Percolation on bipartite scale-free networks
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J a n Percolation on bipartite scale-free networks
H. Hooyberghs a ∗ , B. Van Schaeybroeck a , b , J. O. Indekeu aa Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven,Celestijnenlaan 200D, B-3001 Heverlee, Belgium b Koninklijk Meteorologisch Instituut (KMI),Ringlaan 3, B-1180 Brussels, BelgiumRecent studies introduced biased (degree-dependent) edge percolation as a model for failures in real-lifesystems. In this work, such process is applied to networks consisting of two types of nodes with edgesrunning only between nodes of unlike type. Such bipartite graphs appear in many social networks, forinstance in affiliation networks and in sexual contact networks in which both types of nodes show thescale-free characteristic for the degree distribution. During the depreciation process, an edge betweennodes with degrees k and q is retained with probability proportional to ( kq ) − α , where α is positive so thatlinks between hubs are more prone to failure. The removal process is studied analytically by introducinga generating functions theory. We deduce exact self-consistent equations describing the system at amacroscopic level and discuss the percolation transition. Critical exponents are obtained by exploitingthe Fortuin-Kasteleyn construction which provides a link between our model and a limit of the Pottsmodel.
1. INTRODUCTION
Scale-free self-similar structures occupy a prominent place among Nihat Berker’s research tools. Hepioneered an interesting class of hierarchical sets, commonly called “Berker lattices” [1,2] on which afamily of renormalization group transformations is satisfied exactly. Further, the geometrical criticalphenomenon of percolation [3] and, recently, the cooperative behavior of interacting degrees of freedomon random scale-free networks [4] has enjoyed his thorough attention. In this paper, we discuss a subtlevariant of a percolation process on scale-free graphs of bipartite structure.In recent decades, a detailed analysis of many real-life structures revealed the existence of a commonproperty. Studies of social, technological and biological networks have indicated that the probabilitydistribution P ( k ) of the degree k of a node, i.e., the number of links attached to the node, followsa decreasing power law P ( k ) ∝ k − γ for large values of k . Notable examples of such so-called scale-freenetworks are the network of co-authors of scientific articles, protein interaction networks, power grids andtransportation networks. An accurate determination of the topological exponent (or “degree exponent”) γ in real-life networks is far from easy, but most studies indicate a value of γ in the range 2 < γ < . finite fraction of links isstill present in the network as a whole, the network is called fragile . However, if after the removal of anarbitrary fraction of links the giant cluster is still present, the network is called robust . ∗ E-mail: [email protected] H. Hooyberghs
Studies of percolation on scale-free networks have revealed that the behavior of the giant cluster dependson the topological exponent γ [6,7]. If γ >
3, the giant cluster breaks down at a finite fraction of links,thus the network is fragile. However, when γ <
3, the network is robust. Most real-life networks are thusrobust against random removal of links. Among the numerous applications, the percolation problem hasfor instance been studied as a model for failures in the Internet structure and in terrorist networks [6],or as a model for virus spreading on social networks [8].Not all networks can be described using a single type of vertex. For instance, the network of hetero-sexual contacts can best be modeled by two types of vertices, men and women, where links always runbetween nodes of unlike type. Such networks are called bipartite networks . They appear in many socialstructures, for instance in affiliation networks, i.e., networks of individuals joined by common membershipof communities. The two types of vertices then represent the individuals and the groups, while the linksbetween them indicate group membership [5]. One could for instance model the research interests ofphysicists: the first type of nodes then represents the researchers, while the second type of nodes consistsof the research topics (for instance classified according to the PACS). Most studies concerning bipartitenetworks focussed on the architecture and the building process of networks. This is for example thecase for sexual-contact networks [9], for listening habits and music genres [10] or for general affiliationnetworks [11,12]. Among these studies, it was revealed that in many bipartite networks, the degreedistribution of both types of nodes shows the scale-free characteristic, for instance in the hetero-sexualcontacts network [13]. Some studies also discuss the behavior of the network under random removal ofits links or nodes [14].In recent work, random link removal on monopartite graphs was extended to biased percolation , inwhich links are removed according to the degree of their neighboring nodes [15]. More specifically, toeach link is attached a weight w kq = ( kq ) − α (1)where k and q are the neighboring node degrees and α is the bias exponent which we take to be positive(and <
