Adding links on minimum degree and longest distance strategies for improving network robustness and efficiency

Masaki Chujyo; Yukio Hayashi

doi:10.1371/journal.pone.0276733

. 2022 Oct 26;17(10):e0276733. doi: 10.1371/journal.pone.0276733

Adding links on minimum degree and longest distance strategies for improving network robustness and efficiency

Masaki Chujyo ^1,^*,^#, Yukio Hayashi ^1,^#

Editor: Lazaros K Gallos²

PMCID: PMC9605036 PMID: 36288333

Abstract

Many real-world networks characterized by power-law degree distributions are extremely vulnerable against malicious attacks. Therefore, it is important to obtain effective methods for strengthening the robustness of the existing networks. Previous studies have been discussed some link addition methods for improving the robustness. In particular, two effective strategies for selecting nodes to add links have been proposed: the minimum degree and longest distance strategies. However, it is unclear whether the effects of these strategies on the robustness are independent or not. In this paper, we investigate the contributions of these strategies to improving the robustness by adding links in distinguishing the effects of degrees and distances as much as possible. Through numerical simulation, we find that the robustness is effectively improved by adding links on the minimum degree strategy for both synthetic trees and real networks. As an exception, only when the number of added links is small, the longest distance strategy is the best. Conversely, the robustness is only slightly improved by adding links on the shortest distance strategy in many cases, even combined with the minimum degree strategy. Therefore, enhancing global loops is essential for improving the robustness rather than local loops.

Introduction

Our social, productive, and economical activities are supported by many complex networks, such as the Internet, World Wide Web, electric power, transportation, water supply, and international trading systems. These real-world networks have commonly scale-free (SF) structures characterized by power-law degree distributions [1], and therefore are extremely vulnerable against malicious attacks [2]. On the other hand, the robustness of connectivity is strongly related to both the degree distributions and the types of node removals, e.g. malicious attacks or random failures. By the attacks, nodes are removed in the decreasing order of degrees or other topological properties such as betweenness or closeness centralities, while by random failures, nodes are removed uniformly at random. Unfortunately, many SF networks are vulnerable against malicious attacks but tolerant against random failures in comparison with synthetic Erdős-Rényi random graphs [2]. To overcome the vulnerabilities in real-world networks, we consider how to reconstruct more robust networks against malicious attacks, because a simple recovery only restores the original fragile structure.

In the state-of-the-art, an onion-like structure is known as the optimal robust network against attacks [3–5]. The onion-like structure has a positive degree-degree correlation [6], in which nodes with similar degrees tend to be connected. It is constructed by random rewiring in maximizing a robustness index against high degree adaptive attacks [3]. Moreover, several rewiring methods have been proposed for improving the robustness by selecting rewired nodes to increase the degree-degree correlations [7, 8].

Recently, apart from the degree-degree correlation, we remark that loops are deeply related to robustness against attacks [5, 9, 10]. In particular, the asymptotic equivalence of network dismantling and decycling has been derived [9]. Here, network dismantling problem is to find a minimum fraction of nodes whose removal makes the size of connected components into a given size, while network decycling or Feedback Vertex Set (FVS) problem is to find a minimum fraction of nodes whose removal makes the network no loops. Intuitively, a network without loops is fragile as a tree, because it is easily fragmented by any node removals. Thus, as a network is hard to become a tree by node removals, it is more robust. Furthermore, the relation between enhancing loops and robustness is also discussed for generating onion-like networks [5]. In the incrementally growing methods, a node is added at each time step in connecting to a randomly selected node and to a node with the minimum degree within a few hops from the randomly selected one. Since the attachments make new interwoven loops or bypasses, the incrementally growing method generates an onion-like network with both high robustness and a large size of FVS. Thus, it has been shown that the robustness and the size of FVS are strongly correlated [5]. Moreover, for improving the robustness, an effective rewiring method has been proposed by selecting nodes to increase the size of FVS instead of the degree-degree correlation [11]. However, the effects of the length of loops on the robustness are rarely discussed. Several recent studies suggest that long-range loops contribute to being robust more than short-range loops [12–15]. By a percolation analysis, it has been found that the robustness becomes weak due to short loops with a constant length in geographic networks [12, 13].

This paper considers link addition methods for improving the robustness of real networks in distinguishing the effects of long and short loops. Instead of rewiring which removes existing links, link additions can improve robustness without destructing the existing structures. Previous works [14, 16–20] have been discussed some link addition methods to improve the robustness. The basic methods repeatedly add links to pairs of unconnected nodes chosen uniformly at random. This random additions improve the robustness of SF [16] and geographical networks [17]. However, over the last decade, two effective strategies have been further proposed. Based on the minimum degree [16, 18–20] or the longest distance strategies [14], they select pairs of unconnected nodes to add links. It has been numerically shown that the minimum degree strategy improves the robustness more than the basic random additions in the same number of adding links [16]. On the other hand, it has been numerically shown that the longest distance strategy outperforms other methods on the robustness [14].

In networks with more complicated structures, link addition methods have also been discussed [21–23]. For improving the robustness of both interdependent and multiplex networks, the minimum degree strategy is effective [21]. Furthermore, the minimum degree strategy is also effective for enhancing the robustness against cascade failures induced by both random failures and targeted attacks in interdependent power grids and communication networks [22]. The minimum degree strategy is considered an effective method for interdependent and multiplex networks, while the longest distance strategy is effective for spatial networks. In spatial networks, adding many links or long links can improve robustness against attacks [23]. Although there are several discussions about effective link addition methods in various types of networks, they are basically based on either the minimum degree or longest distance strategies.

Thus, we investigate the contributions of the minimum degree and longest distance strategies to improving robustness against attacks by adding links, respectively. We point out that the effects of the minimum degree and longest distance strategies on the robustness tend to be overlapping, because the selected nodes are similar to each other in some cases. To distinguish the effects as much as possible, we introduce some link addition methods based on the two strategies. Through numerical simulations, we show that the robustness is the most improved by adding links on the minimum degree strategy for synthetic and real networks. As an exception, when the number of added links is small to an initial tree, the robustness is the most improved by adding links on the longest distance strategy. On the other hand, even combined with the minimum degree strategy, the robustness is only slightly improved by adding links on the shortest distance strategy for synthetic networks. Thus, adding links to pairs of the minimum degree nodes between longer distances is the most effective for improving robustness. However, when the minimum degree strategy selects nodes with longer distances on some real networks, the robustness is improved by adding links on both the minimum degree and shortest distance strategies. Moreover, we discuss the relation between improving the robustness and enhancing loops. From the obtained results, we emphasize that enhancing long loops is important for improving the robustness rather than short loops.

