Effective Self-Healing Networks against Attacks or Disasters in Resource Allocation Control
aa r X i v : . [ phy s i c s . s o c - ph ] S e p Effective Self-Healing Networks against Attacks orDisasters in Resource Allocation Control
Yukio Hayashi
Graduate School of AdvancedInstitute of Science and Technology,Japan Advanced Institute ofScience and TechnologyNomi-city, Ishikawa 923-1292Email: [email protected]
Atsushi Tanaka
Department of Informatics and ElectronicsFaculty of Engineering,Yamagata UniversityYonezawa-city, Yamagata 992-8510Email: [email protected]
Jun Matsukubo
Department of Creative Engineering,National Institute of TechnologyKitakyushu CollegeKitakyushu-city, Fukuoka 802-0985Email: [email protected]
Abstract —With increasing threats by large attacks or disasters,the time has come to reconstruct network infrastructures suchas communication or transportation systems rather than torecover them as before in case of accidents, because many realnetworks are extremely vulnerable. Thus, we consider self-healingmechanisms by rewirings (reuse or addition of links) to besustainable and resilient networks even against malicious attacks.In distributed local process for healing, the key strategies are theextension of candidates of linked nodes and enhancing loops byapplying a message-passing algorithm inspired from statisticalphysics. Simulation results show that our proposed combinationof ring formation and enhancing loops is particularly effective incomparison with the conventional methods, when more than halfdamaged links alive or are compensated from reserved ones.
Keywords – Self-Healing; Complex Network Science; Connec-tivity; Enhancing Loops; Message-Passing Algorithm; ResourceAllocation; Resilience.
I. I
NTRODUCTION
In contemporary world, network infrastructures, such ascommunication, trading, transportation, energy and water sup-ply systems are crucial for supporting social activities, econ-omy, industrial production, etc., while increasing the frequencyof large disasters or military conflicts turn threats by destroyingthe functions into reality. To confront the serious problems, anew supple approach resilience [1], [2] attracts much atten-tion in system engineering, biology, ecology, and sociology.Resilience means the ability to sustain basic objective andintegrity even in encountering with the extreme change ofsituations or environments (e.g., by disasters or malicious at-tacks) for technological system, organization, or individual [1].However, to be resilient system, the concept of safety againstaccidents or disruptions should be extended from
Safety-I to Safety-II [3]: from “as few things as possible go wrong” to“as many things as possible go right”, from “reactive, respondwhen something happens” to “proactive, continuously tryingto anticipate developments and events”, and so on, in these twocomplementary views, which do not conflict. Safety-II requiresto adjust, adapt, develop, and design better processes withtechnological or human resource allocations. Moreover, theconcept of resilience includes reorganization or reconstructionof the system with adaptive capacity beyond the conventionalrecovery [4], as shown in Table I. Such paradigm shift has affinity to self-adaptive mechanism or system. Thus, we focuson developing new mechanism to lead to (social-ecological)resilience with reconstruction as far as accepting innovationrather than finding and decreasing weak or wrong parts in anetwork system.In this paper, we study how to reconstruct a sustainablenetwork under limited resource, and propose effective self-healing methods based on enhancing loops through a localprocess around damaged parts. The motivations for enhancingloops are as follows. There is a common topological struc-ture called Scale-Free (SF) in many social, biological, andtechnological networks [5], [6]. Although the SF networkshave an extreme vulnerability against malicious attacks [7], ithas been found that onion-like structure with positive degree-degree correlations gives the optimal robustness of connectivity[8], [9]. Onion-like structure can be generated by wholerewiring [8], [10] in enhancing the correlations under a givendegree distribution. Moreover, since dismantling and decyclingproblems are asymptotically equivalent at infinite graphs ina large class of random networks with light-tailed degreedistribution [11], a tree remains without loops at the criticalstate before the complete fragmentation by node removals.Dismantling (or decycling) problem known as NP-hard [12]is to find the minimum set of nodes in which removal leavesa graph with the largest connected cluster whose size is atmost a constant (or a graph without loops). It is suggestedthat the robustness becomes stronger as many loops exist aspossible. In fact, to be onion-like networks, enhancing loopsby copying or intermediation is effective for improving therobustness in incrementally growing methods [13], [14] basedon a local distributed process. Thus, we remark that loops makebypasses and may be more important than the degree-degreecorrelations in order to improve the connectivity in a net-work reconstruction after large disasters or attacks. However,identifying the necessary nodes to form loops is intractabledue to combinatorial NP-hardness, we effectively apply anapproximate calculation based on a statistical physics approachin our proposed self-healing. We assume that rewirings (reuseof undestroyed links) are performed by changing directions orranges of flight routes or wireless beams in the healing process,though we do not discuss the detail realization that dependson the current or future technologies and target systems.
