More Tolerant Reconstructed Networks by Self-Healing against Attacks in Saving Resource
aa r X i v : . [ phy s i c s . s o c - ph ] J a n More Tolerant Reconstructed Networks by Self-Healingagainst Attacks in Saving Resource
Yukio Hayashi
Japan Advanced Institute of Science and Technology,Ishikawa, 923-1292, Japan
Atsushi Tanaka
Yamagata UniversityYonezawa-city, Yamagata 992-8510, Japan
Jun Matsukubo
National Institute of Technology Kitakyushu CollegeKitakyushu-city, Fukuoka 802-0985, Japan (Dated: January 11, 2021) bstract Complex network infrastructure systems for power-supply, communication, and transportationsupport our economical and social activities, however they are extremely vulnerable against thefrequently increasing large disasters or attacks. Thus, a reconstructing from damaged network israther advisable than empirically performed recovering to the original vulnerable one. In order toreconstruct a sustainable network, we focus on enhancing loops so as not to be trees as possibleby node removals. Although this optimization is corresponded to an intractable combinatorialproblem, we propose self-healing methods based on enhancing loops in applying an approximatecalculation inspired from a statistical physics approach. We show that both higher robustness andefficiency are obtained in our proposed methods with saving the resource of links and ports thanones in the conventional healing methods. Moreover, the reconstructed network by healing canbecome more tolerant than the original one before attacks, when some extent of damaged links arereusable or compensated as investment of resource. These results will be open up the potential ofnetwork reconstruction by self-healing with adaptive capacity in the meaning of resilience.
PACS numbers: 89.75.Fb, 89.20.-a, 02.60.-x, 05.65.+bKeywords: Self-Healing, Network Science, Resource Allocation, Enhancing Loops, Belief Propagation, Ro-bustness of Connectivity, Efficiency of Paths, Resilience . INTRODUCTION Unfortunately, the frequencies of large disasters or malicious attacks increase dueto climate exchange, crustal movements, military conflicts, cyber-terrorism, and mega-urbanization in our world day by day. For example, it is well known that a little accidentinvolved the large area’s power collapse in North America [1] or Italian peninsula [2] in2003, and that enormous destruction of social infrastructure systems was happen by thegreat earthquake in Japan in 2011 [3]. While there exists a surprisingly common topologicalstructure called scale-free (SF) in many real networks [4], such as power-grid, airline, com-munication, transportation systems, and so on, which support our social activities, economy,industrial production, etc. The SF structure is considered to be generated by a selfish rule:preferential attachment [5], and consisted of many low degree nodes and a few (high degree)hubs, heterogeneously. Here, degree means the number of links at a node. Moreover, by theheterogeneity, a SF network has extreme vulnerability against hub attacks [6]. These vul-nerable infrastructures appear everywhere and are interdependent on each other. Exactly,since a node prefers to connect high degree nodes in the efficiency bias to shorten the pathlengths counted by hops, the preferential attachment encourages the heavy concentration oflinks to hubs. In many real networks, once hubs are damaged and removed as malfunction,the remaining nodes are fragmented and lost the basic function for communication or trans-portation. It is a plausible scenario for our network infrastructures that the weak points ofhubs are involved in a large disaster.Therefore, when large-scale failures or attacks occur, recovery to the original vulnera-ble network is inadvisable. Rather reconstruction by healing is required. In changing thestructure instead of recovering to the original one, a question arises as to how a sustainablenetwork should be reconstructed to maintain the network function. However, the resourcesof links (wire cables, wireless communication or transportation lines between two nodes, etc.)and ports (channels or plug sockets at a node, etc.) are usually limited, the allocation shouldbe controlled at the same time in the rewiring or additional investment for healing. Such areconstruction conforms with the concept of resilience in system engineering or ecology asa new supple approach to sustain basic objective and integrity even in encountering withthe extreme change of situations or environments (e.g., by disasters or malicious attacks)for technological system, organization, or individual [7][8][9].3n this paper, through numerical simulation, we study how to reconstruct a sustain-able network under limited resource, and propose effective self-healing methods based onenhancing loops through a local process around damaged parts. In addition, we show thesignificant improvement form the previous study [10] to reduce the additional ports preparedin advance besides reusable ports. The motivations for enhancing loops are as follows. Inpercolation analysis, as a part of network science, it has been found that onion-like