It has been well-known that many social, technological, and biological networks are commonly represented by a scale-free (SF) network whose degree distribution follows a power law [1]. However, a SF network is extremely vulnerable against targeted attacks to high-degree hub nodes [2, 3]. In infrastructure networks such as airlines, the Internet, and power-grids, these weak structures are frequently exposed to natural and man-made disasters. For example, an accident of drones by unknown attackers at an airport in England disrupted a thousand flights in Europe [4], while lots of airport shutdowns resulted from bad weather conditions and natural phenomena [5]. Also, the backbone of communication system, the Internet, is consistently threatened by distributed denial-of-service (DDoS) attacks [6]. Major causes of blackouts in North America are due to natural disasters [7]. In addition, the global supply chain network was damaged by the earthquakes and tsunamis that occurred in Japan in 2011 [8]. Since such infrastructures are necessary for our sustainable society, structural weakness should be overcome in order to maintain connectivity. Thus, a resilience-based system design has attracted attention, especially from system engineering and environmental science fields. The concept of resilience includes not only recovering the original state from disturbances but also reconstructing systems with adaptive capacity [9, 10]. We emphasized the reconstruction of damaged networks by healing for improving the robustness of connectivity because if a damaged network is recovered to the original structure, the extremely vulnerable SF structure still remains. For example, as a recovery method, several self-healing methods have been proposed recently [11, 12]. Comparing the healing methods [11, 12] with our method, these methods aim to recover the original structure, whereas our method aims to reconstruct a new structure. Therefore, the objectives are intrinsically different between recovery and reconstruction, although the aforementioned methods and our method are based on a similar strategy of healing by adding links and controlling the number of added links by a parameter. As shown in figures of Section 3, the networks reconstructed using our method have both higher robustness and efficiency than the original SF network (denoted by a black dashed line), with much less incomplete recovery by these methods in controlling fewer added links.

In a state-of-the-art research study, it has been found that an onion-like structure has the optimal robustness against malicious attacks [13, 14]. Within reserving a given degree distribution, this structure can be formed by whole rewiring for increasing degree–degree correlations [13, 15]. However, it is suggested that enhancing loops is more effective than increasing the correlations [16–19]. This is supported from a fact that network dismantling and decycling problems are asymptotically equivalent in random networks with light-tail degree distributions [16]. When the network becomes a tree without loops, we can intuitively understand that it is easily fragmented by any node removal. Here, the dismantling problem is finding the minimum set of nodes whose removal makes a graph fragmented into at most a given size, while the decycling problem is finding the minimum set of nodes whose removal makes a graph acyclic. The decycling set is also called as the feedback vertex set (FVS) in computer science, although finding the FVS belongs to a class of NP-hard in combinatorial optimization problems. To generate an onion-like structure, enhancing loops by copying [17] and intermediation [18] are effective in incrementally growing networks. Moreover, rewirings by enhancing loops with/without preserving a degree distribution generate the onion-like structure or robust networks with negative degree–degree correlations [19]. Thus, by enhancing loops, a network becomes more robust, as it is hard to be a tree. In other words, it corresponds to maximize the size of FVS approximately.

On the other hand, based on enhancing loops, a new self-healing method for network reconstruction has been proposed recently [20]. In order to get stronger connectivity, this method applies an extended connection range for new links as the key idea, which has been introduced for interdependent two-layered square lattices [21]. It has been shown [20] that the reconstructed network has higher robustness than the ones using conventional heuristic methods: simple local repair [22], in which healing links are added by priority of more damaged nodes, and bypass rewiring [23], in which a damaged node is randomly connected only one time with the other one. By Forgiving Tree algorithm [24] in computer science, although the connections between a removed node and its nearest neighbors are replaced as a tree to reduce the diameter of the reconstructed network, this algorithm does not consider the robustness of connectivity. Despite the advantages [20], it may be expected that the healing method based on enhancing loops is further improved by focusing on the network by the rewirings which has stronger robustness due to smaller gaps between the maximum and minimum degrees [19]. For reducing the gap in the healing after attacks, adding links for low-degree nodes is effective because high-degree nodes tend to be removed. This is the reason for connecting between low-degree nodes, as mentioned later.