ABLE I. A
SEQUENCE OF RESILIENCE CONCEPTS WHICH ARE PARTIALLYEXTRACTED FROM [4].
Resilience concept Characteristics Focus on
Engineering resilience Return time, Recovery,efficiency constancyEcological/ecosystem Buffer capacity, Persistence,resilience withstand shock, robustnessmaintain functionSocial − ecological Interplay disturbance Adaptive capacity ,resilience and reorganization , transformability,sustaining and learning,developing innovation
II. R
ELATED W ORKS
We briefly review recent progress of typical methods forrecovery and healing of a network in complex network science(inspired from fractal statistical physics) and computer science.In complex network science, several recovery and heal-ing methods have been proposed. As one of the recoverymethods, the strategies of random, greedy (for regaining thelargest connectivity), and preferential recovery weighted bypopulation have been considered in taking into account theorder of recovered links [15]. The effectiveness of recoveryfrom localized attacks is investigated on a square lattice.Against link failures, a simple recovery method has been alsointroduced to reconstruct an active tree for delivering from asource node by using back-up links [16]. However, it is unclearwhich pairs of two nodes should be prepared for back-up linksin advance.On the other hand, a self-healing method has been proposedby establishing new random links on interdependent (two-layered) networks of square lattices [17], and the effect againstnode attacks is numerically studied. In particular, for addinglinks by the healing process, the candidates of linked nodesare incrementally extended from only the direct neighborsof the removed node by attacks until no more separation ofcomponents occurs. In other words, the whole connectivityis maintained except the isolating removed parts (known asinduced sub-graphs for removed nodes in computer science).Note that such an extension of the candidates of linked nodesis a key idea in our proposed self-healing method as mentionedlater.Furthermore, the following self-healing methods, whoseeffects are investigated for some data of real networks, areworthy to note. One is a distributed local repair in order ofa priority to the most damaged node [18]. In the repair bylinking from the most damaged node to a randomly chosennode from the unremoved node set in its next-nearest neighborsbefore attacks, the order of damaged nodes is according to thesmaller fraction k dam /k orig of its remained degree k dam andthe original degree k orig before the attacks. The selectionsare repeated until reaching a given rate f s controlled by thefraction of nodes whose k dam /k orig falls below a threshold.Another is a bypass rewiring [19] on more limited resource oflinks (wire cables, wireless communication or transportationlines between two nodes) and ports (channels or plug socketsat a node). To establish links between pair nodes, a nodeis randomly chosen only one time in the neighbors of eachremoved node. When k i denotes the degree of removed node i , only ⌊ k i / ⌋ links are reused. Note that a degree represents the number of using ports at the node. In the bypass rewiring,reserved additional ports are not necessary: they do not exceedthe original one before attacks. Moreover, greedy bypassrewiring [19] is proposed in order to improve the robustness,the selection of pair nodes is based on the number of the linksnot yet rewired and the size of the neighboring components.In computer science, ForgivingTree algorithm has beenproposed [20]. Under the repeated attacks, the following self-healing is processed one-by-one after each node removal,except when the removed node is a leaf (whose degree is one).It is based on both distributed process of sending messages anddata structure, furthermore developed to an efficient algorithmcalled as compact routing [21]. In each rewiring process, aremoved node and its links are replaced by a binary tree.Note that each vertex of the binary tree was the neighbors ofthe removed node, whose links to the neighbors are reusedas the edges of the binary tree. Thus, additional links forhealing is unnecessary. It is remarkable for computation (e.g.in routing or information spreading) that the multiplicativefactor of diameter of the graph after healing is never morethan O (log k max ) , where k max is the maximum degree in theoriginal network, because of the replacing by binary trees.However, the