structurewith positive degree-degree correlations gives the optimal robustness of connectivity evenfor a SF network with a power-law degree distribution [11][12]. The name of onion-likecomes from that it is visualized by the correlations when similar degree nodes are set on aconcentric circle arranged in decreasing order of degrees from core to peripheral. Onion-likestructure can be generated by whole rewiring [11][13] in enhancing the correlations undera given degree distribution. On the other hand, since dismantling and decycling problemsare asymptotically equivalent at infinite graphs in a large class of random networks withlight-tailed degree distribution [14], trees remain without loops at the critical state beforethe complete fragmentation by node removals. Dismantling (or decycling) problem knownas NP-hard [15] is to find the minimum set of nodes in which removal leaves a graph brokeninto connected components whose maximum size is at most a constant (or a graph withoutloops). It is suggested from the equivalence that the robustness becomes stronger as manyloops exist as possible. In fact, to be the optimal onion-like networks at the same level tothe rewired ones [13], enhancing loops by copying [16] or intermediation [17][18] is effectivefor improving the robustness in incrementally growing methods based on a local distributedprocess as self-organization. Similar effect is also obtained in preserving or non-preservingthe degrees at nodes after the other rewiring based on enhancing loops instead of correlations[19]. Thus, we remark that loops make bypasses and may be more important than the degree-degree correlations in order to improve the connectivity in a network reconstruction afterlarge disasters or attacks. It is predicted as the top priority to maximize the decycling set(or called Feedback Vertex Set (FVS) in computer science [15]) so as not to be tree withoutloops as possible even by the worst case of node removals. In other words, enhancing loopscorrespond to optimizing the tolerance of connectivity in graphs (but not in the contentsof general computing or problem solving). Off course, increasing the path lengths betweennodes and wasteful resource should be avoided in the reconstruction by healing. However,even identifying the necessary nodes to form loops is intractable due to combinatorial NP-4ardness [15], we effectively apply an approximate calculation by Belief Propagation (BP)based on a statistical physics approach in our self-healing through rewirings (or additionalinvestment instead) as mentioned later. We describe the healing methods as sequential pro-cesses for computer simulation in envisioning the further development of distributed controlalgorithms.
II. METHODSA. Outline of Healing Process
Almost simultaneously attacked nodes are not recoverable immediately, therefore areremoved from the network function for a while or long time. In such case of emergency forhealing, unconnected two nodes are chosen and rewired as the reconstruction assistance orreuse of links emanated from removed qN nodes, when the fraction of attacks is q and N denotes the total number of nodes (as the network size). Some of disconnected links maybe reusable at the neighbor’s sides according to the damage level. Although we call thereuse rewiring, removal of nodes is a different problem setting to that in the so-called usualrewiring methods [11][13][19]. The outline of healing process is as follows. Step0:
Detection and initiationAfter detecting a removal as malfunction at a nearest neighbor of the attacked node,the healing process is initiated autonomously.
Step1:
Selection of two nodesSince the neighbor loses links at least temporary before rewiring, the damaged onebecomes an attached candidate for healing. Thus, unconnected two nodes are chosenfrom neighbors of removed nodes by attacks. The selections are different in our pro-posed and the conventional healing methods. Moreover, neighbors are extended in ourproposed methods.
Step2:
Rewiring for healingThe chosen two nodes are connected as rewiring for healing. The above process isrepeated for M h def = r h × ˜ P i ∈ D q k i links.5ere, ˜ P i ∈ D q k i means the number of disconnected links by attacks without multiple counts. D q denotes the set of removed nodes, | D q | = qN . M h includes the number of reused andadditionally invested links. When reusable links are insufficient, we assume to add links asinvestment until to the considered M h for a parameter 0 < r h ≤ k j ports are reusable atthe undamaged neighbor node j ∈ ∂i of a removed node i by attacks, where ∂i denotes aset of the nearest connecting neighbors of i . Thus, there exist active (reusable) ports of anode at least as many as its degree in the original network before attacks. B. Proposed Healing Methods
Basically, in our proposed healing methods, there are two phases: ring formation andenhancing loops by applying BP in the next subsection. Moreover, they (RingRecal,RingLimit1,5,10, RingLimit5Recal) are modified to reduce the additional ports from theprevious results [10] by avoiding the concentration of links at some nodes.
