A new design principle of robust onion-like networks self-organized in growth

Abstract Today's economy, production activity, and our life are sustained by social and technological network infrastructures, while new threats of network attacks by destructing loops have been found recently in network science. We inversely take into account the weakness, and propose a new design principle for incrementally growing robust networks. The networks are self-organized by enhancing interwoven long loops. In particular, we consider the range-limited approximation of linking by intermediations in a few hops, and show the strong robustness in the growth without degrading efficiency of paths. Moreover, we demonstrate that the tolerance of connectivity is reformable even from extremely vulnerable real networks according to our proposed growing process with some investment. These results may indicate a prospective direction to the future growth of our network infrastructures.


I. INTRODUCTION
Social and technological networks for communication, collaboration, trading, travel, or supply chain become more and more important, since their systems support our daily life and economy.The connections between nodes facilitate information deliveries, physical logistics, and energy supplies.Moreover, through some intermediations, the connections sometimes lead to new business chances, acquaintanceship, or remote control of the infrastructures efficiently.Some case studies in organization theory: the rapid recovery of Toyota group's supply chain from a large fire accident of their subcontract plants [14,15], worldwide economic networks with expanding business chances by Wenzhou people in China [15], and the brain circulation system known as Silicon Valley (SV) model for developing innovational high-tech industry with market opportunities by immigrant engineers [19] has been suggested the importance of long-distance relations for both robustness of connectivity and efficiency of path in a network.The established connections via intermediations probably work well for managing cross-border operations.
On the other hand, many social, technological, and biological infrastructural networks have a common scale-free (SF) structure [2] generated by the selfish preferential attachment referred to as rich-get-richer rule in consciously/unconsciously considering efficiency of paths between two nodes connected within a few hops.The SF networks also have an extreme vulnerability against intentional attacks [1].However, in these several years by percolation analyses, it has been clarified that onion-like topological structure with positive degree-degree correlations gives the optimal robustness even for the attacks in SF networks [20,21].Based on a natural but unselfish rule, onion-like networks can be incrementally grown by applying cooperative partial copying and adding shortcut [6,7] instead of the expensive whole rewiring [24] or hierarchically expanding outer ring [18] for enhancing the positive degree-degree correlations.One of the drawback is that the robustness is weak in early stage of the growth [7].While none of incremental generation of networks has been so far based on interwoven loops, new threats of network attacks by destructing loops have been found recently [11,12].They give severer damage than the conventional intentional attacks [1], and can be easily performed.One is Collective Influence (CI) attack [11] considered for a global optimization to identifying the most influence nodes called influencers in information spreading.Another is Belief Propagation (BP) attack [12] derived from a message-passing approximation algorithm rooted by the spin glass model in statistical physics for the Feedback Vertex Set (FVS) problem in belonging to NP-hard [8,9].
Inversely taking into account the weakness caused by the CI and BP attacks, we propose a new design principle for generating robust onion-like networks in focusing on enhancing of long loops, whose key factor is long-distance relation inspired from the organization theory [15].Furthermore, we consider a practical approximation of the network generation with moderately long loops, which is based on range-limited intermediations for finding linked nodes without large costs or efforts in the growth.

MENTS
We propose a self-organized growing network by enhancing long loops.After explaining the basic model, we consider the realistic range-limited approximation in 2.1.We estimate the degree distribution in our proposed network in 2.2.

A. Basic model and the practical approximation
We explain a basic model of self-organized growing network by enhancing long loops.At each time step of growing, a new node is added and connects to existing nodes.As the connection rule for even number m links emanated from the new node, we introduce a pair of attachments referred to as random and long distance attachments (RLD-A) or preferential and long distance attachments (PLD-A).The difference of the connection rule from that in the well-known Barabási-Albert (BA) model [2] is a pair of attachments with long distance attachment.As shown in Fig. 1a, the following pair of attachments is repeated in m/2 times at each time step.