Materials and methods

We explain the similarity of nodes selected by the minimum degree and longest distance strategies in considering a tree with many leaf nodes as shown in Fig 1A. The leaf nodes are the end of trees and have the minimum degree. The longest distance strategy selects nodes with the maximum distance, while these nodes are leaf nodes with the minimum degree. Conversely, the minimum degree strategy selects leaf nodes with the minimum degree. Many leaf nodes are located far away rather than close together. Thus, the minimum degree strategy tends to select nodes with longer distances. Since these strategies select similar nodes on a tree, the effects on the robustness are very overlapping. However, the similarity becomes weak on a network with loops. Fig 1B shows an extreme example in which these strategies select different nodes. In the network, the longest distance strategy selects nodes with the maximum degree, while the minimum degree strategy selects nodes with a short distance. First of all, we set a tree as the initial network for adding links, and then set a real network to investigate the effects of strategies in tree and network with loops.

Fig 1 — (A) The minimum degree and longest distance strategies select similar nodes on a tree, while (B) these strategies select different nodes on a network with loops. Red and green links show added links on the minimum degree and longest distance strategies. Blue nodes are leaf nodes.

We introduce link addition methods with two-step selections for investigating the effects of the minimum degree and longest distance strategies on the robustness against attacks. At the first step for linking, we consider two types: candidates are all nodes or selected from nodes with the minimum degree. Here, we omit to select nodes with the maximum degree, because it does not obviously contribute to improving the robustness. At the second step, we consider three types: a pair of nodes is selected uniformly at random, by the longest, or by the shortest distances in the candidates. In repeating these two steps, we add links to the selected pairs in the combinations as shown in Table 1. In Table 1, the minimum degree strategy is applied to the min-k RandAdd, min-k LongAadd, and min-k ShortAdd methods. The longest distance strategy is applied to the LongAdd and min-k LongAdd methods. In addition, we consider the shortest distance strategy for distinguishing the effects of long and short loops. The shortest distance strategy is applied to the ShortAdd and min-k ShortAdd methods. The RandAdd method is the basic random additions. By applying the six link addition methods, we investigate the effects of the minimum degree and longest strategies on the robustness.

Table 1. Six link addition methods in the combinations of the first step by degree and the second step by distance.

First step candidates are	all nodes	selected by the minimum degree
Random distance	random additions method (RandAdd method) [16, 17]	min-k random additions method (min-k RandAdd method) [16, 18–20]
Longest distance	longest additions method (LongAdd method) [14]	min-k longest additions method (min-k LongAdd method)
Shortest distance	shortest additions method (ShortAdd method)	min-k shortest additions method (min-k ShortAdd method)

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In general, as increasing the number of links, the robustness is more improved. For example, the complete graph with N nodes and the maximum N(N − 1)/2 links has the optimal robustness against any attacks (and failures). Thus, we consider effective link addition methods to improve the robustness by adding $O (N)$ links. We apply the above link addition methods to synthetic and real-world networks for investigating the effects of these strategies on the robustness.

We first introduce synthetic networks and then explain real networks in the later paragraph. As synthetic networks, we consider random trees. Since trees are very fragile against attacks, we can clearly show the effects of improvement by adding a small number of links. In particular, because degree distribution strongly affects the robustness [2], we consider random trees with degree distributions ranging from power-law to exponential or narrower distributions. For generating random trees with the different degree distributions, we use a growing network (GN) model [24–26] and an inverse preferential attachment (IPA) model [27], which are extensions of the Barabási-Albert model [1]. In these models, a new node is added and connects a link to the existing node at each time step. By repeated attachments from the initial configuration of two connected nodes, the network becomes a tree with N nodes and N − 1 links. The degree distributions vary according to a value of parameter ν or β as follows. In the GN model, the connection probability from a new node to the existing node i is proportional to $k_{i}^{ν}$ , ν ≥ 0, where k_i denotes the degree of node i. The degree distribution is a power-law distribution for ν = 1, a power-law distribution with an exponential cut-off for 0 < ν < 1, and an exponential distribution for ν = 0 [1]. In the IPA model, the connection probability is proportional to $k_{i}^{- β}$ , β ≥ 0. As β becomes larger, the degree distribution changes from exponential (β = 0) to narrower distributions. Note that the GN model with ν = 0 and the IPA model with β = 0 are the same as uniformly random attachments. By using the GN and IPA models, we can generate random trees with degree distributions ranging from power-law to exponential or narrower distributions. Moreover, we also use configuration models [28, 29] for the random trees, because networks generated by the IPA model tend to have special chain-like structures [27]. In the configuration models without chain-like structures, it is numerically found that the robustness increases as the degree distributions become narrower [15]. Similar results are obtained for configuration models (See S1–S5 Figs).

As real networks, we consider an airline network [30], e-mail interchanges [31], and protein-protein interactions in budding yeast [32]. These networks are categorized as technological, social, and biological networks. Table 2 shows the number N of nodes, the number M of links, and the degree-degree correlation r [6] for these networks.

Table 2. Properties of real networks.

Name		N	M	r
AirTraffic	[30]	1226	2408	-0.015
Email	[31]	1133	5451	0.078
Yeast	[32]	2224	6594	-0.11

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For the link addition methods to synthetic and real-world networks, we compare the robustness index against malicious attacks [3]. As malicious attacks, we consider two types of attacks: the typical high degree adaptive (HDA) attacks [3] and the state-of-the-art most destructive belief propagation (BP) attacks [10]. The HDA attacks remove nodes in increasing order of degrees with recalculating degrees, while the BP attacks destroy loops by removing nodes with the highest probability of belonging to the FVS. For two types of attacks, we measure the robustness index $R = 1 / N \sum_{q = 1 / N}^{1} S (q)$ , where S(q) denotes the number of nodes in the largest connected components after qN nodes are removed [3]. The values of R range from 0 to 0.5. In the distinction by suffixes, R_HDA and R_BP denote the robustness index against the HDA and BP attacks, respectively. We also compare the size of FVS. However, it is difficult to obtain the exact solution of the minimum FVS for a large network, because to find the minimum FVS is NP-hard in combinatorial optimization problems. Therefore, we apply an approximate method by message-passing based on statistical physics [33].

