Bypass rewiring and robustness of complex networks
aa r X i v : . [ phy s i c s . d a t a - a n ] S e p Bypass Rewiring and Robustness of Complex Networks
Junsang Park ∗ and Sang Geun Hahn
1, 21
Graduate School of Information Security,Korea Advanced Institute of Science and Technology,291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea Department of Mathematical Sciences,Korea Advanced Institute of Science and Technology,291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea (Dated: October 10, 2018)
Abstract
A concept of bypass rewiring is introduced and random bypass rewiring is analytically andnumerically investigated with simulations. Our results show that bypass rewiring makes networksrobust against removal of nodes including random failures and attacks. In particular, randombypass rewiring connects all nodes except the removed nodes on an even degree infinite networkand makes the percolation threshold 0 for arbitrary occupation probabilities. In our example, theeven degree network is more robust than the original network with random bypass rewiring whilethe original network is more robust than the even degree networks without random bypass. Wepropose a greedy bypass rewiring algorithm which guarantees the maximum size of the largestcomponent at each step, assuming which node will be removed next is unknown. The simulationresult shows that the greedy bypass rewiring algorithm improves the robustness of the autonomoussystem of the Internet under attacks more than random bypass rewiring.
PACS numbers: 89.75.Hc, 64.60.ah, 05.10.-a ∗ [email protected] . INTRODUCTION Many real world systems (the Internet, electric power grids, the World Wide Web, socialnetworks, urban streets, airline routes, subway, and others) are represented by complex net-works with many nodes and many links between nodes [1–13]. A network (graph) breaksinto small disconnected parts when nodes are deleted. Complex networks are robust againstrandom failures or errors (random removal of nodes) but fragile and vulnerable to (inten-tional) attacks (targeted removal of nodes in decreasing order of degree from the highestdegree) [1–3, 5, 7–10, 13–16]. There are various mitigation methods which make networksmore robust [12, 17–19]. However, there are geographical, economic, and technical problemsto implement the mitigation methods known so far. Therefore, we propose a concept ofbypass rewiring to make networks robust against random failures and attacks.
II. A CONCEPT OF BYPASS REWIRING
A node in Fig. 1(a) is removed by random failures or attacks and turns into the removednode in Fig. 1(b). Bypass rewiring is to directly connect each pair of links of the removednode like Fig. 1(c). Each pair of links for rewiring can be chosen in various ways includingrandom selection like random bypass rewiring and heuristic methods like greedy bypassrewiring algorithm. If the degree of the removed node is odd, one link remains open. Forexample, an engineer or equipment can simply rewire cables (links) of a router (node) onthe Internet (network) and relay (and sometimes amplify) the signals directly when therouter does not work under random failures or attacks. Since repair is generally harderthan rewiring, bypass rewiring would be a more useful and simpler way to improve theconnectivity of the other part of the network except the broken router while the router doesnot work or is under repair. 2
II. RANDOM BYPASS REWIRING