In this study, several infrastructure networks imagined with adaptive wireless communication [25] or future microwave power transfer [26] are subjected to target systems for healing. Although the detail realizations are not discussed because of the dependence on the progress of device technologies, we described a distributed algorithm for controlling an autonomous network system, in which nodes work together for reconstructing from damaged systems after attacks or disasters. Since such infrastructures are large-scale and complex, a distributed control is more desirable than a central control because the distributed control network systems provide several advantages [27–29] in contrast to the central control. Each node has a same algorithm which can be applied to a network independently of its size or topological structure in a scalable manner. Since some tasks are processed concurrently at different locations, the execution times are shorter than those processed sequentially by a central control. A distributed control has high reliability to protect the whole down cooperatively by the remaining parts after attacks or disasters, while the malfunctions of control centers are fatal. Moreover, local resource allocation of healing links is available in a distributed algorithm for rapid reconstruction of a network. We designed a distributed healing algorithm with the advantages of scalability, concurrency, reliability, and local resource allocation.

Our article is organized as follows: in Section 2, we introduce the outline of our self-healing method in contrast to the previous method [20]. In Section 3, the effects of our method are numerically evaluated by measuring connectivity, robustness of connectivity, and efficiency of paths for typical three real networks. In each of them, the structural change of a healed network from the original one before attacks is investigated by both the number of neighbor nodes and its degree distribution. In Section 4, the autonomous distributed algorithm for our method is described in detail, although the simulation results are shown in Section 3 beforehand. In Section 5, we summarize the obtained results from Section 3 and Section 4.

Usually, attacked nodes are not recovered immediately in natural or man-made disasters, and the malfunctions may continue for a long time, and their neighbors are damaged by losing healing links. Under such urgent situation, our healing method compensates the lost connectivity. Without losing generality, we assumed that some links emanated from qN-attacked nodes can be reused for healing, where q denotes the fraction of simultaneously attacked nodes, and N denotes the original network size before attacks. This assumption is feasible: some parts of disconnected cable lines or wireless beams are reusable in a communication network, while directions of stopped flight routes can be changed in an airline network. However, the amount of reusable links depends on the damages on a network. If reusable links are insufficient, we also assume the additional support of links as an investment. Thus, we define the total number of links M _h for healing as follows. M h = def r h × ∑ i ∈ D q ∼ k i . (1) Here, k _i denotes the degree of node i, D _q denotes a set of attacked nodes, ∑ ∼ means the sum of attacked nodes’ degrees without duplicating counts, and r _h is a control parameter whose range is 0 < r _h ≤ 1 according to the damaged situation. For M _h -healing links, it is repeated to add a healing link between two damaged nodes.

Based on ring formation and enhancing loops, our method is slightly different from the previous method [20]. After describing each step in our method, we explain the reason why our method differs.

• Step 1 Initiation for healing

Before initiation, damaged nodes as neighbors of attacked nodes already have a local map in the normal mode. Remember that the local map stores the node IDs within three hops. The reason within three hops is as follows. Figure 1 shows the possible healing links between damaged nodes (blue circles) after attacks (red circles). In Figure 1B, node 1 finds only node 4 as the candidate for a healing link, when the range of local map is until two hops. Node 2 also finds only one node. Therefore, although nodes 1 and 2 create a healing link, the parts are still fragmented after healing. However, when the range is within three hops, as shown in Figure 1C, each node finds more candidates. In this example, node 1 initially has a set of candidates denoted as CN ₁ = (2, 4, 5) within three hops from node 1. Since node 2 belongs in CN ₁, CN ₂ = (1, 3, 4, 6) is merged as CN ₁ ← CN ₁ ∪ CN ₂. Moreover, since node 3 belongs in CN ₂, CN ₃ is also merged as CN ₁ ← CN ₁ ∪ CN ₂ ∪ CN ₃. Consequently, CN ₁ for node 1 is extended as (1, 2, 3, 4, 5, 6, 7). For nodes 2, 3, 4, 5, 6, and 7, same candidates are obtained. Thus, gradually extending the candidates by communicating control signals is reasonable in distributed local healing.