As another measure of network topology that differs from the robustness index, we consider network efficiency. There is a trade-off between robustness and network efficiency in SF networks [34], whereas high robustness coexists with network efficiency in incrementally growing onion-like networks [35]. In link addition methods, several previous methods have been proposed for improving only the efficiency [20, 36]. Therefore, we investigate the impact of the minimum degree and longest distance strategies on not only the robustness but also network efficiency E and diameter D defined as the maximum length of shortest paths in the network. Here, network efficiency E is given by E = 1/(N(N − 1))∑_ij 1/d_ij, where d_ij denotes the shortest path length counted by the hops between nodes i and j [37].

Results

We investigate the robustness against the typical HDA and the worst-case BP attacks in comparing the corresponding size of FVS by applying six link addition methods. First, we show the results on random trees and SF networks as synthetic networks. Next, we show the results on real networks. As random trees, we set three random trees generated by the IPA model with β = 5 or the GN model with ν = 0 or 1. The random trees have degree distributions following narrower, exponential, or power-law distributions. The following results are averaged over 100 samples.

Fig 2A–2C show R_HDA after adding links to the initial random trees with narrower, exponential, and power-law degree distributions. It is trivial that R_HDA is increased, as the number of added links increases. Thus, we compare the slopes of lines corresponding to these methods in increasing R_HDA. Fig 2A–2C show that red and green solid lines (denote min-k RandAdd and min-k LongAdd methods) have the largest slopes among the six lines. Moreover, the difference between red and green solid lines is very small. From these results, both min-k RandAdd and min-k LongAdd methods are the most effective for improving R_HDA on a tree. Therefore, the minimum degree strategy contributes to improving the robustness. As an exception, green dotted line (denotes LongAdd method) is higher than the other five lines, when the number of added links is less than 170, 380, and 530 in Fig 2A–2C, respectively. Thus, only when the number of added links is small to an initial tree, the longest distance strategy is effective for improving the robustness. On the other hand, blue dotted and solid lines (denote ShortAdd and min-k ShortAdd methods) have smaller slopes in Fig 2A–2C. In particular, blue dotted lines are lower than the other five lines. Therefore, the contribution is small to improving the robustness from initial random trees on the shortest distance strategy.

Fig 3 shows R_BP after adding links to the random trees. Although the values of R_BP are slightly smaller than the values of R_HDA, the whole trend is almost same as shown in Fig 2. Once again, the min-k RandAdd and min-k LongAdd methods are the most effective for improving R_BP. As an exception, the LongAdd method is the most effective for improving R_BP, only when the number of added links is small in the range of less than 130, 380, and 540 added links in Fig 3A–3C, respectively. On the other hand, the ShortAdd method has the smallest slope among six methods in increasing R_BP.

Fig 4A–4C show the size of FVS after adding links to the initial random trees. Similar to the behavior of R_HDA, it is trivial that as the number of added links increases, the size of FVS is increased. Thus, we compare the slopes of lines in increasing the size of FVS. Fig 4A–4C show that red and green solid lines (denote min-k RandAdd and min-k LongAdd method) have larger slopes than the other three lines except for the blue solid line (denotes min-k ShortAdd method). Red and green solid lines are higher than the three lines in the range of more than 780, 460, and 890 added links in Fig 4A–4C, respectively. We remark that red and green solid lines outperform the other lines for increasing R_HDA and R_BP in Figs 2 and 3. Thus, the min-k RandAdd and min-k LongAdd methods are effective for both improving the robustness and increasing the size of FVS. Conversely, as shown in the blue solid line, the effect of the min-k ShortAdd method is strong for increasing the size of FVS but weak for improving the robustness. Blue solid line has the largest slope among the six lines for increasing the size of FVS in Fig 4, although this line has a small slope in increasing R_HDA as shown Fig 2. The reason of a large slope in increasing the size of FVS is considered as that the min-k ShortAdd method generates many short loops. By the min-k ShortAdd method, triangles are generated between leaf nodes with a same neighbor node. Such triangles increase the size of FVS but do not contribute to improving the robustness.

Table 3 summarizes the results for improving the robustness and the size of FVS from initial random trees. The min-k RandAdd and min-k LongAdd methods are the most effective for increasing both R_HDA and R_BP from ones in random trees. On the other hand, the effect of the min-k Short method is strong for increasing the size of FVS but weak for increasing the robustness. From these results, the minimum degree strategy contributes significantly to improving the robustness from initial random trees. However, even on the minimum degree strategy, adding links to pairs of nodes with short distances only slightly improves the robustness.

Table 3. The most effective methods for improving R_HDA, R_BP, and the size of FVS to random trees.

		β = 5	ν = 0	ν = 1
R _HDA	Fig 2	min-k RandAdd	min-k RandAdd	min-k RandAdd
R _HDA	Fig 2	min-k LongAdd	min-k LongAdd	min-k LongAdd
R _BP	Fig 3	min-k RandAdd	min-k RandAdd	min-k RandAdd
R _BP	Fig 3	min-k LongAdd	min-k LongAdd	min-k LongAdd
Size of FVS	Fig 4	min-k ShortAdd	min-k ShortAdd	min-k ShortAdd

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In SF networks, we investigate the effects of node size N and average degree 〈k〉 on the improvement of robustness by link addition methods. The SF networks are generated by the GN model with ν = 1, which is the same as Barabási-Albert model. Fig 5 shows R_HDA after adding links on SF networks with N = 500, 1000, and 5000 and 〈k〉 = 2 and 4. Note that the SF network with 〈k〉 = 2 is a tree. Fig 5 shows similar results even for different node sizes N. In comparing the ratios of added and existing links, the trend of improving the robustness of each strategy is almost the same. The trend means that, as shown in Fig 5, the orders of lines from top to bottom are similar at the same ratio of added links for N = 500, 1000, and 5000. These results show that the effect of node size is negligible. When the number of added links is 10 times the number of existing links, the five lines except the blue dotted line (ShortAdd method) have almost similar slopes, as shown in Fig 5A for N = 500. For sufficiently large numbers of added links, the ratio of added links strongly affects the improvement of the robustness more than the differences between methods. As the impact of the initial average degree 〈k〉 on the improvement of robustness, Fig 5A–5C for 〈k〉 = 2 show that blue solid line (denotes min-k ShortAdd method) has smaller slope than red and green solid lines (denote min-k RandAdd and min-k LongAdd methods), whereas Fig 5D–5F for 〈k〉 = 4 show that the blue solid line has a similar or larger slope than red and green solid lines. From these results, the min-k ShortAdd method is effective for non-tree SF networks. The reason is considered as follows. For non-tree networks, such triangles are not generated, since there are no leaf nodes. However, the min-k ShortAdd method is ineffective for trees. For trees, adding links by the min-k ShortAdd method generates triangles between leaf nodes that is not related to improve the robustness. Thus, it is considered that the min-k ShortAdd method improves the robustness effective by making the degree distribution homogeneous without constructing local triangles. These results are supported from that the effect of the minimum degree strategy becomes greater as the average degree 〈k〉 increases and networks become further different from trees.

