A loop enhancement strategy for network robustness
AA LOOP ENHANCEMENT STRATEGY FOR NETWORK ROBUSTNESS
A P
REPRINT
Masaki Chujyo ∗ Japan Advanced Institute of Science and TechnologyIshikawa, 923-1292 Japan [email protected]
Yukio Hayashi
Japan Advanced Institute of Science and TechnologyIshikawa, 923-1292 Japan [email protected]
January 1, 2021 A BSTRACT
Many real systems are extremely vulnerable against attacks, since they are scale-free networksas commonly existing topological structure in them. Thus, in order to improve the robustness ofconnectivity, several edge rewiring methods have been so far proposed by enhancing degree-degreecorrelations. In fact, onion-like structures with positive degree-degree correlations are optimallyrobust against attacks. On the other hand, recent studies suggest that the robustness and loopsare strongly related to each other. Therefore, we focus on enhancing loops as a new approach forimproving the robustness. In this work, we propose edge rewiring methods and evaluate the effect onthe robustness by applying to real networks. Our proposed methods are two types of rewirings inpreserving degrees or not for investigating the effect of the degree modification on the robustness.Numerical results show that our proposed methods improve the robustness to the level as same ormore than the state-of-the-art methods. Furthermore, our work shows that the following two pointsare more important for further improving the robustness. First, the robustness is strongly relatedto loops more than degree-degree correlations. Second, it significantly improves the robustness byreducing the gap between the maximum and minimum degrees. K eywords edge-rewiring · robustness against attacks · enhancing loops Improving robustness against malicious attacks has been one of the important issues in network science. Because manyreal networks are scale-free whose degree distributions follow power-law, and their connectivity is extremely vulnerableagainst removal of targeted nodes [1, 2, 3]. This vulnerability causes the loss of essential functions as networks in manyreal systems, which operate on the assumption that all nodes are connected.For example, the following infrastructures were damaged by disasters, and caused significant impacts on our society.A massive ice storm in Eastern Canada caused a long-term blackout in 1998 [4]. It reveals the strong dependence onelectric power. Also, a power outage in the USA and Canada continued for two days in 2003, and caused transportationand economic disruptions [5]. In 2008, a power grid network in China broken down due to heavy snowfall [6]. Onthe other type of disasters, the eruption of Icelandic volcano Eyjafjallajökull in 2010 affected European air traffic andstranded thousands of passengers [7]. The earthquakes and tsunamis that struck Japan in 2011 caused a terrible loss oflife and property further disrupting the global supply chain network [8]. In 2012, Hurricane Sandy destroyed the largeareas in New York and New Jersey [9]. After that, the blackout in several months affected the transportation network,and caused multiple damages [10]. Thus, these infrastructures directly connected to our life have potential risks, and itis necessary to design a new structure to mitigate the outage.They are also seriously damaged by attacks with targeting a specific part. For example, the North American powernetwork is robust against failures, but its function is greatly reduced for targeted attacks [11]. Moreover, an assessmentof the urban rail transport network indicates that the Shanghai Metro is vulnerable to degree-based attacks [12]. In ∗ Corresponding author a r X i v : . [ phy s i c s . s o c - ph ] D ec PREPRINT - J
ANUARY
1, 2021investigating the robustness of the global air transport network against intentional attacks, the weak points are discussedin a viewpoint from each airport’s centrality [13]. Furthermore, the robustness is analyzed against both failures andattacks of airline network routes in combining Low Cost Carriers (LCCs) and Full Service Carriers (FSCs) [14]. It isconcluded that route networks of LCC are more robust than ones of FSC.Since many of the above infrastructures in daily life have vulnerability against attacks, several methods should bedeveloped for improving the robustness of connectivity in network systems. We remark that such systems can be morerobust by partial edge rewiring without adding new resources of edges [15]. Thus, we aim to improve the robustnesswithout adding any resources in assuming that the number of both nodes and edges is constant. Particularly, the edgesin the airline network or the wireless communication network can be easily changed as rewiring. The rewiring withpreserving degrees in the airport network or wireless communication is possible by changing the destination of theairport or the direction of the wireless beam. In contrast, it may be difficult to change the network structure when edgesare spatially embedded, such as road networks, water supply networks, and power grid networks. However, even onsuch systems, it will be useful for maintaining network functions to make the robust structure by adding new resourcesor renovating and rebuilding.Although we discuss the robustness of connectivity by attacks, we may consider other attacks. Some edge rewiringalgorithms have also been proposed as adversarial attacks against link prediction [16] and community detection [17].While they aim to rewire the connections for privacy protections, we aim to enhance the tolerance of connectivityagainst node removals.On the other hand, an onion-like structure with positive degree-degree correlations [18] is optimally robust againsttargeted attacks under a given degree distribution [15, 19]. The degree-degree correlations r is defined as the Pearsoncorrelation coefficient for degrees at both ends of an edge [18]. The onion-like structure is visualized by arrangingsimilar degree nodes on a concentric circle in decreasing order of degrees from the core to the peripheral. Sincesimilar degree nodes tend to be connected by the positive degree-degree correlations, they