RingBP
Previous our combination method of ring formation and enhancing loops [10].After making rings on the extended neighbors of removed nodes as shown in Figure1, enhancing loops on the rings is performed by applying the BP algorithm [23] (seesubsection II C). However, in the BP, a set of { p i } as probability of node i to benecessary for loops is calculated only once just after attacks. Note that a ring is thesimplest loop by using the least number of links. RingRecal
Modified our method with recalculations of BP. After making rings, a set of6 p i } is recalculated one-by-one through each rewiring in the remaining links within M h for enhancing loops. RingLimit1,5,10
Modified our method with limited rewirings. After making rings, inenhancing loops, the number of rewiring links is limited at node i to its degree k i +1,+5, or +10. RingLimit5Recal
Modified our method by a combination of RingRecal and RingLimit5.After making rings, a set of { p i } is recalculated one-by-one through each rewiring inthe remaining links within M h for enhancing loops. Moreover, the number of rewiringlinks is limited at node i to k i + 5.First, in ring formation (see Figure 1), the order of process is basically according tothe order of the removed nodes i , i , . . . , i qN . Thus, rings are made for the neighbors ∂i , ∂i , . . . , ∂i qN in this order. However, if there is i k ′ ∈ ∂i k , k ′ > k , it is extended asthe union ∂i k ← ∂i k ∪ ∂i k ′ . In addition, if there is i k ′′ ∈ ∂i k ′ , k ′′ > k ′ , it is also extended asthe union ∂i k ← ∂i k ∪ ∂i k ′ ∪ ∂i k ′′ . Such extensions of neighbors are repeated until that a ringencloses the induced subgraph of removed nodes and their links. To make a ring, a nodeis chosen u.a.r and connect to a subsequent similarly chosen node in a set of the extendedneighbors. This is repeated without multi-selections until return to the first chosen nodefrom the last chosen node.Next, in enhancing loops on each ring for remained rewirings in M h , a node j with theminimum p j is chosen in all of the neighbors of removed nodes, and connected to othernode j ′ with the second minimum p j ′ on the ring to which j belongs. For each rewiring,a set of { p i } is recalculated one-by-one in RingRecal and RingLimit5Recal methods. Inaddition, the number of rewiring links is limited at node i to k i + 5 (or +1, +10) accordingto its degree k i in RingLimit5Recal and RingLimit5 (or RingLimit1, RingLimit10) methods.If the condition is unsatisfied, other node with the second, third, forth, and subsequentminimum is chosen as a candidate for healing. Although a node j ′′ with small p j ′′ tendsto not contribute to making loops because of not included in FVS, it is expected that thenumber of loops is increased by connecting such nodes. This is the reason for the aboveselection. 7 IG. 1: Schematic illustration for ring formation and enhancing loops. Red nodes and their linksare removed by attacks. Gray filled nodes are the neighbors. Blue lines make rings, and green linesare rewirings for enhancing loops on rings.