RLD-A:
One of link destination is uniformly randomly chosen as encountering, and another link destination is the furthest node from the chosen node.When there are several candidates of the furthest with a same distance counted by hops, one of them is randomly selected.Some kind of randomness is useful to avoid fixed weak-points in the growth.
PLD-A: For the comparison with RLD-A, instead of uniformly random selection, one of the pair is preferentially chosen node with a probability proportional to its degree [2].
Another link destination is the furthest node from the preferentially chosen node.
For attached even number m links, m/2 loops through the pair of nodes are created at each time step.The interwoven loops via new node are significant for m ≥ 4 as shown in Sections 3 and 4. The minimum m = 4 is corresponded to the least effort of attachment linking to be strongly robust network in our growing method.Such connection rule in Fig. 1a was not noticed because of the lack of emphasis on loops, but the importance of the part of long distance relation was covertly suggested in organization theory [15].Moreover, since range-limited approach is useful for efficiently investigating global property of network such as influencer [11], centrality [3], or random percolation [17], we apply it for generating robust networks.We consider a range-limited approximation of RLD-A as random and intermediated attachments (MED).
MED: Instead of the furthest node, we select a distant node to the extent of a few hops via intermediations from the randomly chosen pair node.Intermediations in one hop mean attachments to the 2nd neighbors of the randomly chosen node, intermediations in two hops mean ones to the 3rd neighbors, and intermediations in µ hops mean ones to the µ+1-th neighbors.When µ is small, the attachments to a few hops-th neighbors have reality without large connection costs or efforts.
If a same destination node in RLD-A, PLD-A, or MED is chosen, other selection of pair is tried due to the prohibition of multiple links between two nodes.

B. Estimation of degree distribution
We consider growing networks with a same condition of the total number M = m(N − m) + m(m − 1)/2 of links for size N: total number of nodes at time step t = N − m.As the initial configuration, we set a complete graph of N 0 = m nodes and M 0 = m(m − 1)/2 links at t = 0. Figure 1b shows onion-like structure in which older nodes form the core while younger nodes surround it.Figure 2c justifies that older nodes get more links.Moreover, we can derive exponential tails of degree distributions p(k) by the asymptotic approximation [10] as follows.The invariant ordering Once a link is generated, it is undirected.In PLD-A, the destination node of blue line is chosen with a probability proportional to its degree, instead of random selection in RLD-A.In MED, the destination node of green line is chosen in the µ + 1-th neighbors from the blue node, instead of the furthest node in RLD-A.b) Example of onion-like structure by RLD-A for m = 4 at N = 200.
The circle size of node is proportional to its degree.The structure is visualized at the positions as node degrees become smaller from core to peripheral.parallel curves in Fig. 2c.Since the time course of degree of node i follows k i (t) ∼ log(t)/β as a monotone increasing function of t with a constant β > 0, we obtain Indeed, the orange and cyan lines guide log(t) and 2 log(t) in Fig. 2c, the estimated same color lines of e −k and e −k/2 are fitting with the tails of p(k) in Fig. 2a.The largest degree is bounded around 20 ∼ 35 without heavy connection load as on hub nodes in SF structure of many real networks.Figure 2b shows that the range-limited cases of MED in µ = 2, 3, 4 intermediations have slightly deviated but similar exponential tails of p(k).

III. STRONG ROBUSTNESS AND THE SMALL-WORLD EFFECT
For our proposed networks, we investigate the robustness index where S(q) denotes the number of nodes included in the giant component (GC as the largest cluster) after removing qN nodes, q is a fraction of removed nodes by High Degree Adaptive (HDA), CI for l = 3 layer [11], and BP [12] attacks.As in Appendix or [11,12], the highest value of CI l (i) in Eq. ( 4) or q 0 i in Eqs. ( 5)-( 8) to be removed is recalculated after each node removal.Note that the maximum R ≥ 0 is 0.5 in general.The following results are insensitive for varying values of inverse temperature x = 7 and 100 rounds of the messagepassing [12], and there is no difference for l ≥ 3 in CI attacks.Figure 3a shows that our networks by RLD-A for m = 4 have strong robustness R > 0.3 even in the early stage of growth, while Figure 3b shows that R is lower in the conventional SF networks by BA model.The networks by PLD-A show the intermediate R values.In Fig. 3c for m = 2, these lines fall in overall, but it is invariant that the ordering of damage by attacks is BP > CI 3 > HDA whose differences are very small.Each value of R is almost constant in the growing at least from the initial complete graph.In the range-limited cases of MED in µ = 2, 3, 4 intermediations, we obtain 0.31 < R < 0.35 and 0.28 < R < 0.34 against HDA and BP attacks, respectively.Figure 4ab show the relative size S(q)/N with the sudden breakdowns by BP attacks (bluish lines) as mentioned in [12].Each of the robustness in Fig. 4ab for m = 4 is improved from the corresponding one in Fig. 4c for m = 2, although larger m requires more links.
We also investigate the assortativity −1 ≤ r ≤ 1 as the Pearson correlation coefficient for degrees [13].
where k e and k ′ e denote degrees at both end-nodes of link e, M is the total number of links.Figure 5a shows that our networks by RLD-A for m = 4 (red line) have high assortativity r > 0.2 as similar to the copying model [6,7].However PLD-A is insufficient to create strong correlations.Figure 5b shows that the range-limited cases of MED in µ = 3, 4 (purple and cyan lines) are close to the case of RLD-A (red line).Although there is no clear criteria for the value of r in order to be an onion-like network with necessary positive degree-degree correlations, too large r is unsuitable [21].We do not discuss the optimally robust onion structure, but concern about incrementally growing proper good onion-like networks selforganized by natural and reasonable attachments.From Figs. shortest path length is O(log(N)) as shown in Fig. 6, even though half links in RLD-A or MED are created by random attachment without intention to be efficiency.In the growing from the initial complete graph, the number µ ≈ 3 of intermediations is at the similar level of the average path length.Averagely the length of simple one-round loop (as shown in Fig. 1a, it consists of the path between blue and green nodes + the corresponding blue and green links) generated by the pair of RLD-A or MED becomes short and inexpensive as O(log(N)).