draw a circle. It can begenerated by greedy rewiring to maximize a robustness index R hub , which accumulates the size of the largest connectedcomponent after attacks [15]. The robustness index R hub is defined as R hub = N P q =1 /N S ( q ) , where S ( q ) denotesthe number of nodes included in the largest connected component after removing qN nodes, and q is the fraction ofremoval nodes by high degree adaptive attacks. However, there is no strict definition of an onion-like structure for thethresholds of R hub or r , since too high degree-degree correlations rather decrease the robustness [15, 19, 20]. Thus,through numerical simulations, it is considered that the onion-like networks have R hub > . and r > . because notonion-like scale-free networks by Barabási-Albert model [1] have R hub < . and r ≈ at the same size [21]. Thevalues of R hub and r for onion-like networks are also obtained by rewiring for enhancing degree-degree correlations[22]. Note that the R hub represents the area under the curve S ( q ) /N versus q = N , N , ... N − N , , and can take differentvalues for networks with the same critical point q c for whole fragmentation. In other words, the R hub takes a largevalue when S ( q ) decreases steeply at q c , whereas it takes a small value when S ( q ) decreases gently even at the same q c .Based on enhancing the degree-degree correlations, several rewiring methods have been proposed for improving therobustness [23, 22]. However, in recent years, an incrementally growing method is also proposed for constructing anonion-like network by enhancing loops (or cycles in graph theory) instead of the degree-degree correlations [24, 21]. Ithas been suggested that there is a strong relation between robustness and loop structure.In this work, we propose new rewiring methods to enhance loops, and discuss the topological structures in improvingthe robustness for real data of the infrastructure networks. We emphasize the relation between robustness and looprather than the conventional degree-degree correlations. We explain our motivations for the rewiring strategy in enhancing loops. Several methods have been so far proposed forimproving the robustness to be an onion-like structure by increasing the degree-degree correlations [23, 22]. However, anetwork with the extremely high degree-degree correlations is not the best [15, 19, 20]. Therefore, for the improvementof robustness, there may exist other approaches instead of the degree-degree correlations.We remark a strong relation of robustness and loops from some suggesting works [25, 21]. One of them is theequivalence of network dismantling and decycling problems [25]. Here, network dismantling problem is finding aminimum set of nodes that removal makes the network broken into connected components at most a given size. Networkdecycling problem is finding a minimum set of nodes that removal makes the network without loops. The decyclingset is named as Feedback Vertex Set (FVS) in computer science. The equivalence means that networks become a treestructure at the critical point before the whole fragmentation. Therefore, in order to avoid fragmentation, it is necessarynot to be a tree as long as possible against node removals. On the other hand, the relation of the robustness and loops is2
PREPRINT - J
ANUARY
1, 2021also discussed in generating the onion-like structure with the optimal tolerance of connectivity against attacks. In thegeneration based on a pair of random and intermediation attachments, a new node links to a randomly selected node andthe minimum degree node in distant neighbors through a few hops of intermediation from the randomly selected pairnode [21]. We intuitively understand that many loops with bypasses are formed by pairs of attachments as shown in Fig.1. In fact, it is found that the robustness index R hub and the size of FVS have strong correlations in the networks. A new nodeRandom attach.
Random attach.intermediation. intermediation.Bypass in existing network.
Figure 1: Illustration of pairs of intermediation attachments. Black bold and blue dashed lines denote added edges andexisting paths in the network. Orange line denote paths of intermediations from randomly selected nodes.Focusing the above correlations, we propose a new edge rewiring strategy for enhancing loops in two types: Preservingand Non-Preserving with modification of the degree distribution. As similar to the conventional methods, in Preserving,the rewiring does not change each node’s degree under the original degree distribution. However, in Non-Preserving,the rewiring changes degrees and the degree distribution in order to investigate the effect of changes in the degreedistribution on robustness.
Spanning trees and the fundamental cycles
The fundamental system of cycles (loops) in a spanning tree is known in graph theory. A spanning tree is a subgraphthat all nodes of the network are connected without loops. The chords are edges not belonging in the spanning tree.Each chord and a loop called a fundamental cycle are one-to-one correspondings [26]. In other words, a spanning treehas M − N + 1 fundamental cycles as a linearly independent basis, where M and N denote the numbers of edgesand nodes. Therefore, any loop is represented as a combination of the basis. It is expected that there are many loopsindependently on networks with a large number of spanning trees. Independently from the above our explanation, arewiring method for increasing the number of spanning trees has been proposed by applying the perturbation theory ofthe Laplacian matrix [27]. The authors consider a rewiring in Preserving by the addition and removal of edges basedon the Kirchhoff’s matrix-tree theorem [28]. In contrast, we consider the edge rewiring strategy to enhance loops byincreasing the size of Feedback Vertex Set instead of the number of spanning trees. Feedback Vertex Set
Since the Feedback Vertex Set (FVS) is the minimum set of nodes whose removal makes the network acyclic, theremaining trees after removing the FVS are easily fragmented by further removal. The attack method to a node estimatedin the FVS by Belief Propagation is proposed [29]. Inversely, it is expected that increasing the size of FVS leads toimprove the robustness of connectivity against node removal.3
