Percolation of networks with directed dependency links

Transcription

1 Percolation of networs with directed dependency lins 9] to study the percolation phase transitions in a random networ A with both connectivity and directed dependency lins. Randomly removing a fraction p of nodes in networ A causes (i) connectivity lins to be disconnected, causing some nodes and clusters to fail due to the disconnection to the networ giant component (percolation process), and (ii) failing nodes to mae their dependent nodes to also fail even though they are still connected to the networ giant component via connectivity lins (dependency process). Thus, the removal of nodes in the percolation process leads to the failure of dearxiv:63.375v [physics.soc-ph] Mar 6 Dunbiao Niu,, Xin Yuan, 3 Minhui Du, H. Eugene Stanley, 3 and Yanqing Hu, 4 School of Data and Computer Science, Sun Yat-sen University, Guangzhou 56, China School of Mathematics, Southwest Jiaotong University, Chengdu 63, China 3 Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 5 USA 4 Big Data Research Center, University of Electronic Science and Technology of China, Chengdu 673, China (Dated: September 9, 8) The self-consistent probabilistic approach has proven itself powerful in studying the percolation behavior of interdependent or multiplex networs without tracing the percolation process through each cascading step. In order to understand how directed dependency lins impact criticality, we employ this approach to study the percolation properties of networs with both undirected connectivity lins and directed dependency lins. We find that when a random networ with a given degree distribution undergoes a second-order phase transition, the critical point and the unstable regime surrounding the second-order phase transition regime are determined by the proportion of nodes that do not depend on any other nodes. Moreover, we also find that the triple point and the boundary between first- and second-order transitions are determined by the proportion of nodes that depend on no more than one node. This implies that it is maybe general for multiplex networ systems, some important properties of phase transitions can be determined only by a few parameters. We illustrate our findings using Erdős-Rényi (ER) networs. PACS numbers: Hc, Fb, 64.6.ah I. INTRODUCTION Complex networs science has become an effective tool for modeling complex systems. It treats system entities as nodes and the mutually supporting or cooperating relations between the entities as connectivity lins [ ]. In many systems, nodes that survive and fail together form dependency groups through dependency lins. Dependency lins denote the damaging or destructive relations among entities [3 ]. Compared to ordinary networs [5, 6, ], networs with dependency groups or lins are more vulnerable and subject to catastrophic collapse [, 3]. The previous wors have studied the networ system in which the dependency groups, with sizes either fixed at two [] or characterized by different classic distributions [3], are formed through undirected dependency lins. The outcome when the dependency lins are directed, however, is more general. For example, in a financial networ where each company has trading and sales connections (connectivity lins) with other companies, the connections enable the companies to interact with others and function together as a global financial maret, and companies that belong to the same corporate group strongly depend on the parent company (i.e. there are directed dependency lins), but the reverse is not true [4]. Another example is in a social networ in which people (followers) follow trends set by celebrities (pioneers), e.g., popular singers and actors but the reverse is not true [5]. We use a self-consistent probabilistic framewor [6 Corresponding author. yanqing.hu.sc@gmail.com (a) (b) (c) Delete (d) (e) (f) FIG. : (Color online) Demonstration of the synergy between the percolation process and the dependency process that leads to a cascade of failures. The networ contains two types of lins: connectivity lins (solid blac lines) and directed dependency lins (dashed red arrows). (a) (b) Initial failure: a random node is removed. (c) (e) Synergy between percolation process and dependency process: nodes cut-off from the giant component or depending on failed nodes are removed. (f) Steady state: the surviving giant component contains four nodes.

