≥ π ( b ) ofbredge probabilities is slightly more involved, as accord-ing to Eqs. (18) and (19) it involves a natural deconvolu-tion on contributions depending on the degrees of bothend-nodes of a bredge. Exploiting once more the factthat ˜ π (˜ g ) ≃ δ (˜ g − g ∗ ) for ER networks with large meandegree, we obtain bredge probabilities as functions of thedegrees k, k ′ of terminal nodes as b = b ( k, k ′ ) = 1 − (cid:2) − (1 − pg ∗ ) k − (cid:3)(cid:2) − (1 − pg ∗ ) k ′ − (cid:3) (31)Noting that for a Poisson degree distribution kc p k = p k − we relabel k ← k − k ′ ← k ′ − x = k/c and y = k ′ /c are normally distributed randomvariables of mean 1 and variance 1 /c . Rewriting Eq. (31)in terms of scaled relabeled degrees gives b = b ( x, y ) = 1 − (cid:2) − (1 − pg ∗ ) cx (cid:3)(cid:2) − (1 − pg ∗ ) cy (cid:3) . (32)We can solve this, for instance, for y to obtain y = y ( b | x ) = 1 c ln(1 − pg ∗ ) ln " − − b − (1 − pg ∗ ) cx , (33)provided that b > (1 − pg ∗ ) cx , giving y for any given valueof x as a function of the bredge probability b . This allowsus to transform the Gaussian pdf π ( y ) = p c π exp (cid:2) − c ( y − (cid:3) of the scaled degree y — using an analogoustransformation of variables identity for pdfs, albeit nowconditioned on x — into π ( b | x ) = N ( x ) π ( y ) (cid:12)(cid:12)(cid:12) ∂y∂b (cid:12)(cid:12)(cid:12) Θ (cid:16) b − (1 − pg ∗ ) cx (cid:17) = N ( x ) exp h − c (cid:16) y ( b | x ) − (cid:17) i √ πc | ln(1 − pg ∗ ) | (cid:2) b − (1 − pg ∗ ) cx (cid:3) × Θ (cid:16) b − (1 − pg ∗ ) cx (cid:17) , (34)with y ( b | x ) given by Eq. (33) and Θ( x ) the Heaviside step-function. Here N ( x ) is a normalization factor that isneeded due to the x dependent restriction on the allowedrange of b values, to ensure that the π ( b | x ) are normalizedpdfs for all x . It cannot be evaluated in closed form andhas to be obtained numerically. This finally gives π ( b ) = Z d x π ( b | x ) π ( x ) (35)with π ( x ) the pdf of an N (1 , /c ) normal random variablegiven by Eq. (29). The integral in Eq. (35), too, has to bedone numerically.In Sec.V below we show that even for moderate values ofthe mean degree c these results already provide a decentapproximation for the distributions of articulation pointprobabilities and bredge probabilities. V. RESULTS
Below we present results both for synthetic networksin the configuration model class in the thermodynamiclimit, and for single instances of a real world network.