In this paper, we use generating functions based on the generating function formalismintroduced in Refs. [4, 13, 14, 20]. We define G ( x ) = ∞ X k =0 p k x k , (1) G ( x ) = P ∞ k =1 kp k x k − P ∞ k =1 kp k = ∞ X k =0 q k x k , (2) H ( x ) = ∞ X k =0 h k x k , (3)where p k is the probability that a randomly chosen node has degree k and h k is the probabilitythat a randomly chosen link reaches a small component which has k nodes. In Eq. (2), q k isthe probability that a randomly chosen link reaches a node with degree k + 1. Since nodesof the giant component do not belong to any small component which has a fixed number ofnodes on an infinite network, the probability that a randomly chosen node belongs to thegiant component is S = ∞ X k =0 p k φ k { − [ H (1)] k } = ∞ X k =0 p k φ k (1 − u k ), (4)for H ( x ) = ∞ X k =0 q k { − φ k +1 + φ k +1 [ H ( x )] k } , (5) H (1) = u = f ( u ) = ∞ X k =0 q k (1 − φ k +1 + φ k +1 u k ), (6)where φ k is the occupation probability that a randomly chosen node with degree k is notremoved and u is the smallest non-negative real solution of Eq. (6), that is, u is the averageprobability that a randomly chosen link is not connected to the giant component [3, 13, 14].The average occupation probability is φ = ∞ X k =0 p k φ k . (7)3ased on the idea seen in Fig. 2, H ( x ) and u satisfy H ( x ) = q φ x + q (1 − φ ) + q φ xH ( x ) + q (1 − φ ) H ( x ) + q φ x [ H ( x )] + 23 q (1 − φ ) H ( x ) + 13 q (1 − φ ) + · · · = ∞ X k =0 q k φ k +1 x [ H ( x )] k + H ( x ) ∞ X k =0 q k (1 − φ k +1 )+[1 − H ( x )] ∞ X k =0 p k +1 (1 − φ k +1 ) P ∞ k ′ =1 k ′ p k ′ , (8) u = f ( u ) = ∞ X k =0 q k φ k +1 u k + u ∞ X k =0 q k (1 − φ k +1 ) + (1 − u ) ∞ X k =0 p k +1 (1 − φ k +1 ) P ∞ k ′ =1 k ′ p k ′ , (9)when random bypass rewiring is applied to an infinite network. In the case of randomfailures ( φ k = φ ), Eq. (9) corresponds to u = f ( u ) = φ ∞ X k =0 q k u k + (1 − φ ) u + (1 − u )(1 − φ ) ∞ X k =0 p k +1 P ∞ k ′ =1 k ′ p k ′ . (10)The self-consistent equations like Eqs. (6) and (9) can be solved as follows by the fixed-point iteration, which is a numerical method [21]. Iterating u i +1 = f ( u i ), (11a) v i +1 = f ( v i ), (11b)for u = v = 0, u i and v i approaches to ¯ u and ¯ v , respectively, as i goes to infinity, for¯ u = f (¯ u ), (12a)¯ v = f (¯ v ). (12b)Since the right-hand side of Eq. (6) is equal to or larger than the right-hand side of Eq. (9)for 0 ≤ u ≤ u i ≥ v i , (13)is satisfied for all i . Therefore, S with random bypass rewiring is always equal to or largerthan without random bypass rewiring; that is, the percolation threshold with random bypassrewiring is always equal to or smaller than without random bypass rewiring.To simulate attacks, a node with the highest degree is firstly removed and nodes areremoved one by one in decreasing order of degree while randomly chosen nodes are removed4ne by one in case of random failures. In the simulation, the degree of each node is notrecalculated while nodes are removed. To simulate random bypass rewiring, each pair oflinks of the removed node are randomly chosen and rewired until no or one link remains.For p k +1 = 0, (14)the smallest non-negative real solution of Eq. (9) is u = 0 since the last term of the rightside of Eq. (9) is 0. The smallest non-negative real solution of Eq. (10) is also u = 0 forthe same reason. Therefore, S is equal to φ , and the percolation threshold is 0 on an evendegree infinite network with random bypass rewiring for arbitrary φ k . In other words, evendegree networks randomly generated are extremely robust against removal of nodes includingrandom failures and attacks with random bypass rewiring. Figure 3(b) shows that almost allthe nodes except the removed nodes on the even degree network are connected by randombypass rewiring. Every percolation threshold with random bypass rewiring in Fig. 3(b) is 0,while every percolation threshold in Fig. 3(a) is not.The even degree network for Fig. 3(b) is randomly generated by degree distribution p ′ k = p k + p k +1 where p k is the degree distribution of the original network for Fig. 3(a).For this reason, the original network has more links and larger average degree than the evendegree network has. Without random bypass rewiring, the size of the largest componentand S on the original network is larger than on the even degree network, respectively. Onthe other hand, with random bypass rewiring, the size of the largest component and S onthe even degree network are larger than on the original network, respectively, as seen inFig. 3. In other words, the even degree network is more robust than the original networkwith random bypass rewiring, while the original network is more robust than the even degreenetwork without random bypass rewiring. IV. GREEDY BYPASS REWIRING