2 pendent nodes in the dependency process, which in turn initiates a new percolation process, which further sets off a dependency process, and so on. We show that this synergy between the percolation process and the dependency process leads to a cascade of failures that continues until no further nodes fail (See Fig. ). To fully capture the structure of networ A, we introduce the degree distribution P () and, in addition, the directed dependency degree distribution Q( o ), which is the probability that a randomly chosen node has o directed dependency lins connecting to o nodes which are supporting this chosen node. In our model, when i depends on o nodes, we assume that if any one of these o nodes fails, node i will fail too (see Fig. ). Usually, this ind of multiplex has both first- and second-order phase transitions [, 3]. Here we find that Q( o ) strongly affects the robustness of networ A. Specifically, the percolation threshold c, at which networ A disintegrates in a form of second-order phase transition, is determined solely by Q() for a given Q( o ), and Q()+Q() characterizes the boundary between the first-order phase transition and the second-order phase transition regime. This paper is organized as the follows. In Sec.II we introduce the general framewor and develop the analytic formulae to solve the influence of Q( o ) on the percolation properties of a random networ. In Sec.III, we demonstrate these influences using an ER networ. II. GENERAL FRAMEWORK For a random networ A of size N with both connectivity lins and directed dependency lins (see Fig. (a)), as in Ref. [5], we introduce the generating function G (z) of the degree distribution P (), G (z) = P ()z. () Analogously, we have the generating function of the related branching processes [5], G (z) = G (z) G () = P () z. () Similarly, we introduce the generating function for the directed dependency degree distribution Q( o ) as D(z) = o Q( o )z o. (3) We designate h(s) the probability distribution of the number of nodes approachable along the directed dependency lins starting from a randomly chosen node in networ A. This allows us to write the generating function H(z) for h(s), i.e., H(z) = s h(s)z s. (4) According to Ref.[3], H(z) also satisfies a self-consistent condition of the form H(z) = z D(H(z)). (5) A random removal of a fraction p of nodes triggers a cascade of failures. When no more nodes fail, networ A reaches its final steady state. At this steady state, we use the probabilistic approach [9] and define x to be the probability that a randomly chosen connectivity lin leads to the giant component at one of its ends. If we randomly choose a connectivity lin l and find an arbitrary node n by following l in an arbitrary direction, the probability that node n has degree is P () P () =. (6) P () For node n, the root of a directed cluster of size s, to be part of the giant component, at least one of its other out-going connectivity lins (other than the lin first chosen) leads to the giant component, provided that every other s node is also in the giant component because the disconnection of any one of these s nodes to the giant component will cause node n to lose support and fail. Computing this probability, we can write out the self-consistent equation for x as x = p{ P () [ ( x) ]} {h(s){p P ()[ ( x) ]} s }, (7) s where p is the probability that a node survives the initial removal process, ( x) is the probability that at least one of the other connectivity lins of node n leads to the giant component, h(s) is the probability that node n is the root of a directed cluster of size s, and {p P ()[ ( x) ]} s is the probability that every other s node in the directed cluster supporting node n is also in the giant component. Using the generating functions defined in Eqs. (), () and (4), we transform Eq. (7) into the compact form x = G ( x) G ( x) H(p[ G ( x)]). (8) which, by viewing p[ G ( x)] as a whole and using the property of H(z) outlined in Eq. (5), can also be written as x = p[ G ( x)] D[H(p( G ( x))] F (x, p). (9) For a given p, x can be numerically calculated through iteration with a proper initial value. Correspondingly, using similar arguments, the probability P (p) that a randomly chosen node n in the steady