A. Synthetic Networks
Figures 3 and 4 show heat-maps of the p -dependent prob-ability density functions π ( a ) of articulation point proba-bilities and and π ( b ) of bredge probabilities, respectively.Results in Fig. 3 are for ER networks of mean degree c = 2, those in Fig. 4 for scale free networks with degreedistribution p k ∝ k − for k ≥
2. The first prominentdifference between the two networks is a different perco-lation threshold, namely p c = 0 . p c ≃ . k min = 2 andthus it has, at p = 1, no finite isolated clusters or evensets of nodes that form tree-like structures. This featureexplains the main qualitative difference of the heat-mapsof AP probabilities in the ER and the scale-free network:non-leaf nodes on trees are always APs; these are abun-dant in the ER network, explaining the fact that, for large p , the pdf π ( a ) of AP probabilities has significant massfor a ≃ p close to 1, the pdfof AP probabilities is very close to zero near a = 1 in thescale free network where trees do not exist right at p = 1.There are, in both figures, sharp p -dependent structuresthat are clearly visible sufficiently far above the perco-lation threshold. They can be rationalized, both for theheat maps of π ( a ) and π ( b ) in terms of local neighbor-hoods of nodes and edges, as explained in Sec. III A forarticulation points, and in Sec III B for bredges respec-tively.For articulation points there is a family of curves con-necting the points ( p, a ) = (0 ,
0) and ( p, a ) = (1 ,
1) de-scribed by Eq. (12) for finite clusters and by Eq. (13) with k t > p range for the ER net-work with mean degree c = 2 as finite clusters and treesattached to the GCC exist at all p . For the scale-freenetwork with k min = 2 there are no finite clusters andtree-like structures at p = 1. They are, however createdby random bond removal, so features with locations de-scribed by these equations become more and more promi-nent as p is lowered. For articulation points which are not root nodes of trees, Eq. (13) with k t = 0 defines afamily of curves which connect the points ( p, a ) = (0 , p, a ) = (1 , p dependent peaks in pdfs of AP probabilities which areclearly visible close to p = 1 as they require that the prob-ability to have cavity percolation probabilities g ( i ) j ≃ a p 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 b p 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (a) (b) FIG. 3: (Color online) (a) : Heat-map of the pdf π ( a ) of articulation point probabilities for an ER network of mean degree c = 2. (b) : heat-map of the pdf π ( b ) of bredge probabilities for the same system. To achieve a color-code that remains discriminatealso at relatively low values of pdfs, a nonlinear mapping of pdfs into the interval [0,1] of the form π ( · ) → p π ( · ) / (0 . p π ( · ) )is adopted. a p 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 b p 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (a) (b) FIG. 4: (Color online) (a) : Heat-map of the pdf π ( a ) of articulation point probabilities for a scale-free network with degreedistribution p k ∝ k − , with k ≥ (b) : heat-map of the pdf π ( b ) of bredge probabilities for the same system. As in Fig. 3, thecolor code is generated using the non-linear mapping π ( · ) → p π ( · ) / (0 . p π ( · ) ). respond to maxima in pdfs of bredge probabilities, which— in analogy to the corresponding family of curves de-scribing AP probabilities — will be clearly visible close to p = 1 as they, too, require the probability to have cavitypercolation probabilities g ( i ) j ≃ not captured by these families of curves, asthey require an analysis of local environments of nodesand edges beyond first coordination shells, as discussedand exploited before in [24, 33] to rationalize prominentfeatures in pdfs of percolation probabilities.To the extent that there is some degree of similarity of theresults for articulation points and bredges in both cases, it is due to the intimate connection which exists betweenbredges and articulation points, as each end-node of abredge is an articulation point if it is not a leaf node andeach articulation point of degree k ≥ π ( a ) and π ( b ) at given values of bond retention probabilities p , which are bet-ter suited to demonstrate quantitative details. Figure 6shows results for ER networks of Fig. 3 at p = 0 .