C. Applying Belief Propagation Algorithm
To calculate the probability p i of belonging to FVS, the following BP algorithm [23] areapplied. It is based on a cavity method in statistical physics. We review the outline derivedfor approximately estimating FVS known as NP-hard problem [15]. In the cavity graph, it isassumed that nodes j ∈ ∂i are mutually independent of each other when node i is removed(the exception is denoted by \ i ). Then the joint probability is P \ i ( A j : j ∈ ∂i ) ≈ Π j ∈ ∂i p A j j → i by the product of independent marginal probability p A j j → i for the state A j as the node indexof j ’s root or empty 0: it belongs to FVS. The corresponding probabilities are representedby p i def = 1 z i ( t ) , (1) p i → j = 1 z i → j ( t ) , (2) p ii → j = e x Π k ∈ ∂i ( t ) \ j (cid:2) p k → i + p kk → i (cid:3) z i → j ( t ) , (3)where ∂i ( t ) denotes node i ’s set of connecting neighbor nodes at time t , and x > z i ( t ) def = 1 + e x X k ∈ ∂i ( t ) − p k → i p k → i + p kk → i Π j ∈ ∂i ( t ) (cid:2) p j → i + p jj → i (cid:3) , (4) z i → j ( t ) def = 1 + e x Π k ∈ ∂i ( t ) \ j (cid:2) p k → i + p kk → i (cid:3) × X l ∈ ∂i ( t ) \ j − p l → i p l → i + p ll → i , (5)8o be satisfied for any node i and link i → j as p i + p ii + X k ∈ ∂i p ki = 1 , p i → j + p ii → j + X k ∈ ∂i p ki → j = 1 . We repeat these calculations of message-passing until to be self-consistent in principle butpractically to reach appropriate rounds from initial setting of (0 ,
1) random values. The unittime from t to t + 1 for calculating a set { p i } consists of a number of rounds by updatingequations (1)-(5) in order of random permutation of the total N nodes. Since the sums orproducts in equations (1)-(5) are restricted in the nearest neighbor, they are local processes.The distributed calculations can be also considered. As included in FVS, a node k with themaximum p k is chosen. After removing the chosen node, { p i } is recalculated at next time.Such process is repeated until to be acyclic for finding the FVS. However, in our healingmethod, { p i } is used for selecting attached two nodes on a ring by rewiring. D. Conventional Healing Methods
We briefly explain the following typical healing methods in network science (inspired fromfractal statistical physics) and computer science.
RBR
Conventional Random Bypass Rewiring (RBR) method [24] (corresponded to r h =0 . GBR
Greedy Bypass Rewiring (GBR) method improved from RBR heuristically [24].
SLR
Conventional Simple Local Repair (SLR) method [25] with priority of rewirings tomore damaged nodes.In network science, a self-healing method by adding new random links on interdepen-dent two-layered networks of square lattices has been proposed, and the effect against nodeattacks is numerically studied [26]. In particular, for adding links by the healing process,the candidates of linked nodes are incrementally extended from only the direct (nearestconnecting) neighbors of the removed node by attacks until no more separation of com-ponents occurs. In other words, the whole connectivity is maintained except the isolatingremoved parts. Such extension of the candidates of linked nodes is a key idea in our proposedself-healing method. 9urthermore, the following self-healing methods, whose effects are investigated for somedata of real networks, are worthy to note. One is a distributed SLR [25] with the repair by alink between the most damaged node and a randomly chosen node from the unremoved nodeset in its next-nearest neighbors before attacks. The priority of damaged nodes is accordingto the smaller fraction k dam /k orig of its remained degree k dam and the original degree k orig before the attacks. The selections are repeated until reaching a given rate f s controlled bythe fraction of nodes whose k dam /k orig falls bellow a threshold. Another is RBR [24] onmore limited resource of links and ports. To establish links between pair nodes, a node israndomly chosen only one time in the neighbors of each removed node. When k i denotesthe degree of removed node i , only ⌊ k i / ⌋ links are reused. Note that reserved additionalports are not necessary: they do not exceed the original one before attacks. Moreover, GBR[24] is proposed in order to improve the robustness, the selection of pair nodes is based onthe number of the links not yet rewired and the size of the neighboring components.In computer science, ForgivingTree algorithm has been proposed [27]. Under the repeatedattacks, the following self-healing is processed one-by-one after each node removal, exceptwhen the removed node is a leaf (whose degree is one). It is based on both distributed processof sending messages and data structure, furthermore developed to an efficient algorithmcalled as compact routing [28]. In each rewiring process, a removed node and its links arereplaced by a binary tree. Note that each vertex of the binary tree was the neighbors ofthe removed node, whose links to the neighbors are reused as the edges of the binary tree.Thus, additional links for healing is unnecessary, but not controllable. It is remarkablefor computation (e.g., in routing or information spreading) that the multiplicative factorof diameter of the graph after healing is never more than O (log k max ), where k max is themaximum degree in the original network because of the replacing by binary trees. However,the robustness of connectivity is not taken into account in the limited rewiring based onbinary trees, since a tree structure is easily fragmented into subtrees by any attack to thearticulation node. Thus, this healing method is excluded from compared ones. III. RESULTS