IV. VIRTUAL TEST FOR PROSPECTIVE GROWTH OF REAL NETWORKS
As a virtual test for exploring future design of networks, we study the robustness of our model in growing to onion-like structure from the initial configuration of real networks [26] in Table I.For these social and technological networks, long distance connections will be somewhat required in order to seek solution strategies to undeveloped relationship or inconvenience, and realizable by intermediation or investment (e.g. for low cost carrier or innovation of power transmission) in a trade-off between the benefit and the cost.This network design in growth is different task from healing or recovering by rewirings e.g. between second neighbors [4,16] in almost constant numbers of nodes and links for a damaged network by earthquakes or terrorist attacks, etc.Because we focus on a structural change of network from almost uncorrelated SF to onion-like without hubs in the growth rather than topologically partial changes by rewirings.We also investigate dependence of the initial network structure not complete graph and the initial size N 0 on the robustness and degree-degree correlations for our growing method.Of course, some investments may be required for the growing network in larger size than the initial real one, however the virtual test will give a prospective insight.
In the following, the cases of intermediately destructive CI 3 attacks and not very effective PLD-A are omitted to simplified the discussion.Figures 7ab and 8ab show that high robustness against both HDA and BP attacks is obtained with increasing to R > 0.3 from R ≈ 0.1 in initial vulnerable real networks of Facebook and USair.In Figs.7ce and 8ce, the cases of m = 10 and m = 12 are investigated for checking the emergence of onion-like structure with high assortativity r > 0.2.We remark that some range-limited cases of MED in µ = 3 intermediations (bluish lines) have higher R than the cases of RLD-A with the attachments to the furthest nodes, but the effect is weak in the cases of MED in µ = 2 intermediations.If we do not insist onion-like networks, USpower can be already grown with high robustness before around N ≈ 10000 of double size of the initial as shown in Figs.7de and 8de.It suggests a possibility for incrementally growing other robust networks with r < 0 or r ≈ 0 (see Fig. 9de) instead of onion-like networks.Figure 9abc shows the enhancing of degree-degree correlations in increasing values of r in the growth of Facebook which induce large diameter, while Facebook and USair are compact with small diameter in  These distributions of path lengths are bell-shaped with the peak around the average length for each size N.In addition, the average path length in USpower is substantially greater than the number µ of intermediations at least in the early stage of the growth as shown in Table I. for increasing assortativity r with enhancing the correlations in Fig. 9.It is also interesting that the robustness in Figs.7 and 8 is improved in spite of decreasing average degree k in the growing from Facebook and USair for m = 4 (red and purple lines) as shown in Fig. 10a.Note that the average degree approaches to 2 × m for N → ∞ in Fig. 10.