75. Thereare δ -peaks at 0 as well as a set of other δ -peaks at lo-cations given by a FC k ( p = 0 .
75) in π ( a ) for k ≥
2. Thepeak at zero is due to isolated nodes and nodes that areleaf nodes after random bond removal. The continuumpart of π ( a ) is due to nodes originally on the 2-core of theGCC. Individual contributions π ( a, k ) for some k are alsoshown and are seen to contribute to different features inthe overall pdf. The pdf π ( b ) of bredge probabilities hasa δ -peak at 1, originating from finite tree-like clustersor trees attached to the GCC on the original network,0 p a (2, 2)(2, 3)(2, 4)(3, 2)(3, 3)(3, 4)(4, 2)(4, 3)(4, 4) 0.0 0.2 0.4 0.6 0.8 1.0 p b (k,1)(2,2)(2,3)(2,4)(2,5)(3,3)(3,4)(3,5)(4,4)(4,5) (a) (b) FIG. 5: (Color online) (a) : Families of p -dependent AP probabilities a FC k ( p ) (upper unlabeled set of curves, given by Eq. (12))for k = 2 , . . . , a k ; k t , k ( p ) (lower set of curves, given by Eq. (13)); the lower set of curves corresponds to asub-family corresponding to a set of nodes of degree k i = k , with k t = 0, and k j = k ′ for all j ∈ ∂i , and we use the label ( k, k ′ )members of this sub-family. (b) : Families of p -dependent bredge probabilities of the form b k,k ′ ( p ), given by Eq. (22). a π ( a ) π(a)π(a, 2)π(a, 3)π(a, 4)π(a, > 5) b π ( b ) π(b)π(b, 2)π(b, 3)π(b, 4)π(b, > 5) (a) (b) FIG. 6: (Color online) (a) : Distribution π ( a ) of articulation point probabilities (black full line) and contributions π ( a, k ) for aselection of different degrees, shown here for k = 2 , , k >
5, for the ER network of mean degree c = 2 at bond retentionprobability p = 0 .
75, with individual contributions decreasing with increasing k . (b) : distribution π ( b ) of bredge probabilities(black full line) and contributions π ( b, k ) = P k ′ π ( b, k, k ′ ) for k = 2 , ,
4, and k > b values for increasing k . For π ( a ) the δ -peak at a = 0 originates from nodes with k = 0 and k = 1, whereas the δ -peakat a = 1 is due to nodes on finite clusters and on tree branches that are not leaf nodes. For π ( b ) the δ -peak at b = 1 is due toedges which reside on tree branches. whereas the continuous part of π ( b ) is due to nodes onthe giant cluster that were part of one or several loops onthe GCC of the original network. Any marked featuresthat would occur at locations given by b k,k ′ ( p = 0 . π ( b ) accord-ing to the degree k of one of the terminal nodes showsthat larger values of k entail that bredge probabilities areshifted towards smaller values.Figure 7 shows results for the scale-free network of Fig. 4at p = 0 .
5. The scale free network was constructed with k min = 2, so there are no isolated nodes nor leaves, nor fi-nite tree-like clusters or trees attached to the giant cluster at p = 1. As a consequence the δ -peaks in π ( a ) and π ( b )are absent in this system. The deconvolution accordingto degree reveals that nodes of degree k = 2 are predom-inantly responsible for the larger π ( a ) values for a ≤ . a = 0 . a = 0 .
125 predictedby Eq. (13), while others require to include further co-ordination shells in the analysis. The sharp maxima in π ( b ) are mainly due to edges, with one terminal node ofdegree k = 2. The peaks at b = 0 .
75 and at a = 0 . b (2 , , p = 0 .
5) andthe latter as an accumulation point of the b ( k, k ′ , p = 0 . k ′ → ∞ . For the scale-free network, large degrees dooccur with sufficient probability to give sufficient weight1 a π ( a ) π(a)π(a, 2)π(a, 3)π(a, 4)π(a, > 6) b π ( b ) π(b)π(b, 2)π(b, 3)π(b, 4)π(b, > 6) (a) (b) FIG. 7: (Color online) (a) : Distribution π ( a ) of articulation point probabilities (black line) and contributions π ( a, k ) for aselection of different degrees, shown here for k = 2 , , k >
6, for the scale free network at bond retention probability p = 0 .