We propose a greedy bypass rewiring algorithm to improve robustness of networks againstremoval of nodes including random failures and attacks. The algorithm chooses a pair oflink, based on the number of the links not yet rewired and the size of the neighboringcomponents. 5 removed node with degree k has k neighbor nodes (neighbor 1, neighbor 2, . . . , neighbor k ) and k links (link 1, link 2, . . . , link k ). R i denotes whether link i is rewired ( R i = 1) ornot yet rewired ( R i = 0). Initially, set R i = 0 for all i at each step. T i,j denotes whetherneighbor i and neighbor j belong to the same component ( T i,j = 1) or do not ( T i,j = 0),that is, there exists a path from neighbor i to neighbor j without going through the removednode or does not. T i,i = 1 is trivially satisfied for all i , and Q i = k X j =1 T i,j (1 − R j ) (15)denotes how many links in the component to which neighbor i belongs are not yet rewired. S i denotes the size of the component to which neighbor i belongs. At t -th step for 1 ≤ t ≤ ⌊ k ⌋ ,choose α ′ which satisfies R α ′ = 0 and Q α ′ ≥ Q i with R i = 0 for all i . From chosen α ′ ,choose α which satisfies R α = 0, Q α = Q α ′ , and S α ≥ S i with R i = 0 and Q i = Q α ′ for all i . Update R α = 1. If T i,j = 1 is satisfied for all i and j , choose randomly β which satisfies R β = 0 without choice of β ′ . Otherwise, choose β ′ which satisfies R β ′ = 0, T α,β ′ = 0, Q β ′ ≥ Q i with R i = 0 and T α,i = 0 for all i . From chosen β ′ , choose β which satisfies R β = 0, T α,β = 0, Q β = Q β ′ , and S β ≥ S i with R i = 0, T α,i = 0, and Q i = Q β ′ for all i .Update R β = 1. When neighbor α and neighbor β do not belong to the same component( T α,β = 0), update the size of the component to which neighbor α and neighbor β belong,that is, S i = [1 − (1 − T α,i )(1 − T β,i )]( S α + S β ) for all i . Update T i,j = T j,i = 1 if there exist i and j which satisfy T i,α T β,j = 1, that is, T i,j = T j,i = 1 − (1 − T i,j )(1 − T i,α T β,j ). Repeateach step of the algorithm ⌊ k ⌋ times whenever a node is removed.If there exists i ( = α, β ) which satisfies Q i >
1, the maximum size of the largest compo-nent is not guaranteed for Q α ≤ Q β ≤
1. From this aspect, we choose α ′ and β ′ = α which maximize Q α ′ and Q β ′ in the algorithm. If Q i ≤ i ( = α ), themaximum size of the largest component is not guaranteed when there exists i which satisfies S β < S i . If Q i ≤ i , the maximum size of the largest component is notguaranteed when there exists i which satisfies S α < S i or S β < S i . From this point of view,we choose α and β ( = α ) which maximize S α and S β in the algorithm for Q α = Q α ′ and Q β = Q β ′ . Therefore, the algorithm guarantees the maximum size of the largest componentat each step where which node will be removed next is unknown.Figure 4 shows that the greedy bypass rewiring algorithm improves the robustness of theInternet and the electrical power grid under attacks more than random bypass rewiring.6ince we ignore and eliminate self links and double links for the simulation, the number oflinks on the Internet is 12572 where the network originally has 13895 links. V. CONCLUSIONS
In summary, we have introduced a concept of bypass rewiring and analytically and numer-ically investigated random bypass rewiring with simulations. The results have shown thatrandom bypass rewiring improves robustness of networks under removal of nodes includingrandom failures and attacks. With random bypass rewiring, all nodes except the removednodes on an even degree infinite network are connected for arbitrary occupation probabili-ties, and then the percolation threshold is 0. With (without) random bypass rewiring, thesize of the largest component and S on the original network are smaller (larger) than on theeven degree network randomly