3 3 state of networ A is in the giant component is P (p) = p{ P ()[ ( x) ]} h(s) {p P ()[ ( x) ]} s s = H(p[ G ( x)]), () where ( x) is the probability that at least one of the connectivity lins of node n leads to the giant component. Note that P (p) is also the normalized size of the giant component of networ A at the steady state. We find that there is no giant component at the steady state of networ A, i.e., P (p) = when p is smaller than a critical probability c and above the threshold, the giant component appears and its size increases continuously from as p increases. This is typical second-order phase transition behavior and as p c, P ( c ) = H( c [ G ( x)]), which suggests x. Thus we can tae the Taylor expansion of Eq. (9) with x to obtain c as (see Appendix A), c = Q()G () = Q() ( ), () which is consistent with our previous result reported in Ref.[5] and depends on Q() only but not any other terms from Q( o ). In some cases, however, there is no giant component at the steady state of networ A, i.e., P (p) = when p is smaller than a critical probability p I c but above the threshold, the giant component suddenly appears and its size increases abruptly from as p increases. This is typical first-order phase transition behavior. When p = p I c, the straight line y = x and the curve y = F (x, p) from Eq. (9) will tangentially touch each other at (x c, x c ) [5]. Thus, the condition corresponding to the first-order transition is that the derivatives of both sides of Eq. (9) with respect to x are equal, = df (x, p) x = x c, p = p I dx c. () Due to the complexity of Eqs. (9) and (), numeric methods are generally used to get p I c. Note p I c = c corresponds to the case where the phase transition changes from first-order to second-order when the conditions for both the first- and second-order transitions are satisfied simultaneously. By substituting c from Eq. () into Eq. () and further evaluating x, we obtain the boundary between the first-order and secondorder phase transitions, which is characterized by (see Appendix B), Q() = Q()G () Q() ( )( ) G () =. (3) Thus, the boundary between first- and second-order transitions is determined only by the proportion of nodes that do not depend on more than one node, i.e., the boundary is solely determined by Q() and Q() but not any other terms from Q( o ). This implies that the triple point the intersection of first order phase transition, second order phase transition and the unstable regime is also determined by Q() and Q(). When removing any fraction of nodes results in the total collapse of networ A, i.e., when c, the networ is unstable. By requiring c = and using Eq. (), we can obtain the boundary between the second-order phase transition and the unstable state, Q() = G () = ( ), (4) which depends solely on the proportion of nodes that do not depend on other nodes at all, i.e., Q(). Similarly, by requiring p I c = in Eq. (), we use numerical calculations to find the boundary between the first-order phase transition and the unstable state. Therefore, the complete boundary between the unstable state and the phase transition state is achieved by joining these two boundaries together. Moreover, substituting Eq. (3) into Eq. (4), we could obtain the explicit formula of the triple point which is the intersection of these two boundaries: ( )( ) Q() = (5) ( ) Note that for scale-free networs with power law degree distribution P () γ and γ (, 3], both ( ) and ( )( ) are divergent. This implies that c = for any Q() according to Eq. () and the regime of the second order phase transition shrins towards the origin. Thus for scale free networs, the situation becomes a little bit simple. Therefore, if Q() > one could always see the second-order phase transition with c = and if Q() = the system undergoes unstable or first-order phase transition. III. RESULTS ON ER NETWORKS Section II provided the general framewor for random networs with an arbitrary degree distribution P (). We here illustrate it using an ER networ [3 33] with a Poisson degree distribution P () = e /! where is the average degree. We choose this networ because it is representative of random networs, and the generating function corresponding to the degree distribution P () is G (z) = e (z ). A. Second-order phase transitions Plugging G () = into Eq. (), we get the secondorder phase transition point c, c = Q(). (6)

4 4.8.6 Q()=.4;Q()=;Q()=.6 Q()=.4;Q()=.;Q()=.4 Q()=.4;Q()=.4;Q()=. Q()=.4;Q()=.6;Q()= theory.8.6 Q()=.4,Q()=.6,<>=5 Q()=.,Q()=.8,<>=5 Q()=.,Q()=.7,Q()=.,<>= Q()=.,Q()=.8,Q()=.,<>= theory P P.4.4. c p p FIG. : (Color online) The size of the giant component P (p), as a function of the fraction of nodes that remain after random removal, p, for ER networs with = 8 and D(z) = Q() + Q()z + Q()z. The symbols represent simulation results of 4 nodes and the dashed lines show the theoretical predictions from Eq. (). The percolation threshold is uniquely determined by Q(). c Therefore, for ER networs, the critical point of secondorder phase transition is indeed determined solely by Q() and its average degree. We support our analytical results by simulations. We choose = 8 and D(z) = Q() + Q()z + Q()z with Q() fixed at.4 and Q(), Q() tunable. Fig. shows the size of the giant component P (p) as a function of p with the given and D(z). Note that