PREPRINT - J
ANUARY
1, 2021For increasing the size of FVS, we propose a new rewiring strategy by enhancing loops to improve the robustness. Sinceto find the FVS belongs to a class of NP-hard combinatorial optimization problems, the exact solution is intractable fora large network. Therefore, we apply an approximation algorithm of Belief Propagation (BP) in statistical physics [30].The algorithm estimates the probability q i belonging to FVS for node i . Here, q A i i denotes the marginal probability fornode i ’s root: A i = 0 (empty) or A i = i (the root is itself). When the node is empty, it is unnecessary as a root so that itis estimated as belonging to the FVS. Based on a cavity method [30, 29], the explicit formulas are q i = 1 z i , (1) z i = 1 + e x " X k ∈ ∂i − q k q k → i + q kk → i j ∈ ∂i [ q j → i + q jj → i ] , (2)where ∂i denotes node i ’s set of neighbor nodes, x > is a parameter of inverse temperature, and z i is normalizationconstant. The q i → j and q ii → j are calculated from the following self-consistent BP equations, q i → j = 1 z i → j , (3) q ii → j = e x Q k ∈ ∂i \ j [ q k → i + q kk → i ] z i → j , (4) z i → j = 1 + e x Y k ∈ ∂i \ j [ q k → i + q kk → i ] × X l ∈ ∂i \ j − q l → i q l → i + q ll → i , (5)where ∂i \ j denotes node i ’s set of neighbor nodes except node j , and z i → j is normalization constant. Equations(1)-(5) are iterated from an initial set of random values in (0 , until given rounds in practically. In each round, a set { q i | i = 1 , ..., N } are updated in order of random permutation of the all N nodes. To obtain the FVS, we remove anode i with a higher q i and recalculate a set { q i } for all existing nodes until given rounds. The removed nodes areestimated as the FVS. We repeat them until the network without loops. A node i with a smaller q i is less related toloops. In other words, a node i with a smaller q i tends not to belong to FVS. Using this { q i } , we consider the edgeaddition and deletion for increasing the size of FVS as follows. a b l k j i lkj i lk ji lk ji Figure 2: Illustration of our proposed methods. (a) BP Non-Preserving, (b) BP Preserving. The nodes with larger q i and q j and the edges between them are filled with red, while the nodes with smaller q k and q l and the edges betweenthem are filled with blue. 4 PREPRINT - J
ANUARY
1, 2021
BP Non-Preserving
The following rewiring without preserving degrees is called BP Non-Preserving. Here this rewiring modifies degreesunder a constant number of nodes and edges. In BP Non-Preserving, to increase the size of FVS, we add an edge ( k, l ) between nodes with smaller q k and q l in the all unconnected nodes pairs. It is expected to increase the size of FVS byadding the edge, since these nodes tend not to belong to any loops, and their connection makes a new loop. To keepthe number of edges, we remove an edge ( i, j ) between nodes with larger q i and q j in the all connected nodes pairs.Removing the edge ( i, j ) has little impact on the size of FVS, since the nodes i and j are on many loops because oflarge q i and q j as candidates of FVS. As shown in Fig. 2a, the following steps are repeated in BP Non-Preserving.Note that we may exchange steps 1 and 2 because they are independent processes. Step 1.
Add a non-existing edge ( k, l ) with the minimum q k and q l . Step 2.
Remove an edge ( i, j ) with the maximum q i and q j . Step 3.
Recalculate { q i | i = 1 , ..., N } . BP Preserving
The following rewiring with preserving degrees is called BP Preserving. As similar to [27], we apply the three steps:( i )add a non-existing edge ( i, j ) , ( ii ) remove edges ( i, k ) and ( j, l ) , and ( iii ) add a non-existing edge ( k, l ) . We illustratethe steps before and after rewiring at the top and bottom in Fig. 2b, respectively. It consists of the additions of twoedges ( i, j ) and ( k, l ) , and the removal of two edges ( i, k ) and ( j, l ) in order to preserve degrees.For increasing the size of FVS, it is effective to add two edges between nodes with smaller q i . However, it can notapply to BP Preserving, since the removal for preserving degrees in the above step ( ii ) makes fragmentation. Nodeswith smaller q i tend to have smaller degrees and belong to a tree. In the worst-case, when we select nodes i and j withdegree one or on a dangling tree, they are isolated by the removal. To avoid it, we select two unconnected nodes i and j with larger q i and q j in all nodes at first, since they tend to have large degrees and removing edges emanated fromthem is not likely to decrease the connectivity. Then, we select unconnected nodes k and l with smaller q k and q l in theneighbors of nodes i and j , respectively, in order to enhance loops by connecting them in the above step ( iii ). Since thenodes k and l are selected in the neighbors of nodes i or j , not in all nodes, they may be a little contained in loops,which is not the worst-case and prevents isolation. In this way, we first add an edge ( i, j ) between nodes i and j withlarger q i and q j in avoiding fragmentation by rewiring as much as possible. These selections possibly increase the sizeof FVS, since the nodes k and l with smaller q k and q l are expected to be included in new loops. For the removal, weselect edges ( i, k ) between nodes with larger q i and smaller q k , and ( j, l ) between nodes with larger q j and smaller q l .The removals have little impact on the size of FVS, since the edges linked to nodes with smaller q k or q l tend to belongto fewer loops. As shown in Fig. 2b, the following steps are repeated in BP Preserving. Step 1.
Let ( i, j ) be a non-existing edge with the maximum q i and q j in a network. Step 2.
Let k be a node with the minimum q k in the neighbor of either node i . Let l be a node with the minimum q l inthe neighbor of node j but not the neighbor of node k . Step 3.
Add non-existing edges ( i , j ) and ( k , l ) and remove edges ( i , k ) and ( j , l ). Step 4.
