Statistical analysis of edges and bredges in configuration model networks
aa r X i v : . [ c ond - m a t . d i s - nn ] S e p Statistical analysis of edges and bredgesin configuration model networks
Haggai Bonneau, Ofer Biham, Reimer K¨uhn, and Eytan Katzav Racah Institute of Physics, The Hebrew University, Jerusalem 9190401, Israel Department of Mathematics, King’s College London, Strand, London WC2R 2LS, UK
A bredge (bridge-edge) in a network is an edge whose deletion would split the network compo-nent on which it resides into two separate components. Bredges are vulnerable links that play animportant role in network collapse processes, which may result from node or link failures, attacks orepidemics. Therefore, the abundance and properties of bredges affect the resilience of the networkto these collapse scenarios. We present analytical results for the statistical properties of bredges inconfiguration model networks. Using a generating function approach based on the cavity method,we calculate the probability b P ( e ∈ B) that a random edge e in a configuration model network withdegree distribution P ( k ) is a bredge (B). We also calculate the joint degree distribution b P ( k, k ′ | B)of the end-nodes i and i ′ of a random bredge. We examine the distinct properties of bredges onthe giant component (GC) and on the finite tree components (FC) of the network. On the finitecomponents all the edges are bredges and there are no degree-degree correlations. We calculate theprobability b P ( e ∈ B | GC) that a random edge on the giant component is a bredge. We also calcu-late the joint degree distribution b P ( k, k ′ | B , GC) of the end-nodes of bredges and the joint degreedistribution b P ( k, k ′ | NB , GC) of the end-nodes of non-bredge (NB) edges on the giant component.Surprisingly, it is found that the degrees k and k ′ of the end-nodes of bredges are correlated, whilethe degrees of the end-nodes of non-bredge edges are uncorrelated. We thus conclude that all thedegree-degree correlations on the giant component are concentrated on the bredges. We calculatethe covariance Γ(B , GC) of the joint degree distribution of end-nodes of bredges and show it is nega-tive, namely bredges tend to connect high degree nodes to low degree nodes. We apply this analysisto ensembles of configuration model networks with degree distributions that follow a Poisson distri-bution (Erd˝os-R´enyi networks), an exponential distribution and a power-law distribution (scale-freenetworks). The implications of these results are discussed in the context of common attack scenariosand network dismantling processes.
I. INTRODUCTION
Network models provide a useful conceptual frameworkfor the study of a large variety of systems and processesin science, technology and society [1–5]. These modelsconsist of nodes and edges, where the nodes representphysical objects, while the edges represent the interac-tions between them. Unlike regular lattices in which allthe nodes have the same coordination number, networkmodels are characterized by a degree distribution P ( k ).The backbone of a network often consists of high degreenodes or hubs, which connect the different branches andmaintain the integrity of the network. In some applica-tions, such as communication networks, it is crucial thatthe network will consist of a single connected compo-nent. However, mathematical models also produce net-works that combine a giant component and isolated finitecomponents, as well as fragmented networks that consistonly of isolated finite components [6].Networks are often exposed to the loss of nodes andedges, which may severely affect their functionality. Suchlosses may occur due to inadvertent node or edge failures,propagation of epidemics or deliberate attacks. Startingfrom a single connected component, as nodes or edgesare deleted they may lead to the separation of networkfragments from the giant component. As a result, thesize of the giant component decreases until it completelydisintegrates. The ultimate failure, when the network fragments into isolated finite components was studied ex-tensively using percolation theory [7–10].A major factor in the sensitivity of networks to nodeor edge deletion processes is the fact that the deletionof a single node or a single edge may separate a wholefragment from the giant component. This fragmentationprocess greatly accelerates the disintegration of the net-work. Using iterative search algorithms one can identifythe nodes whose deletion would break the component onwhich they reside into two or more components [11–13]Such nodes, called articulation points (APs), were re-cently studied in the context of network resilience and op-timized attack strategies [14]. Using similar methods onecan also identify the edges whose deletion would breakthe component on which they reside into two separatecomponents [15, 16]. Such edges are called bridge-edgesor cut-edges [17]. Here we use the term bredges , whichprovides a shorthand for bridge-edges, and avoids a po-tential confusion with many other technical terms involv-ing the word ‘bridge’. Moreover, the word ’bredge’ wasused in ancient English as a synonym to the word ‘bridge’[18]. In fact, an edge that does not participate in any cy-cle is a bredge (B). Thus, in network components thatexhibit a tree structure, such as the finite tree compo-nents of configuration model networks, all the edges arebredges.In Fig. 1(a) we present a schematic illustration of abredge e (marked by a thick line) in a tree network and (cid:1861) (cid:1857) (cid:1861) (cid:4593) (cid:1861) (cid:1857) (cid:1861) (cid:4593) (cid:1861) (cid:1857) (cid:1861) (cid:4593) (cid:1853)(cid:1854)(cid:1855) FIG. 1: Schematic illustration of bredges and their surround-ing network components: (a) A bredge e (marked by a thickline) in a finite tree component. Deletion of the bredge e would split the tree component into two separate tree compo-nents. The end-node i will reside on one of the tree compo-nents and the end-node i ′ will reside on the other tree compo-nent; (b) A bredge e (thick line) where one of its end nodes, i ′ , resides on a cycle. Deletion of the bredge would split thenetwork into two separate components; (c) Here the edge e ,marked by a thick line, is not a bredge because the end nodesof this edge are connected by another path. As a result, upondeletion of the marked edge its two end nodes remain on thesame network component. its end-nodes i and i ′ (full circles). Deletion of the bredgewould split the network into two separate tree compo-nents. In Fig. 1(b) we show a bredge e (thick line), whereone of its end-nodes, i ′ , resides on a cycle. Deletion of thebredge would split the network into two separate compo-nents. The component that includes the end-node i ′ rep-resents the giant component of the reduced network fromwhich e is removed, while the component that includesthe end-node i represents the finite tree component thatis detached from the giant component upon deletion of e . The edge e marked by a thick line in Fig. 1(c) is nota bredge because its end-nodes are connected by a paththat does not go through e . As a result, upon deletionof e its end-nodes i and i ′ remain on the same networkcomponent. Since the paths connecting the end-nodes ofan edge e may be long, the determination of whether e is a bredge or not cannot be done locally and requiresaccess to the large-scale structure of the whole network[11, 12].In practice, the functionality of most networks relieson the integrity of their giant components. Therefore,it is particularly important to study the properties ofbredges and APs that reside on the giant component.These bredges and APs are vulnerable spots in the struc-ture of a network, because the deletion of a single bredge may detach an entire tree branch from the giant compo-nent while the deletion of a single AP may detach oneor several tree branches. This vulnerability is exploitedin network attack strategies, which generate new bredgesand AP via decycling processes and then attack them todismantle the network [14, 19–22]. While bredges and APmake the network vulnerable to attacks, they are advan-tageous in fighting epidemics. In particular, maintainingisolation between nodes connected by bredges preventsthe spreading of epidemics between the network compo-nents connected by these bredges. Similarly, in commu-nication networks the party in possession of an AP or abredge may control, screen, block or alter the commu-nication between the network components connected bythe AP or the bredge.There is an intricate connection between bredges andAPs. On the one hand, each one of the end-nodes i and i ′ of a bredge e is either an AP (if its degree satisfies k ≥
2) or a leaf node (if its degree is k = 1). On theother hand, if a node i of degree k ≥ k edges must be a bredge. Moreover,in the case of a node i of degree k = 2, both edges of i are bredges. The statistical properties of APs in config-uration model networks were studied in a recent paper[23]. The probability P ( i ∈ AP) that a random node i ina configuration model network with degree distribution P ( k ) is an AP was calculated. Moreover, closed formexpressions were obtained for the conditional probability P ( i ∈ AP | k ) that a random node of a given degree k is anAP and for the conditional degree distribution P ( k | AP).An important property of an AP is the articulation rank r , which is the number of components that are added tothe network upon deletion of the AP. For each node in thenetwork the articulation rank satisfies 0 ≤ r ≤ k , where k is the degree of the node. The articulation rank of anode which is not an AP is r = 0, while the articulationranks of APs satisfy r ≥
1. In fact, the articulation rankof an AP is the number of bredges connected to it. Thedistribution P ( r ) of articulation ranks was calculated inRef. [23].In this paper we present analytical results for the sta-tistical properties of bredges in configuration model net-works. In order to quantify the abundance of bredges, wecalculate the probability b P ( e ∈ B), that a random edge e in a configuration model network with degree distri-bution P ( k ) is a bredge. To characterize the statisticalproperties of bredges, we derive a closed form expressionfor the joint degree distribution b P ( k, k ′ | B) of the end-nodes i and i ′ of a random bredge. We also examine thedistinct properties of bredges on the giant component(GC) and on the finite tree components (FC) of the net-work. On the finite components all the edges are bredges,namely b P ( e ∈ B | FC) = 1. We calculate the probability b P ( e ∈ B | GC) that a random edge that resides on thegiant component is a bredge and the joint degree distri-bution b P ( k, k ′ | B , GC) between the end-nodes of bredgeson the giant component. It is found that the degrees k and k ′ of the end-nodes of a bredge that resides on thegiant component are correlated. This is in contrast tothe end-nodes of random edges in the network and to theend-nodes of non-bredge (NB) edges on the giant com-ponent, which exhibit no degree-degree correlations. Wethus conclude that all the degree-degree correlations onthe giant component are concentrated on the bredges.We calculate the covariance Γ(B , GC) and show that itis negative, which means that bredges on the giant com-ponent tend to connect high degree nodes to low degreenodes. We apply these results to ensembles of configura-tion model networks with degree distributions that followa Poisson distribution (Erd˝os-R´enyi networks), an expo-nential distribution and a power-law distribution (scale-free networks).The paper is organized as follows. In Sec. II we de-scribe the configuration model network and its construc-tion. In Sec. III we present the generating functions ofthe degree distribution. In Sec. IV we present a statis-tical analysis of nodes on the giant component and onthe finite components. In Sec. V we present a statisti-cal analysis of edges on the giant and finite components.In Sec. VI we present a detailed statistical analysis ofbredges. In Sec. VII we apply these results to config-uration model networks with a Poisson degree distribu-tion (ER networks), exponential degree distribution andpower-law degree distribution (scale-free networks). Theresults are discussed in Sec. VIII and summarized in Sec.IX.