We evaluate the effect of healing by four measures: the ratio S ( q ) /N q [25] for theconnectivity, the robustness index R ( q ) def = P q ′ S q ( q ′ ) /N q , the efficiency of paths E ( q ) def =10 N q ( N q − P i = j L ij , and the average degree k avg ( q ) in N q def = (1 − q ) N nodes, where S ( q ) and S q ( q ′ ) denote the sizes of GC (giant component or largest connected cluster) after removing qN nodes by attacks from the original network and removing q ′ N q nodes by further attacksfrom the surviving N q nodes, respectively. Here, a removed node is chosen with recalcula-tion of the highest degree node as the target. Remember that q = 1 /N, /N, . . . , ( N − /N (or q ′ = 1 /N q , /N q , . . . , ( N q − /N q ) is a fraction of attacks. While S ( q ) or S q ( q ′ ) rep-resents the size of GC after attacks to qN or q ′ N q nodes, R ( q ) is a measure of toleranceof connectivity against further attacks. L ij denote the length of the shortest path countedby hops between i - j nodes in the surviving N q nodes. The ranges are 0 < S ( q ) /N q ≤ < R ( q ) ≤ .
5, and 0 < E ( q ) ≤
1. We investigate the four measures before or after healingfor OpenFlights between airports, Internet AS Oregon, and US PowerGrid as examples oftypical infrastructure of SF networks [29] after extracting from each of them to a connectedand undirected graph without multiple links (see Table I). We compare the results shownby color lines with marks in figures for the conventional RBR, GBR, SLR, and our proposedRingBP, RingRecal, RingLimit1,5,10, RingLimit5Recal methods.
TABLE I: Basic properties for the original networks. N and M denote the numbers of nodesand links. k avg = 2 M/N , k min , and k max are the average, minimum, and maximum degrees. L avg , D, R and E denote the average path length, diameter, robustness index, and efficiency ofpaths, respectively. Network
N M k avg k min k max L avg D R E
OpenFlight 2905 15645 10.77 1 242 4.097 14 0.080912 0.266934AS Oregon 6474 12572 3.883 1 1458 3.705 9 0.012500 0.290399PowerGrid 4941 6594 2.669 1 19 18.989 46 0.052428 0.062878
In each Figure 2,3,4,5, no-healing, conventional, and previous our methods are comparedin (a), previous our and RingRecal or RingLimit methods are compared in (c)(d), previousour and the best combination RingLimit5Recal methods are compared in (b). Red, green,blue, orange, and purple lines denote the rate r h = 0 . , . , . , . .
0, respectively,for the number M h of rewirings. The results for the original and no-healing networks areshown by dashed magenta and solid black lines. The following results are averaged over 100samples with random process for tie-breaking in a node selection or ordering of nodes on a11 ingBP r h r h r h r h FIG. 2: Ratio S ( q ) /N q of connectivity vs fraction q of attacks for the rate r h in rewirings. ring.Figure 2 shows the ratio S ( q ) /N q of connectivity in the surviving N q nodes. Remem-ber that S ( q ) is the size of GC after healing (or no-healing) against attacks to qN nodes.Higher ratio means larger connectivity as maintaining the network function for communi-cation or transportation, S ( q ) /N q < M h . As shown in Figure 2(a), the ratio rapidly decreases in the conventional SLR methodmarked by open circles for OpenFlights and PowerGrid, while it is moderately higher around S ( q ) /N q ≈ . S ( q ) /N q ≈ . S ( q ) /N q = 0 are the results without the network function for no-healing. Thus,previous our RingBP method marked by open squares has higher ratio than the conven-tional methods in comparison with same color lines. Figure 2(b)(c) shows that the ratioin RingBP method marked by open squares almost coincide with ones in RingLimit5Recalmethod marked by open diamonds and RingRecal method marked by filled diamonds. Sim-ilarly, Figure 2(d) shows that the ratio in RingBP method marked by open squares almostcoincide with ones in RingLimit5,10 methods marked by lower triangles and asterisks. How-ever it is slightly lower in RingLimit1 method marked by open upper triangles. Therefore,RingLimit5Recal, RingRecal, and RingLimit5,10 methods are the best at the same level toRingBP in maintaining the connectivity. The constraint to the number of additional portsis slightly too strong as only one in RingLimit1 method.Figure 3 shows the robustness index R ( q ) as the tolerance of connectivity against furtherattacks to the surviving N q nodes after healing. Note that a major part of N q nodes belongto the GC but other parts belong to isolated clusters. In Figure 3(a) for OpenFlights, ASOregon, and PowerGrid, the values of R ( q ) rapidly decrease to very low level ≤ . R ( q ) (on purple and orange lines for r h ≥ .