Recalculate { q i | i = 1 , ..., N } . In this section, we evaluate the effects of enhancing loops on the improvement of robustness in our proposed methods.Also, we discuss a relation between the robustness and the size of FVS. For comparison, Degree, WuHolme [22], andSP [27] are investigated. Degree is a modification of our rewiring strategy between nodes with smaller degrees insteadof smaller q i of BP. It has two types: Degree Preserving and Degree Non-Preserving, corresponding to BP Preservingand BP Non-Preserving. We consider WuHolme as a baseline in Preserving, because it is the best conventional methodfor improving the robustness by increasing the degree-degree correlations. SP is the previously mentioned method forincreasing the number of spanning trees, and it also has two types: SP Preserving and SP Non-Preserving. We compareit as a different approach in order to enhance loops.We apply our proposed and conventional methods for several real networks including social, biological, and technologicalnetworks, for which similar results are obtained (See Additional File). As a typical result, Figure 3 shows the robustnessindex R hub , the approximate size of FVS, and the degree-degree correlations r versus the number of rewirings for a real5 PREPRINT - J
ANUARY
1, 2021Figure 3: The robustness index, the approximate size of FVS, and the degree-degree correlations vs. the number ofrewiring. (Left: a, c, e) Rewirings in Preserving. Violet, green, and light blue solid lines denote the result by Degree, BP,and SP Preserving, respectively. The red dot line indicates a baseline of the conventional best. (Right: b, d, f) Rewiringsin Non-Preserving. Violet, green, and light blue solid lines denote the result by Degree, BP, and SP Non-Preserving,respectively.network: an airline network named OpenFlights with N = 2905 nodes and M = 15 , edges [31]. The robustnessindex R hub is the sum of the fraction of nodes in the largest connected component against high degree adaptive attackswith recalculation of degrees [15]. The degree-degree correlations r is the Pearson correlation coefficient for degrees[18]. In the following, | FVS | denotes the size of FVS by Belief Propagation [30], and is the number ofrewirings. Note that almost all edges are rewired at , which is nearly half of M .First, we show the results for our proposed BP Preserving. At the left in Fig. 3c, BP Preserving (denoted by the greenline) increases up to | FVS | = 698 from the original | FVS | = 528 before rewiring at . The rate of FVSrises from 18% to 24%. Since the baseline (denoted by the red dot line) is | FVS | = 646 , BP Preserving increases | FVS | over the baseline. Furthermore, at the left in Fig. 3a, BP Preserving (denoted by the green line) improves R hub to almost the same as the baseline (denoted by the red dot line). The maximum R hub is 0.175 in BP Preserving (denotedby the green line) and 0.174 in the baseline (denoted by the red dot line) at the left in Fig. 3a. The maximum R hub in Degree Preserving (denoted by the violet line) is 0.136, which is smaller than that in BP Preserving (denoted by6 PREPRINT - J
ANUARY
1, 2021the green line). Therefore, BP Preserving increases | FVS | more than other methods of Degree and SP Preserving andimproves R hub to almost the same level as the conventional best in Preserving. Enhancing the loop effectively improvesthe robustness.Next, we show the results in BP Non-Preserving. At the right in Fig. 3d, BP Non-Preserving (denoted by the greenline) increases to the maximum | FVS | = 1549 , which is 53% of nodes are included in FVS. It is about twice largerthan | FVS | = 698 for the baseline (denoted by the red dot line) at the right in Fig. 3d. In addition, other rewirings inNon-Preserving also increase | FVS | at the same level as BP Non-Preserving. The maximum | FVS | is 1566 in DegreeNon-Preserving (denoted by the violet line) and 1527 in SP Non-Preserving (denoted by the light blue line) at theright in Fig. 3d. At the right in Fig. 3b, BP Non-Preserving (denoted by the green line) increases to the maximum R hub = 0 . . It is also about twice larger than R hub = 0 . for the baseline (denoted by the red dot line). As similarto the result for | FVS | , other methods also increase R hub to almost the same level as BP Non-Preserving. The maximum R hub is 0.405 in Degree Non-Preserving (denoted by the violet line) and 0.392 in SP Non-Preserving (denoted bythe light blue line) at the right in Fig. 3b. Therefore, in comparison with the baseline, BP Non-Preserving is moreeffective for improving both R hub and | FVS | . Moreover, from the difference between the results in Non-Preserving andPreserving, it suggests that modification of the degree distribution significantly affects both R hub and | FVS | .We find that the rewirings in Non-Preserving commonly make the network more homogeneous and reduce the fractionof the high degree nodes as follows. Figure 4a shows the initial degree distribution for OpenFlights and the modifieddegree distribution after rewired 7800 times by each method.Figure 4: Change of the degree distributions by the rewirings in Non-Preserving. (a) Degree distributions in originaland after rewiring networks, (b) Maximum and minimum degrees vs. the number of rewiring in Degree, BP, and SPNon-Preserving for OpenFlights. The above three lines show the maximum degrees. The below three lines show theminimum degrees. Violet, green, and light blue denote Degree, BP, and SP Non-Preserving. Orange denotes the originaldegree distribution.The initial distribution has a long-tailed distribution characterized as a scale-free property, in which the minimum andmaximum degrees are 1 and 242. After rewiring by the three methods, a gap between the minimum and maximumdegree becomes smaller. In particular, SP Non-Preserving modifies to the very narrow gap with the minimum andmaximum degree of 9 and 41 in which the 61% occupy the nodes with degree 10. Figure 4b shows that the maximumdegree is lower than 100. The methods of Degree, BP, and SP Non-Preserving also decrease the gap between theminimum and maximum degrees. Thus, it is suggest that reducing the gap of degrees leads to improve both R hub and | FVS | significantly.The increases in R hub and | FVS | by BP Non-Preserving are partly due to changing the degree distributions as reducinglarge degree nodes. On the other hand, the increases in them by BP Preserving are only due to enhancing loopswith preserving degrees. However, the values of R hub and | FVS | in BP Non-Preserving are larger than ones in BPPreserving. 7 PREPRINT - J