generated by the degree distribution p ′ k = p k + p k +1 , respec-tively, where p k is the degree distribution of the original network. This means that randombypass rewiring makes even degree networks extremely robust. Based on the number of thelinks not yet rewired and the size of the neighboring components, we have proposed a greedybypass rewiring algorithm which guarantees the maximum size of the largest component ateach step, assuming that which node will be removed next is unknown. The simulationresult has shown that the algorithm improves robustness of the autonomous system of theInternet more than random bypass rewiring. We hope that bypass rewiring equipment isimplemented and added on the existing routers on the Internet. More applications of variouskinds and studies of bypass rewiring in many fields are expected. [1] R. Albert, H. Jeong, and A.-L. Barabasi, Nature (London) , 378 (2000).[2] R. Cohen, K. Erez, D. ben-Avraham, and S. Havlin, Phys. Rev. Lett. , 4626 (2000).[3] R. Cohen, K. Erez, D. ben-Avraham, and S. Havlin, Phys. Rev. Lett. , 3682 (2001).[4] M. E. J. Newman, S. H. Strogatz, and D. J. Watts, Phys. Rev. E , 026118 (2001).[5] S. N. Dorogovtsev and J. F. F. Mendes, Adv. Phys. , 1079 (2002).[6] V. Latora and M. Marchiori, Physica A , 109 (2002).[7] P. Holme, B. J. Kim, C. N. Yoon, and S. K. Han, Phys. Rev. E , 056109 (2002).[8] R. Albert and A.-L. Barabasi, Rev. Mod. Phys. , 47 (2002).
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Numerical Analysis , 9th ed. (Brooks/Cole, Boston, MA, 2011).[22] J. Leskovec and A. Krevl, http://snap.stanford.edu/data .[23] D. J. Watts and S. H. Strogatz, Nature (London) , 440 (1998). a) (b) (c) FIG. 1. A circle is for a node and squares are for components. (a) Before removal of the node, onenode and five components are connected. (b) After removal of the node, the network fragmentsinto five smaller components without bypass rewiring. (c) After removal of the node, the networkfragments into two larger components and one smaller component with bypass rewiring. φ q
23 1 ( ) −φ q
13 1 ( ) −φ q φ q ( )1 −φ q φ q ( )1 −φ q FIG. 2. A schematic diagram to calculate the probability that a component (square) is reached bya randomly chosen link with random bypass rewiring under removal of a node (circle). t h e s i z e o f t h e l a r g e s t c o m p o n e n t a n d S (a) t h e s i z e o f t h e l a r g e s t c o m p o n e n t a n d S (b) FIG. 3. The size of the largest component with respect to the number of removed nodes underrandom failures [circles (plus signs)] and attacks [squares (crosses)] without (with) random bypassrewiring. The solid lines are for numerically calculated S with respect to N (1 − φ ) from Eqs. (4), (6),(9), and (10) on an infinite network with the same degree distribution. (a) On the undirected scale-free network randomly generated by the configuration model with degree distribution p k ∼ k − , N = 20000 nodes, and M = 30719 links. (b) On the undirected even degree scale-free networkrandomly generated by the configuration model with degree distribution p ′ k = p k + p k +1 , N =20000 nodes, and M = 28160 links. Two straight lines for random bypass rewiring are overlapped.
500 1000 1500 2000 2500the number of removed nodes01000200030004000500060007000 t h e s i z e o f t h e l a r g e s t c o m p o n e n t (a)
500 1000 1500 2000 2500 3000the number of removed nodes010002000300040005000 t h e s i z e o f t h e l a r g e s t c o m p o n e n t (b) FIG. 4. The circles are for the case without bypass rewiring. The squares (crosses) are for thecase with random (greedy) bypass rewiring. (a) The size of the largest component with respect tothe number of removed nodes on the autonomous system (AS-733) from [22] with N = 6474 nodesand M = 12572 links under attacks. (b) The size of the largest component with respect to thenumber of removed nodes on the electrical power grid of the western United States from [23] with N = 4941 nodes and M = 6594 links under attacks.= 6594 links under attacks.