in all cases simulation results (symbols) agree well with numerical results (dotted lines) and the curves of P (p) converge at a fixed value of c =.35 as predicted by Eq. (6). This convergence of P (p) curves is possible because c is determined solely by Q(), which is fixed to be.4 in Fig.. Note that if there is no directed dependency lins in the networ, i.e., Q() =, we will get c = /, which is consistent with the well-nown result obtained in Ref. [3]. FIG. 3: (Color Online) The size of the giant component P as a function of the fraction of nodes that remain after random removal, p, for ER networs. Here we used D(z) = Q() + Q()z with = 5 ( and ) and D(z) = Q() + Q()z + Q()z with = ( and ). The symbols represent simulation results of 4 nodes and the dashed lines are the theoretical predictions from Eq. (). With a relatively larger Q(), the networ undergoes a second-order phase transition at c, which only depends on Q(). However for relatively smaller Q() and larger Q() and Q(), the networ undergoes a first-order phase transition. comparing simulation results and theoretical predictions. Note that they agree with each other very well. Fig. 3 shows that with = 5 and Q() + Q() =, when Q() =.4, P (p) undergoes a second-order phase transition at c =.5 ( ), but when Q() =., P (p) exhibits behavior of a first-order phase transition at p I c, satisfying Eq. (7) ( ). In addition, when =, Q() =., Q() =.7 and Q() =., P (p) undergoes a second-order phase transition at c =.5 ( ), but when Q() =., Q() =.8 and Q() =., P (p) undergoes a first-order phase transition at p I c predicted by Eq. (7) ( ). B. First-order phase transitions When networs have a greater proportion of directed dependency lins, an abrupt transition can occur instead of a continuous transition demonstrated in Fig.. To get the p I c for the onset of this abrupt transition, we equate the derivatives of both sides of Eq. (9) with respect to x, i.e., = d{p( e x ) D[H(p( e x )} dx x=xc,p=p I, c (7) where we used the equtions G (z) = G (z) = e (z ). Using Eqs. (9) and (7), we apply numerical methods to get p I c. With D(z) = Q() + Q()z + Q()z, Fig. 3 shows the size of the giant component P (p) as a function of p by C. Boundaries of phase diagram We fix the average degree and from Eq. (6) we conclude that the smaller Q() in the networ, the bigger the c value. If Q() is properly small that c, which corresponds to the case in which the removal of any fraction of nodes causes a second-order phase transition that totally disintegrates networ A. Thus, by requiring c =, and using Eq. (6) we obtain the boundary between the second-order phase transition and the unstable state, =. (8) Q() In addition, using Eq. (3), we obtain the boundary between the first-order and second-order phase transitions

5 5.5. Q()+Q()=.9,Q()=.,Q(3)= Q()+Q()=.9,Q()=.5,Q(3)=.5 Q()+Q()=.9,Q()=,Q(3)=..8.6 Unstable First Order P (Pc).5. Q().4 Triple Point Second Order.5 Boundary of first and second order phase transitons Q().78.8 FIG. 4: (Color online) Comparison between simulation (symbols) and theory (lines) for P (p c) as a function of Q() for different D(z) (D(z) = Q() + Q()z + Q()z ) while eeping Q() + Q() =.9 and =. At the first-order phase transition point p I c, P (p c) is nonzero; whereas at the second-order phase transition point c and P (p c) is zero. From Eq. (9) the boundary between first-order phase transition and second-order phase transition is only dependent on Q() + Q(), thus at Q() c =.75 for this case.. Q()+Q()=.9;Q()=.5;Q(3)=.5 Q()+Q()=.9;Q()=;Q(3)= < > FIG. 5: (Color online) The fraction of nodes that have one dependent node Q() as a function of the average degree with Q() + Q() = 9. The dashed lines are theoretical results obtained from Eqs. () (green) and () (red) with intersection points at (, ). The dashed blue line is the M boundary between first-order phase transition and unstable system, obtained numerically. Here the dashed red and green lines only depend on m whereas the blue lines (both solid and dashed) depend on the specific details of Q( o) other than M. of networ A, Q() = Q(). (9) Using D(z) = Q() + Q()z + Q()z + Q(3)z 3 where Q() + Q() =.9 and =, Fig. 4 plots P (p c ) as a function of Q() by comparing simulation and numerical results. The critical value of Q() c falls onto Q() c =.75 as predicted by Eq. (9), delimiting two different transition regimes. Specifically, if Q() <.75, P (p c ) =, which indicates the presence of a secondorder phase transition, but if Q() >.75, P (p c ) >, which indicates the presence of a first-order phase transition. We also consider a special case in which Q() + Q() = M and use Eq. (9) to determine the boundary between the first-order phase transition and the second-order phase transition, Q() = M +. () In addition, in terms of M, Eq. (8) delivers