Fig 5 — The min-k RandAdd and min-k LongAdd methods (red and green solid lines) are effective for SF trees, while the min-k RandAdd, min-k ShortAdd, and min-k LongAdd methods (three solid lines) are effective for non-tree SF networks. The initial SF networks are trees with (A) N = 500, (C) 1000, or (E) 5000, and are non-trees with (B) N = 500, (D) 1000, or (F) 5000 nodes.

Since it is found that the homogeneous degree distribution strongly affects the improvement of the robustness, we investigate the changes of degree distributions by link addition methods. Fig 6 show the variances of degree distributions after adding links to SF networks with N = 1000 and 〈k〉 = 2 and 4. For the min-k RandAdd, min-k ShortAdd, and min-k LongAdd methods, variances are the same value, since they select the minimum degree nodes. In the other three methods, the LongAdd method decreases variance, while the ShortAdd method increases the variance. This means that the longest distance strategy makes the degree distribution homogeneous. Fig 7 shows the degree distributions after adding links to SF networks. In the min-k RandAdd, min-k ShortAdd, and min-k LongAdd methods, lower degrees are rightly shifted with unchanged higher degrees. Similar changes can be seen for the LongAdd method. Compared to degree distributions in the RandAdd methods, lower degrees are rightly shifted with unchanged higher degrees in the LongAdd method. The longest degree strategy tends to add links to nodes with lower degrees and makes the degree distribution homogeneous. On the other hand, in the ShortAdd method, lower degrees change little, and medium degrees are rightly shift. The shortest distance strategy tends to add nodes with higher degrees to increase the variance of degree distribution.

Fig 6 — The LongAdd method (green dotted line) decreases the variance, while the ShortAdd method (blue dotted line) increases the variance. The min-k RandAdd, min-k ShortAdd, and min-k LongAdd methods (three solid lines) are the same. The initial networks are SF networks with N = 1000 and (A) 〈k〉 = 2 or (B) 4.

Fig 7 — The LongAdd method adds links to nodes with lower degrees, while the ShortAdd method adds links to nodes with middle to higher degrees. The results are shown for by (A) ShortAdd, (B) min-k ShortAdd, (C) RandAdd, (D) min-k RandAdd, (E) LongAdd, and (F) min-k LongAdd methods, respectively. The initial networks are SF networks with N = 1000 and 〈k〉 = 2.

We discuss that the distance between selected nodes for adding links, since there is no difference in degree distributions for three min-k methods. Fig 8 show the distance between selected nodes for adding links to SF networks with N = 1000 and 〈k〉 = 2 and 4. It is obvious that the LongAdd method selects node pairs with the longest distance and the ShortAdd method selects node pairs with the shortest distance. The RandAdd method selects node pairs with about middle distance nodes between the LongAdd and ShortAdd methods. For three min-k methods, the min-k LongAdd method tends to select node pairs with longer distance and the min-k ShortAdd method tends to select node pairs with shorter distance. However, the LongAdd method selects node pairs with longer than the min-k LongAdd methods, which select node pairs from nodes with the minimum degree. Moreover, for the min-k ShortAdd method and min-k LongAdd method, the distances between selected node pairs sharply change when the value of the minimum degree changes.

Fig 8 — The LongAdd method (green dotted lines) selects node pairs with the longer distances, while the ShortAdd method (blue dotted lines) selects node pairs with the shorter distances. The LongAdd method (green dotted lines) selects node pairs with longer than the min-k LongAdd methods (green solid lines). The initial networks are SF networks with N = 1000 and (A) 〈k〉 = 2 or (B) 4.

In summary, we investigate that node size has a small effect, but the average degree has a strong effect on the improvement of the robustness. In particular, the min-k ShortAdd method is effective for improving the robustness for non-tree SF networks, while it is ineffective for trees. There is no difference between the three min-k methods in improving the robustness for non-tree SF networks, although these methods select nodes with different distances. Therefore, for non-tree SF networks, it is suggested that the effect of adding links to the minimum degree nodes and that homogeneous degree distribution is more important for improving the robustness than the effect of distance.

Next, we explain the results on real networks. Fig 9A–9C show R_HDA after adding links to initial real networks: AirTraffic, Email, and Yeast. Similar to the results of R_HDA on random trees in Fig 2, the min-k RandAdd and min-k LongAdd methods are the most effective for improving R_HDA. Unlike the results on random trees, the min-k ShortAdd method is the best on real networks. Fig 9A–9C show that the slope of blue solid line (denotes min-k ShortAdd method) is close to those of red and green solid lines, as the number of added links increases. From these results, the minimum degree strategy significantly contributes to improving the robustness on real networks. In other words, by comparing to the minimum degree strategy, the effect of the longest distance strategy is weaker, especially when the number of added links is large. This is different from the result on the tree. We later discuss the strong effect of the minimum degree strategy on the distance between nodes in real networks. It is considered that the minimum degree strategy selects pairs of nodes with sufficient longer distances for real networks even on the minimum distance strategy.

Fig 10A–10C show R_BP after adding links to the real networks. The whole trend is almost same as shown in Fig 9. Once again, the min-k RandAdd, min-k LongAdd, and min-k ShortAdd methods are the most effective for improving R_BP. In particular, as the number of added links increases, the min-k ShortAdd method becomes more effective. Similarly, Fig 11A–11C show the size of FVS after adding links to the real networks. As similar to results of the size of FVS on random trees in Fig 4, the min-k ShortAdd method is the most effective for increasing the size of FVS. Blue solid line is higher than the other five lines in Fig 11A–11C. Thus, on the real networks, the min-k ShortAdd method is effective for increasing both the robustness and the size of FVS.