1) for our present purposes [34]. During the depreciation process, the probability to retain a linkis proportional to the weight of the link. Since α >
0, links between hubs are more prone to failure.Therefore, the percolation process is called centrally biased . Such a process is inherent to many socialnetworks, where friendships between people with many acquaintances are expected to be weaker andlast less long than friendships involving people with few connections [16]. In previous articles, Refs. [15]and [17], we showed that the critical behavior of such a biased percolation on a scale-free network can bemapped onto a random removal process on another scale-free network. More precisely, biased percolationwith bias exponent α > γ belongs to the same universalityclass as random percolation ( α = 0) on a network with topological exponent γ , given by γ = γ − α − α . (2)Therefore, the network is, in the macroscopic limit, robust against biased removal of links as long as2 < γ <
3. If γ >
3, the network is fragile for the central bias process. Note that γ > γ .The paper at hand discusses biased percolation on bipartite scale-free networks. In Sect. 2 we introducebipartite scale-free networks and present a basic description of a link removal process on those networks.The third section introduces the theory of generating functions, which we extend so as to describe bipartitenetworks. The percolation threshold and the scaling of the critical point are extracted from the theory.In Sect. 4, the equivalence between percolation and a limit of the Potts model is elaborated. Usingthis equivalence, we construct a finite-size scaling theory which enables the calculation of the criticalexponents for the percolation transition. In Sect. 5 we present our conclusions.
2. THE MODEL
We start from a bipartite network with N nodes, divided in N A nodes of type A and N B nodes of type B , each type with its own degree distribution P A ( k ) and P B ( k ). Both degree distributions are assumedto follow a decreasing power law and thus are scale free. More precisely, P i ( k ) = C i k − γ i (3) ercolation on bipartite scale-free networks k between the minimal and maximal degrees, m i and K i , respectively, for i = A, B . InEq. (3), C i is a normalization constant and the exponent γ i is assumed to be larger than two to ensurea finite mean degree. Links exist only between nodes of unlike type. Moreover, we assume that nodegree correlations or mixing patterns between the nodes, as for instance assortative mixing, occur. Theprobability P An ( k ) that a randomly chosen edge emerging from a node of type A leads to a type- B nodewith degree k thus only depends on the degree distribution P B ( k ). More precisely, P An ( k ) = kP B ( k ) h k i B , (4)where h·i i denotes the average over the nodes of type i , obtained by using the degree distribution P i ( k ).Note that the total number of links attached to the nodes of type A equals the total number of linksattached to type- B nodes, i.e. h k i A N A = h k i B N B .In our percolation process, a fraction f of the links is removed in a single sweep; we call this the simultaneous process. An edge between nodes with degrees k and q is retained with probability ρ kq = f w kq h w i e , (5)and is removed with probability 1 − ρ kq . The weight w kq is defined in Eq. (1) and h w i e denotes theaverage weight of an edge, h w i e = h k − α i A h k − α i B h k i A h k i B . (6)Note that the random link removal process is recovered if α = 0. The positive bias exponent α shouldbe smaller than one in order for the depreciated network to be scale free. Moreover, the depreciationprocess is only well-defined if ρ kq < k and q . Therefore, our percolation canonly be used correctly for values of f for which [35] f < f u = h w i e ( m A m B ) α . (7)We now introduce the marginal distribution ρ ik ( f ) as the mean probability that an edge connected toa node of type i with degree k is present when a fraction f of links is reincluded in the network. Onefinds ρ Ak ( f ) = f h k i A h k − α i A k − α . (8)Note that the marginal distribution ρ Ak does not depend on the distribution of the nodes of type B andvice versa. Using the marginal distribution, the degree distribution P i ( k ) and nearest-neighbor degreedistribution P in ( k ) of nodes of type i in the diluted network can be deduced. One arrives at P i ( k ) = K i X k = k P i ( k ) (cid:18) kk (cid:19) ( ρ ik ) k (1 − ρ ik ) k − k , (9a) P in ( k ) = kP j ( k ) f h k i j . (9b)The importance of these expressions will become clear when introducing the generating functions in thenext section. Moreover, it can be shown that f ρ kq = ρ Ak ρ Bq , (10)and thus, according to the arguments in Ref. [17], the diluted network is still uncorrelated. H. Hooyberghs
3. GENERATING FUNCTIONS
Percolation is often studied using the generating functions approach. By this method, self-consistentequations for the size of the giant cluster can be obtained easily. Moreover, the method is exact if thediluted network is uncorrelated and if loops in the finite clusters can be ignored, which is justified forscale-free networks [18]. We first briefly introduce the general scheme, closely following the approach ofNewman [19]. Then, the generating functions method is exploited for the case of biased percolation onbipartite networks.