ANUARY
1, 2021
Relation between the robustness and the size of FVS
In more detailed comparisons, we discuss a relation between R hub and | FVS | in Preserving and Non-Preserving. Atthe left in Figs. 3ace for Preserving, we compare the ordering of BP, Degree, SP Preserving, and WuHolme by themaximum value of each index. It is BP > WuHolme > Degree > SP for R hub , BP > Degree > WuHolme > SP for | FVS | , and Degree > BP > WuHolme > SP for r . The order for R hub and | FVS | are almost the same, only theorder of Degree and WuHolme are exchanged. As shown in Table 1, R hub and | FVS | have a very strong correlationcoefficient of 0.970 in Preserving. On the other hand, the correlation coefficient between R hub and r is 0.762. It islower than that of R hub and | FVS | . These values suggest that | FVS | is more strongly related to R hub than r .The difference in the ordering is more remarkable in Non-Preserving. At the right in Figs. 3bdf for Non-Preserving,in BP, Degree, SP Non-Preserving and WuHolme, it is Degree > BP > SP > WuHolme for both R hub and | FVS | ,while Degree > BP > WuHolme > SP for r . Furthermore, as shown in Table 1, the correlation coefficient inNon-Preserving is 0.980 for R hub and | FVS | . It is slightly larger than that in Preserving. On the other hand, thecorrelation between R hub and r is 0.527, which is smaller than that in Preserving. From these results, the correlationbetween R hub and r in Non-Preserving becomes weaker than that in Preserving. Thus, | FVS | is more strongly relatedto R hub than r .Table 1: The correlation coefficient between R hub and r , and R hub and | FVS | after rewiring. R hub and r R hub and | FVS | Preserving 0.762 0.970Non-Preserving 0.527 0.980
Strongly robust networks with negative degree-degree correlations
As known in the onion-like structure, it has been considered that networks with the moderate degree-degree correlationstend to be more robust [15, 19]. However, from the obtained results, we find that networks with the negative degree-degree correlations are possible to be highly robust. Figures 3bf show that SP Non-Preserving (denoted by the light blueline) decreases r , while increases R hub . SP Non-Preserving modifies it negatively up to -0.187, showing as the lightblue line at the right in Fig. 3f. However, at the right in Fig. 3b, SP Non-Preserving increases R hub at the almost samelevel as both Degree and BP Non-Preserving. Note that both Degree and BP Non-Preserving make the degree-degreecorrelations positive over the baseline. These obtained results are commonly found for other networks (See AdditionalFile). This study proposes a strategy for enhancing loops in increasing the size of FVS to improve the robustness of connectivity.We consider two kinds of rewirings for enhancing loops as BP Preserving and BP Non-Preserving. The rewiring inPreserving does not change each node’s degree, while the rewiring in Non-Preserving changes the degree. We obtainsimilar results in applying our proposed and conventional rewirings to several real networks (See the Additional Files).From the results, BP Preserving increases the size of FVS effectively. It also improves the robustness index to the levelas the same or more than the conventional best. Thus, enhancing loops is a useful strategy for improving robustness. Onthe other hand, BP Non-Preserving increases the robustness index and the size of FVS much more than the conventionalbest in Preserving. Moreover, the other rewirings in Non-Preserving also increase them as the same, and commonlyreduce the gap of maximum and minimum degrees. Therefore, it is suggested that reducing the difference in degreesstrongly affects increasing the robustness index and the size of FVS. Note that the results in BP Non-Preserving arepartly due to changing the degree distributions, while ones in BP Preserving are only due to enhancing loops.In addition, we discuss the relation between the robustness and the size of FVS. The size of FVS is more stronglyrelated to the robustness than the degree-degree correlations in both Preserving and Non-Preserving. We also find thatexisting of strongly robust networks with the negative degree-degree correlations is possible. Therefore, we suggest thatenhancing loops is more essential for improvements of robustness than the degree-degree correlations.
Acknowledgements
This research is supported in part by JSPS KAKENHI Grant Number JP.17H01729.8
PREPRINT - J
ANUARY
1, 2021