II. THE CONFIGURATION MODEL
The configuration model is an ensemble of uncorre-lated random networks whose degree sequences are drawnfrom a given degree distribution P ( k ). The first moment(mean degree) and the second moment of P ( k ) are de-noted by h K n i , where n = 1 and 2, respectively, while thevariance is given by V [ K ] = h K i − h K i . The supportof the degree distribution of random networks is oftenbounded from below by k min ≥ P ( k ) = 0for 0 ≤ k ≤ k min −
1, with non-zero values of P ( k ) onlyfor k ≥ k min . For example, the commonly used choice of k min = 1 eliminates the possibility of isolated nodes inthe network. Choosing k min = 2 also eliminates the leafnodes. One may also control the upper bound by im-posing k ≤ k max . This may be important in the case offinite networks with heavy-tail degree distributions suchas power-law distributions. The configuration model net-work ensemble is a maximum entropy ensemble underthe condition that the degree distribution P ( k ) is im-posed [24–26]. Here we focus on the case of undirectednetworks.To generate a network instance drawn from an ensem-ble of configuration model networks of N nodes, with agiven degree distribution P ( k ), one draws the degrees ofthe N nodes independently from P ( k ). This gives rise toa degree sequence of the form k , k , . . . , k N . For the dis- cussion below it is convenient to list the degree sequencein a decreasing order of the form k ≥ k ≥ · · · ≥ k N .It turns out that not every possible degree sequence isgraphic, namely admissible as a degree sequence of a net-work. Therefore, before trying to construct a networkwith a given degree sequence, one should first confirmthe graphicality of the degree sequence. To be graphic,a degree sequence must satisfy two conditions. The firstcondition is that the sum of the degrees is an even num-ber, namely P i k i = 2 L , where L is an integer thatrepresents the number of edges in the network. Thesecond condition is expressed by the Erd˝os-Gallai the-orem, which states that an ordered sequence of the form k ≥ k ≥ · · · ≥ k N that satisfies the first condition isgraphic if and only if the condition n X i =1 k i ≤ n ( n −
1) + N X i = n +1 min( k i , n ) (1)holds for all values of n in the range 1 ≤ n ≤ N − N nodes such that each node i is connected to k i half edges or stubs [2]. At each stepof the construction, one connects a random pair of stubsthat belong to two different nodes i and j that are notalready connected, forming an edge between them. Thisprocedure is repeated until all the stubs are exhausted.The process may get stuck before completion in case thatall the remaining stubs belong to the same node or topairs of nodes that are already connected. In such caseone needs to perform some random reconnections in orderto complete the construction.In the dense-network limit, configuration model net-works consist of a single connected component, while inthe dilute-network limit they consist of many finite treecomponents. At intermediate densities they exhibit acoexistence between a giant component, which is exten-sive in the network size, and many non-extensive finitetree components. Some commonly studied configurationmodel networks can be described in terms of single pa-rameter families of degree distributions. A particularlyconvenient choice of the parameter is the mean degree c = h K i . In this case, the degree distribution can beexpressed by P ( k ) = P c ( k ), such that small values of c correspond to the dilute network limit while large val-ues of c correspond to the dense network limit. At somevalue c , referred to as the percolation threshold, thereis a percolation transition below which the network con-sists of finite tree components and above which a giantcomponent emerges. The percolation transition is a sec-ond order phase transition, whose order parameter is thefraction g of nodes that reside on the giant component.Below the transition, where c < c , the order parameteris g = 0, while for c > c the function g = g ( c ) graduallyincreases. III. THE GENERATING FUNCTIONS OF THEDEGREE DISTRIBUTION
Consider a configuration model network with a givendegree distribution P ( k ). To obtain the probability g that a random node in the network belongs to the giantcomponent, one needs to first calculate the probability˜ g , that a node i selected via a random edge e belongs tothe giant component of the reduced network, from whichthe edge e is removed. The probability ˜ g is determinedby [1, 2] 1 − ˜ g = G (1 − ˜ g ) , (2)where G ( x ) = ∞ X k =1 x k − e P ( k ) (3)is the generating function of the distribution e P ( k ) = k h K i P ( k ) , (4)which is the degree distribution of nodes that are sampledvia random edges. The solution of Eq. (2) is an attractivefixed point (Sec. 13.8 in Ref. [2]). Using ˜ g , one can thenobtain the probability g from the equation g = 1 − G (1 − ˜ g ) , (5)where G ( x ) = ∞ X k =0 x k P ( k ) (6)is the generating function of the degree distribution P ( k ).The two generating functions are related to each other by G ( x ) = G ′ ( x ) /G ′ (1), where G ′ ( x ) is the derivative of G ( x ).From the definitions of G ( x ) and G ( x ) in Eqs. (6)and (3), respectively, we find that 0 < G ( x ) , G ( x ) < < x < G (1) = G (1) = 1 This meansthat x = 1 is a fixed point for both generating func-tions. Therefore, g = ˜ g = 0 is a solution of Eqs. (2)and (5). This solution corresponds to the case of sub-critical networks, in which there is no giant component.In some networks there are no isolated nodes (of degree k = 0) and no leaf nodes (of degree k = 1). In such net-works P (0) = 0 and P (1) = 0, while P ( k ) > k ≥
2. The generating functions associated with thesenetworks satisfy G (0) = 0 and G (0) = 0. This impliesthat in such networks both x = 0 and x = 1 are fixedpoints of both G ( x ) and G ( x ) and there are no otherfixed points with 0 < x <
1. The coexistence of a giantcomponent and non-trivial finite tree components (thatconsist of more than a single node) appears only in casethat the degree distributions P ( k ) supports a non-trivialsolution of Eq. (2), in which 0 < ˜ g <
1. This requires a non-zero probability of leaf-nodes, namely P (1) >
0, andthus occurs only when k min = 0 or k min = 1. In largeconfiguration model networks in which k min ≥ c >
2, the giant com-ponent encompasses the whole network and g = ˜ g = 1[29].Here we focus on configuration model networks withdegree distributions P ( k ), which are bounded from be-low by k min = 0 or 1. Under suitable conditions, suchnetworks may exhibit a coexistence between a giant com-ponent and finite tree components. The condition for theexistence of a giant component can be expressed in theform h K ih K i − > , (7)which is known as the Molloy-Reed criterion [24, 25]. Inorder to discuss this condition, consider a node i thatis sampled via a random edge e . The excess degree k ex of i is the number of other edges apart from the edge e ,namely k ex = k −
1, where k is the degree of i . In essence,the condition of Eq. (7) states that a giant componentexists if the expectation value of the excess degree ofnodes sampled via a random edge exceeds 1. Thus, thepercolation threshold c is the value of the mean degree h K i at which h K i = 2 h K i . IV. STATISTICAL ANALYSIS OF NODES
Below we analyze the statistical properties of randomlysampled nodes in configuration model networks. We cal-culate the probability that a random node resides onthe giant component (and the complementary probabilitythat it resides on one of the finite components). We alsoanalyze the distinct statistical properties of the nodesthat reside on the giant component and on the finite com-ponents.
A. The fraction of nodes that reside on thegiant/finite components
The probability that a random node i in a configu-ration model network resides on the giant component is[35, 36] P ( i ∈ GC) = g, (8)where g is given by Eq. (5), while the probability that itresides on one of the finite components is P ( i ∈ FC) = 1 − g. (9)A node i of a given degree k resides on the giant com-ponent if at least one of its k neighbors resides on thegiant component of the reduced network from which i is removed [Fig. 2(a)]. Using the theoretical framework (cid:1853)(cid:1854)(cid:1855) (cid:1861) (cid:1842) (cid:1861) (cid:1488) GC (cid:3560)(cid:1842) (cid:1861) (cid:1488) GC (cid:3552)(cid:1842) (cid:1857) (cid:1488) GC (cid:1861) (cid:1857) FIG. 2: (a) A random node i (empty circle) of degree k ina configuration model network (left). The probability that i does not reside on the giant component is equal to the proba-bility that none of its k neighbors (full circles) resides on thegiant component of the reduced network (right) from which i is removed, together with its links (dashed lines). (b) A node i (empty circle) of degree k sampled via a random edge (left),which is marked as a dashed line. We are interested in theprobability that i does not reside on the giant component ofthe reduced network from which the sampled edge (dashedline) is removed. This probability is equal to the probabilitythat none of its k − i resides on thegiant component of the further reduced network (right) fromwhich the node i is removed together with its links (dashedlines). (c) A random edge e with end-nodes i and i ′ of degrees k and k ′ , respectively (left). The probability that e does notreside on the giant component is equal to the probability thatnone of its two end-nodes resides on the giant component ofthe reduced network from which e is removed. This probabil-ity is equal to the probability that none of the k − i and none of the k ′ − i ′ resides on the giant component of the further reduced net-work (right) from which i and i ′ are removed together withtheir links (dashed lines). of the cavity method [30–33], each neighbor of i can beconsidered as a node selected via a random edge. There-fore, the probability that each one of the neighbors of i resides on the giant component of the reduced networkfrom which i is removed is given by ˜ g . Moreover, due tothe locally tree-like structure of configuration model net-works, the probabilities of different neighbors of i to re-side on the giant component of the reduced network fromwhich i is removed are independent of each other. There-fore, the probability that a node i selected randomly fromall the nodes of degree k in the network resides on thegiant component, is given by [35, 36] P ( i ∈ GC | k ) = 1 − (1 − ˜ g ) k , (10)where ˜ g is given by Eq. (2), while the probability that itresides on one of the finite tree components is given by P ( i ∈ FC | k ) = (1 − ˜ g ) k . (11) P (cid:1861) (cid:1488) GCP (cid:1863) |GC (cid:3560) P (cid:1861) (cid:1488) GC (cid:3560) P (cid:1863) |GCP (cid:1861) (cid:1488) FCP (cid:1863) |FC (cid:3560) P (cid:1861) (cid:1488) FC (cid:3560) P (cid:1863) |FC Sampling Mechanism via random node via random edge N e t w o r k C o m p o n e n t F i n i t e c o m p o n e n t s G i a n t c o m p o n e n t FIG. 3: Illustration of the four categories of nodes consideredin this paper, presented in the form of a two by two matrixdiagram. The horizontal axis accounts for the two samplingprocedures, namely random node sampling and node sam-pling via random edges. The vertical axis accounts for thelocation of a node in the network, which can be either on thegiant component or on one of the finite tree components. Eachone of the four categories of nodes exhibits different statisticalproperties.
Clearly, the probability of i to reside on the giant compo-nent is an increasing function of the degree k , while theprobability of i to reside on one of the finite componentsis a decreasing function of k .The different categories of nodes in configurationmodel networks, in terms of the sampling procedure andtheir location in the network, are illustrated in Fig. 3 inthe form of a two by two matrix diagram. The horizontalaxis accounts for the two sampling procedures, namelyrandom node sampling and node sampling via randomedges. The vertical axis accounts for the location of anode in the network, which can be either on the giantcomponent or on one of the finite tree components. Eachone of the four categories of nodes exhibits different sta-tistical properties. Such 2 × B. The degree distributions of nodes on thegiant/finite components
The micro-structure of the giant component of config-uration model networks was recently studied [35, 36]. Itwas shown that the degree distribution, conditioned onthe giant component, is given by P ( k | GC) = 1 − (1 − ˜ g ) k g P ( k ) , (12)while the degree distribution conditioned on the finitecomponents is given by P ( k | FC) = (1 − ˜ g ) k − g P ( k ) , (13)where k ≥ k min . In the analysis below we focus on de-gree distributions whose support is bounded from belowby either k min = 0 or k min = 1, which enable the coex-istence between the giant component and the finite treecomponents. The derivations apply to both cases. Thespecific value of k min is not specified in each equation,but it is implicitly assumed that in the case of k min = 1the probability P (0) = 0.As expected, Eq. (12) satisfies P (0 | GC) = 0 even for k min = 0, namely there are no isolated nodes on the giantcomponent. Isolated nodes are considered as finite treecomponents of size s = 1. The probability that a ran-dom node on the finite components is an isolated nodeis given by P (0 | FC) = P (0) / (1 − g ), namely the fractionof isolated nodes on the finite components is higher thanin the whole network. Regarding leaf nodes of degree k = 1, their fraction on the giant component, given by P (1 | GC) = (˜ g/g ) P (1), is higher than in the whole net-work in case that g < ˜ g and lower in case that g > ˜ g .Since g > ˜ g [1 − P (0)] the former case may occur only innetworks that include isolated nodes, in which P (0) > < g, ˜ g < g and ˜ g satisfy the condition [23](1 − ˜ g ) − g < , (14)the fraction of nodes of degrees k ≥ − ˜ g )˜ gg > . (15)Note that the numerator on the left hand side of Eq. (15)satisfies (2 − ˜ g )˜ g < . (16)To show this we define ˜ h = 1 − ˜ g and obtain (2 − ˜ g )˜ g =1 − ˜ h <
1. The degree distribution of the whole networkis recovered by P ( k ) = P ( k | GC) P ( i ∈ GC) + P ( k | FC) P ( i ∈ FC) , (17)where P ( i ∈ GC) and P ( i ∈ FC) are given by Eqs. (8)and (9), respectively.The giant component of a configuration model networkconsists of a 2-core which is decorated by tree branches.The 2-core (2-CORE) is a connected component, suchthat each node on the 2-core has links to at least twoother nodes that reside on the 2-core [37–40]. Moreover,each node on the 2-core of a configuration model net-work resides on at least one cycle. The nodes on the tree branches belong to the 1-core of the giant component butnot to the 2-core. This is expressed by i ∈ GC ∩ X represents the complementary set of X and X ∩ Y is the intersection of X and Y . The degree distri-bution of the nodes on the 2-core of the giant componentis given by P ( k | − (1 − ˜ g ) k − k ˜ g (1 − ˜ g ) k − g − ˜ g (1 − ˜ g ) h K i P ( k ) , (18)while the degree distribution of the nodes on the treebranches of the giant component is given by P ( k | GC ∩ − ˜ g ) k − e P ( k ) . (19)The probability that a random node on the giant com-ponent resides on the 2-core is given by P ( i ∈ | GC) = 1 − ˜ g (1 − ˜ g ) g h K i , (20)while the probability that it resides on one of the treebranches is given by P ( i ∈ | GC) = ˜ g (1 − ˜ g ) g h K i . (21) C. The mean degrees of nodes on the giant/finitecomponents
The mean degree of the nodes that reside on the gi-ant component is given by E [ K | GC] = P k kP ( k | GC).Inserting P ( k | GC) from Eq. (12) and carrying out thesummation, we obtain E [ K | GC] = (2 − ˜ g )˜ gg h K i . (22)Using Eq. (15) we conclude that E [ K | GC] > h K i ,namely the mean degree of the nodes that reside on thegiant component is larger than the mean degree of thewhole network.The mean degree of the nodes that reside on the finitetree components is denoted by E [ K | FC]. Using P ( k | FC)from Eq. (13), we obtain E [ K | FC] = (1 − ˜ g ) − g h K i . (23)Using Eq. (14) we conclude that the mean degree of thenodes that reside on the finite tree components is smallerthan the mean degree of the whole network, namely E [ k | FC] < h K i . The mean degree of the whole networkis recovered by h K i = E [ K | GC] P ( i ∈ GC) + E [ K | FC] P ( i ∈ FC) , (24)where P ( i ∈ GC) and P ( i ∈ FC) are given by Eqs. (8)and (9), respectively.