After extending the candidates, a ring between damaged nodes is formed by generating healing links (green lines in Figure 2). In the ring formation, every node on the ring has the same extended candidates which are on different sets of nodes on other rings. At several locations, rings can be formed in a healed network. Instead of a random order for connections as a ring in the previous method [20], we chose the candidates by decreasing order of their connected component sizes in order to maintain larger connectivity. As a tie-breaker, if more than two nodes belong to connected components with the same size, one node is randomly chosen in the order of connections for the ring.

Moreover, two nodes with the lowest degrees are selected on each ring (yellow lines in Figure 2) for enhancing loops. Our idea of connection between nodes with the lowest degrees is inspired from recent studies [30, 31]. This connection is created up to the number of residual healing links after ring formation. On the other hand, the previous method [20] applied a belief propagation (BP) algorithm [32] which estimates the feedback vertex set (FVS) as the necessary minimum nodes to form loops. We have shown that our method is slightly better than the previous method [20] in the next section.

For a distributed process, in the next section, we considered local resource control instead of global resource control for a sequential process in usual computation. In the local control, healing links are gathered and used only in each extended neighbor, whereas they are used in the whole remaining network in the global control. The numbers are calculated by modifying Eq. 1 with i ∈ D q l o c a l : a set of attacked nodes in an extended neighbor instead of D _q attacked nodes in the whole network. For example, if the value of r _h is 0.8 in Figure 2, the number of usable healing links is eight and four in the left and right, respectively, by the local control. However, four links are insufficient to form a ring completely in the right, while one link is residual in the left after ring formation. Thus, an incomplete ring is formed in the right.

On the other hand, by the global control, the number of usable healing links is 8 + 4 = 12. Note that seven and five links are necessary in the left and right, respectively. Since 12 links are usable in the whole network, a complete ring can be formed in both the left and right. In Section 3, we have shown almost same results of robustness and efficiency obtained by local and global controls. In Section 4, the distributed algorithm for local resource control is described.

We numerically evaluated our method in comparison with the previous method [20]. After reconstruction by both methods after malicious attacks, the following three measures were investigated for the typical infrastructure networks [33–35] related to the SF structure; it is numerically estimated that OpenFlight and AS Oregon have power-law degree distribution, while PowerGrid has an exponential degree distribution [36]. The basic properties of these networks are shown in Table 1. We chose the high degree adaptive attack (HDA) as malicious attacks so that the qN highest degree nodes with recalculation of degrees are removed from the networks.

• Ratio S(q)/(1−q)N, where S(q) denotes the number of nodes in the largest connected component (LCC) in the reconstructed network after attacks to qN nodes.

The ranges of these measures are 0 < S(q)/(1−q) N ≤ 1, 1 N < R ≤ 0.5 , and 0 < E ≤ 1. Table 1 shows the basic properties for the original networks extracted as connected graphs with undirected edges.

We investigated the three measures as averaged values of over 100 realizations for reconstructed networks by our and the previous method [20]. In our method, random selections are applied not only for tie-breaking in ring formation and enhancing loops but also for asynchronous processing, as mentioned in the next section.

Before explaining the results, we have summarized the most important point: since these measures for reconstructed networks by our method (marked by upper triangle in Figure 3) are higher than those by the previous method (marked by lower triangle in Figure 3), it is obvious that our method is effective to reconstruct a network efficiently. Each of the results is explained as follows.