Generating functions are used in a plethora of problems concerning series. The generating function ofa sequence is the power series which has as coefficients the elements of the sequence [20]. In percolationproblems, generating functions that generate the probability distributions characterizing the network arewidely used [19,21]. The most important functions are those that generate the degree distribution andthose that generate the nearest-neighbor distribution. To study the percolation problem, both generatingfunctions will be defined in the diluted network, for both vertex types.The generating function for the degree distribution P i ( k ) is defined as F i ( h ) = K i X k =1 P i ( k ) e − hk , (11)while the distribution of the residual edges in the diluted network is generated by F i ( h ) = K i X k =1 P in ( k ) e − h ( k − , (12)where i = A, B . Substituting Eqs. (9) and (4), we obtain F i ( h ) = K i X k = m i P i ( k )(1 − ρ ik + e − h ρ ik ) k , (13a) F i ( h ) = K i X k = m i P in ( k )(1 − ρ ik + e − h ρ ik ) k − , (13b)with i = A, B . For our interest, the most relevant generating functions for the percolation problem arethe ones associated with the probability distribution of the size of the finite clusters, since these quantitiescan be related to the size of the giant cluster. Let H i generate the probability that a randomly chosennode of type i belongs to a cluster of a given finite size. Furthermore, we introduce H i as the generatingfunction for the probability that upon following a randomly chosen edge emerging from a node i towardsthe endnode of type j , a cluster of given (finite) size is reached. If the finite clusters can be treated astrees and the diluted network is uncorrelated, these generating functions satisfy coupled self-consistencyequations, analogous to those derived for monopartite graphs in Ref. [19]. For bipartite graphs, we obtain H A ( h ) = e − h F B [ H B ( h )] , (14a) H B ( h ) = e − h F A [ H A ( h )] , (14b) H A ( h ) = e − h F A [ H A ( h )] , (14c) H B ( h ) = e − h F B [ H B ( h )] . (14d)Here the function F i [ H i ( h )] denotes the function F i wherein e − h is replaced by H i ( h ) with i = A, B . Thepercolation threshold can now be derived with the aid of these generating functions and self-consistentrelations.