the boundary between the second-order phase transition and unstable state, Q() = M. () Thus, in the coordinate system of -Q(), using Eqs. () and () we can plot the phase diagram of networ A under random failures, with these two boundaries converging at the triple point ( M, ) (the solid red dot in Fig. 5). Because > always holds, when M this intersection point is non-physical, indicating that the networ will not be subject to first-order phase transitions under attac irregardless of the form of P (), but if M >, the networ will be subject to first-order phase transitions. Fig. 5 shows the boundaries in the phase diagram with M = 9 >, where the boundaries between first-order phase transitions and the unstable state are determined numerically. Note that, when M is fixed, the boundary between the second-order phase transition and the unstable state (dashed red line) as well as the boundary between the first-order and second-order phase transitions (dashed green line) are also fixed because they depend only on M, but the boundary between the firstorder phase transition and the unstable state (dashed blue line) is subject to the details of Q( o ). For example, when Q() + Q() = 9, a shuffle of the remaining terms in Q( o ) causes a shift in the boundary line, shown as the displacement of the solid blue line to the dashed blue line in Fig. 5. IV. CONCLUSIONS In summary, we present an analytical formalism for studying random networs with both connectivity lins and directed dependency lins under random node failures. Using a probabilistic approach, we find that the directed dependency lins greatly reduce the robustness of a networ. We show that the system disintegrates in a form of second-order phase transition at a critical threshold and the boundary between second-order phase transition and unstable regimes solely determined by the

6 6 proportion of nodes that do not depend on other nodes. Our framewor also provides the solution for the boundary between the first-order and second-order phase transitions, which is characterized by the proportion of nodes that depend on no more than one node. As x, taing the limits of both sides of Eq. (A5) we get c = Q()G (). (A6) Acnowledgments This wor is partially supported by the NSFC grant no The Boston University wor is supported by NSF grant no. CMMI 59 and DTRA grant no. HDTRA and Appendix A If p c, x. From Eq. (9) we have G ( x) = G ()x G () x + O(x 3 ), G ( x) = G ()x G () x + O(x 3 ), (A) (A) D{H{p[ G ( x)]}} = Q() + pq()h()g ()x +O(x ). (A3) Using Eqs. (A), (A) and (A3), we can write Eq. (9) as x = pq()g ()x + p[pq()h()g ()G () Q()G () ]x + O(x 3 ). (A4) Since x (, ), we can divide both sides of Eq. (A4) by x, and obtain = pq()g () + p[pq()h()g ()G () Q()G () ]x + O(x ). (A5) Appendix B Putting Eq. (A6) bac into Eq. (A5), we get Q()h()G () x = Q()G () x + O(x ). (B) Q() To simplify Eq. (B), we first tae the derivatives of both sides of Eq. (5) with respect to x and obtain H (z) = D(H(z)) z (D(H(z))) H (z). (H(z)) (B) Plugging z = into Eq. (B), we get H () = D(H()) = D() = Q(). Using Eq. (4), we easily obtain H () = h() and thus h() = Q(), which would reduce Eq. (B) as Q()x = Q()G () G () x + O(x ). (B3) Up to this point, if x x t =, networ A undergoes a second-order phase transition and thus Eq. (B3) clearly holds, but if x x t, networ A undergoes a firstorder phase transition. On the boundary between the first-order and the second-order phase transitions, we get a nonzero x t, but it is negligibly small. Here, we can treat O(x t ) and obtain the condition characterizing this boundary as Q() = Q()G () G (). (B4) [] D. J. Watts and S. H. Strogatz, Nature 393, 44 (998). [] R. Albert, H. Jeong, and A.-L. Barabási, Nature 46, 378 (). [3] R. Cohen, K. Erez, D. Ben-Avraham, and S. Havlin, Phys. Rev. Lett. 85, 466 (). [4] D. S. Callaway, M. E. J. Newman, S. H. Strogatz, and D. J. Watts, Phys. Rev. Lett. 85, 5468 (). [5] M. E. J. Newman, Phys. Rev. E 66, 68 (). [6] M. E. J. Newman, SIAM Rev. 45, 67 (3). [7] V. Rosato, L. Issacharoff, F. Tiriticco, S. Meloni, S. Porcellinis, and R. Setola, Int. J. Crit. Infrastruct. 4, 63 (8). [8] A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, Phys. Rep. 469, 93 (8). [9] S. D. Reis, Y. Hu, A. Babino, J. S. Andrade Jr, S. Canals, M. Sigman, and H. A. Mase, Nature Physics (4). [] R. Cohen and S. Havlin, Complex networs: structure, robustness and function (Cambridge University Press, ). [] M. E. J. Newman, Networs: An Introduction (Oxford University Press, ). [] Y. Hu, Y. Wang, D. Li, S. Havlin, and Z. Di, Physical Review Letters 6, 87 (). [3] S. V. Buldyrev, R. Parshani, G. Paul, H. E. Stanley, and S. Havlin, Nature 464, 5 (). [4] J. Gao, S. V. Buldyrev, H. E. Stanley, X. Xu, and

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