References [1] Barabási, A.-L., Albert, R.: Emergence of scaling in random networks. Science (5439), 509–512 (1999)[2] Albert, R., Jeong, H., Barabási, A.-L.: Error and attack tolerance of complex networks. Nature (6794), 378–382(2000)[3] Cohen, R., Erez, K., Ben-Avraham, D., Havlin, S.: Breakdown of the internet under intentional attack. PhysicalReview Letters (16), 3682 (2001)[4] Chang, S.E., McDaniels, T.L., Mikawoz, J., Peterson, K.: Infrastructure failure interdependencies in extremeevents: power outage consequences in the 1998 ice storm. Natural Hazards (2), 337–358 (2007)[5] Minkel, J.R.: The 2003 northeast blackout–five years later. Scientific American (2008). Accessed 26 Aug. 2020[6] Zhou, M., Liu, J.: A two-phase multiobjective evolutionary algorithm for enhancing the robustness of scale-freenetworks against multiple malicious attacks. IEEE transactions on cybernetics (2), 539–552 (2016)[7] Brooker, P.: Fear in a handful of dust: aviation and the icelandic volcano. Significance (3), 112–115 (2010)[8] MacKenzie, C.A., Santos, J.R., Barker, K.: Measuring changes in international production from a disruption: Casestudy of the japanese earthquake and tsunami. International Journal of Production Economics (2), 293–302(2012)[9] Manuel, J.: The long road to recovery: environmental health impacts of hurricane sandy. Environmental healthperspectives (5), 152–159 (2013)[10] Lipton, E.: Cost of storm-debris removal in city is at least twice the us average. The New York Times (2013).Accessed 26 Aug. 2020[11] Albert, R., Albert, I., Nakarado, G.L.: Structural vulnerability of the north american power grid. Physical ReviewE (2), 025103 (2004)[12] Sun, D.J., Zhao, Y., Lu, Q.-C.: Vulnerability analysis of urban rail transit networks: A case study of shanghai,china. Sustainability (6), 6919–6936 (2015)[13] Lordan, O., Sallan, J.M., Simo, P., Gonzalez-Prieto, D.: Robustness of the air transport network. TransportationResearch Part E: Logistics and Transportation Review (1), 155–163 (2014)[14] Lordan, O., Sallan, J.M., Escorihuela, N., Gonzalez-Prieto, D.: Robustness of airline route networks. Physica A:Statistical Mechanics and its Applications (1), 18–26 (2016)[15] Schneider, C.M., Moreira, A.A., Andrade, J.S., Havlin, S., Herrmann, H.J.: Mitigation of malicious attacks onnetworks. Proceedings of the National Academy of Sciences (10), 3838–3841 (2011)[16] Yu, S., Zhao, M., Fu, C., Zheng, J., Huang, H., Shu, X., Xuan, Q., Chen, G.: Target defense against link-prediction-based attacks via evolutionary perturbations. IEEE Transactions on Knowledge and Data Engineering, 1–1 (2019).doi: [17] Chen, J., Chen, L., Chen, Y., Zhao, M., Yu, S., Xuan, Q., Yang, X.: Ga-based q-attack on community detection.IEEE Transactions on Computational Social Systems (3), 491–503 (2019). doi: [18] Newman, M.E.: Assortative mixing in networks. Physical Review Letters (20), 208701 (2002)[19] Tanizawa, T., Havlin, S., Stanley, H.E.: Robustness of onionlike correlated networks against targeted attacks.Physical Review E (4), 046109 (2012)[20] Murakami, M., Ishikura, S., Kominami, D., Shimokawa, T., Murata, M.: Robustness and efficiency in intercon-nected networks with changes in network assortativity. Applied Network Science (1), 6 (2017)[21] Hayashi, Y., Uchiyama, N.: Onion-like networks are both robust and resilient. Scientific Reports (1), 1–13(2018)[22] Wu, Z.-X., Holme, P.: Onion structure and network robustness. Physical Review E (2), 026106 (2011)[23] Xulvi-Brunet, R., Sokolov, I.M.: Evolving networks with disadvantaged long-range connections. Physical ReviewE (2), 026118 (2002)[24] Hayashi, Y.: A new design principle of robust onion-like networks self-organized in growth. Network Science (1), 54–70 (2018)[25] Braunstein, A., Dall’Asta, L., Semerjian, G., Zdeborová, L.: Network dismantling. Proceedings of the NationalAcademy of Sciences (44), 12368–12373 (2016)[26] Bollobás, B.: Modern Graph Theory vol. 184. Springer, New York (2013)9 PREPRINT - J
ANUARY
1, 2021[27] Chan, H., Akoglu, L.: Optimizing network robustness by edge rewiring: a general framework. Data Mining andKnowledge Discovery (5), 1395–1425 (2016)[28] Buekenhout, F., Parker, M.: The number of nets of the regular convex polytopes in dimension ≤
4. DiscreteMathematics (1), 69–94 (1998). doi: [29] Mugisha, S., Zhou, H.-J.: Identifying optimal targets of network attack by belief propagation. Physical Review E (1), 012305 (2016)[30] Zhou, H.-J.: Spin glass approach to the feedback vertex set problem. European Physical Journal B (11), 455(2013)[31] Kunegis, J.: Konect: The koblenz network collection. In: Proceedings of the 22nd International Conference onWorld Wide Web. WWW ’13 Companion, pp. 1343–1350. Association for Computing Machinery, New York, NY,USA (2013). doi: . https://doi.org/10.1145/2487788.2488173 upplementary Information forA loop enhancement strategyfor network robustness M. Chujyo ∗ , and Y. Hayashi † Appendix
For 10 real networks shown in Table S1, we apply edge-rewiring methods: ourproposed methods (BP), a method with smaller degrees instead of smaller q i of BP (Degree), SP [1], and WuHolme [2]. As a preprocessing, we transformthe network data into undirected, unweighted, and no self-loop and no multipleedges, and extract the giant component. In the Figures, we compare the effectson the robustness index R hub [3], the approximate size of FVS by Ref [4], andthe degree-degree correlations r [5] versus the number of rewiring. In addition,to show the modifications in the degrees by rewiring in Non-Preserving, weshow the gap between the maximum and minimum degrees versus the numberof rewiring in original and after rewiring networks.The robustness and the size of FVS are more strongly related to each otherthan the degree-degree correlations. There is an exception for Power Grid shownin Figs. S5ac. BP Preserving (denoted by the green line at the left) increasesthe robustness over the baseline but decreases the size of FVS. We consider thatit is caused by a special property of Power Grid with a smaller average degreeand maximum degree, but a larger diameter shown in Table S1. ∗ [email protected] † [email protected] a r X i v : . [ phy s i c s . s o c - ph ] D ec etwork N M r
Min k < k >
Max k D
Figs. Refs. URLAirTraffic 1226 2408 -0.015 1 3.9 34 17 S1, S2 [6] urlE-mail 1133 5451 0.078 1 9.6 71 8 S3, S4 [7] urlPowerGrid 4941 6594 0.003 1 2.7 19 46 S5, S6 [8] urlYeast 2224 6609 -0.105 1 5.9 64 11 S7, S8 [9] urlJapanese 2698 7995 -0.259 1 5.9 725 8 S9, S10 [10] urlHamster 1788 12476 -0.089 1 14.0 272 14 S11, S12 [6] urlGRQC 4158 13422 0.639 1 6.5 81 17 S13, S14 [11] urlUCIrvine 1893 13835 -0.188 1 14.6 255 8 S15, S16 [6, 12] urlOpenFlights 2905 15645 0.049 1 10.8 242 14 S17, S18 [6, 13] urlPolblogs 1222 16714 -0.221 1 27.4 351 8 S19, S20 [14] urlTable S1: Basic properties for real networks after preprocessing. From theleft, we note the name of the network, the number of nodes, the number ofedges, the degree-degree correlations, the minimum degree, the average degree,the maximum degree, the diameter, figures, references, and available URL todownload the data. 2 irTraffic