5) in RingBP method marked by open squaresthan the horizontal dashed magenta lines in the original network. The results for no-healingare at the bottom as R ( q ) ≈ S q ≈ R ( q ) than RingBP marked by open squaresin comparison with same color lines for OpenFlights and AS Oregon, while these methodshave almost same values of R ( q ) to ones in RingBP for PowerGrid. Similarly, as shownin Figure 3(d) for OpenFlights and AS Oregon, RingLimit1,5,10 methods marked by openlower, upper triangles and asterisks have higher values of R ( q ) than RingBp method markedby open squares in comparison with same color lines. However, the difference becomessmaller in green and red lines for r h ≤ .
1. In Figure 3(d) for PowerGrid, similar values of13 ingBP r h r h r h r h OriginalCommon to Figure 3.
FIG. 3: Robustness index as the tolerance of connectivity against further attacks to the surviving N q nodes vs fraction q of attacks for the rate r h in rewirings. R ( q ) are obtained on each color lines regardless of marks for different methods. Partially,for OpenFlights and AS Oregon, purple, orange and blue line ( r h ≥ .
2) in RingLimit5Recalmarked by open diamonds are slightly higher than ones in Ringlimit1,5,10 marked by openupper, lower triangles and asterisks as shown in Figure 3(b)(d). Thus, the reconstructednetworks by our proposed healing methods can become more stronger with higher values of R ( q ) than the original network against further attacks. In particular, the improvement isremarkable from R ( q ) < . R ( q ) > . E ( q ) of shortest paths between two nodes in the surviving14 ingBP r h r h r h r h on to Figure 4. FIG. 4: Efficiency of paths in the surviving N q nodes vs fraction q of attacks for the rate r h inrewirings. N q nodes. Note that E ( q ) = 0 . , . , .
25 is corresponded to 10 , , L avg ( q ) from L avg ( q ) ≈ /E ( q ) in the arithmetic and the harmonic means of pathlengths. The following results are common for OpenFlights, AS Oregon, and PowerGrid. Assimilar to Figures 2(a) and 3(a), Figure 4(a) shows that the values of E ( q ) rapidly decreasein the conventional SLR method marked by open circles, RBR and GBR methods denotedby light-blue and brown dashed lines, while the values are higher in RingBP method markedby squares in comparison with same color lines. In Figure 4(b)(c), RingLimit5Recal methodmarked by open diamonds and RingRecal method marked by filled diamonds have similar or15lightly lower values of E ( q ) than ones in RingBP method marked by squares in comparisonwith same color lines. In Figure 4(d) for OpenFlights and AS Oregon, the values are slightlylower in RingLimit1,3,5 methods marked by open upper, lower triangles and asterisks thanones in RingBP method marked by squares, while for PowerGrid the values are similarregardless of these methods in comparison with same color lines. RingBP r h r h r h r h FIG. 5: Average degree k avg ( q ) in the surviving N q nodes vs fraction q of attacks for the rate r h in rewirings. Figure 5 shows the average degree k avg ( q ) in the surviving N q nodes. This value indicateshow much links are effectively used for hearing. In other words, a small value of k avg ( q )16eans that rewirings are restricted and not fully used until the possible number M h of linksespecially in the conventional methods, by the constraints on linking between not the ex-tended but the nearest neighbors of attacked nodes or the limitation (see the subsectionII D). The following results are common for OpenFlights, AS Oregon, and PowerGrid. Asshown in Figure 5(a), it is remarkable that the values are small k avg ( q ) <