The percolation threshold is most easily studied by introducing the average cluster size in the dilutednetwork, i.e., the average fraction of nodes of type i in a cluster, denoted by S i . Using the properties of ercolation on bipartite scale-free networks S i can be related to H i [20]: S i = − ˙ H i (0) , (15)where the dot represents differentiation with respect to h . An expression for the average cluster size S ,i.e., the average fraction of nodes of type A and B in a finite cluster, is then easily found: S = − X i = A,B ˙ H i (0) . (16)The average cluster size S in the diluted network can be further worked out by differentiating Eqs. (13a)and (13b) with respect to h : S = 1 + f h k i A (1 − ˙ F B (0)) + h k i B (1 − ˙ F A (0))1 − ˙ F A (0) ˙ F B (0) . (17)Hence the average cluster size diverges when1 = ˙ F A (0) ˙ F B (0) . (18)The percolation criterion (18) is the extension for bipartite graphs of the Molloy-Reed criterion, whichprovides a condition for the existence of a giant cluster in a network [6,22]. A similar expression forthe percolation threshold was already found in studies concerning percolation on general multipartitenetworks [14].Using Eqs. (8) and (13b) an explicit criterion for the critical fraction f c can be found, f c = q f Ac f Bc , (19)where f ic denotes the critical fraction of a monopartite graph consisting only of nodes of type i withdegree distribution P i ( k ), f ic = h k − α i i h k i i ( h k − α i i − h k − α i i ) . (20)Note that Eq. (19) reduces to the criterion for monopartite graphs if γ A = γ B , as it should be. Moreover,the critical fraction of a bipartite graph can easily be found if the critical fraction of the monopartitegraphs consisting of only nodes of type A and B is known. As a consequence, also the resilience againstbiased failures of the bipartite network in the macroscopic limit is known completely if the behavior ofthe different subgraphs is known. If one of the two subgraphs is robust against biased percolation, alsothe bipartite network will be robust. The robustness criterion is therefore:min ( γ A , γ B ) < , (21)where γ i is defined as in Eq. (2). This is illustrated in Fig. 1, where the robust regimes are indicated bythe blank (undotted) regions. Only if both γ A and γ B are larger than 3, the network is fragile, indicatedby the regimes dotted in red on Fig. 1.For the robust regimes, we can quite easily determine the scaling relation of f c as a function of thenetwork size, f c ∝ N ς , by explicitly evaluating the expectation values in Eq. (20). Replacing the sumsover the degree distibution by integrals, it can be seen that the moments h k i i and h k − α i i never divergeif central bias is applied to a network with γ i >
2. The moments h k − α i i and h k − α i i , on the otherhand, may diverge, but the former will always grow faster than the latter. The behavior of the criticalfraction as a function of the maximal degrees, K i for i = A, B , thus stems from the first term, which for2 < γ i < K − α − γ i i . Since K i ∝ N / ( γ i − i [6], the scaling of the critical point as a functionof the network size can be determined. If both γ A and γ B are smaller than 3 (regimes IV and IV’ onFig. 1), ς is given by ς = 12 (cid:18) − γ A − γ A + 3 − γ B − γ B (cid:19) . (22) H. Hooyberghs
Figure 1. Overview of the different universality regimes as a function of the scale free exponents γ A and γ B . The red line indicates the division between a fragile network (red dotted regimes) and a robustnetwork (blank regimes). g g III
II’
III III’IV IV’ AB Note that this reduces to the result for monopartite graphs, determined in Ref. [15], if γ A = γ B . If2 < γ A < < γ B (regime III on Fig. 1), the exponent ς is the same as the exponent for a robustmonopartite graph of type A: ς = 3 − γ A − γ A . (23)Note that the exponent in the regime III’ (2 < γ B < < γ A ) can be found easily by interchanging γ A and γ B in Eq. (23).In sum, we have now provided an extension of the Molloy-Reed criterion for the critical threshold ofbiased percolation on bipartite scale-free networks. Moreover, the scaling of the critical fraction as afunction of the network size was determined for robust graphs. Our results show that the critical fractionof a bipartite network is governed by the critical percolation behavior of the scale-free graphs consistingof only one type of nodes.