robustness of connectivity is not taken intoaccount in the limited rewiring based on binary trees, sincea tree structure is easily disconnected into subtrees by anyattack to the joint node. In other research, a recovery strategywith resource allocation of bandwidth in a communicationnetwork is discussed at several service levels (from full topartial service) w.r.t what and how optimization [22], althoughconsidering the link’s thickness (e.g. defined by bandwidth ortransportation amounts) is out of our current scope.The characteristics of resource allocation are summarizedin Table II for the above conventional and our proposedmethods, although there has been no discussion about resourceof links and ports. ForgivingTree or bypass rewiring methodsis not controllable but strongly depending on the reuse ofall or half links before attacks. We assume that some linksemanated from a removed node i can be reused for healingby local rewiring between the neighbors. Some links (cablelines) may work at the neighbor’s sides, even though theyare disconnected at the removed node’s side. As a controlparameter in our simulation, we set the reusable rate r h according to the damage, on the assumption k i (1 − r h ) linksdo not work in the removed node’s degree k i . In the two kindsof resource, we consider that ports work independently fromconnection links, as similar to a relation of airport runaway(or plug socket) and flight by airplane (or cable line). TABLE II. R
ESERVED RESOURCE AT A NODE IN SELF - HEALING METHODS
Method Additional links Additional ports
ForgivingTree [20] Unnecessary, Two or three at mostenough by the original in a binary treeunder the assumption of reuseBypass Rewiring [19] Unnecessary, Unnecessary,if about half is reusable enough by the originalfrom the originalSimple Local Repair [18] Controllable Necessary f s (1 − q ) N according to f s and attack rate q Our Proposed Method Controllable Necessary M h according to r h and attack rate q II. E
FFECTIVE S ELF -H EALING
A. Outline of Proposed Methods
We assume that almost simultaneously attacked nodes arenot recoverable immediately, therefore are removed from thenetwork function for a while. In case of emergency for healing,unconnected two nodes are chosen and rewired as the recon-struction assistance or reuse of links emanated from removed qN nodes, when the fraction of attacks is q . The healingprocess in each of the following 1), 2), and 3) is initiated justafter detecting attacks and repeated by M h def = r h × ˜ P i ∈ D q k i links. Here, ˜ P i ∈ D q k i means the number without multiplecounts of lost links by attacks. D q denotes the set of removednodes, | D q | = qN . The key strategies are 1) enhancingloops contributes to improve the robustness [14], [23], 2)forming rings that encloses damaged parts is able to maintainthe connectivity on the edges of extended neighbors, and 3)complementary effects of 1) and 2) in the limited resourceof M h links. Any one of them is performed as the healingprocess.1) Enhancing loops for smaller q j + q j ′ To select two nodes in the neighbor nodes j, j ′ ∈ ∂i in the increasing order of q j + q j ′ for all i ∈ D q ,as shown in Fig. 1.2) Extended ringTo make rings of simple cycles without crossing,the neighbors are extended from the first damaged,the second damaged, . . . , to the last damaged areain this order, as shown in Fig. 2.3) Combination of extended rings & enhancing loopsfor smaller q j + q j ′ After using M r ≤ ˜ P i ∈ D q k i links for the rings, if M h > M r then the selections of two nodes inthe extended neighbors on ring are repeated in theincreasing order of q j + q j ′ for M h − M r links.Enhancing loops is performed by applying the values of q i (introduced in next subsection) for estimating Feedback VertexSet (FVS) whose nodes are necessary to form loops. Since anode i with small q i belongs to a dangling subtree with highprobability, by connecting such nodes, it is expected that a newloop on which a part of the subtree is included is added. Fromleft to right in Fig. 1, the original red links emanated from theremoved node i (marked by filled circle) are reused as the blueones for the healing. When there is at least a path between thenodes j and j ′ in Fig. 1, a new loop is created. Note that theattacked node i is isolatedly removed as breakdown.A ring is generated as follows. In Fig. 2, the process isinitiated in order of removals of three nodes from left