D. The variance of the degree distributions on thegiant/finite components
The second moment of the degree distribution P ( k | GC) of the nodes that reside on the giant compo-nent is given by E [ K | GC] = 1 g (cid:26) h K i− (1 − ˜ g ) (cid:2) G ′ (1 − ˜ g ) (cid:3) h K i (cid:27) , (25)where G ′ ( x ) is the derivative of G ( x ). Since the fixedpoint of Eq. (2) is a stable fixed point, the derivativesatisfies G ′ (1 − ˜ g ) <
1. Writing G ′ (1 − ˜ g ) explicitly, inthe form G ′ (1 − ˜ g ) = 1 h K i ∞ X k =2 k ( k − − ˜ g ) k − P ( k ) , (26)we find that it satisfies G ′ (1 − ˜ g ) < min (cid:26) h K ih K i − , (cid:27) . (27)Inserting this result into Eq. (25) and using Eq. (15), itis found that in the coexistence phase, where 0 < g, ˜ g < E [ K | GC] > h K i . In the dilute network regime of0 < ˜ g ≪
1, just above the percolation transition, one canexpand the right hand side of Eq. (26) to first order in ˜ g and obtain G ′ (1 − ˜ g ) ≃ h K ih K i − − (cid:18) h K ih K i − h K ih K i + 2 (cid:19) ˜ g + O (cid:0) ˜ g (cid:1) . (28)The variance of P ( k | GC) is given by V [ K | GC] = 1 g (cid:26) h K i − (1 − ˜ g ) (cid:2) G ′ (1 − ˜ g ) (cid:3) h K i (cid:27) − [(2 − ˜ g )˜ g ] g h K i . (29)While both the first and second moments of P ( k | GC)are larger than the corresponding moments of P ( k ), thevariance V [ K | GC] may be either larger or smaller than V [ K ], depending on the specific properties of the degreedistribution.The second moment of the degree distribution P ( k | FC)of the nodes that reside on the finite tree components isdenoted by E [ K | FC]. Using P ( k | FC) from Eq. (13), weobtain E [ K | FC] = (1 − ˜ g ) − g [1 + G ′ (1 − ˜ g )] h K i . (30)Using Eqs. (14) and (27) one can show that E [ K | FC] < h K i . The variance of P ( k | FC) is denoted by V [ K | FC].Using the first and second moments from Eqs. (23) and(30), respectively, we obtain V [ K | FC] = (1 − ˜ g ) − g h K i (cid:26) G ′ (1 − ˜ g ) − (1 − ˜ g ) − g h K i (cid:27) . (31) While both the first and second moments of P ( k | FC) aresmaller than the corresponding moments of P ( k ), thevariance V [ K | FC] may be either larger or smaller than V [ K ], depending on the specific properties of the degreedistribution. (cid:3552) P (cid:1857) (cid:1488) GC (cid:3552) P (cid:1863) , (cid:1863) (cid:4593) |GC (cid:3552) P (cid:1857) (cid:1488) B, GC (cid:3552) P (cid:1863) , (cid:1863) (cid:4593) |B, GC (cid:3552) P (cid:1857) (cid:1488) FC (cid:3552) P (cid:1863) , (cid:1863) (cid:4593) |FC (cid:3552) P (cid:1857) (cid:1488) B, FC (cid:3552) P (cid:1863) , (cid:1863) (cid:4593) |B, FC Sampling Mechanism via random Edge via random Bredge N e t w o r k C o m p o n e n t F i n i t e c o m p o n e n t s G i a n t c o m p o n e n t FIG. 4: Illustration of the four categories of edges consideredin this paper, presented in the form of a two by two matrixdiagram. The horizontal axis accounts for the two samplingprocedures, namely random sampling from all the edges inthe network or random sampling restricted to those edgeswhich are bredges. The vertical axis accounts for the locationof an edge in the network, which can be either on the giantcomponent or in one of the finite tree components. Each oneof the four categories of edges exhibits different statisticalproperties.
V. STATISTICAL ANALYSIS OF EDGES
Below we analyze the statistical properties of randomlyselected edges in configuration model networks. We cal-culate the probability that a random edge resides onthe giant component (and the complementary probabil-ity that it resides on one of the finite components). Wealso analyze the distinct statistical properties of the edgesthat reside on the giant component and of those that re-side on the finite components.The different categories of edges in terms of the sam-pling procedure and their location in the network areillustrated in Fig. 4. The horizontal axis accounts forthe two sampling procedures, namely random samplingfrom all the edges in the network or random samplingrestricted to those edges which are bredges. The verticalaxis accounts for the location of an edge in the network,which can be either on the giant component or in one ofthe finite tree components. Each one of the four cate-gories of edges exhibits different statistical properties.
A. The fraction of edges that reside on thegiant/finite components
Consider a randomly chosen end-node i of a randomedge e [Fig. 2(b)]. The probability that i resides on thegiant component of the reduced network from which e isremoved is e P ( i ∈ GC) = ˜ g, (32)where ˜ g is given by Eq. (2), while the probability that i resides on one of the finite components of the reducednetwork is e P ( i ∈ FC) = 1 − ˜ g. (33)Consider a random edge e [Fig. 2(c)]. The probabilitythat e resides on one of the finite tree components of thenetwork amounts to the probability that both its end-nodes reside on finite components of the reduced networkfrom which e is removed. It is thus given by b P ( e ∈ FC) = (1 − ˜ g ) . (34)Therefore, the complementary probability that a randomedge e resides on the giant component is b P ( e ∈ GC) = (2 − ˜ g )˜ g. (35)The degrees of end-nodes satisfy k ≥ k min = 0. The probability that the end-node i belongs tothe giant component of the reduced network from which e is removed, is given by e P ( i ∈ GC | k ) = 1 − (1 − ˜ g ) k − , (36)while the probability that it belongs to one of the finitecomponents of the reduced network is e P ( i ∈ FC | k ) = (1 − ˜ g ) k − , (37)where k ≥ e whose end-nodes i and i ′ areof degrees k ≥ k ′ ≥
1, respectively. The probabilitythat such an edge resides on the giant component is givenby b P ( e ∈ GC | k , k ′ ) = 1 − (1 − ˜ g ) k + k ′ − , (38)while the probability that it resides on one of the finitetree components is b P ( e ∈ FC | k , k ′ ) = (1 − ˜ g ) k + k ′ − . (39)Interestingly, these probabilities depend only on the sumof k and k ′ rather than on each one of them separately.For k = k ′ = 1 one obtains b P ( e ∈ GC | ,
1) = 0 and b P ( e ∈ FC | ,
1) = 1. This is due to the fact that in thiscase i and i ′ form a dimer, which is isolated from therest of the network. As the sum k + k ′ increases, theprobability that the edge e resides on one of the finitecomponents decays exponentially while the probabilitythat it resides on the giant component converges towards1. B. The marginal degree distributions of end-nodes
The degree distribution of the end-nodes of randomedges is given by Eq. (4), where k ≥
1. The degreedistribution of the end-nodes of random edges that resideon the giant component is given by e P ( k | GC) = k E [ K | GC] P ( k | GC) . (40)Inserting P ( k | GC) from Eq. (12) and E [ K | GC] from Eq.(22), we obtain e P ( k | GC) = 1 − (1 − ˜ g ) k (2 − ˜ g )˜ g e P ( k ) , (41)where k ≥
1. From Eq. (41) one finds that the fraction ofend-nodes of degree k = 1 on the giant component, givenby e P (1 | GC) = e P (1) / (2 − ˜ g ), is lower than in the wholenetwork. Interestingly, the fraction of end-nodes of de-gree k = 2 on the giant component, given by e P (2 | GC) = e P (2), is identical to their fraction in the whole network.For k ≥ e P ( k | GC) > e P ( k ), namely nodesof degrees k ≥ k → ∞ e P ( k | GC) → e P ( k ) / [(2 − ˜ g )˜ g ], where (2 − ˜ g )˜ g < e P ( k | FC) = k E [ K | FC] P ( k | FC) . (42)Inserting P ( k | FC) from Eq. (13) and E [ K | FC] from Eq.(23), we obtain e P ( k | FC) = (1 − ˜ g ) k − e P ( k ) , (43)where k ≥
1. The degree distribution e P ( k ) of the end-nodes of random edges in the network is recovered by e P ( k ) = e P ( k | GC) b P ( e ∈ GC) + e P ( k | FC) b P ( e ∈ FC) , (44)where e P ( k | GC) is given by Eq. (41), e P ( k | FC) is given byEq. (43), b P ( e ∈ GC) is given by Eq. (35) and b P ( e ∈ FC)is given by Eq. (34).From Eq. (43) one finds that the fraction of end-nodes of degree k = 1 on the finite components, givenby e P (1 | FC) = e P (1) / (1 − ˜ g ), is higher than in the wholenetwork. The fraction of end-nodes of degree k = 2 onthe finite components is identical to their fraction in thewhole network. For any value of k ≥ k on the finite components is lower thanin the whole network. The ’phase separation’ between thegiant component and the finite components may thus beconsidered as a distillation process, in which high-degreenodes tend to concentrate on the giant component whilelow-degree nodes end up in the finite components. C. The mean degrees of end-nodes
The mean degree of end-nodes of random edges is de-noted by e E [ K ]. Using e P ( k ) from Eq. (4), we obtain e E [ K ] = h K ih K i . (45)The mean degree of end-nodes of random edges that re-side on the finite tree components is denoted by e E [ K | FC].Using e P ( k | FC) from Eq. (43), we obtain e E [ K | FC] = 1 + G ′ (1 − ˜ g ) , (46)where G ′ ( x ) is the derivative of G ( x ). Interestingly, thisimplies that G ′ (1 − ˜ g ) can be interpreted as the meanexcess degree e E [ K ex | FC] = e E [ K | FC] − R ofinfectious diseases [41] and to the neutron multiplicationfactor of nuclear chain reactions [42]. Using Eq. (27) itis found that e E [ K | FC] < e E [ K ].The mean degree of end-nodes of random edges thatreside on the giant component is denoted by e E [ K | GC].Using e P ( k | GC) from Eq. (41), we obtain e E [ K | GC] = 1(2 − ˜ g )˜ g (cid:26) h K ih K i − (1 − ˜ g ) [1 + G ′ (1 − ˜ g )] (cid:27) . (47)Using Eq. (27) it is found that e E [ K | GC] > e E [ K ]. Notethat in heavy tail degree distributions the mean degree e E [ K | GC] of the end-nodes that reside on the giant com-ponent may diverge even under conditions in which h K i is finite. This is due to the fact that in such distributionsthe second moment h K i that appears on the right handside of Eq. (47) may diverge, leading to the divergenceof e E [ K | GC]. The mean degree e E [ K ] can be recovered by e E [ K ] = e E [ K | GC] b P ( e ∈ GC) + e E [ K | FC] b P ( e ∈ FC) . (48) D. The variance of the degree distribution ofend-nodes