For reconstructing a sustainable network, a higher ratio S(q)/(1−q) N is crucial to maintain larger connectivity. Figure 3A shows relatively high ratios for OpenFlight and AS Oregon in both the methods (upper triangle and lower triangle marks for each color line). Especially, the ratio has the highest value for r _h ≥ 0.5 (overlapped purple and yellow lines for OpenFlight and AS Oregon). If it is lower than 1, there exist isolated nodes even after healing. Moreover, high robustness of connectivity and short path length between nodes are also important for both the tolerance of healed network against further attacks and effective communications or transportations. Figures 3B,C show that our method has both higher values of R(q) and E(q) than those of the previous one for OpenFlight and AS Oregon (yellow and purple lines marked by upper triangle for r _h ≥ 0.5). In addition, the values of R(q) are higher than those for the original network (black dashed lines). As a result, Figure 3A suggests that both methods have similar ratios regardless of the connecting order of nodes on a ring. However, as shown in Figures 3B,C, the selection of the nodes with the minimum degrees for enhancing loops improves both values of R(q) and E(q) instead of applying BP in the previous method [20]. As an exception, slightly different results are partially obtained for PowerGrid with a large diameter in Table 1. For PowerGrid, in the right of Figure 3A, the ratio S(q)/(1−q) N rapidly decreases at q = 0.5 in our method (yellow and purple lines marked by upper triangle). In addition, as shown in the right of Figures 3B,C, the values of R(q) and E(q) by our method (marked by upper triangle) become lower than those by the previous method [20] (marked by lower triangle on yellow and purple lines for q ≥ 0.7).

We showed the size of neighbor nodes and the degree distribution as the structural changes of healed networks from the original ones in Figures 4, 5, respectively. The size of neighbor nodes is measured as the maximum ratio (MR) of local map. Remember that local map at each node consists of node IDs within three hops from it. The MR of the local map is defined as the ratio of the number of node IDs included in the local map and the original network size N. If this value is close to 1, it means that a node has almost all node IDs with a large memory space. Figure 4 shows the MR of the local map in a healed network by our method. As shown in Figure 4 except for PowerGrid, the reconstructed network by our method has the smaller MR than 0.6 for OpenFlight and 0.2 for AS Oregon. For AS Oregon and OpenFlight, the maximum RS is close to 1 (black dashed lines) because of existing hubs with huge degrees in those original networks before attacks, while the MR becomes smaller because of no hubs in the healed network, as shown in Table 2 in comparison with k _max in Table 1. However, in some cases (yellow and purple lines) for PowerGrid, the reconstructed network has a slightly higher MR than the original network (the right of Figure 4). Figure 5 shows the degree distribution in reconstructed networks by our method. For OpenFlight and AS Oregon, the degrees of most nodes are within a small range. In other words, the gap between maximum and minimum degrees in reconstructed networks becomes smaller rather than the gaps in the original SF networks with power-law degree distributions. Note that even if links are added for healing, the reconstructed networks do not become a complete graph because the total number of healing links even at r _h = 1.0 is not enough to form a complete graph, as shown in Table 3.

In the previous method [20], it is assumed that a node i can reuse its ports as much as the degree k _i in the original network at most. If healing links are added over k _i at a node i, additional ports are necessary. Here, ports are corresponded to channels or plug sockets at a node. Note that ports work independently of links just like airport runways or plug sockets and airplane flights or cables. Under these assumptions, the self-healing methods with a limitation of additional ports have been considered to avoid the intensive healing links at some nodes [20]. However, such limitations are out of scope in our method. The reasons are as follows. The degree distributions of the reconstructed networks by the previous method [20] follow an exponential distribution, and the maximum numbers of additional ports are less than the maximum degrees in the original networks before attacks. In contrast, the degrees of nodes in the reconstructed networks by our method are bounded within a narrow range, as shown in Figure 5. Moreover, the maximum degrees are decreased than ones in the original networks (Table 1 and 2). In particular, the maximum degrees by our method become 15, five, and seven for OpenFlight, AS Oregon, and PowerGrid at q = 0.1, respectively, while those by the previous method are 65, 19, and four [20]. Ever if the degrees of such nodes are one in the original networks, the numbers of additional ports are very few for our method. Thus, since our method saves the number of ports without the limitation of additional ports, we do not strictly take into account such limitation but consider that the amount of additional ports should be equipped in advance for healing.