V. CONCLUSION
We have proposed a second method for incrementally growing strongly robust onionlike networks self-organized by more natural and reasonable pair of attachments than the copying model [6,7].In addition, it becomes robust even in the early stage of the growth, and there is no huge hub whose largest degree is bounded.Since random attachments make an exponential degree distribution [2], the random ones are dominant in the tail for high degrees, while another intermediated attachments mainly work for low degrees and the positive correlations among them.In a virtual test for the growing from real networks with extreme vulnerability, we have shown that the proposed growing networks have reformable robustness to be future prospective infrastructures.It is also expected that the range-limited intermediations in a few hops reduce the Euclidean distances of links embedded on a space.
We emphasize the emergence of robust onion-like networks in enhancing moderately long loops by range-limited MED in a few hops without both degrading efficiency of paths and large connection costs or efforts.We should remember that the coexistence of robustness and efficiency has not been realized in many real networks, and the threat against attacks [11,12] is never decreased rather increased more and more, unless the dependence on selfish preferential attachment [2] is changed by ourselves.Therefore, our study suggests that we should discontinue the dependence on selfish rule and develop the potential of distant connections for the half of links, which may mean necessary investment for highly reliable connectivity in our network infrastructures even against intelligent attacks.

FIG. 1 :
FIG. 1: Topological properties of the proposed networks.a) In the case of m = 4, there are two pairs of attachments represented by green and blue lines from a new node added at each time step.The green node is at the end of the longest path represented by dashed-line in the shortest paths counted by hops from the blue node.The furthest node is easily findable by a labeling method.
FIG. 2: Estimation of exponential tail of degree distribution.a) Degree distribution p(k) in the average over 100 samples of our networks at N = 5000.Thin orange and cyan lines guide the exponential tails.b) the cases of MED for µ = 2, 3, 4. c) Time course of degree k i (t) of node i in the average over 100 samples of our networks.The thick lines from top to bottom (red, green, light blue for m = 4, or cyan, yellow, blue for m = 2) denote k i (t) of node i = 1, 10, and 100 inserted at the birth times t i = i − m > 0. Thin orange and cyan lines guide O(log(t)).

2 FIG. 3 :
FIG. 3: Robustness in the growing networks.Robustness index R against HDA, CI 3 , and BP attacks vs size N in our networks by a) RLD-A or PLD-A, b) SF networks by BA model for m = 4, and c) RLD-A or PLD-A for m = 2.

2 FIG. 4 :
FIG. 4: Relative size S(q)/N vs fraction q of removed nodes in the networks by a) RLD-A, PLD-A, b) BA for m = 4, and c) RLD-A, PLD-A, BA for m = 2 at N = 5000.The reddish curves against HDA attacks are gradually decreased, while bluish ones against BP attacks are suddenly dropped.The yellow-greenish or black curves against CI 3 attacks are the intermediate.Note that R is defined by the area under the line of S(q)/N .
FIG. 5: Degree-degree correlations in the growing networks.Assortativity r as the measure of correlations for size N in comparison with the networks by a) RLD-A, PLD-A, and BA model for m = 2 or m = 4, b) RLD-A and MED0∼4 for m = 4. MED0 ∼ 4 denote the case of MED for µ = 0, 1, 2, 3, 4.
FIG. 6: Average path length on the shortest paths counted by hops in our growing networks by RLD-A.The purple and blue thin lines guide O(log(N )) as the small-world effect.These results are averaged over 100 samples.

12 FIG. 7 : 12 FIG. 8 :
FIG. 7: Drastically improved robustness against HDA attacks in growing networks from the initial configuration of real networks.Robustness index R vs N in the growing networks from the initial a) Facebook for m = 4, b) US Airport Network for m = 4, c) Facebook for m = 10, d) US power grid for m = 4, and e) US power grid for m = 12.

12 FIG. 9 :
FIG. 9: Assortativity r vs N in growing networks from the initial a) Facebook for m = 4, b) US Airport Network for m = 4, c) Facebook for m = 10, d) US Power Grid for m = 4, and e) US Power Grid for m = 12.
FIG. 10: Average degree k vs N in the growing networks from the initial a) Facebook for m = 4 and m = 10 and US Airport Network for m = 4, b) US Power Grid for m = 4 and m = 12.

TABLE I :
Basic data of the real networks after converting from each of them to an undirected graph without multiple links.USair and USpower are abbreviations of US Airport Network and US Power Grid, respectively.The average path length and diameter are defined by the averaged length of the shortest paths with the minimum number of hops between two nodes and the longest length in a network.

Table I
. We note that the average path length is monotonously increasing to 4.13 and 4.21 at N = 10000 in Facebook and USair, but decreasing to 4.97 at N = 40000 in USpower.