Figure S1: AirTraffic [6]. Comparison of the robustness index R hub , the sizeapproximate of FVS, and the degree-degree correlation coefficient r vs. thenumber of rewiring. (Left: a, c, e) Rewirings in Preserving. Violet, green,and light blue solid lines denote the result by Degree, BP, and SP Preserving,respectively. The red dot line indicates a baseline of the conventional best.(Right: b, d, f) Rewirings in Non-Preserving. Violet, green, and light blue solidlines denote the result by Degree, BP, and SP Non-Preserving, respectively.3 irTraffic Figure S2: AirTraffic [6]. (a) Degree distributions in original and after rewiringnetworks, (b) Maximum and minimum degrees vs. the number of rewiring inDegree, BP, and SP Non-Preserving. The above three lines show the maximumdegrees. The below three lines show the minimum degrees. Violet, green, andlight blue denote Degree, BP, SP Non-Preserving. Orange denotes the originaldegree distribution. 4 -mail
Figure S3: E-mail [7]. Comparison of the robustness index R hub , the size ap-proximate of FVS, and the degree-degree correlation coefficient r vs. the numberof rewiring. (Left: a, c, e) Rewirings in Preserving. Violet, green, and light bluesolid lines denote the result by Degree, BP, and SP Preserving, respectively.The red dot line indicates a baseline of the conventional best. (Right: b, d, f)Rewirings in Non-Preserving. Violet, green, and light blue solid lines denotethe result by Degree, BP, and SP Non-Preserving, respectively.5 -mail Figure S4: E-mail [7]. (a) Degree distributions in original and after rewiringnetworks, (b) Maximum and minimum degrees vs. the number of rewiring inDegree, BP, and SP Non-Preserving. The above three lines show the maximumdegrees. The below three lines show the minimum degrees. Violet, green, andlight blue denote Degree, BP, SP Non-Preserving. Orange denotes the originaldegree distribution. 6 ower Grid
Figure S5: Power Grid [8]. Comparison of the robustness index R hub , the sizeapproximate of FVS, and the degree-degree correlation coefficient r vs. thenumber of rewiring. (Left: a, c, e) Rewirings in Preserving. Violet, green,and light blue solid lines denote the result by Degree, BP, and SP Preserving,respectively. The red dot line indicates a baseline of the conventional best.(Right: b, d, f) Rewirings in Non-Preserving. Violet, green, and light blue solidlines denote the result by Degree, BP, and SP Non-Preserving, respectively.7 ower Grid Figure S6: Power Grid [8]. (a) Degree distributions in original and after rewiringnetworks, (b) Maximum and minimum degrees vs. the number of rewiring inDegree, BP, and SP Non-Preserving. The above three lines show the maximumdegrees. The below three lines show the minimum degrees. Violet, green, andlight blue denote Degree, BP, SP Non-Preserving. Orange denotes the originaldegree distribution. 8 east
Figure S7: Yeast [9]. Comparison of the robustness index R hub , the size approx-imate of FVS, and the degree-degree correlation coefficient r vs. the number ofrewiring. (Left: a, c, e) Rewirings in Preserving. Violet, green, and light bluesolid lines denote the result by Degree, BP, and SP Preserving, respectively.The red dot line indicates a baseline of the conventional best. (Right: b, d, f)Rewirings in Non-Preserving. Violet, green, and light blue solid lines denotethe result by Degree, BP, and SP Non-Preserving, respectively.9 east Figure S8: Yeast [9]. (a) Degree distributions in original and after rewiringnetworks, (b) Maximum and minimum degrees vs. the number of rewiring inDegree, BP, and SP Non-Preserving. The above three lines show the maximumdegrees. The below three lines show the minimum degrees. Violet, green, andlight blue denote Degree, BP, SP Non-Preserving. Orange denotes the originaldegree distribution. 10 apanese
Figure S9: Japanese [10]. Comparison of the robustness index R hub , the sizeapproximate of FVS, and the degree-degree correlation coefficient r vs. thenumber of rewiring. (Left: a, c, e) Rewirings in Preserving. Violet, green,and light blue solid lines denote the result by Degree, BP, and SP Preserving,respectively. The red dot line indicates a baseline of the conventional best.(Right: b, d, f) Rewirings in Non-Preserving. Violet, green, and light blue solidlines denote the result by Degree, BP, and SP Non-Preserving, respectively.11 apanese Figure S10: Japanese [10]. (a) Degree distributions in original and after rewiringnetworks, (b) Maximum and minimum degrees vs. the number of rewiring inDegree, BP, and SP Non-Preserving. The above three lines show the maximumdegrees. The below three lines show the minimum degrees. Violet, green, andlight blue denote Degree, BP, SP Non-Preserving. Orange denotes the originaldegree distribution. 12 amster
Figure S11: Hamster [6]. Comparison of the robustness index R hub , the sizeapproximate of FVS, and the degree-degree correlation coefficient r vs. thenumber of rewiring. (Left: a, c, e) Rewirings in Preserving. Violet, green,and light blue solid lines denote the result by Degree, BP, and SP Preserving,respectively. The red dot line indicates a baseline of the conventional best.(Right: b, d, f) Rewirings in Non-Preserving. Violet, green, and light blue solidlines denote the result by Degree, BP, and SP Non-Preserving, respectively.13 amster Figure S12: Hamster [6]. (a) Degree distributions in original and after rewiringnetworks, (b) Maximum and minimum degrees vs. the number of rewiring inDegree, BP, and SP Non-Preserving. The above three lines show the maximumdegrees. The below three lines show the minimum degrees. Violet, green, andlight blue denote Degree, BP, SP Non-Preserving. Orange denotes the originaldegree distribution. 14