to right.Filled and open circles denote removed and active nodes, redand magenta lines denote removed and virtually added links,respectively. From top left to top right in Fig. 2, a red node andits links are damaged, a ring formation around the 1st removalnode at the left is tried to the direct neighbors of it. A greenlink is established, while virtual magenta links are consideredby sending messages to active neighbors. From middle leftto middle right in Fig. 2, the ring formation around the 2nd If M h < M r , a ring is incomplete and an open chain is generated amongthe extended neighbors. In this case, additional rewirings based on smaller q j + q j ′ are not performed due to lack of links. removal node at the center is tried again to the neighborswhich include the extended ones by the virtual links. Lightblue links are added, but virtual magenta links are considered.From bottom left to bottom right in Fig. 2, the ring formationaround the 3rd removal node at the right is tried similarly.Finally, a ring is established by green, light blue, and bluelinks. The connections between neighbors on a ring are inrandom order except through the extension process. i j j’ Figure 1. Rewiring between nodes with small q j + q j ′ .Figure 2. Generation process of a ring. B. Applying Belief Propagation Algorithm
We review the following approximation algorithm [24],[25] derived for estimating FVS known as NP-hard problem[12]. It is based on a cavity method in statistical physics inthe assumption that nodes j ∈ ∂i are mutually independent ofeach other when node i is removed. The joint probability is P \ i ( A j : j ∈ ∂i ) ≈ Π j ∈ ∂i q A j j → i by the product of independentmarginal probability q A j j → i for the state A j as the index of i ’sroot. In the cavity graph, if all nodes j ∈ ∂i are either empty( A j = 0 ) or roots ( A j = j ), the added node i can be a root( A i = i ). There are the following exclusive states.1) A i = 0 : i is empty (removed). Since i is unnecessaryas a root, it belongs to FVS.2) A i = i : i becomes its own root. The state A j = j of j ∈ ∂i is changeable to A j = i when node i isadded.) A i = k : one node k ∈ ∂i becomes the root of i whenit is added, if k is occupied and all other j ∈ ∂i areeither empty or roots.The corresponding probabilities to the above three states arerepresented by q i def = 1 z i ( t ) , (1) q ii def = e x Π j ∈ ∂i ( t ) h q j → i + q jj → i i z i ( t ) ,q ki def = e x (1 − q k → i ) q k → i + q kk → i Π j ∈ ∂i ( t ) h q j → i + q jj → i i z i ( t ) ,q i → j = 1 z i → j ( t ) , (2) q ii → j = e x Π k ∈ ∂i ( t ) \ j (cid:2) q k → i + q kk → i (cid:3) z i → j ( t ) , (3)where ∂i ( t ) denotes node i ’s set of connecting neighbor nodesat time t , and x > is a parameter of inverse temperature. Thenormalization constants are z i ( t ) def = 1+ e x X k ∈ ∂i ( t ) − q k → i q k → i + q kk → i Π j ∈ ∂i ( t ) h q j → i + q jj → i i , (4) z i → j ( t ) def = 1 + e x Π k ∈ ∂i ( t ) \ j (cid:2) q k → i + q kk → i (cid:3) (5) × X l ∈ ∂i ( t ) \ j − q l → i q l → i + q ll → i , (6)to be satisfied for any node i and link i → j as q i + q ii + X k ∈ ∂i q ki = 1 ,q i → j + q ii → j + X k ∈ ∂i q ki → j = 1 . The message-passing iterated by Eqs. (1)-(6) is called beliefpropagation (BP). These calculations of q i , q ii , q ki , q i → j , q ii → j ,and q ki → j are locally executed through the message-passinguntil to be self-consistent in principle but practically to reachappropriate rounds from initial setting of (0 , random values.The unit time from t to t +1 for calculating a set { q i } consistsof a number of rounds by updating Eqs. (1)-(6) in order ofrandom permutation of the total N nodes. The distributedcalculations can be also considered.IV. S IMULATION R ESULTS
We evaluate the effect of healing by two measures: theratio S ( q )(1 − q ) N [18] for the connectivity and the efficiency E def = N ( N − P i = j L ij , where S ( q ) and L ij denote the sizeof GC (giant component or largest connected cluster) and thelength of the shortest path counted by hops between i - j nodes,respectively, for a network after removing qN nodes by attackswith recalculation of the highest degree node as the target.We investigate them for Open Flight between airports andInternet AS Oregon as examples of real networks [26], whose TABLE III.
NUMBER OF ADDITIONAL PORTS IN OUR PROPOSEDCOMBINATION METHOD . Open Flight: kmax = 242
PPPP rh q kmax = 1458
PPPP rh q
TABLE IV.