The second moment of the degree distribution e P ( k )of the end-nodes of random edges is denoted by e E [ K ].Using e P ( k ) from Eq. (4), we obtain e E [ K ] = h K ih K i . (49)The variance of e P ( k ) is denoted by e V [ K ]. Using the firstmoment e E [ K ] from Eq. (45) and the second moment e E [ K ] from Eq. (49), we obtain e V [ K ] = h K ih K i − h K i h K i . (50) The second moment of the degree distribution e P ( k | FC)of end-nodes of random edges that reside on the finitetree components is denoted by e E [ K | FC]. Using e P ( k | FC)from Eq. (43), we obtain e E [ K | FC] = (1 − ˜ g ) G ′′ (1 − ˜ g ) + 3 G ′ (1 − ˜ g ) + 1 , (51)where G ′′ ( x ) is the second derivative of G ( x ). Writing G ′′ (1 − ˜ g ) explicitly in the form G ′′ (1 − ˜ g ) = 1 h K i ∞ X k =3 k ( k − k − − ˜ g ) k − P ( k ) , (52)we find that it satisfies G ′′ (1 − ˜ g ) ≤ h K ih K i − h K ih K i + 2 , (53)where equality is obtained for ˜ g = 0. Combining thisresult with Eq. (27), it is found that in the coexistencephase, where 0 < g, ˜ g <
1, the second moment satisfies e E [ K | FC] < e E [ K ]. In the dilute network regime of 0 < ˜ g ≪
1, just above the percolation transition, one canexpand the right hand side of Eq. (52) to first order in ˜ g and obtain G ′′ (1 − ˜ g ) ≃ h K ih K i − h K ih K i + 2 − h K ih K i − h K ih K i +11 h K ih K i − ! ˜ g + O (cid:0) ˜ g (cid:1) . (54)Using Eq. (53) it is found that e E [ K | FC] < e E [ K ].The variance of the degree distribution e P ( k | FC) of theend-nodes that reside on the finite components is denotedby e V [ K | FC]. Using the first and second moments fromEqs. (46) and (51), respectively, we obtain e V [ K | FC] = (1 − ˜ g ) G ′′ (1 − ˜ g ) + G ′ (1 − ˜ g ) (cid:2) − G ′ (1 − ˜ g ) (cid:3) . (55)While both the first and second moments of e P ( k | FC) aresmaller than the corresponding moments of e P ( k ), thevariance e V [ K | FC] may be either larger or smaller than e V [ K ], depending on the specific properties of the degreedistribution.The second moment of the degree distribution e P ( k | GC) of the end-nodes that reside on the giant com-ponent is denoted by e E [ K | GC]. Using e P ( k | GC) fromEq. (41), we obtain e E [ K | GC] = 1(2 − ˜ g )˜ g ( h K ih K i − (1 − ˜ g ) (cid:2) (1 − ˜ g ) × G ′′ (1 − ˜ g ) + 3 G ′ (1 − ˜ g ) + 1 (cid:3)) . (56)Using Eq. (53) it is found that e E [ K | GC] > e E [ K ]. Thevariance of the degree distribution e P ( k | GC) of the end-nodes that reside on the giant component is denoted by0 e V [ K | GC]. Using the first and second moments from Eqs.(47) and (56), respectively, we obtain e V [ K | GC] = 1(2 − ˜ g )˜ g ( h K ih K i − (1 − ˜ g ) (cid:2) (1 − ˜ g ) × G ′′ (1 − ˜ g ) + 3 G ′ (1 − ˜ g ) + 1 (cid:3)) − − ˜ g )˜ g ] (cid:26) h K ih K i − (1 − ˜ g ) (cid:2) G ′ (1 − ˜ g ) (cid:3)(cid:27) . (57)While both the first and second moments of e P ( k | GC)are larger than the corresponding moments of e P ( k ), thevariance e V [ K | GC] may be either larger or smaller than e V [ K ], depending on the specific properties of the degreedistribution. The variance e V [ K | GC] is used below as anormalization factor for the covariance of the joint degreedistribution of edges that reside on the giant component,which yields the Pearson correlation coefficient.
E. The joint degree distribution of end-nodes
The joint degree distribution b P ( k, k ′ ) of the end-nodes i and i ′ of a random edge in a configuration model net-work with degree distribution P ( k ) is given by b P ( k, k ′ ) = e P ( k ) e P ( k ′ ) , (58)where e P ( k ) is given by Eq. (4). Note that in Eq. (58)the degrees satisfy k, k ′ ≥
1. The end-nodes i and i ′ areconsidered as two distinguishable objects. Thus, b P ( k, k ′ )is the probability that i is of degree k and i ′ is of degree k ′ . The probability that i is of degree k ′ and i ′ is ofdegree k is given by b P ( k ′ , k ) = b P ( k, k ′ ). Therefore, theprobabilities b P ( k, k ′ ), k, k ′ ≥ b P ( k, k ′ | FC) = e P ( k | FC) e P ( k ′ | FC) , (59)where k, k ′ ≥
1. Inserting e P ( k | FC) from Eq. (43) intoEq. (59) we obtain b P ( k, k ′ | FC) = (1 − ˜ g ) k + k ′ − e P ( k ) e P ( k ′ ) . (60)The joint degree distribution b P ( k, k ′ ) can be expressedas a weighted sum of the joint degree distribution of end-nodes of edges that reside on the giant component and on the finite components in the form b P ( k, k ′ ) = b P ( k, k ′ | GC) b P ( e ∈ GC)+ b P ( k, k ′ | FC) b P ( e ∈ FC) , (61)where b P ( e ∈ GC) and b P ( e ∈ FC) are given by Eqs. (35)and (34), respectively. Extracting b P ( k, k ′ | GC) from Eq.(61), we obtain b P ( k, k ′ | GC) = b P ( k, k ′ ) − b P ( k, k ′ | FC) b P ( e ∈ FC) b P ( e ∈ GC) . (62)Note that nodes that reside on the giant component sat-isfy k, k ′ ≥
1. Moreover, the giant component doesnot include edges for which k = k ′ = 1 (dimers), thus b P (1 , | GC) = 0. As a result, the lowest possible de-grees of the end-nodes of an edge that resides on thegiant component are ( k, k ′ ) = (1 ,
2) or ( k, k ′ ) = (2 , k + k ′ ≥ k, k ′ ≥
1. Inserting the joint degree dis-tributions b P ( k, k ′ ) and b P ( k, k ′ | FC) from Eqs. (58) and(60), respectively, and the probabilities b P ( e ∈ FC) and b P ( e ∈ GC) from Eqs. (34) and (35), respectively, intoEq. (62), we obtain b P ( k, k ′ | GC) = 1 − (1 − ˜ g ) k + k ′ − (2 − ˜ g )˜ g e P ( k ) e P ( k ′ ) , (63)where k, k ′ ≥ k + k ′ ≥
3. Extracting e P ( k ) fromEq. (41), we obtain e P ( k ) = (2 − ˜ g )˜ g − (1 − ˜ g ) k e P ( k | GC) . (64)Inserting e P ( k ) and e P ( k ′ ) from Eq. (64) into Eq. (63),we obtain b P ( k, k ′ | GC) = (2 − ˜ g )˜ g − (1 − ˜ g ) k + k ′ − [1 − (1 − ˜ g ) k ][1 − (1 − ˜ g ) k ′ ] × e P ( k | GC) e P ( k ′ | GC) . (65)Inserting k = k ′ = 1 into Eq. (65), we confirm that b P (1 , | GC) = 0. In the opposite limit of k, k ′ → ∞ , itis found that b P ( k, k ′ | GC) → (2 − ˜ g )˜ g e P ( k | GC) e P ( k ′ | GC).Since (2 − ˜ g )˜ g <
1, the probability that both end-nodesof an edge that resides on the giant component will beof high degree is suppressed. We thus conclude that thedegree-degree correlations between end-nodes of randomedges on the giant component are negative, namely thegiant component is disassortative [44–48].
F. The covariance of the joint degree distributionof end-nodes of edges
The covariance of the joint degree distribution of end-nodes of edges in a configuration model network is de-noted by Γ = b E [ KK ′ ] − e E [ K ] e E [ K ′ ] , (66)1where b E [ KK ′ ] = ∞ X k =1 ∞ X k ′ =1 kk ′ b P ( k, k ′ ) , (67)is the mixed second moment of b P ( k, k ′ ). In configurationmodel networks there are no degree-degree correlationsand therefore Γ = 0. Moreover, the sub-network thatconsists of all the finite tree components is also a con-figuration model network. Therefore, the covariance ofthe joint degree distribution of end-nodes of edges thatreside on the finite tree components satisfies Γ(FC) = 0.The covariance of the joint degree distribution of end-nodes of edges that reside on the giant component is givenby Γ(GC) = b E [ KK ′ | GC] − e E [ K | GC] e E [ K ′ | GC] , (68)where b E [ KK ′ | GC] = ∞ X k =1 ∞ X k ′ =1 kk ′ b P ( k, k ′ | GC) (69)is the mixed second moment of b P ( k, k ′ | GC) and themean degree e E [ K | GC] is given by Eq. (47). Inserting b P ( k, k ′ | GC) from Eq. (63) into Eq. (69), carrying outthe summations, and inserting the result into Eq. (68),we obtainΓ(GC) = − (1 − ˜ g ) [(2 − ˜ g )˜ g ] (cid:20) h K ih K i − − G ′ (1 − ˜ g ) (cid:21) . (70)It is found that Γ(GC) <
0, namely the giant componentof a configuration model network is always disassortative [44–48]. This means that on the giant component highdegree nodes tend to connect to low degree nodes andvice versa.In the dilute network regime of 0 < ˜ g ≪
1, just abovethe percolation transition, the giant component is smallbut it exhibits strong degree-degree correlations. UsingEq. (28), it is found that in this regimeΓ(GC) ≃ − (cid:18) h K ih K i − h K ih K i + 2 (cid:19) + O (cid:0) ˜ g (cid:1) . (71)In the opposite limit of ˜ g → − , in which the giant com-ponent expands to encompass the whole network (apartfrom any isolated nodes), Γ(GC) →
0. More precisely, inthe regime of 1 − ˜ g ≪ ∼ − (1 − ˜ g ) . The Pearson correla-tion coefficient for pairs of end-nodes of edges that resideon the giant component is given by R (GC) = Γ(GC) e V [ K | GC] , (72)where e V [ K | GC] is given by Eq. (57). Unlike the co-variance Γ(GC), the Pearson correlation coefficient isbounded in the range − ≤ R (GC) ≤
1. It is thus amore suitable measure for the comparison of the correla-tions between the degrees of pairs of end-nodes in differ-ent populations of edges and bredges.