Fig 11 — In each figure of(A)(B)(C), the min-k ShortAdd method (a blue line) outperforms the other methods. The min-k RandAdd and min-k LongAdd methods (red and green solid lines) are also effective for increasing the size of FVS. The initial networks are (A) AirTraffic, (B) Email, and (C) Yeast. The correspondence between lines and methods is the same as shown in Fig 2.

Table 4 summarizes the results for improving the robustness and the size of FVS from initial real networks. The min-k RandAdd, min-k LongAdd, and min-k ShortAdd methods are the most effective for increasing both R_HDA and R_BP for AirTraffic and Email. For Yeast, the min-k RandAdd and min-k LongAdd methods are the most effective. As a larger number of added links, the min-k ShortAdd method becomes more effective for all three networks. From these results, the contribution is stronger on the minimum degree strategy to improving the robustness, as the number of added links increases. Conversely, the contribution is weaker on the longest distance strategy.

Table 4. The most effective methods for improving R_HDA, R_BP, and the size of FVS to real networks.

		AirTraffic	Email	Yeast
R _HDA	Fig 9	min-k RandAdd	min-k RandAdd	min-k RandAdd
		min-k LongAdd	min-k LongAdd	min-k LongAdd
		min-k ShortAdd	min-k ShortAdd
R _BP	Fig 10	min-k RandAdd	min-k RandAdd	min-k RandAdd
		min-k LongAdd	min-k LongAdd	min-k LongAdd
		min-k ShortAdd	min-k ShortAdd
Size of FVS	Fig 11	min-k ShortAdd	min-k ShortAdd	min-k ShortAdd

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For investigating the effects of link addition methods on the distances between nodes, we show the changes of the topological structure measured by diameter D and network efficiency E. Remember that there is a trade-off between robustness and network efficiency in scale-free networks [34], while high robustness coexists with network efficiency in incrementally growing onion-like networks [35]. Thus, it is not trivial that the distances become smaller when the robustness is strong. In general, as the number of added links increases, D decreases while E increases. Thus, we compare the slopes of lines in decreasing D or increasing E. Fig 12A–12C show the diameter D after adding links to the initial random trees with narrower, exponential, and power-law degree distributions. Fig 12A–12C show that green dotted line (denotes LongAdd method) has the largest slope in decreasing D among six lines. Since D is the maximum length in all shortest paths, it is clear that D is the most decreased by adding the longest links. Moreover, the slope of red solid line (denotes min-k RandAdd methods) is close to that of green dotted line. Therefore, the minimum degree strategy contributes to decreasing D. On the other hand, blue solid line (denotes min-k ShortAdd method) has a smaller slope in decreasing D than the other four lines. The slope of blue solid line is close to that of blue dotted lines (denotes ShortAdd method). For initial random trees, the effect of the minimum shortest strategy is weak in decreasing D even on the minimum degree strategy.

Fig 12 — The LongAdd method (green dotted line) is the most effective for decreasing D among the six methods. The ShortAdd and min-k ShortAdd method (blue dotted and solid lines) are higher than other four methods. The initial random tree has (A) narrower, (B) exponential, and (C) power-law degree distributions. The correspondence between lines and methods is the same as in Fig 2.

Fig 13A–13C show the diameter D after adding links to initial real networks: AirTraffic, Email, and Yeast. Similar to the results on random trees in Fig 12, the longest distance strategy is the most effective for decreasing D in real networks. Furthermore, the minimum degree strategy contributes to decreasing D. Unlike the results on random trees, the min-k ShortAdd method is more effective for decreasing D in AirTraffic and Email. In Fig 13A and 13B, the slope of blue solid line (denotes min-k ShortAdd method) is close to that of red dotted line (denotes RandAdd method) rather than blue dotted line (denotes ShortAdd method). Thus, it is considered that the min-k ShortAdd method selects pairs of nodes with sufficient distances for decreasing D for AirTraffic and Email. We remark that the min-k ShortAdd method is the most effective for improving the robustness for AirTraffic and Email as shown in Table 4. On the other hand, the effect of min-k ShotAdd method is a little weak in decreasing D for Yeast. In Fig 13C, the slope of blue solid line is close to that of blue dotted line.

Similarly, Figs 14 and 15 show network efficiency E after adding links to the random trees and real networks, respectively. We also compare the slopes of lines in increasing E. In general, the average distance between nodes becomes smaller, as E increases. Fig 14A–14C show that red dotted line (denotes RandAdd method) is higher than the other five lines for initial random trees. Moreover, the slopes of green and red solid lines (denote min-k RandAdd and min-k LongAdd methods) are close to that of red dotted line. Thus, the min-k RandAdd and min-k LongAdd methods are effective for increasing both the robustness and E for initial random trees. On the other hand, blue solid line (denotes min-k ShortAdd method) is lower than the other five lines in Fig 14A–14C. As mentioned above, since the min-k ShortAdd method generates triangles between leaf nodes on random trees, the contribution is small to increasing both E and the robustness. Also for initial real networks, the RandAdd method is the most effective for increasing E as shown in Fig 15A–15C. Furthermore, the min-k RandAdd and min-k LongAdd methods are effective for increasing both the robustness and E, especially for AirTraffic and Email as shown in Fig 15A and 15B. For Yeast, E is slightly increased by the min-k RandAdd and min-k LongAdd methods as shown in Fig 15C. In contrast, although the min-k ShortAdd method is effective for improving the robustness, E is only slightly increased by the min-k ShortAdd method as shown in blue solid lines of Fig 15A–15C. Table 5 summarizes the most effective methods for decreasing D and increasing E from initial random trees and real networks.

Fig 14 — The RandAdd method (red dotted line) is the most effective for increasing E among six methods. The min-k ShortAdd method (blue solid lines) is lower than the other five methods. The initial random tree has (A) narrower, (B) exponential, and (C) power-law degree distributions. The correspondence between lines and methods is the same as shown in Fig 2.

Fig 15 — The RandAdd method (red dotted line) is the most effective for increasing E among six methods. The min-k ShortAdd method (blue solid line) is lower than the other five methods. The initial networks are (A) AirTraffic, (B) Email, and (C) Yeast. The correspondence between lines and methods is the same as shown in Fig 2.

Table 5. The most effective methods for decreasing D and increasing E to random trees and real networks.