NUMBER OF ADDITIONAL PORTS IN THE CONVENTIONALSIMPLE LOCAL REPAIR METHOD . Open Flight: kmax = 242
PPPP rh q kmax = 1458
PPPP rh q number of nodes and links are N = 2905 , M = 15645 , and N = 6474 , M = 12572 . The following results are averagedover samples with random process for tie-breaking in anode selection or ordering of nodes on a ring.Figure 3 shows the ratio of GC in the surviving nodeswhich may be divided into isolated clusters after attacks. Thenumber M h of rewiring is controlled by a parameter r h in thehealing. Red, green, blue, orange, and purple lines correspondto r h = 0 . , . , . , . , and . . Black line shows theresult for no healing. In comparison with same color lines, ourproposed combination method (marked by square) of extendedring and enhancing loops is superior with higher ratio thanthe conventional simple local repair [18] method (marked bycircle), whose healing works only for weak attacks in small q . We remark that in our proposed combination method thecases of r h ≥ . (overlapped orange and purple lines markedby square) maintains the almost whole connectivity in thesurviving (1 − q ) N nodes. In other words, the network functioncan be revived completely, if more than half of links emanatedfrom removed nodes are active. In r h ≤ . (green and redlines marked by square), making a ring is unfinished, the ratiois dropped. Moreover, since the ratio in Fig. 4 is lower than theratio marked by square in Fig. 3, only enhancing loops (marked R a t i o S ( q ) / (( - q ) N ) o f G C Fraction q of attacksOpen Flight 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 R a t i o S ( q ) / (( - q ) N ) o f G C Fraction q of attacksAS Oregon
Figure 3. Communicable or transportable size with healing by our proposedcombination (square) and conventional simple local repair (circle). by up-pointing triangle) or extended ring (marked by down-pointing triangle) has weaker effect than the combination.However, enhancing loops increase the ratio of GC moderatelyin r h ≤ . for q ≤ . (green and red lines marked by up-pointing triangle) in Fig. 4.As shown in Figs. 5 and 6, our proposed combinationmethod (marked by square) has higher efficiency than theconventional simple local repair method (marked by circle)in comparison with same color lines, although the effect inthe method by only enhancing loops (marked by up-pointingtriangle) or extended ring (marked by down-pointing triangle)becomes weaker with E < . . Dotted line shows the effi-ciency in the original network before attacks.On the other hand, we consider the additional ports whichshould be prepared besides reusable ports. The original portsat neighbors of a removed node remain and can be reused,even if the links at the neighbor’ sides are disconnected. Thus,there exist active ports of a node at least as many as itsdegree in the original network before attacks. Note that theminimum, average, and maximum degrees are k min = 1 , h k i = 10 . , and k max = 242 in Open Flight, k min = 1 , h k i = 3 . , and k max = 1458 in AS Oregon. Table IIIshows the maximum number of reserved additional ports inour proposed combination method. The number tends to belarger ranging from a few to nearly k max ∼ k max , as the R a t i o S ( q ) / (( - q ) N ) o f G C Fraction q of attacksOpen Flight 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 R a t i o S ( q ) / (( - q ) N ) o f G C Fraction q of attacksAS Oregon
Figure 4. Communicable or transportable size with healing by onlyenhancing loops (up triangle) or extended ring (down triangle). fraction q of attacks and the reusable rate r h increase. Shownin parentheses are averaged values over the nodes that requireadditional ports in each network with healing. The averagednumber of additional ports is small, thus the cost of resourceis so much inexpensive. While the number is small even forthe maximum in the conventional simple local repair methodas shown in Table IV. It is almost constant for varying r h and q . V. C ONCLUSION
We have proposed self-healing methods for reconstructinga sustainable network by rewirings against attacks or disastersin the meaning of resilience with adaptive capacity. The fun-damental rewiring mechanisms are based on maintaining theconnectivity on rings and enhancing loops for improving therobustness by applying BP algorithm inspired from statisticalphysics. As the resource allocation, the rewirings are controlledby a parameter r h for reuse or addition of links betweenthe extended neighbors of attacked nodes. We have shownthat our proposed method is better than the conventionalsimple local repair method [18] with a priority of rewiringsto more damaged nodes, although reserved additional portsare required much more. In particular, the whole connectivitycan be revived with high efficiency of paths in our proposedmethod, when more than half links emanated from attackednodes alive. Thus, such amount of links are necessary for E ff i c i en cy E o f pa t h s Fraction q of attacksOpen Flight 0 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 E ff i c i en cy E o f pa t h s Fraction q of attacksAS Oregon
Figure 5. Efficiency in the network with healing by our proposedcombination (square) and conventional simple local repair (circle). sustaining network function. If there is lack of the resource,the shortage parts should be compensated according to thedamages. A
CKNOWLEDGMENT
This research is supported in part by JSPS KAKENHIGrant Number JP.17H01729.R
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