VI. STATISTICAL ANALYSIS OF BREDGESA. The probability that a random edge is a bredge
Consider a random edge e in a configuration model network of N nodes with degree distribution P ( k ). Theprobability b P ( e ∈ B) that e is a bredge is given by [49] b P ( e ∈ B) = 1 − ˜ g , (73)where ˜ g is given by Eq. (2). This is due to the fact that in order that a random edge will not be a bredge, its end-nodes i and i ′ should both belong to the giant component of the reduced network from which the edge e was removed. Theprobability for each one of these nodes to belong to the giant component of the reduced network is ˜ g . Thus, theprobability that both of them belong to the giant component of the reduced network is ˜ g . The probability that atleast one of them does not belong to the giant component of the reduced network is thus 1 − ˜ g , which leads to Eq.(73). The complementary probability, that a random edge e is a non-bredge (NB) edge is given by b P ( e ∈ NB) = ˜ g .Therefore, in the dilute network regime of 0 < ˜ g ≪
1, just above the percolation transition, almost every edge is abredge.Consider a random edge e whose end-nodes i and i ′ are of known degrees, k and k ′ , where k, k ′ ≥
1. In order thatthe edge e will not be a bredge, both i and i ′ should reside on the giant component of the reduced network from which e is removed. Therefore, the probability that e is a bredge is given by b P ( e ∈ B | k, k ′ ) = 1 − [1 − (1 − ˜ g ) k − ][1 − (1 − ˜ g ) k ′ − ] , (74)2where k, k ′ ≥
1. This result can also be expressed in the form b P ( e ∈ B | k, k ′ ) = (1 − ˜ g ) k − + (1 − ˜ g ) k ′ − − (1 − ˜ g ) k + k ′ − . (75)The probability b P ( e ∈ B) that a random edge is a bredge can be expressed in the form b P ( e ∈ B) = ∞ X k,k ′ =1 b P ( e ∈ B | k, k ′ ) b P ( k, k ′ ) . (76)Inserting the conditional probability b P ( e ∈ B | k, k ′ ) from Eq. (75) into Eq. (76) and carrying out the summation, onerecovers Eq. (73).The probability b P ( e ∈ B) can be expressed as a sum of two terms, where one term accounts for nodes that resideon the giant component and the other accounts for nodes that reside on the finite components. It takes the form b P ( e ∈ B) = b P ( e ∈ B | GC) b P ( e ∈ GC) + b P ( e ∈ B | FC) b P ( e ∈ FC) . (77)Extracting the conditional probability b P ( e ∈ B | GC) that a random edge on the giant component is a bredge, oneobtains b P ( e ∈ B | GC) = b P ( e ∈ B) − b P ( e ∈ B | FC) b P ( e ∈ FC) b P ( e ∈ GC) . (78)Since all the edges on the finite tree components are bredges, b P ( e ∈ B | FC) = 1. Inserting b P ( e ∈ B) from Eq. (73), b P ( e ∈ GC) from Eq. (35) and b P ( e ∈ FC) from Eq. (34) into Eq. (78), we obtain b P ( e ∈ B | GC) = 2(1 − ˜ g )2 − ˜ g . (79)Therefore, the complementary probability that a random edge on the giant component is not a bredge is given by b P ( e ∈ NB | GC) = ˜ g − ˜ g . (80)To calculate the fraction of bredges that belong to the giant component one can use Bayes’ theorem, and obtain b P ( e ∈ GC | B) = b P ( e ∈ B | GC) b P ( e ∈ GC) b P ( e ∈ B) . (81)Inserting b P ( e ∈ B | GC) from Eq. (79), b P ( e ∈ GC) from Eq. (35) and b P ( e ∈ B) from Eq. (73) into Eq. (81), we obtain b P ( e ∈ GC | B) = 2˜ g g . (82)Therefore, the fraction of bredges that reside on the finite components is b P ( e ∈ FC | B) = 1 − ˜ g g . (83)The conditional probability P ( e ∈ B | k, k ′ ), given by Eq. (75), can be expressed as a sum of two terms, where oneterm accounts for nodes that reside on the giant component and the other accounts for nodes that reside on the finitecomponents. It takes the form b P ( e ∈ B | k, k ′ ) = b P ( e ∈ B | GC , k, k ′ ) b P ( e ∈ GC | k, k ′ )+ b P ( e ∈ B | FC , k, k ′ ) b P ( e ∈ FC | k, k ′ ) . (84)The conditional probability b P ( e ∈ GC | k, k ′ ), given by Eq. (38), takes non-zero values only for degrees k, k ′ ≥ k + k ′ ≥
3, while b P ( e ∈ FC | k, k ′ ) is given by Eq. (39), where k, k ′ ≥
1. Since all the edges that reside on3the finite components are bredges, b P ( e ∈ B | FC , k, k ′ ) = 1. Extracting the conditional probability b P ( e ∈ B | GC , k, k ′ )from Eq. (84), one obtains b P ( e ∈ B | GC , k, k ′ ) = b P ( e ∈ B | k, k ′ ) − b P ( e ∈ B | FC , k, k ′ ) b P ( e ∈ FC | k, k ′ ) b P ( e ∈ GC | k, k ′ ) . (85)Since all the edges on the finite tree components are bredges, b P ( e ∈ B | FC , k, k ′ ) = 1, where k, k ′ ≥
1. Evaluating theright hand side of Eq. (85), we obtain b P ( e ∈ B | GC , k, k ′ ) = (1 − ˜ g ) k − + (1 − ˜ g ) k ′ − − − ˜ g ) k + k ′ − − (1 − ˜ g ) k + k ′ − , (86)where k, k ′ ≥ k + k ′ ≥
3. The probability that an edge connecting end-nodes of degrees k and k ′ on the giantcomponent is not a bredge is thus given by b P ( e ∈ NB | GC , k, k ′ ) = 1 − (1 − ˜ g ) k − − (1 − ˜ g ) k ′ − + (1 − ˜ g ) k + k ′ − − (1 − ˜ g ) k + k ′ − , (87)where k, k ′ ≥ k + k ′ ≥ B. The joint degree distribution of the end-nodes of bredges
The joint degree distribution of the nodes on both sides of a bredge can be expressed in the form b P ( k, k ′ | B) = b P ( e ∈ B | k, k ′ ) b P ( k, k ′ ) b P ( e ∈ B) . (88)Inserting b P ( e ∈ B | k, k ′ ) from Eq. (75), b P ( k, k ′ ) from Eq. (58) and b P ( e ∈ B) from Eq. (73) into Eq. (88), we obtain b P ( k, k ′ | B) = 11 + ˜ g h (1 − ˜ g ) k − + (1 − ˜ g ) k ′ − − (1 − ˜ g ) k + k ′ − i e P ( k ) e P ( k ′ ) , (89)where k, k ′ ≥
1. Below we consider the joint degree distributions b P ( k, k ′ | B , GC) and b P ( k, k ′ | B , FC) of the end-nodesof random bredges on the giant component and on the finite components, respectively. Since all the edges on thefinite components are bredges, the joint degree distribution of the end nodes of random bredges that reside on thefinite components satisfies b P ( k, k ′ | B , FC) = b P ( k, k ′ | FC), where b P ( k, k ′ | FC) is given by Eq. (60).The conditional probability b P ( k, k ′ | B) can be expressed in the form b P ( k, k ′ | B) = b P ( k, k ′ | B , GC) b P ( e ∈ GC | B)+ b P ( k, k ′ | B , FC) b P ( e ∈ FC | B) . (90)Extracting b P ( k, k ′ | B , GC) from Eq. (90) we obtain b P ( k, k ′ | B , GC) = b P ( k, k ′ | B) − b P ( k, k ′ | B , FC) b P ( e ∈ FC | B) b P ( e ∈ GC | B) . (91)Inserting b P ( e ∈ FC | B) from Eq. (83) and b P ( e ∈ GC | B) from Eq. (79) into Eq. (92), we obtain b P ( k, k ′ | B , GC) = 12˜ g h (1 − ˜ g ) k − + (1 − ˜ g ) k ′ − − − ˜ g ) k + k ′ − i e P ( k ) e P ( k ′ ) , (92)where k, k ′ ≥ k + k ′ ≥ b P ( k, k ′ | GC) = b P ( k, k ′ | B , GC) b P ( e ∈ B | GC) + b P ( k, k ′ | NB , GC) b P ( e ∈ NB | GC) , (93)4where the first term on the right hand side accounts for the bredges and the second term account for all the edgesthat are not bredges. Note that the joint degree distribution b P ( k, k ′ | NB , GC) may take non-zero values only underconditions in which both k ≥ k ′ ≥
2. Extracting the joint degree distribution on the edges that are not bredges,we obtain b P ( k, k ′ | NB , GC) = b P ( k, k ′ | GC) − b P ( k, k ′ | B , GC) b P ( e ∈ B | GC) b P ( e ∈ NB | GC) . (94)Inserting b P ( k, k ′ | GC) from Eq. (63), b P ( k, k ′ | B , GC) from Eq. (92), b P ( e ∈ B | GC) from Eq. (79) and b P ( e ∈ NB | GC)from Eq. (80), we obtain b P ( k, k ′ | NB , GC) = 1˜ g h − (1 − ˜ g ) k − − (1 − ˜ g ) k ′ − + (1 − ˜ g ) k + k ′ − i e P ( k ) e P ( k ′ ) , (95)where k, k ′ ≥
2. Eq. (95) can be written as a product of the form b P ( k, k ′ | NB , GC) = (cid:20) − (1 − ˜ g ) k − ˜ g (cid:21) e P ( k ) " − (1 − ˜ g ) k ′ − ˜ g P ( k ′ ) , (96)which means that the degrees k and k ′ of the end-nodes of non-bredge edges on the giant component are uncorrelated.Therefore, the degree distribution e P ( k | NB , GC) of end-nodes of non-bredge edges that reside on the giant componentis given by e P ( k | NB , GC) = 1 − (1 − ˜ g ) k − ˜ g e P ( k ) , (97)where k ≥
2. We thus conclude that all the degree-degree correlations in the giant component of a configurationmodel network are concentrated in the bredges.A special property of bredges on the giant component is that they are ‘polarized’ in the sense that each bredge e has one end-node that resides on the giant component of the reduced network from which e is removed, while theother end-node resides on the finite tree component that is detached from the giant component. These two end-nodesexhibit different statistical properties. The conditional probability that the end-node i (of degree k ) resides on thegiant component and the end-node i ′ (of degree k ′ ) resides on the detached finite tree is given by b P ( K GC = k, K FC = k ′ | k, k ′ , B , GC) = (1 − ˜ g ) k ′ − (cid:2) − (1 − ˜ g ) k − (cid:3) (1 − ˜ g ) k − + (1 − ˜ g ) k ′ − − − ˜ g ) k + k ′ − , (98)for k = k ′ and b P ( K GC = k, K FC = k ′ | k, k ′ , B , GC) = 1 for k = k ′ . The joint degree distribution b P ( K GC = k, K FC = k ′ | B , GC) can thus be written in the form b P ( K GC = k, K FC = k ′ | B , GC) = (2 − δ k,k ′ ) b P ( K GC = k, K FC = k ′ | k, k ′ , B , GC) b P ( k, k ′ | B , GC) , (99)where δ k,k ′ is the Kronecker delta symbol and b P ( k, k ′ | B , GC) is given by Eq. (92). Inserting the right hand side ofEq. (98) into Eq. (99), we find that Eq. (99) can be written as a product of the form b P ( K GC = k, K FC = k ′ | B , GC) = e P ( K GC = k | B , GC) e P ( K FC = k ′ | B , GC) , (100)where the the degree distribution of the end-node on the giant component side is e P ( K GC = k | B , GC) = 1 − (1 − ˜ g ) k − ˜ g e P ( k ) , (101)and the degree distribution of the end-node on the finite component side is e P ( K FC = k ′ | B , GC) = (1 − ˜ g ) k ′ − e P ( k ′ ) . (102)Eq. (100) implies that once we recognize that each bredge on the giant component has one end-node whose degree issampled from e P ( K GC = k | B , GC), while the degree of the other end-node is sampled from e P ( K FC = k ′ | B , GC), thecorrelation between the degrees of the two end-nodes vanishes. The correlation found in the analysis above, betweenthe degrees k and k ′ of the end-nodes i and i ′ , in the joint degree distribution b P ( k, k ′ | B , GC) [Eq. (92)] is due to thefact that if i ends up on the giant component of the reduced network, then i ′ must end up on a finite component andvice versa.5 C. The marginal degree distribution of the end-nodes of bredges