5, with individual contributions decreasing with increasing k . Note that typical probabilities of nodes to be APs appearto increase with the degree k . (b) : distribution π ( b ) of bredge probabilities and contributions π ( b, k ) = P k ′ π ( b, k, k ′ ) for k = 2 , , k > b values for increasing k . to this peak. Other sharp peaks at larger values of b canbe rationalized by including higher coordination shells inthe analysis.One can harness techniques of [35] used originally to dis-entangle contributions to sparse random matrix spectracoming from the giant connected components and from fi-nite clusters to investigate probabilities of nodes to be ar-ticulation points and probabilities of edges to be bredges conditioned on these nodes and edges having belonged tothe GCC prior to any percolation experiment. This isobviously the most relevant issue to study when thinkingof maintaining functionality of a network or conversely ofattack strategies which would efficiently undermine suchfunctionality. Technically this is done by analysing themessage passing equations for two ‘replica’ of indicatorvariables, one for the system without random bond re-movals, and one for the same system with random bondremovals. This analysis quantitatively confirms the at-tribution of features in π ( a ) and π ( b ) discussed aboveaccording to whether they are due to nodes or edges orig-inally on the GCC or on finite clusters.Figure 8 finally demonstrates that the large mean de-gree approximation is remarkably efficient already for thefairly moderate value c = 10 for the mean degree. B. Single-Instance Cavity for a Real WorldNetwork
If we compare results for synthetic networks shown abovewith those for a real world network, we can state the fol-lowing. There are p -dependent structures — with the same p -dependence — in the real world and syntheticnetworks. In Fig. 9 we present results for one such realworld network, a symmetrized version of the Gnutellafile-sharing network[36] with N = 62 ,
586 nodes. It isnotable that the clarity with which individual featuresshow up depends very much on the system, and differ-ences between the networks are clearly visible. For ex-ample, except in the vicinity of the percolation transitionat p c ≃ .
1, the pdf of bredge probabilities is concen-trated at significantly lower values of b in the Gnutellanetwork than in the two examples of synthetic randomnetworks. The same appears to be true for the pdf of ar-ticulation point probabilities generated by nodes that areon the 2-core of the original network, for which therefore a i ( p ) → p → average bredge probabilities at all valuesof the bond retention probability p , and to a lesser de-gree of similarity for average AP probabilities. However,despite the closeness of values of average AP and bredgeprobabilities in the original network and its randomizedversion, there remain marked differences at the level ofthe full distributions π ( a ) and π ( b ) of AP and bredgeprobabilities, respectively.2 a π ( a ) b π ( b ) (a) (b) FIG. 8: (Color online) (a) : Comparing the distribution π ( a ) of articulation point probabilities (blue full line) with its largemean degree approximation red (dashed line) for an ER graph of mean degree c = 10 and bond-retention probability p = 0 . π ( b ) of bredge probabilities is in panel (b) . The agreement between the resultsof the full cavity analysis and its large mean degree approximation is surprisingly good already for the moderate value of themean degree considered here. a p 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 b p 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (a) (b) FIG. 9: (Color online) (a) : Heat-map of the pdf π ( a ) of articulation point probabilities for the Gnutella file-sharing network. (b) : heat-map of the pdf π ( b ) of bredge probabilities for the same system. The non-linear map to produce colour-codes is thesame as in Figs. 3 and 4. C. Single-Instance Cavity Algorithm to IdentifyArticulation Points and Bredges
The message passing approach can also be used to iden-tify individual APs and bredges in a given network, ratherthan local AP probabilities and bredge probabilities inensembles of systems affected by random bond or siteremoval. To this end, one returns to the message pass-ing equations for the indicator variables n i and the cav-ity indicator variables n ( i ) j for a single realization of thebond-occupancy variables x ij , i.e., Eqs. (1) and (2). Aniterative solution of Eqs. (2) for x ij ≡ x ij resulting from random or targeted bond re-moval one could repeat that analysis and find bredgesand APs in the same manner after bond removal. We have tested this idea for the Gnutella file-sharing net-work. Although the message passing algorithm will beexact only on trees, we found it to be extremely efficientand accurate for this real world network, despite the factthat the network contains a large number of loops. Forbredges, comparison with exact recursive algorithms re-vealed that the results of the message passing analysiswere in fact exact : there were neither false positives norfalse negatives. Of the 147,892 edges in this network, wecorrectly identified all 28,759 bredges in the network. Inthe case of APs, comparison with exact recursive algo-rithms revealed just a single false negative: we identified12,253 out of 62,586 nodes to be APs, missing only oneadditional AP found by the exact algorithm. Althoughthis demonstrates performance only on a single example,results are certainly encouraging.The complexity of the algorithm is estimated to scale3 p a GnutellaRandomized p b GnutellaRandomized a π ( a ) GnutellaRandomized b π ( b ) GnutellaRandomized (a) (b)(c) (d)
FIG. 10: (Color online) (a)
Average AP probabilities for the Gnutella file sharing network undergoing a percolation processas a function of bond retention probability p , compared with the p -dependent AP probabilities for a randomized version ofthe same network. (b) The same comparison for the p -dependent average bredge probabilities. Panels (c) and (d) comparefull distributions π ( a ) of AP probabilities and π ( b ) of bredge probabilities for the original and the randomized version of theGnutella network at bond retention probability p = 0 .