		β = 5	ν = 0	ν = 1	AirTraffic	Email	Yeast
D	Figs 12 & 13	LongAdd	LongAdd	LongAdd	LongAdd	LongAdd	LongAdd
		min-k LongAdd	min-k LongAdd	min-k LongAdd	min-k LongAdd
		min-k RandAdd	min-k RandAdd		min-k RandAdd
		RandAdd
E	Figs 14 &15	RandAdd	RandAdd	RandAdd	RandAdd	RandAdd	RandAdd
		LongAdd	LongAdd	LongAdd	LongAdd	LongAdd
		min-k LongAdd	min-k LongAdd		min-k LongAdd	min-k LongAdd
		min-k RandAdd	min-k RandAdd		min-k RandAdd

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Conclusion

We consider the contributions of the minimum degree and longest distance strategies to improving the robustness by adding links. To distinguish the effects of degrees and distances as much as possible, we introduce link addition methods with two-step selections based on degrees or distances. It is expected that two strategies select similar nodes on a tree, while they select different nodes on a network with loops. Moreover, since the robustness is strongly affected by the degree distribution, we use initial random trees with different degree distributions following power-law, exponential, and narrower distributions generated by the GN [24–26] and IPA models [27]. Our results show that the min-k RandAdd and min-k LongAdd methods are the most effective for improving the robustness by adding links to an initial random tree. As an exception, only when the number of added links is small, the longest distance strategy is the best. Note that the min-k RandAdd and min-k LongAdd methods are also effective for increasing the size of FVS. On the other hand, the effect of min-k ShortAdd method is weak for improving the robustness by connections between leaves locally. Thus, it is suggested that enhancing long (global) loops is important for improving the robustness rather than short (local) loops for random trees. Similarly, for real networks, the min-k RandAdd, min-k LongAdd, and min-k ShortAdd methods are the most effective for improving the robustness. Even on the shortest distance strategy, the minimum degree strategy significantly contributes to improving the robustness on real networks. The reason may be considered as that the minimum degree strategy selects nodes with sufficient long distances in decreasing the diameter even on the shortest distance strategy. However, its detailed analysis requires further studies. In conclusion, adding links to pairs of the minimum degree nodes between longer distances is the most effective for improving the robustness on both synthetic trees and real networks. In other words, enhancing global loops is essential for improving the robustness and efficiency. For future studies, it may be useful to extend our methods of adding links for more complicated networks, such as multilayer networks [38], temporal networks [39], multi-fractal networks [40, 41], and brain connectomes [42, 43].

Supporting information

S1 Fig. Robustness index against typical HDA attacks after adding links to configuration models of random trees.

The min-k RandAdd and min-k LongAdd methods (red and green solid lines) are the most effective for improving R_HDA. For small numbers of added links less than (A) 170, (B) 380, and (C) 530, the LongAdd method (green dotted lines) is higher than other five methods. The initial network is configuration model of random tree with (A) narrower, (B) exponential, and (C) power-law degree distributions. Note that the network is randomly rewired for connecting all nodes.

(EPS)

Click here for additional data file.^{(228KB, eps)}

S2 Fig. Robustness index against the worst-case BP attacks after adding links to configuration models of random trees.

The min-k RandAdd and min-k LongAdd methods (red and green solid lines) are the most effective for improving R_BP. For small numbers of added links less than (A) 170, (B) 380, and (C) 530, the LongAdd method (green dotted lines) is higher than other five methods. The initial network is configuration model of random tree with (A) narrower, (B) exponential, and (C) power-law degree distributions. Note that the network is randomly rewired for connecting all nodes.

(EPS)

Click here for additional data file.^{(226.9KB, eps)}

S3 Fig. Size of FVS after adding links to configuration models of random trees.

The min-k RandAdd and min-k LongAdd methods (red and green solid lines) are effective for increasing the size of FVS. In each figure of(A)(B)(C), the min-k ShortAdd method (a blue line) outperforms the other methods. The initial network is configuration model of random tree with (A) narrower, (B) exponential, and (C) power-law degree distributions. Note that the network is randomly rewired for connecting all nodes.

(EPS)

Click here for additional data file.^{(214.1KB, eps)}

S4 Fig. Diameter after adding links to configuration models of random trees.

The LongAdd method (green dotted line) is the most effective for decreasing D among the six methods. The ShortAdd and min-k ShortAdd method (blue dotted and solid lines) are higher than other four methods. The initial network is configuration model of random tree with (A) narrower, (B) exponential, and (C) power-law degree distributions. Note that the network is randomly rewired for connecting all nodes.

(EPS)

Click here for additional data file.^{(215.4KB, eps)}

S5 Fig. Network efficiency after adding links to configuration models of random trees.

The RandAdd method (red dotted line) is the most effective for increasing E among six methods. The min-k ShortAdd method (blue solid lines) is lower than the other five methods. The initial network is configuration model of random tree with (A) narrower, (B) exponential, and (C) power-law degree distributions. Note that the network is randomly rewired for connecting all nodes.

(EPS)

Click here for additional data file.^{(211.4KB, eps)}

Data Availability

AirTraffic is available from http://konect.cc/networks/maayan-faa/. Email is available from https://deim.urv.cat/~alexandre.arenas/data/welcome.htm. Yeast is available from http://vlado.fmf.uni-lj.si/pub/networks/data/bio/Yeast/Yeast.htm.

Funding Statement

YH is partially supported by a Grant-in-Aid for Scientific Research (Grant Number JP.21H03425) from the Japan Society for the Promotion of Science. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. There was no additional external funding received for this study.

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PLoS One. doi: 10.1371/journal.pone.0276733.r001

Decision Letter 0

Lazaros K Gallos

9 Aug 2022

PONE-D-22-09998Adding links on minimum degree and longest distance strategies for improving network robustness and efficiencyPLOS ONE

Dear Dr. Chujyo,

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Reviewer #2: No

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Reviewer #1: The authors start from the premise that manu real-world networks characterized by power-law degree distributions and claim that those are extremely vulnerable against malicious attacks. To obtain higher robustness some link addition methods were proposed. Precisely, two effective strategies for selecting nodes to add links are: the minimum degree and longest distance strategies.