The degree distribution e P ( k | B ) of an end-node of a random bredge can be obtained as the marginal distributionof the joint degree distribution b P ( k, k ′ | B ) by tracing over k ′ , namely e P ( k | B) = ∞ X k ′ =1 b P ( k, k ′ | B) . (103)Inserting b P ( k, k ′ | B) from Eq. (89) and carrying out the summation, we obtain e P ( k | B) = 1 + ˜ g (1 − ˜ g ) k − g e P ( k ) . (104)Extracting e P ( k ) from Eq. (104), we obtain e P ( k ) = 1 + ˜ g g (1 − ˜ g ) k − e P ( k | B) . (105)Inserting e P ( k ) and e P ( k ′ ) from Eq. (105) into Eq. (89), we obtain b P ( k, k ′ | B) = (1 + ˜ g ) (1 − ˜ g ) k − + (1 − ˜ g ) k ′ − − (1 − ˜ g ) k + k ′ − [1 + ˜ g (1 − ˜ g ) k − ] [1 + ˜ g (1 − ˜ g ) k ′ − ] e P ( k | B) e P ( k ′ | B) . (106)Since on the finite tree components all the edges are bredges, the degree distribution e P ( k | B , FC) of the end-nodesof bredges that reside on the finite components satisfies e P ( k | B , FC) = e P ( k | FC), where e P ( k | FC) is given by Eq. (43).The degree distribution e P ( k | B , GC) of end-nodes of bredges that reside on the giant component can be obtained bymarginalizing b P ( k, k ′ | B , GC), given by Eq. (92), over k ′ . This yields e P ( k | B , GC) = 1 + (2˜ g − − ˜ g ) k − g e P ( k ) , (107)where k ≥
1. Extracting e P ( k ) from Eq. (107), we obtain e P ( k ) = 2˜ g g − − ˜ g ) k − e P ( k | B , GC) . (108)Inserting e P ( k ) from Eq. (108) into Eq. (92), we obtain b P ( k, k ′ | B , GC) = 2˜ g h (1 − ˜ g ) k − + (1 − ˜ g ) k ′ − − − ˜ g ) k + k ′ − i [1 + (2˜ g − − ˜ g ) k − ] [1 + (2˜ g − − ˜ g ) k ′ − ] e P ( k | B , GC) e P ( k ′ | B , GC) , (109) D. The mean degree of end-nodes of bredges
The mean degree of end-nodes of bredges is denoted by e E [ K | B]. Using the degree distribution e P ( k | B), given byEq. (104), we obtain e E [ K | B] = 11 + ˜ g (cid:26) h K ih K i + ˜ g (cid:2) G ′ (1 − ˜ g ) (cid:3)(cid:27) . (110)The mean degree e E [ K | B , FC] of the end-nodes of random bredges that reside on the finite components is identical to e E [ K | FC], which is given by Eq. (46). Using Eq. (27) it is found that e E [ K | B , FC] < e E [ K ]. Using Eq. (107) we obtainthe mean degree of the end-nodes of bredges that reside on the giant component, which is given by e E [ K | B , GC] = 12˜ g (cid:26) h K ih K i + (2˜ g − (cid:2) G ′ (1 − ˜ g ) (cid:3)(cid:27) . (111)6Using Eq. (27) it is found that e E [ K | B , GC] ≥ e E [ K ]. Note that in heavy tail degree distributions the mean degree e E [ K | B , GC] of end-nodes on the giant component may diverge even under conditions in which h K i is finite. Thisis due to the fact that the second moment h K i appears on the right hand side of Eq. (47). In heavy-tail degreedistributions h K i may diverge, leading to the divergence of e E [ K | B , GC].Below we evaluate the means of the degree distributions of the end-nodes of bredges e on the giant component,which reside on the giant and on the finite components of the reduced network from which e is removed. Using Eq.(101) we obtain the mean degree of the end-nodes that reside on the giant component of the reduced network, whichis given by e E [ K GC | B , GC] = 1˜ g (cid:26) h K ih K i − (1 − ˜ g ) (cid:2) G ′ (1 − ˜ g ) (cid:3)(cid:27) . (112)Using Eq. (102) we obtain the mean degree of the end-nodes that reside on a finite component of the reduced network,which is given by e E [ K FC | B , GC] = 1 + G ′ (1 − ˜ g ) . (113)Similarly, the mean degree of the end-nodes of random non-bredge edges that reside on the giant component, obtainedusing Eq. (97), is given by e E [ K | NB , GC] = 1˜ g (cid:26) h K ih K i − (1 − ˜ g ) (cid:2) G ′ (1 − ˜ g ) (cid:3)(cid:27) . (114) E. The variance of the degree distribution of end-nodes of bredges
The second moment of the degree distribution e P ( k | B) of the end-nodes of bredges, obtained using Eq. (104), isgiven by e E [ K | B] = 11 + ˜ g (cid:26) h K ih K i + ˜ g (cid:2) (1 − ˜ g ) G ′′ (1 − ˜ g ) + 3 G ′ (1 − ˜ g ) + 1 (cid:3)(cid:27) . (115)Using e E [ K | B] from Eq. (110) and e E [ K | B] from Eq. (115), we obtain the variance e V [ K | B] = 11 + ˜ g (cid:26) h K ih K i + ˜ g (cid:2) (1 − ˜ g ) G ′′ (1 − ˜ g ) + 3 G ′ (1 − ˜ g ) + 1 (cid:3)(cid:27) − g ) (cid:26) h K ih K i + ˜ g (cid:2) G ′ (1 − ˜ g ) (cid:3)(cid:27) . (116)Since all the edges on the finite components are bredges, it is clear that e E [ K | B , FC] = e E [ K | FC], which is givenby Eq. (51). Similarly, e V [ K | B , FC] = e V [ K | FC], which is given by Eq. (55). The second moment of the degreedistribution e P ( k | B , GC) of nodes selected via random edges that reside on the giant component, obtained using Eq.(104), is given by e E [ K | B , GC] = 12˜ g (cid:26) h K ih K i + (2˜ g − (cid:2) (1 − ˜ g ) G ′′ (1 − ˜ g ) + 3 G ′ (1 − ˜ g ) + 1 (cid:3)(cid:27) . (117)Using the first and second moments from Eqs. (111) and (117), respectively, we obtain the variance e V [ K | B , GC] = 12˜ g (cid:26) h K ih K i + (2˜ g − (cid:2) (1 − ˜ g ) G ′′ (1 − ˜ g ) + 3 G ′ (1 − ˜ g ) + 1 (cid:3)(cid:27) − g (cid:26) h K ih K i + (2˜ g − (cid:2) G ′ (1 − ˜ g ) (cid:3)(cid:27) (118)Below we evaluate the second moments of the degree distributions and of the end-nodes of bredges e on the giantcomponent, which end up on the giant and on the finite components of the reduced network from which e is removed.7The second moment of the degree distribution of the end-nodes that end up on the giant component, obtained usingEq. (101), is given by e E [ K | B , GC] = 1˜ g (cid:26) h K ih K i − (1 − ˜ g ) (cid:2) G ′′ (1 − ˜ g ) + 3 G ′ (1 − ˜ g ) + 1 (cid:3)(cid:27) . (119)Using the first and second moments from Eqs. (112) and (119), respectively, we obtain the variance e V [ K GC | B , GC] = 1˜ g (cid:26) h K ih K i − (1 − ˜ g ) (cid:2) G ′′ (1 − ˜ g ) + 3 G ′ (1 − ˜ g ) + 1 (cid:3)(cid:27) − g (cid:26) h K ih K i − (1 − ˜ g ) (cid:2) G ′ (1 − ˜ g ) (cid:3)(cid:27) . (120)The second moment of the degree distribution of the end-nodes that end up on the finite tree that is detached fromthe giant component of the reduced network, obtained using Eq. (102), is given by e E [ K | B , GC] = (1 − ˜ g ) G ′′ (1 − ˜ g ) + 3 G ′ (1 − ˜ g ) + 1 . (121)Using the first and second moments from Eqs. (113) and (121), respectively, we obtain the variance e V [ K FC | B , GC] = (1 − ˜ g ) G ′′ (1 − ˜ g ) + G ′ (1 − ˜ g ) (cid:2) − G ′ (1 − ˜ g ) (cid:3) . (122)The second moment of the degree distribution e P ( k | NB , GC) of the end-nodes of non-bredge edges that reside on thegiant component, obtained using Eq. (97), is given by e E [ K | NB , GC] = 1˜ g (cid:26) h K ih K i − (1 − ˜ g ) (cid:2) (1 − ˜ g ) G ′′ (1 − ˜ g ) + 3 G ′ (1 − ˜ g ) + 1 (cid:3)(cid:27) . (123)using e E [ K | NB , GC] from Eq. (114) and e E [ K | NB , GC] from Eq. (123), we obtain the variance e V [ K | NB , GC] = 1˜ g (cid:26) h K ih K i − (1 − ˜ g ) (cid:2) (1 − ˜ g ) G ′′ (1 − ˜ g ) + 3 G ′ (1 − ˜ g ) + 1 (cid:3)(cid:27) − g (cid:26) h K ih K i − (1 − ˜ g ) (cid:2) G ′ (1 − ˜ g ) (cid:3)(cid:27) . (124) F. The covariance of the joint degree distribution of end-nodes of bredges
The covariance of the joint degree distribution of end-nodes of random bredges is given byΓ(B) = b E [ KK ′ | B] − e E [ K | B] e E [ K ′ | B] (125)where b E [ KK ′ | B] is the mixed second moment of the joint degree distribution b P ( k, k ′ | B) and the mean degree e E [ K | B]of the marginal degree distribution is given by Eq. (110). Evaluating the right hand side of Eq. (125), we obtainΓ(B) = − g ) (cid:20) h K ih K i − − G ′ (1 − ˜ g ) (cid:21) . (126)As expected, below the percolation transition, where ˜ g = 0, the correlation coefficient is zero. In the dilute networkregime of 0 < ˜ g ≪
1, just above the percolation transition,Γ(B) ≃ − (cid:18) h K ih K i − h K ih K i + 2 (cid:19) ˜ g + O (˜ g ) . (127)In the opposite limit of ˜ g → − the covariance Γ(B) converges towards an asymptotic value that depends on thedegree distribution. It is given by Γ(B) → − (cid:20) h K ih K i − − P (2) h K i (cid:21) . (128)8The Pearson correlation coefficient for pairs of end-nodes of bredges in configuration model networks is given by R (B) = Γ(B) e V [ K | B] , (129)where e V [ K | B] is given by Eq. (116).The covariance of the joint degree distribution of end-nodes of bredges that reside on the giant component is givenby Γ(B , GC) = b E [ KK ′ | B , GC] − e E [ K | B , GC] e E [ K ′ | B , GC] , (130)where b E [ KK ′ | B , GC] is the mixed second moment of b P ( k, k ′ | B , GC). Evaluating the right hand side of Eq. (130), weobtain Γ(B , GC) = − g (cid:20) h K ih K i − − G ′ (1 − ˜ g ) (cid:21) . (131)In the dilute network regime of 0 < ˜ g ≪
1, just above the percolation transition, the covariance is given byΓ(B , GC) ≃ − (cid:18) h K ih K i − h K ih K i + 2 (cid:19) + O (˜ g ) . (132)In the opposite limit of ˜ g → − the covariance Γ(B , GC) converges towards an asymptotic value that depends on thedegree distribution. It is given by Γ(B , GC) → − (cid:20) h K ih K i − − P (2) h K i (cid:21) , (133)which is identical to Γ(B) in that limit. The Pearson correlation coefficient for pairs of end-nodes of bredges thatreside on the giant component is given by R (B , GC) = Γ(B , GC) e V [ K | B , GC] , (134)where e V [ K | B , GC] is given by Eq. (118). The end-nodes of non-bredge edges that reside on the giant componentare actually independent, as expressed by Eq. (96), and in particular they exhibit no degree-degree correlations.Therefore, R (NB , GC) = 0.
VII. APPLICATIONS TO SPECIFIC NETWORKMODELS
Here we apply the approach presented above to severalexamples of configuration model networks with given de-gree distribution. More specifically, we consider the casesof the Poisson degree distribution (ER networks), the ex-ponential degree distribution and the power-law degreedistribution (scale-free networks).