RQC
Figure S13: GRQC [11]. Comparison of the robustness index R hub , the sizeapproximate of FVS, and the degree-degree correlation coefficient r vs. thenumber of rewiring. (Left: a, c, e) Rewirings in Preserving. Violet, green,and light blue solid lines denote the result by Degree, BP, and SP Preserving,respectively. The red dot line indicates a baseline of the conventional best.(Right: b, d, f) Rewirings in Non-Preserving. Violet, green, and light blue solidlines denote the result by Degree, BP, and SP Non-Preserving, respectively.15 RQC
Figure S14: GRQC [11]. (a) Degree distributions in original and after rewiringnetworks, (b) Maximum and minimum degrees vs. the number of rewiring inDegree, BP, and SP Non-Preserving. The above three lines show the maximumdegrees. The below three lines show the minimum degrees. Violet, green, andlight blue denote Degree, BP, SP Non-Preserving. Orange denotes the originaldegree distribution. 16
CIrvine
Figure S15: UCIrvine [6, 12]. Comparison of the robustness index R hub , thesize approximate of FVS, and the degree-degree correlation coefficient r vs. thenumber of rewiring. (Left: a, c, e) Rewirings in Preserving. Violet, green,and light blue solid lines denote the result by Degree, BP, and SP Preserving,respectively. The red dot line indicates a baseline of the conventional best.(Right: b, d, f) Rewirings in Non-Preserving. Violet, green, and light blue solidlines denote the result by Degree, BP, and SP Non-Preserving, respectively.17 CIrvine
Figure S16: UCIrvine [6, 12]. (a) Degree distributions in original and afterrewiring networks, (b) Maximum and minimum degrees vs. the number ofrewiring in Degree, BP, and SP Non-Preserving. The above three lines showthe maximum degrees. The below three lines show the minimum degrees. Violet,green, and light blue denote Degree, BP, SP Non-Preserving. Orange denotesthe original degree distribution. 18 penFlights
Figure S17: OpenFlights [6, 13]. Comparison of the robustness index R hub ,the size approximate of FVS, and the degree-degree correlation coefficient r vs.the number of rewiring. (Left: a, c, e) Rewirings in Preserving. Violet, green,and light blue solid lines denote the result by Degree, BP, and SP Preserving,respectively. The red dot line indicates a baseline of the conventional best.(Right: b, d, f) Rewirings in Non-Preserving. Violet, green, and light blue solidlines denote the result by Degree, BP, and SP Non-Preserving, respectively.19 penFlights Figure S18: OpenFlights [6, 13]. (a) Degree distributions in original and afterrewiring networks, (b) Maximum and minimum degrees vs. the number ofrewiring in Degree, BP, and SP Non-Preserving. The above three lines showthe maximum degrees. The below three lines show the minimum degrees. Violet,green, and light blue denote Degree, BP, SP Non-Preserving. Orange denotesthe original degree distribution. 20 olBlogs
Figure S19: PolBlogs [14]. Comparison of the robustness index R hub , the sizeapproximate of FVS, and the degree-degree correlation coefficient r vs. thenumber of rewiring. (Left: a, c, e) Rewirings in Preserving. Violet, green,and light blue solid lines denote the result by Degree, BP, and SP Preserving,respectively. The red dot line indicates a baseline of the conventional best.(Right: b, d, f) Rewirings in Non-Preserving. Violet, green, and light blue solidlines denote the result by Degree, BP, and SP Non-Preserving, respectively.21 olBlogs Figure S20: PolBlogs [14]. (a) Degree distributions in original and after rewiringnetworks, (b) Maximum and minimum degrees vs. the number of rewiring inDegree, BP, and SP Non-Preserving. The above three lines show the maximumdegrees. The below three lines show the minimum degrees. Violet, green, andlight blue denote Degree, BP, SP Non-Preserving. Orange denotes the originaldegree distribution. 22 eferences [1] Chan, H., Akoglu, L.: Optimizing network robustness by edge rewiring: ageneral framework. Data Mining and Knowledge Discovery (5), 1395–1425 (2016)[2] Wu, Z.-X., Holme, P.: Onion structure and network robustness. PhysicalReview E (2), 026106 (2011)[3] Schneider, C.M., Moreira, A.A., Andrade, J.S., Havlin, S., Herrmann, H.J.:Mitigation of malicious attacks on networks. Proceedings of the NationalAcademy of Sciences (10), 3838–3841 (2011)[4] Zhou, H.-J.: Spin glass approach to the feedback vertex set problem. Eu-ropean Physical Journal B (11), 455 (2013)[5] Newman, M.E.: Assortative mixing in networks. Physical Review Letters (20), 208701 (2002)[6] Kunegis, J.: Konect: the koblenz network collection. In: Proceedings of the22nd International Conference on World Wide Web, pp. 1343–1350 (2013)[7] Guimera, R., Danon, L., Diaz-Guilera, A., Giralt, F., Arenas, A.: Self-similar community structure in a network of human interactions. PhysicalReview E (6), 065103 (2003)[8] Watts, D.J., Strogatz, S.H.: Collective dynamics of ‘small-world’networks.Nature (6684), 440 (1998)[9] Bu, D., Zhao, Y., Cai, L., Xue, H., Zhu, X., Lu, H., Zhang, J., Sun, S.,Ling, L., Zhang, N., et al. : Topological structure analysis of the protein–protein interaction network in budding yeast. Nucleic Acids Research (9),2443–2450 (2003)[10] Milo, R., Itzkovitz, S., Kashtan, N., Levitt, R., Shen-Orr, S., Ayzenshtat,I., Sheffer, M., Alon, U.: Superfamilies of evolved and designed networks.Science (5663), 1538–1542 (2004)[11] Leskovec, J., Kleinberg, J., Faloutsos, C.: Graph evolution: Densificationand shrinking diameters. ACM Transactions on Knowledge Discovery fromData (TKDD) (1), 2 (2007)[12] Opsahl, T., Panzarasa, P.: Clustering in weighted networks. Social Net-works (2), 155–163 (2009)[13] Opsahl, T., Agneessens, F., Skvoretz, J.: Node centrality in weighted net-works: Generalizing degree and shortest paths. Social Networks32