So, the authors investigate the contributions of these strategies to improving the robustness by adding links in distinguishing the effects of degrees and distances as much as possible. Through

numerical simulation, they find that the robustness is effectively improved by adding links on the minimum degree strategy for both synthetic trees and real networks. Here are some suggestions for improvement:

(1) Is there any reason that the authors only consider trees in the synthetic study? Even though they are fragile against attacks, non-tree graph structures should also be analyzed since real-world networks are usually not trees.(2) If more links are continuously added to the attacked graph (after exceeding 2000 and 3000 as the limit of x-axis in the figures), can we observe convergence of the robustness?(3) In the synthetic experiment section, the authors only consider graphs with 1000 nodes. I suggest also adding an experiment which investigates how the graph size and density influences the robustness and efficiency in the same setting. If such behaviors are expected to be consistent across graphs which have different sizes and densities, please further explain it.(4) Real world networks display complex topologies and there has been significant work recently to characterize the network properties from many angles such as dealing with the power laws and fractal features and exploit them for network reconstruct, or for inferring network generating rules, or for quantifying the degree of network complexity. Here are some such network models "Reconstructing missing complex networks against adversarial interventions." Nature communications 10, no. 1 (2019): 1-12. "Network science characteristics of brain-derived neuronal cultures deciphered from quantitative phase imaging data." Scientific reports 10, no. 1 (2020): 1-13. "Hidden network generating rules from partially observed complex networks." Communications Physics 4, no. 1 (2021): 1-12. "Deciphering the generating rules and functionalities of complex networks." Scientific reports 11, no. 1 (2021): 1-15. that match real world networks and should be discussed at least if not considered as case studies as more realistic network models.

(5) It's good that the authors mention trees are more fragile against attacks. I wonder how this is reflected in the experiments?This is another reason why the authors can also consider non-tree and more complex simulated graphs. It would be great if the authors can measure and quantify the similarity/dissimilarity of the arbitrary simulated graphs with tree-like graphs, and further show how the tree structure influences the link addition methods and the corresponding robustness.

Reviewer #2: The current work investigates the improvement of network robustness and efficiency via adding links based on minimum degree and longest distance. Concretely, it proposes serval strategies of adding links for selecting nodes with minimum-k degree and longest distance, and improves the network density, which makes the varying network be more robustness and efficiency. In my view, the strategy of adding links is common method for improving network robustness and efficiency, and has be developed for a long time. In the current work, the related works are out-of-date (most of them before 2019, except of the authors’ works [11] and [15]), as well as the comparison methods. Thus, the authors should survey more recent works, and introduce them to compare with the proposed method. Additionally, it does not illustrate the difference of the proposed method from the authors’ works [11] and [15]. Most importantly, my comments involve the figures and the corresponding results, shown as follows as:

(1) Figures are blurred especially when they are enlarged.

(2) In Figures 2-7, and 9-10 the strategies of min-k RandAdd and min-k LongAdd have the same ability of improving network robustness and efficiency. These results suggest that the min-k selecting strategy play the key operation, thus I think that the authors should check the results and independently discus the min-k selecting strategy.

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Reviewer #1: No

Reviewer #2: No

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PLoS One. 2022 Oct 26;17(10):e0276733. doi: 10.1371/journal.pone.0276733.r002

Author response to Decision Letter 0

27 Sep 2022

Prof. Lazaros K. Gallos

Academic Editor

PLOS ONE

Dear Prof. Lazaros K. Gallos,

Re: Manuscript reference No. PONE-D-22-09998

Thank you for your ongoing consideration of our manuscript entitled “Adding links on minimum degree and longest distance strategies for improving network robustness and efficiency” for publication in PLOS ONE.

We also appreciate the time and effort you and the reviewers have dedicated to providing insightful feedback on ways to improve our paper. We have incorporated changes that reflect the detailed suggestions. We hope that our revised manuscript and the responses satisfactorily address all the issues and concerns you and the reviewers have pointed out.

In the revised manuscript with track changes, we marked added text in blue and removed ones in red. In accordance with suggestions, we added several results on non-tree scale-free networks as more realistic networks (described in 241-308 lines of the revised manuscript with track changes). We inserted new Figures 4-9, and the numbers of figures are updated.

To facilitate your review of our revisions, the following pages are our point-by-point responses to the questions and comments.

Again, thank you for inviting us the opportunity to improve our manuscript with your valuable comments and queries. We have worked hard to incorporate your feedback and hope that these revisions persuade you to accept our submission.

Sincerely,

Masaki Chujyo

Ph. D. student

Japan Advanced Institute of Science and Technology

1-1 Asahidai, Nomi, Ishikawa 923-1292 Japan

E-mail: mchujyo@jaist.ac.jp

On behalf of all authors.

For the comments of Reviewer 1

Our answer (1)

Thank you for the suggestion. Yes, there is no reason for the restriction on trees in the synthetic study. Moreover, we added some results of non-tree scale-free networks in 241-308 lines (blue texts) of the revised manuscript with track changes. We discussed that the min-k ShortAdd method improves the robustness more effectively for non-tree networks.

(2) If more links are continuously added to the attacked graph (after exceeding 2000 and 3000 as the limit of x-axis in the figures), can we observe convergence of the robustness?

Our answer (2)

Yes. We observe the convergence of the robustness for scale-free networks (Fig 5), in comparing the ratios of added and existing links. As the number of added links increases, the slope of improved robustness becomes smaller. For a sufficiently large number of added links, the differences between the methods become small through improving the robustness. We described it in 244-254 lines (blue texts) of the revised manuscript with track changes.

(3) In the synthetic experiment section, the authors only consider graphs with 1000 nodes. I suggest also adding an experiment which investigates how the graph size and density influences the robustness and efficiency in the same setting. If such behaviors are expected to be consistent across graphs which have different sizes and densities, please further explain it.

Our answer (3)

We added the results of the initial scale-free networks with 500, 1000, and 5000 nodes (Fig 5). When we compared the robustness for the ratios of added and existing links, there was little difference according to node size. We described it in 244-254 lines (blue texts) of the revised manuscript with track changes.

As density, we considered both the initial average degree and the ratio of added links. For the initial average degree, we added the results of the initial scale-free networks with average degrees 2 and 4 (Fig 5). We found that the min-k ShortAdd method is effective for non-tree scale-free networks (the initial average degree 4), although it is inefficient for trees (the initial average degree 2). We described it in 254-268 lines (blue texts) of the revised manuscript with track changes.

For the ratio of added links, in general, the robustness increases, as the number of added links increases. In comparing the slopes of increasing robustness, the difference of slopes for each method becomes smaller, as the number of added links increases (Fig 5A). For sufficiently large numbers of added links, the ratios of added links are more dominant for the improvement of robustness more the differences between methods. We described it in 248-250 lines (blue texts) of the revised manuscript with track changes.