3. While at the level of average AP and bredge probabilities the Gnutellanetwork and its randomized version are very similar, the full pdfs π ( a ) and π ( b ) exhibit remarkable differences. as N ln N with system size N for sparse graphs. Herethe factor of N accounts for the scaling of the algorithmwith the number of edges in a system with finite meandegree, while the ln N factor accounts for the scaling ofthe algorithm with the diameter of the network (whichscales logarithmically with network size N for small worldnetworks of the type considered in the present paper);the diameter dependence is due to the fact that messagesneed to be passed (a few times) through the network inorder to achieve convergence. VI. SUMMARY AND DISCUSSION
In summary, we demonstrated that the message pass-ing approach to percolation — apart from its originalpurpose to compute heterogeneous node-dependent per-colation probabilities [25, 33] — can also be utilized toevaluate heterogeneous node dependent probabilities ofvertices in a complex networks to be APs as well as het- erogeneous edge dependent probabilities of pairs of neigh-boring vertices in a network to be connected by a bredge.Average probabilities of nodes to be APs and averageprobabilities of edges to be bredges were recently evalu-ated for ensembles of networks in the configuration modelclass in [22] and [23] respectively. In the present paperwe looked at the evolution of these probabilities in per-colation where a certain fraction of edges is randomly re-moved from the network, and we were able to go beyond average probabilities. This provides a significant amountof further detail in the analysis. It recognizes and exploitsthe fact that the probability of a node to be an AP orof an edge to be a bredge, in a node or bond percolationexperiment, will depend on the entire environment of thenode or edge in question, rather than just on its degreeor on the two degrees of the end-nodes of an edge. Forthe sake of definiteness, we restricted our analysis in thepresent paper to the case of bond percolation, though itwould be straightforward to formulate the theory for sitepercolation.4We derived a formulation for single instances of large net-works and used it to obtain a formulation for ensemblesof networks in the configuration model class in the ther-modynamic limit. We also obtained closed-form approx-imations for the large mean degree limit of Erd˝os-R´enyi(ER) networks which we found to be fairly efficient al-ready for rather moderate values of the mean degree. It isworth emphasizing that solving Eqs. (4) for cavity perco-lation probabilities is sufficient to obtain node-dependentpercolation probabilities g i ( p ), using Eq. (3), AP proba-bilities a i ( p ), using Eq. (9), and edge dependent bredgeprobabilities b ij ( p ), using Eq. (15), all in one go . Al-ternatively, solving Eq. (5) for the pdf of cavity percola-tion probabilities of configuration model networks in thethermodynamic limit is sufficient to obtain limiting pdfsof percolation probabilities, AP probabilities and bredgeprobabilities from Eqs. (6), (10), and (18), respectively,once more all in one go .Distributions of AP probabilities and bredge probabili-ties were evaluated for ER networks as well as scale freenetworks in the thermodynamic limit; de-convolutionsof these distributions according to degree were also ob-tained. The single instance theory was applied to ob-tain distributions of AP and bredge probabilities for areal-world network. Finally we also implemented the sin-gle instance formulation of the theory prior to averagingover realizations of a percolation experiment as an algo-rithm to locate APs and bredges in a given network, andfor any given realization of a percolation process, and wefind that it performs surprisingly well, giving only a sin-gle false negative for APs in the Gnutella file-sharing net-work data — a network of N = 62 ,