A. Erd˝os-R´enyi networks
The ER network is a random network in which eachpair of nodes is connected with probability p [50–52]. Themean degree of an ER network is c = ( N − p , where N is the network size, and the degree distribution is a Poisson distribution of the form [6] P ( k ) = e − c c k k ! . (135)Since it exhibits no correlations, the ER network is aconfiguration model network with a Poisson degree dis-tribution. Moreover, it is a maximum entropy networkunder the condition that the mean degree h K i = c is con-strained. Asymptotic ER networks exhibit a percolationtransition at c = 1, such that for c < c > c = ln N , there is a second tran-sition, above which the giant component encompasses theentire network.ER networks exhibit a special property, resulting fromthe Poisson degree distribution [Eq. (135)], which satis-fies e P ( k ) = P ( k − e P ( k ) is given by Eq. (4). This9implies that for the Poisson distribution the two gener-ating functions are identical, namely G ( x ) = G ( x ).Using Eqs. (2) and (5) we obtain that for ER net-works ˜ g = g . Carrying out the summations in Eqs.(6) and (3) with P ( k ) given by Eq. (135), one obtains G ( x ) = G ( x ) = e − (1 − x ) c . Inserting this result in Eq.(5), it is found that g satisfies the equation 1 − g = e − gc [6]. Solving for the probability g as a function of themean degree, c , one obtains g = ˜ g = 1 + W ( − ce − c ) c , (136)where W ( x ) is the Lambert W function [53].In Fig. 5(a) we present analytical results for the proba-bility b P ( e ∈ GC) (dashed line), that a randomly samplededge in an ER network resides on the giant component,as a function of the mean degree c . These results are ob-tained by inserting ˜ g from Eq. (136) into Eq. (35). Wealso present the complementary probability b P ( e ∈ FC)(dotted line) that a randomly sampled edge resides onone of the finite components. In Fig. 5(b) we present ana-lytical reults for the probability b P ( e ∈ B) (solid line) thata randomly sampled edge in an ER network is a bredgeas a function of the mean degree c . The probability b P ( e ∈ B) can be expressed as a sum of two components:the probability b P ( e ∈ B , GC) = b P ( e ∈ B | GC) b P ( e ∈ GC)(dashed line) that a randomly sampled edge is a bredgethat resides on the giant component, and the probability b P ( e ∈ B , FC) = b P ( e ∈ FC) (dotted line) that a randomlysampled edge is a bredge that resides on one of the finitecomponents. The analytical results are in excellent agree-ment with the results of computer simulations (circles),performed for an ensemble of ER networks of N = 10 nodes.In Fig. 6 we present analytical results for the marginaldegree distribution e P ( k | GC) of end-nodes of edges onthe giant component of an ER network (solid line), ob-tained by inserting ˜ g from Eq. (136) into Eq. (41). Wealso present the marginal degree distribution e P ( k | B , GC)of end-nodes of bredges on the giant component (dot-ted line), obtained by inserting ˜ g from Eq. (136) intoEq. (107) and for the marginal degree distribution e P ( k | NB , GC) of non-bredge edges that reside on the giantcomponent (dashed line), obtained by inserting ˜ g fromEq. (136) into Eq. (97). The analytical results are inexcellent agreement with the corresponding results ob-tained from computer simulations (circles). It is foundthat the marginal degree distribution of the end-nodesof bredges decreases monotonically as a function of k ,while the marginal degree distribution of the non-bredgeedges exhibits a peak. Overall, the degrees of end-nodesof non-bredge edges tend to be higher than the degreesof end-nodes of bredges.In Fig. 7 we present analytical results for the corre-lation coefficient R (GC) between the degrees of pairs ofend-nodes of edges on the giant component of an ERnetwork as a function of the mean degree c (solid line), obtained by inserting ˜ g from Eq. (136) into Eq. (72). Wealso present the correlation coefficient R (B , GC) betweenthe degrees of end-nodes of bredges that reside on the gi-ant component of an ER network (dotted line), obtainedby inserting ˜ g from Eq. (136) into Eq. (134). The ana-lytical results are in excellent agreement with the resultsobtained from computer simulations (circles). FIG. 5: (Color online) (a) The probability b P ( e ∈ GC)(dashed line), that a randomly sampled edge in an ER net-work resides on the giant component, as a function of themean degree c = h K i , obtained from Eq. (35); The com-plementary probability b P ( e ∈ FC) (dotted line) that a ran-domly sampled edge resides on one of the finite componentsis also shown. (b) The probability b P ( e ∈ B) (solid line) thata randomly sampled edge in an ER network is a bredge, as afunction of the mean degree c , obtained from Eq. (73); Theprobability b P ( e ∈ B) is equal to the sum of two components:the probability b P ( e ∈ B , GC) (dashed line) that a randomlysampled edge is a bredge that resides on the giant compo-nent, and the probability b P ( e ∈ B , FC) (dotted line) thata randomly sampled edge is a bredge that resides on one ofthe finite components. The analytical results are in excellentagreement with the results of computer simulations (circles),performed for an ensemble of ER networks of N = 10 nodes.
0 2 4 6 8 10 1200.10.20.30.4
FIG. 6: (Color online) Analytical results for the marginaldegree distribution e P ( k | GC) (solid line) of end-nodes ofrandomly sampled edges, the marginal degree distribution e P ( k | B , GC) (dotted line) of end-nodes of randomly sampledbredges and the marginal degree distribution e P ( k | NB , GC)(dashed line) of randomly sampled non-bredge edges on thegiant component of an ER network with mean degree c = 2.The analytical results are in excellent agreement with the re-sults obtained from computer simulations (circles). B. Configuration model networks with exponentialdegree distributions
Consider a configuration model network with an ex-ponential degree distribution of the form P ( k ) ∼ e − αk ,where k min ≤ k ≤ k max . In case that k min ≥ g = ˜ g = 1 and there are no bredges. Here weconsider the case of k min = 1 and k max = ∞ . In this caseit is convenient to parametrize the degree distributionusing the mean degree c in the form P ( k ) = 1 c − (cid:18) c − c (cid:19) k . (137)In order to find the properties of bredges in such net-works, we first calculate the parameters ˜ g and g . In-serting the exponential degree distribution of Eq. (137)into the generating function G ( x ), given by Eq. (3), weobtain G ( x ) = [ c − ( c − x ] − . Inserting the above ex-pression of G ( x ) into Eq. (2) and solving for ˜ g , we findthat for c > / g = 12 " c − c − r c + 3 c − . (138)Inserting the exponential degree distribution of Eq. (137)into Eq. (6), we obtain G ( x ) = x/ [ c − ( c − x ]. In-serting ˜ g from Eq. (138) and the above expression of G (1 − ˜ g ) into Eq. (5), we find that for c > / g = c c − " − r c + 3 c − . (139) FIG. 7: (Color online) Analytical results for the correlationcoefficient R (GC) (solid line) between the degrees k and k ′ of end-nodes of randomly sampled edges and the correlationcoefficient R (B , GC) (dotted line) between the end-nodes ofrandomly sampled bredges that reside on the giant componentof an ER network, as a function of the mean degree c . Theanalytical results are in excellent agreement with the resultsobtained from computer simulations (circles). The correla-tions, which are concentrated on the bredges, are negativeand become stronger as c is increased. Since the fraction ofbredges on the giant component is a decreasing function of c , the correlation coefficient over all the edges on the giantcomponent decreases (in absolute value) as c is increased. Thus, it is found that the configuration model networkwith an exponential degree distribution exhibits a perco-lation transition at c = 3 / b P ( e ∈ GC)(dashed line), that a random edge in a configurationmodel network with an exponential degree distributionresides on the giant component, obtained from Eq. (35),and the probability b P ( e ∈ FC) (dotted line) that a ran-dom edge resides on one of the finite components, as afunction of the mean degree c . In Fig. 8(b) we presentthe probability b P ( e ∈ B) that a random edge in a config-uration model network with an exponential degree dis-tribution is a bredge (solid line), as a function of c , ob-tained from Eq. (73). We also present the probability b P ( e ∈ B , GC) (dashed line) that a randomly selectededge in the network is a bredge that resides in the gi-ant component and the probability b P ( e ∈ B , FC) (dottedline) that a randomly selected edge in the network is abredge that resides in one of the finite components. Theanalytical results are found to be in excellent agreementwith the results of computer simulations (circles), per-formed for an ensemble of configuration model networksof N = 10 nodes.In Fig. 9 we present analytical results for the marginaldegree distribution e P ( k | GC) of end-nodes of randomlyselected edges (solid line), the marginal degree distribu-tion e P ( k | B , GC) of end-nodes of bredges (dotted line) andthe marginal degree distribution e P ( k | NB , GC) of non-1
FIG. 8: (Color online) (a) Analytical results for the prob-ability b P ( e ∈ GC) (dashed line) that a randomly samplededge in a configuration model network with an exponentialdegree distribution resides on the giant component, as a func-tion of the mean degree c ; The complementary probability P ( e ∈ FC) (dotted line) that a randomly sampled edge re-sides on one of the finite tree components is also shown. (b)Analytical results for the probability b P ( e ∈ B) (solid line)that a randomly sampled edge is a bredge, as a function ofthe mean degree c , obtained from Eq. (73); The probability P ( e ∈ B) is equal to the sum of two components: the proba-bility b P ( e ∈ B , GC) (dashed line), that a randomly samplededge is a bredge that resides on the giant component and theprobability b P ( e ∈ B , FC) that a randomly sampled edge isa bredge that resides on one of the finite components. Theanalytical results are in excellent agreement with the resultsof computer simulations (circles), performed for an ensembleof configuration model networks of N = 10 nodes. bredge edges (dashed line) on the giant component ofa configuration model network with an exponential de-gree distribution. The analytical results are in excellentagreement with the results obtained from computer sim-ulations (circles).In Fig. 10 we present analytical results for the cor-relation coefficients R (GC) and R (B , GC) between thedegrees k and k ′ of the end-nodes of edges that resideon the giant component (solid line) and bredges that re-side on the giant component (dotted line), respectively,
0 2 4 6 8 10 1200.10.20.30.4
FIG. 9: (Color online) Analytical results for the marginaldegree distribution e P ( k | GC) (solid line) of end-nodes ofrandomly sampled edges, the marginal degree distribution e P ( k | B , GC) (dotted line) of end-nodes of randomly sampledbredges, and the marginal degree distribution e P ( k | NB , GC)(dashed line) of randomly sampled non-bredge edges, on thegiant component of a configuration model network with an ex-ponential degree distribution and mean degree c = 2 .