4. CRITICAL BEHAVIOR
In the following section, we introduce and calculate the critical exponents of the percolation transition.The first part briefly introduces the Fortuin-Kasteleyn construction, which provides a link between per-colation and a limit of the Potts model. Exploiting this link with the Potts model, we can define criticalexponents for the percolation problem. In the second part of the section, a finite-size scaling theory isconstructed, in order to calculate the critical exponents in the third part. ercolation on bipartite scale-free networks There exists an exact equivalence between random edge percolation and the q → q -statePotts model, originally worked out by Fortuin and Kasteleyn in Ref. [23]. Moreover, their proof can easilybe generalized to incorporate edge-dependent coupling constants in the Potts model and edge-dependentremoval in the percolation model, respectively. The Fortuin-Kasteleyn construction states that the freeenergy of the q → q -state Potts model is the same as the “free energy” of the percolationproblem [24]. The latter is defined as the generating function of the cluster-size distribution n s , F ( f, h ) = *X s n s e − hs + . (24)Here the average is performed over all networks in which the probability to retain the edge between nodeswith degrees k and q is ρ kq . We can immediately identify the probability P ∞ for a node of any type tobe in the infinite cluster and the average cluster size S , as a function of the fraction of removed links: P ∞ ( f ) = 1 + ∂ F ∂h (cid:12)(cid:12)(cid:12)(cid:12) h =0 , (25a) S ( f ) = ∂ F ∂h (cid:12)(cid:12)(cid:12)(cid:12) h =0 . (25b)Since we are interested in the behavior close to criticality, we introduce the parameter ǫ = f − f c . (26)The usual critical exponents α , γ p , β and δ can now be defined for the percolation problem using theanalogy with the Potts model: F ( ǫ, ∼ ǫ − α , (27a) P ∞ ( ǫ ) ∼ ǫ − γ p , (27b) S ( ǫ ) ∼ ǫ β , (27c) ∂ F ∂h (cid:12)(cid:12)(cid:12)(cid:12) ǫ =0 ∼ h /δ − . (27d)Relations among these exponents can be found using a scaling theory. In the following section, we introduce finite-size scaling in order to find critical exponents near thepercolation transition. In order to solve the scaling relation, we use a Landau-like theory which we derivefrom the exact self-consistent relations, Eqs. (14). We closely follow the approach presented in Ref. [17],which is based on Refs. [25,26,27].Our scaling theory consists of two basic scaling relations. The free energy F of a large but finitenetwork with N nodes close to criticality can be written in the general form [28]: F ( ǫ, h ) = N − F (cid:16) ǫN /ν ǫ , hN /ν h (cid:17) , (28)where F is a well-behaved function. Close to the critical point, the free energy then scales as F ( ǫ, ∝ ǫ ν ǫ , (29a) F (0 , h ) ∝ h ν h . (29b)As a second ansatz, the scaling of the cluster size distribution n s can in the thermodynamic limit bewritten as n s ( ǫ ) = s − τ G ( ǫs σ ) . (30) H. Hooyberghs
Using the scaling forms of Eqs. (28) and (30), standard techniques provide us with exponent relations bywhich all critical exponents can be related to ν h and ν ǫ . One arrives at [28]: β = ν ǫ (1 − ν − h ) , (31a) γ p = ν ǫ (2 ν − h − , (31b) α = 2 − β − γ p , (31c) σ = ( β + γ p ) − , (31d) τ = 2 + β ( β + γ p ) − , (31e) δ = ( β + γ p ) /β. (31f)The problem we are left with is to find the exponents ν h and ν ǫ for percolation on bipartite scale-freenetworks. Using the self-consistent equations, the exponents ν h and ν ǫ can be determined in an exact way. Weintroduce the parameters ψ i ( ǫ, h ) = 1 − H i ( ǫ, h ) , (32)where i = A, B . As we are merely interested in the behavior near the transition where ǫ ≪ h ≪ ψ i ≪
1, we can expand Eqs. (14a) and (14b).