10 in the con-ventional SLR method marked by open circles, RBR and GBR methods denoted by dashedlight-blue and brown lines, while the values are higher in RingBP method marked by opensquares in comparison with same color lines. In Figure 5(b)(d), by saving rewired links dueto the limitation of additional ports, the values of k avg ( q ) are not large in RingLimit5Recalmethod marked by open diamonds or in RingLimit1,5,10 methods marked by open up-per, lower triangles and asterisks. In Figure 5(c), on each color line, the values of k avg ( q )in RingBP method marked by open squares are almost coincident with ones in RingRe-cal method marked by filled diamonds. However, in RingLimit5Recal and RingLimit1,5,10methods with saving rewired links, both R ( q ) and E ( q ) are high values as shown in Figures3(b)(d) and 4(b)(d). Therefore, these methods are more effective for healing to improvethe robustness and efficiency to similar levels by using less resource. Figure 6 shows thatthe reconstructed degree distribution P ( k ) in RingLmit5Recal method becomes exponentialapproximately in a semi-logarithmic plot from a power-law in the original network. Themaximum degrees are bounded as 65, 19, and 14 for OpenFlights, AS Oregon, and Power-Grid, respectively. They tend to be smaller as q increases.Moreover, Table II shows the maximum number (or in parentheses, the average valueover the nodes that perform much more rewirings than their degrees of the reusable numberof ports) of additional ports in RingRecal method. Although the values are reduced to lessthan k max from nearly 2 k max ∼ k max in previous our RingBP method [10], they are stilllarge. Here, k max is 242, 1458, or 19 for the original networks: OpenFlights, AS Oregon,or PowerGrid as shown in Table I. Off course, the maximum number of additional portsis significantly restricted to a constant 1 , ,
10 or 5 in RingLimit1,5,10 or RingLimit5Recalmethod. Since additional ports should be stored in advance beside reusable number of itsdegree in the original network, fewer preparing is better within lower investment cost ofresource. Thus, RingBP or RingRecal method is not desirable because of requiring manyadditional ports. 17
ABLE II: Maximum number of additional ports (the average number in parenthesis) for thefraction q of attacks and the rate r h in rewirings. OpenFlights ❅❅❅ r h q ❅❅❅ r h q ❅❅❅ r h q h linestyle 1.00.50.20.10.05 FIG. 6: Degree distribution in surviving N q nodes after healing by RingLimit5Recal method forthe fraction of attacks. IV. DISCUSSION
We have proposed self-healing methods with modifications from the previous one [10] forreconstructing a resilient network through rewirings against attacks or disasters in resourceallocation control of links and ports. The healing strategy is based on maintaining theconnectivity by ring formation on the extended neighbors of attacked nodes and enhancingloops for improving the robustness of connectivity in applying the approximate calculations19f BP [23] inspired from statistical physics in distributed manner. We have taken intoaccount the limitation of additions and the recalculations of BP as modifications to reducethe preparing of additional ports by avoiding the concentration of links at some nodes.Simulation results show that our proposed combination methods of ring formation andenhancing loops are better than the conventional SLR [25], RBR, and GBR [24] methods.Especially, in RingLimit5Recal method, both high robustness of connectivity and efficiencyof paths are obtained in saving the resource of links and ports, even though the number ofadditional ports is significantly restricted to a constant 5 from the previous O ( k max ) ∼ [10]. Moreover, we have found that the reconstructed networks by healing can become morerobust and efficient than the original network before attacks, when some extent of damagedlinks are reusable or compensated as the rate r h ≥ . D (see Table I). The development of distributed algorithms within only localinformation is also important for our self-healing methods. Acknowledgments
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