(4) Real world networks display complex topologies and there has been significant work recently to characterize the network properties from many angles such as dealing with the power laws and fractal features and exploit them for network reconstruct, or for inferring network generating rules, or for quantifying the degree of network complexity. Here are some such network models "Reconstructing missing complex networks against adversarial interventions." Nature communications 10, no. 1 (2019): 1-12. "Network science characteristics of brain-derived neuronal cultures deciphered from quantitative phase imaging data." Scientific reports 10, no. 1 (2020): 1-13. "Hidden network generating rules from partially observed complex networks." Communications Physics 4, no. 1 (2021): 1-12. "Deciphering the generating rules and functionalities of complex networks." Scientific reports 11, no. 1 (2021): 1-15. that match real world networks and should be discussed at least if not considered as case studies as more realistic network models.

Our answer (4)

According to your suggestion, we described that the extension of our methods to more complicated and realistic networks will be important in the future in 418-420 lines (blue texts) of the revised manuscript with track changes.

(5) It's good that the authors mention trees are more fragile against attacks. I wonder how this is reflected in the experiments? This is another reason why the authors can also consider non-tree and more complex simulated graphs. It would be great if the authors can measure and quantify the similarity/dissimilarity of the arbitrary simulated graphs with tree-like graphs, and further show how the tree structure influences the link addition methods and the corresponding robustness.

Our answer (5).

We do not directly show that the trees are vulnerable to attacks in our experiments, but we expected to show the difference of improving robustness by each method with a small number of added links. For trees, we conclude that the min-k RandAdd and min-k LongAdd methods are effective for a larger number of added links, while the LongAdd method is effective for a smaller number of added links.

Although we do not discuss the similarity to trees, we show the difference in improvement in robustness depending on whether the initial network is a tree or not, by comparing the results of trees or non-trees. We described it in 254-268 lines (blue texts) of the revised manuscript with track changes.

For the comments of Reviewer 2

(1) Figures are blurred especially when they are enlarged.

Our answer (1)

We changed the format of the figures from tiff to eps in order to be more clear.

Our answer (2)

Since the min-k RandAdd and min-k LongAdd methods select nodes with the minimum degree, both degree distributions and the variances are the same, as shown in Figs 6 and 7. These methods result in homogenous degree distributions, which are dominant in the improvement of robustness. However, the distances of selected node pairs are different. We show that the distances are larger for the min-k LongAdd method in Fig 8. According to your valuable suggestion, we added the discussion on the minimum degree strategy in 269-308 lines (blue texts) of the revised manuscript with track changes.

Attachment

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PLoS One. doi: 10.1371/journal.pone.0276733.r003

Decision Letter 1

Lazaros K Gallos

13 Oct 2022

Adding links on minimum degree and longest distance strategies for improving network robustness and efficiency

PONE-D-22-09998R1

Dear Dr. Chujyo,

We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements.

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PLOS ONE

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6. Review Comments to the Author

Reviewer #1: Thank you very much for addressing all my comments and thoughts. I think this is an interesting study that should receive more attention from the network science community.

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PLoS One. doi: 10.1371/journal.pone.0276733.r004

Acceptance letter

Lazaros K Gallos

17 Oct 2022

PONE-D-22-09998R1

Adding links on minimum degree and longest distance strategies for improving network robustness and efficiency

Dear Dr. Chujyo:

I'm pleased to inform you that your manuscript has been deemed suitable for publication in PLOS ONE. Congratulations! Your manuscript is now with our production department.

If your institution or institutions have a press office, please let them know about your upcoming paper now to help maximize its impact. If they'll be preparing press materials, please inform our press team within the next 48 hours. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information please contact onepress@plos.org.

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Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Supplementary Materials

S1 Fig. Robustness index against typical HDA attacks after adding links to configuration models of random trees.

(EPS)

Click here for additional data file.^{(228KB, eps)}

S2 Fig. Robustness index against the worst-case BP attacks after adding links to configuration models of random trees.

(EPS)

Click here for additional data file.^{(226.9KB, eps)}

S3 Fig. Size of FVS after adding links to configuration models of random trees.

(EPS)

Click here for additional data file.^{(214.1KB, eps)}

S4 Fig. Diameter after adding links to configuration models of random trees.

(EPS)

Click here for additional data file.^{(215.4KB, eps)}

S5 Fig. Network efficiency after adding links to configuration models of random trees.

(EPS)

Click here for additional data file.^{(211.4KB, eps)}

Attachment

Submitted filename: ResponseToReviewers.pdf

Click here for additional data file.^{(73.5KB, pdf)}

Data Availability Statement

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PERMALINK

Adding links on minimum degree and longest distance strategies for improving network robustness and efficiency

Masaki Chujyo

Yukio Hayashi

Roles

Abstract

Introduction

Materials and methods

Fig 1. Added links on the minimum degree and longest distance strategies.

Table 1. Six link addition methods in the combinations of the first step by degree and the second step by distance.

Table 2. Properties of real networks.

Results

Fig 2. Robustness index against typical HDA attacks after adding links to random trees.

Fig 3. Robustness index against the worst-case BP attacks after adding links to random trees.

Fig 4. Size of FVS after adding links to random trees.

Table 3. The most effective methods for improving RHDA, RBP, and the size of FVS to random trees.

Fig 5. Robustness index against typical HDA attacks after adding links to SF networks.

Fig 6. Variances of degree distributions after adding links to SF networks.

Fig 7. Degree distributions after adding links to SF networks.

Fig 8. Distance between selected node pairs for adding links to SF networks.

Fig 9. Robustness index against typical HDA attacks after adding links to real networks.

Fig 10. Robustness index against the worst-case BP attacks after adding links to real networks.

Fig 11. Size of FVS after adding links to real networks.

Table 4. The most effective methods for improving RHDA, RBP, and the size of FVS to real networks.

Fig 12. Diameter after adding links to random trees.

Fig 13. Diameter after adding links to real networks.

Fig 14. Network efficiency after adding links to random trees.

Fig 15. Network efficiency after adding links to real networks.

Table 5. The most effective methods for decreasing D and increasing E to random trees and real networks.

Conclusion

Supporting information

Data Availability

Funding Statement

References

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Lazaros K Gallos

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Author response to Decision Letter 0

Decision Letter 1

Lazaros K Gallos

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Acceptance letter

Lazaros K Gallos

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Associated Data

Supplementary Materials

Data Availability Statement

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Table 3. The most effective methods for improving R_HDA, R_BP, and the size of FVS to random trees.

Table 4. The most effective methods for improving R_HDA, R_BP, and the size of FVS to real networks.