We first discuss all regimes in which both γ A and γ B are larger than three, i.e., the regimes in whichthe network is fragile and has a finite f c . An expansion of the self-consistent equations, Eqs. (14), yields h − ψ B = − c A ( f c + ǫ ) ψ A + c A ψ A + . . . + c As ( ψ A ) γ A − + . . . , (33a) h − ψ A = − c B ( f c + ǫ ) ψ B + c B ψ B + . . . + c Bs ( ψ B ) γ B − + . . . , (33b)wherein all c -constants are positive and according to the percolation criterion f c c A c B = 1. The existenceof the correction terms, c is ( ψ iA ) γ i − , can be checked numerically. Eqs. (33) are derivable by minimizationof the free energy F f = ψ A ψ B + X i = a,b − hψ i − c i ( f c + ǫ ) ψ i c i ψ i c is ( f c + ǫ ) γ i − ψ γ i − i γ i − ! + . . . , (34)with respect to the parameters ψ i . Using the equations of state, Eqs. (33), we may write at the saddlepoint F f c = X i = a,b (cid:18) − hψ i − c i ψ i c is (3 − γ i )2( γ i −
1) ( f c + ǫ ) γ i − ψ γ i − i (cid:19) . (35)If γ A and γ B are both larger than 4, i.e., in regime I in Fig. 1, the equations of state reduce in lowestorder to h − ψ B = − c A ( f c + ǫ ) ψ A + c A ψ A , (36a) h − ψ A = − c B ( f c + ǫ ) ψ B + c B ψ B . (36b)Solving for ψ A and ψ B , we obtain ψ A | h =0 ∼ ψ B | h =0 ∼ ǫ, (37a) ψ A | ǫ =0 ∼ ψ B | ǫ =0 ∼ h / . (37b)Substitution into Eq. (35) yields F ( ǫ, ∼ ǫ , (38a) F (0 , h ) ∼ h / , (38b) ercolation on bipartite scale-free networks ν ǫ = 3 and ν h = 3 /
2. All other exponents can now be calculated usingEqs. (31). We list the result in the first column of Table 1.Next, we focus on the regime in which one of the topological exponents is larger than four, while theother exponent has a value between three and four. Without loss of generality, we take in the remainderof the text γ A as the smallest of the two exponents. The self-consistent equations now reduce to h − ψ B = − c A ( f c + ǫ ) ψ A + c As ψ γ A − A , (39a) h − ψ A = − c B ( f c + ǫ ) ψ B + c B ψ B , (39b)from which we obtain ν ǫ = 1 γ A − , (40a) ν h = 1 γ A − . (40b)All other critical exponents are given in the second column of Table 1.Finally, we discuss the exponents in the case 3 < γ A < γ B <
4. Although the self-consistent equationsare slightly different from those in the previous paragraph, we can verify that Eqs. (40) are still valid.The exponents in this regime are thus the same as in the regime discussed in the previous paragraph.Therefore, we define region II as the region in which 3 < γ A < γ A < γ B , as is illustrated in Fig. 1.The exponents in that regime can be found in the second column of Table 1. We still have to discuss the exponents for robust bipartite networks. We first focus on regime IV inFig. 1 where both γ A and γ B are smaller than three. Since f c = 0, the self-consistent equations close tothe critical point reduce to h − ψ B = − c As ( ǫψ A ) γ A − , (41a) h − ψ A = − c Bs ( ǫψ B ) γ B − , (41b)where c As > c Bs >
0. These equations can be derived by minimization of F ǫ = ψ A ψ B + X i = A,B − hψ i − c is ǫ γ i − ψ γ i − i γ i − , (42)with respect to the order parameters. At the saddle point, we may write: F ǫ = X i = a,b − hψ i − c is (3 − γ i )2( γ i − ǫ γ i − ψ γ i − i . (43)By solving Eqs. (41) for ψ i , one obtains ψ A | h =0 ∼ ǫ ( γA − γB − − ( γA − γB − , (44a) ψ B | h =0 ∼ ǫ ( γB − γA − − ( γA − γB − , (44b) ψ A | ǫ → ∼ | h | γA − /ǫ, (44c) ψ B | ǫ → ∼ | h | γB − /ǫ. (44d)Substitution into Eq. (43) yields ν ǫ = ( γ A − γ B − − ( γ A − γ B − , (45a) ν h = γ B − γ B − . (45b)The other exponents are listed in the last column of Table 1.0 H. Hooyberghs
Table 1Critical exponents in the different regimes for which γ A < γ B . Note that the exponents for the primedregimes in Fig. 1 can be obtained by interchanging γ A and γ B in the expressions for the regimes withoutprimes. Regime I Regime II Regime III Regime IV ς Fragile Fragile − γ A − γ A (cid:16) − γ A − γ A + − γ B − γ B (cid:17) β γ A − γ A − − γ A γ A − − ( γ A − γ B − τ / γ A − γ A − γ B − γ B − σ / γ A − γ A − − γ A γ A − − ( γ A − γ B − γ A − γ B − α − − − γ A γ A − − γ A − − γ A − γ A γ B − γ A + γ B )1 − ( γ A − γ B − γ p − (3 − γ B )( γ A − − ( γ A − γ B − δ γ A − γ B − γ A < γ B >