5. Theanalytical results are in excellent agreement with the resultsobtained from computer simulations (circles).
FIG. 10: (Color online) Analytical results for the correlationcoefficient R (GC) (solid line) between the degrees k and k ′ ofend-nodes of edges and the correlation coefficient R (B , GC)(dotted line) between the end-nodes of bredges that reside onthe giant component of a configuration model network withan exponential degree distribution, as a function of the meandegree c . The analytical results are in excellent agreementwith the results obtained from computer simulations (circles). as a function of the mean degree c in configuration modelnetworks with exponential degree distributions. The an-alytical results are in excellent agreement with the resultsobtained from computer simulations (circles).2 FIG. 11: (Color online) (a) Analytical results for the proba-bility b P ( e ∈ GC) (dashed line) that a randomly sampled edgein a configuration model network with a power-law degree dis-tribution resides on the giant component, as a function of themean degree c . The complementary probability b P ( e ∈ FC)(dotted line) that a randomly sampled edge resides on one ofthe finite tree components is also shown; (b) Analytical re-sults for the probability b P ( e ∈ B) (solid line) that a randomedge is a bredge, as a function of the mean degree c . Theprobability b P ( e ∈ B) is equal to the sum of two components:the probability b P ( e ∈ B , GC) (dashed line) that a randomlysampled edge is a bredge that resides on the giant compo-nent and the probability b P ( e ∈ B , FC) (dotted line) that arandomly sampled edge is a bredge that resides on one ofthe finite components. The analytical results are in excellentagreement with the results of computer simulations (circles),performed for networks of N = 10 nodes. C. Configuration model networks with power-lawdegree distributions
Consider a configuration model network with a power-law degree distribution of the form P ( k ) ∼ k − γ , where k min ≤ k ≤ k max . For γ ≤ k max → ∞ . For 2 < γ ≤ γ > γ >
2, in which the mean degree, h K i , is bounded evenfor k max → ∞ . We choose k min = 1, for which there is acoexistence phase of the giant component and the finitetree components and k max = 100. The normalized degree distribution is given by P ( k ) = A ( γ, k max ) k − γ , (140)where the normalization factor is A ( γ, k max ) = [ ζ ( γ ) − ζ ( γ, k max + 1)] − , the function ζ ( γ, k ) is the Hurwitzzeta function and ζ ( γ ) = ζ ( γ,
1) is the Riemann zetafunction [53]. The mean degree is given by h K i = A ( γ, k max ) /A ( γ − , k max ) and the second moment of thedegree distribution is given by h K i = A ( γ, k max ) /A ( γ − , k max ). Inserting the degree distribution of Eq. (140)into Eqs. (6) and (3) we obtain G ( x ) = A ( γ, k max ) (cid:2) Li γ ( x ) − x k max +1 Φ( x, γ, k max + 1) (cid:3) , (141)and xG ( x ) = A ( γ − , k max ) h Li γ − ( x ) − x k max +1 Φ( x, γ − , k max + 1) i , (142)where Φ( x, γ, k ) is the Lerch transcendent and Li γ ( x ) isthe polylogarithm function [54]. The values of the pa-rameters ˜ g and g are determined by Eqs. (2) and (5).Unlike the ER network and the configuration model net-work with an exponential degree distribution, here wedo not have closed form analytical expressions for g and˜ g . However, using the expressions above for G ( x ) and G ( x ), the values of g and ˜ g can be easily obtained from anumerical solution of Eqs. (2) and (5). Using the Molloy-Reed criterion [24, 25], we find that for k max = 100 thepercolation threshold is c ≃ . γ ≃ . b P ( e ∈ GC)that a random edge in a configuration model networkwith a power-law degree distribution resides on the giantcomponent (dashed line), obtained from Eq. (35), as afunction of c . We also present the complementary prob-ability b P ( e ∈ FC) that a random edge resides on one ofthe finite components (dotted line). In Fig. 11(b) wepresent the probability b P ( e ∈ B) that a random edge ina configuration model network with a power-law degreedistribution is a bredge (solid line), as a function of c ,obtained from Eq. (73). We also present the probabil-ity b P ( e ∈ B , GC) (dashed line) that a randomly selectededge is a bredge that resides on the giant component andthe probability b P ( e ∈ B , FC) (dotted line) that a ran-domly selected edge is a bredge that resides on one ofthe finite tree components. The analytical results are inexcellent agreement with the results of computer simula-tions (circles), performed for an ensemble of configurationmodel networks of N = 10 nodes.In Fig. 12 we present analytical results for themarginal degree distribution e P ( k | GC) of end-nodes ofrandomly selected edges (solid line), the marginal degreedistribution e P ( k | B , GC) of end-nodes of bredges (dottedline) and the marginal degree distribution e P ( k | NB , GC)of non-bredge edges (dashed line) on the giant compo-nent of a configuration model network with a power-law3
0 2 4 6 8 10 1200.10.20.30.4
FIG. 12: (Color online) Analytical results for the marginaldegree distribution e P ( k | GC) of end-nodes of randomly se-lected edges (solid line), the marginal degree distribution e P ( k | B , GC) of end-nodes of bredges (dotted line) and themarginal degree distribution e P ( k | NB , GC) of end-nodes ofnon-bredge edges (dashed line) on the giant component ofa configuration model network with a power-law degree dis-tribution with an exponent γ = 2 . c = 1 . FIG. 13: (Color online) Analytical results for the correlationcoefficient R (GC), between the degrees k and k ′ of end-nodesof edges (solid line) and the correlation coefficient R (B , GC)between the end-nodes of bredges (dotted line) that reside onthe giant component of a configuration model network with apower-law degree distribution, as a function of the mean de-gree c . The analytical results are in excellent agreement withthe results obtained from computer simulations (circles), ex-cept for the dilute network regime just above the percolationtransition. In this regime the giant component is small andits size fluctuates between different network instances. Thedata points in this regime were averaged over 100 networkinstances, while all the other data points were averaged over20 network instances. degree distribution. The analytical results are in excel-lent agreement with the results obtained from computersimulations (circles).In Fig. 13 we present analytical results for the correla-tion coefficient R (GC), between the degrees k and k ′ ofend-nodes of edges (solid line) and the correlation coeffi-cient R (B , GC) between the end-nodes of bredges (dottedline) that reside on the giant component of a configura-tion model network with a power-law degree distribution,as a function of the mean degree c . The analytical resultsare in excellent agreement with the results obtained fromcomputer simulations (circles) except for the dilute net-work regime where there are noticeable deviations due tofinite size effects.In the case of infinite networks, one may consider thelimit of k max → ∞ . In this limit the expression for thedegree distribution is simplified to P ( k ) = k − γ /ζ ( γ ). For γ > h K i = ζ ( γ − /ζ ( γ )and for γ > h K i = ζ ( γ − /ζ ( γ ). The generating functions are simplified to G ( x ) = Li γ ( x ) /ζ ( γ ) and xG ( x ) = Li γ − ( x ) /ζ ( γ − k max → ∞ the percolation threshold is c ≃ . γ ≃ . VIII. DISCUSSION
Transportation, communication and many other net-works consist of a single connected component, in whichthere is at least one path connecting any pair of nodes.This property is essential for the functionality of thesenetworks. The failure of a node or an edge disconnectsthe paths that go through the failed node/edge. In casethat the failed node is an AP or the failed edge is abredge, the disconnected paths have no substitute. As aresult, a whole patch of nodes becomes disconnected fromthe rest of the network. Networks that do not includeany APs and bredges are called biconnected networks[55, 56]. In such networks, any node i is connected toany other node j by at least two non-overlapping paths.While biconnected networks are resilient to the deletionof a single node or a single edge, they are still vulnera-ble to multiple node/edge deletions. This is due to thefact that the deletion of a node/edge may turn othernodes into APs and other edges into bredges. Their sub-sequent deletion would disconnect other nodes from therest of the network. The properties of APs and bredgesare utilized in optimized algorithms of network disman-tling [19–22]. The first stage of these dismantling pro-cesses is the decycling stage in which one node is deletedin each cycle, transforming the network into a tree net-work. In tree networks all the nodes of degrees k ≥ i and j , the shortestpaths are of particular importance because they are likelyto provide the fastest and strongest interactions. Thestatistical properties of the shortest paths are capturedby the distribution of shortest path lengths (DSPL). TheDSPL can be used to characterize the large scale struc-ture of the network, in analogy to the degree destributionwhich is used to characterize the local structure. Centralmeasures of the DSPL such as the mean distance [6, 57–59] and extremal measures such as the diameter [60] werestudied. However, apart from a few studies [26, 61–66]the DSPL has not attracted nearly as much attentionas the degree distribution. Recently, an analytical ap-proach was developed for calculating the DSPL in the(ER) network [67], followed by more general formulationsthat apply to configuration model networks [43, 68, 69],to modular networks [70] and to networks that form bykinetic growth processes [71–73].The importance of a given edge e in a network maybe quantified by its betweeness centrality, which is thenumber of pairs of nodes i and j , such that of shortestpaths between them pass through e [74, 75]. In general,the calculation of the betweeness centrality of an edgecannot be done locally. It involves the calculation of theshortest paths between all the pairs of nodes in the net-work, which requires access to the structure of the wholenetwork [76]. However, in case that an edge e is a bredge,one can easily obtain its betweeness centrality. Considera bredge e that resides on the giant component whose sizeis N GC . If the deletion of e detaches a tree componentof size N FC from the giant component, the betweenesscentrality of e is given by β e = N FC ( N GC − N FC ).The damage exerted on a network upon deletion of abredge can be evaluated using a centrality measure calledbridgeness [49]. The bridgeness of a bredge e that resideson the giant component is defined as the number of nodesdisconnected from the giant component upon deletion of e . The bridgeness of bredges that reside on the finite components is zero. Using a generating function formu-lation derived earlier to calculate the size distribution ofthe finite tree components [77, 78], Wu et al. obtainedthe bridgeness distribution in configuration model net-works with Poisson, exponential and power-law degreedistributions [49]. It was found that the mean bridgenessdiverges at c → c +0 and and monotonically decreases asthe mean degree is increased.Another useful measure of the importance of an edge e in a network is given by its range ρ , which is the dis-tance between its end-nodes i and i ′ in the reduced net-work from which e is removed [79, 80]. In the specialcase in which e is a bredge, its range is ρ = ∞ , becauseupon deletion of e its end-nodes land on different networkcomponents. For edges that are not bredges the range ρ ≥ i and i ′ in the reduced network. It also satis-fies ρ = ℓ − ℓ is the length of the shortest cyclethat includes the edge e in the original network. Edgeswhose range ρ is large are considered important becauseupon their removal the shortest alternate path between i and i ′ is large. In practical applications, large ρ implieslong and potentially costly delays in communication andtransportation in case that the edge e fails.In Fig. 14 we present ER networks of N = 100 nodeswith mean degrees c = 1 . c = 1 . N T = 11 − G ′ (1 − ˜ g ) , (143)which is the sum of a geometric series whose ratio G ′ (1 − ˜ g ) is the excess degree of the end-nodes of thefinite component side of the bredges, whose degree dis-tribution is given by Eq. (102). Thus, the fraction ofroot end-nodes among the end-nodes on the GC side ofbredges on the giant component is 1 /N T . The degreedistribution of the root end-nodes, which reside on the2-core of the giant component, is given by e P ( K = k | B , GC)= [ − (1 − ˜ g ) k − ] − ( k − g (1 − ˜ g ) k − ˜ g [1 − G ′ (1 − ˜ g )] e P ( k ) . (144)The degree distribution of the end-nodes on the GC sidesof all other bredges, which reside on the 1-core of the5 ( a )( b ) FIG. 14: (Color online) ER networks of N = 100 nodes withmean degree c = 1 . c = 1 . giant component is given by e P ( K GC ∩ = k | B , GC) = ( k − − ˜ g ) k − G ′ (1 − ˜ g ) e P ( k ) . (145)The overall distribution of the degrees K GC , given by Eq. (101), is recovered by e P ( K GC = k | B , GC) = 1 N T e P ( K = k | B , GC)+ (cid:18) − N T (cid:19) e P ( K GC ∩ = k | B , GC) . (146)The distinction between root bredges and all the otherbredges on the giant component may be useful for opti-mized dismantling algorithms and targeted attacks. Thisis due to the fact that the deletion of a root bredge discon-nects the whole tree branch that is held by this bredge.In contrast, random deletion of bredges may require alarge number of deletion steps in order to chop each treebranch from the 2-core of the giant component. IX. SUMMARY
We presented analytical results for the statistical prop-erties of edges and bredges in configuration model net-works. To quantify the abundance of bredges, we cal-culated the probability b P ( e ∈ B) that a random edge e in a configuration model network with a given degreedistribution P ( k ) is a bredge. We also obtained the con-ditional probability b P ( e ∈ B | k, k ′ ) that a random edgewhose end-nodes are of degrees k and k ′ is a bredge.Using Bayes’ theorem, we obtained the joint degree dis-tribution b P ( k, k ′ | B) of the end-nodes of randomly sam-pled bredges. We also studied the distinct properties ofbredges on the giant component and on the finite com-ponents. On the finite components all the edges arebredges, namely b P ( e ∈ B | GC) = 1, and there are nodegree-degree correlations. We calculated the probabil-ity b P ( e ∈ B | GC) that a random edge on the giant com-ponent is a bredge. We also obtained the joint degreedistribution b P ( k, k ′ | B , GC) of the end-nodes of bredgesand the joint degree distribution b P ( k, k ′ | NB , GC) of theend-nodes of non-bredge (NB) edges on the giant compo-nent. Surprisingly, it was found that the degrees k and k ′ of the end-nodes of bredges are correlated, while thedegrees of the end-nodes of non-bredge edges are uncorre-lated. This implies that all the degree-degree correlationson the giant component are concentrated on the bredges.We calculated the covariance Γ(B , GC) and found that itis negative, which means that bredges on the giant com-ponent tend to connect high degree nodes to low degreenodes and vice versa. We applied this analysis to ensem-bles of configuration model networks with degree distri-butions that follow a Poisson distribution (Erd˝os-R´enyinetworks), an exponential distribution and a power-lawdistribution (scale-free networks). The implications ofthese results were discussed in the context of commonattack scenarios and network dismantling processes.This work was supported by the Israel Science Foun-dation grant no. 1682/18.6 [1] S. Havlin and R. Cohen,
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