586 nodes and 147,892edges, and no errors at all when locating bredges.A study of bredges in real world networks [16] recentlyrevealed that the fraction of bredges in such network wasvery close to the fraction observe in randomized versionsof these networks. In the present paper we have seenthat, while this remains true if bonds are randomly re-moved from either the original network or its randomizedversion, this is no longer the case for the full distributionsof AP and bredge probabilities.Articulation points and bredges are exploited in opti-mized algorithms of network dismantling [19–21]. Dis-mantling processes typically begin with a decycling stagein which a node is deleted in every cycle of a network.This process transforms the network into a tree or a for-est of trees, in which all the nodes of degrees k ≥ a i ( p ) and b ij ( p ) curves could be useful to devise efficientdecycling heuristics that take into account the fraction1 − p of edges that the attack is able to delete. In gen- eral, a deep attack that aims at dismantling the networkcompletely should initially target nodes/edges with low a i ( p )/ b ij ( p ) values in order to achieve decycling, as ex-plained below.We believe that the quantitative results obtained in thiswork contain a lot of interesting information about nodesand edges in the network, and could be useful in a broadspectrum of applications. For instance, an interesting as-pect of the b ij ( p ) curves is that they exhibit no crossings,meaning that the order between the b ij ( p ) curves is pre-served in the whole range of p ’s where they differ. Inparticular, looking at the b value of all the edges for anyintermediate value of p can be used to rank them; thelower the b ij ( p ) value of an edge ( i, j ), the more likely itis that the network would maintain its functionality whenthe edge in question is removed in the course of a randomdeletion of of a fraction 1 − p of its edges. In other words,the b ij ( p ) curves can be used as a basis for an edge-basedcentrality measure. We believe that targeting edges witha low b ij ( p ) score would boost the efficiency of the de-cycling phase of a deep attack on a network that has alot of resources, because these edges participate in manycycles. Conversely, targeting high- b edges would resultin chipping off fragments from the network even after ashort or modest attack, albeit leaving a relatively wellconnected 2-core behind.In contrast to the b ij ( p ) curves, the a i ( p ) curves do ex-hibit crossings as p is varied and hence cannot provide a p -independent way to rank nodes. Still, these curves conveya lot of useful information. For example a node i whosecurve a i ( p ) starts at a i (1) = 1 and connects smoothly toits sub-percolating branch sits on a tree branch or on afinite component. However, if it starts at a i (1) = 1 butexhibits a knee before connecting to its sub-percolatingbranch then it must be a root-AP, namely an AP thatsits on the 2-core and glues one or more tree branches tothe 2-core. Finally, if the curve starts at a i (1) = 0, thenthe node sits on the 2-core initially, and a deep attackshould aim at initially deleting nodes i with the lowestpossible a i ( p ) values, where 1 − p is a measure of the effortinvested in an attack.A more sophisticated implementation of these ideas coulddevise an attack with multiple stages, where at each stageonly a set of edges or nodes are deleted according to theprinciples mentioned above. This is followed by a re-assessment of the situation by calculating the updated a i ( p ) and b ij ( p ) curves which could now help decide onthe next set of nodes or edges to delete.Interesting problems to look at in the near future couldinclude generalizing the present analysis to directed net-works, which prevail in many technical applications andcontexts, and to provide a more systematic study of thesingle instance message passing algorithm for locating5bredges and APs, concerning both its accuracy and theprecise scaling of the algorithm with system size. Anotheraspect one might want to look at is the distribution of thesizes of clusters created by removing APs or bredges fromthe net, which would require adapting the techniques of [27] as used in [25] to study heterogeneity in percolationto the problem of APs and bredges.This work was partly supported by the Israel ScienceFoundation grant no. 1682/18. [1] S. Havlin and R. Cohen, Complex Networks: Structure,Robustness and Function (Cambridge University Press,New York, 2010).[2] M. E. J. Newman,
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