3, i.e. regime III on Fig. 1. Although theexpansion of the self-consistent relations, Eqs. (14), depends on the actual value of γ B , the scaling of theorder parameters does not. After some calculations, we obtain ν ǫ = 2( γ A − − γ A , (46a) ν h = 2 . (46b)We list the other exponents in the third column of Table 1. To summarize, we have calculated the critical exponents of the percolation transition for all physicallyrelevant regimes. Note that all exponents in the regimes I, II and IV reduce to the correct expressionsfor monopartite graphs if γ A = γ B [17]. Note also that, as expected, the usual mean-field results forpercolation are recovered only in regime I [21,28]. In all other regimes, we find non-universal exponentswhich depend on the constants γ i .The only dependence on the bias exponent α arises through the exponents γ i as defined in Eq. (2).Biased percolation with bias exponent α on a bipartite scale-free network with topological exponents γ A and γ B thus has the same critical behavior as random percolation ( α = 0) on a network with topologicalexponents γ A and γ B . We thus conclude that the results for bipartite graphs are a generalization of thosefor monopartite graphs.
5. CONCLUSION
As an extension of previous work on scale-free graphs with a single type of nodes [15,17], this articlestudies the biased removal of links in a bipartite network. A bipartite network consists of two types ofnodes, with links only running between nodes of unlike type. We assume that the degrees of both nodesare distributed according to a decreasing power law, without any correlations or mixing patterns betweenthe two types of nodes. This model can be used to describe real-life social networks, such as for instancethe network of hetero-sexual contacts [13]. In our percolation process, we reinstall a fraction f of thelinks in a single sweep. Moreover, the process is biased in the sense that the link removal probabilitydepends on the degrees of the nodes they connect. We attach a weight ( kq ) − α to a link between nodeswith degrees k and q and retain links with a probability proportional to their weights. In the study athand, the bias exponent α is a positive number smaller than one. Therefore, links between hubs are moreprone to failure, thus the process is centrally biased. ercolation on bipartite scale-free networks q → q -state Potts model. The scaling theory was solved by expandingthe self-consistent equations close to the percolation threshold. Results for the exponents are given inTable 1. All exponents only depend on the constant α through the constants γ A and γ B , defined as γ i = ( γ i − α ) / (1 − α ). The critical behavior of biased percolation with bias exponent α on a bipartitenet with topological exponents γ i with i = A, B thus is the same as the critical behavior of randomremoval on a network with exponents γ i . Therefore, we extended the main result of the study concerningmonopartite graphs to bipartite graphs.The generating functions theory we introduced can in principle be extended to include general multi-partite networks with an arbitrary number of types of nodes. In such networks, nodes can share edgeswith different types of nodes. For each type of nodes, a new generating function must be introduced,thereby increasing the complexity of the mathematics greatly. Since every new generating function re-quires an additional self-consistent equation, it becomes impossible to extract specific results, except incertain limiting cases. Much progress on this scheme has already been worked out in Ref. [14], but acalculation of critical exponents still remains an open issue. Further, in Ref. [36] the authors performeda study of the extremum events of scale-free networks, thereby focussing on the statistics of the extremeconnectivities. In this context, it would also be interesting to investigate the evolution of such distribu-tion functions during the process of our network reconstruction and more especially near the point ofpercolation. Acknowledgements
H.H. is Research Assistant and B.V.S. is Post-Doctoral Researcher of the Fund for Scientific Research- Flanders (FWO-Vlaanderen). J.O.I. is immensely grateful to Nihat for his meticulous and spiritedguidance throughout the “polar liquid crystal period” (1984-1989), and, ever since he first met Nihatin 1979, for his scale-free hospitality, his appetite for re-entrant farce and his effervescently percolatingfriendship.
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