Finally, two resource controls are compared as follows. Figure 6 shows the values of three measures for reconstructed networks by our method with local and global resource controls marked by plus and cross, respectively. Note that plus mark on each line in Figure 6 does not represent an error bar. As shown in Figure 6A for OpenFlight, local control has a slightly higher ratio than that of the global one in r _h ≤ 0.1 (red and green lines marked by plus sign), while the ratios for both controls almost coincide in r _h ≥ 0.2 (overlapped purple, blue, and yellow lines). In Figures 6B,C for OpenFlight and AS Oregon, the values of R (q) and E (q) for both controls also coincide in comparison with the same color lines. As an exception for PowerGrid, there exists a slight difference between the ratios in r _h ≤ 0.2 (red, green, and blue lines in Figure 6A). These results suggest that the reconstructed networks by local and global resource controls have almost the same properties for three measures, although local resource control is suitable for the distributed algorithm described in the next section.

This section explains the distributed version of our healing algorithm on the following assumptions in asynchronous processing [28, 29].

1. Each node immediately responds to messages stored in a FIFO queue by arriving in order which is not affected to local clocks at sending nodes.

In order to emulate our distributed algorithm on a PC, we applied the AsyncIO module in Python ^{1 2} . A task at a node consisted of sending and receiving messages between its extended candidates. Each of nodes generates a task concurrently by using AsyncIO. Although the AsyncIO module can emulate the concurrent execution of tasks generated at almost same time, the detail is internally ordered. To implement it with a corresponding short time delay encountered by a physical factor, we randomized the execution order of tasks in the total N nodes. Note that some nodes without generating tasks are excluded from the order. Figure 7 shows the schematic diagram of such task processing. Horizontal and vertical directions indicate N nodes in healing mode and global timeline of our healing process, respectively.

We have proposed a network reconstruction method by self-healing to further improve the previous method [20]. In particular, for enhancing loops on a ring, we have considered healing links between low-degree nodes which reduce the gap between the maximum and minimum degrees. Through computer simulation, we have shown that our method is effective to obtain higher robustness and efficiency than the ones by the previous method [20]. As our key idea inspired from the results [19, 30, 31] that the smaller gap makes stronger robustness, the degree distributions of reconstructed networks by our method are changed as having a narrow range of degrees from a power-law with a fat-tail distribution in the original network. Moreover, we have successfully emulated our method as asynchronous processing by randomizing execution-order of tasks with time delays from encountering physical factors. By our distributed algorithm, healing links can be locally allocated for rapid reconstruction after attacks. Moreover, the local distributed process has almost the same results as the sequential process. These findings provide a basis for how to reconstruct a robust and efficient network by locally distributed healing.

We discussed the limitation of our method. In Sections 2 and 4, several assumptions are established for performing our method. For example, links of a network are reusable or investable for healing. The rollback of the healing process is performed if additional attacks occur in a network, especially the most important assumption is that rapid rewiring in a network is available such as changing directions of flights or wireless beam. Our method works well on these assumptions. However, such rewiring cannot be performed in some systems such as road networks. Moreover, if the assumption of rollback is broken by very quick sequential attacks in a network, the usability of our method is limited. Despite its limitations, we expect that our method is effective for the networks with varying degree-degree correlations because we have considered that enhancing loops is more important than increasing the correlations for improving robustness [16–19]. In fact, even though AS Oregon has a negative correlation, as shown in Table 1, our method works effectively for that network. Moreover, the previous research study [19] shows that a robust network with negative correlations can be created by enhancing loops. However, the detail analysis for networks with degree–degree correlations may remain as future research.

On the other hand, we also assumed that malicious attack is HDA, in which qN nodes are removed simultaneously from the network. Thus, not the order of deletion but the set of attacked nodes affects the results in our study. However, considering quick sequential attacks, the order of deletion of nodes may be meaningful. Although such attacks are beyond our current scope, some discussions may be required as another future research.

Further investigation could be conducted to determine the effectiveness of our method against the more destructive attacks to loops, for example, CoreHD [38], collective influence [39], and BP attacks [40]. Other spatial attacks by earthquakes or floods may be also important.