Universal vulnerability in strong modular networks with various degree distributions between inequality and equality

TL;DR

This paper reveals a universal vulnerability in modular networks with various degree distributions, showing that adding community structures can drastically reduce network robustness against failures.

Contribution

It uncovers a previously unknown vulnerability in well-structured modular networks, highlighting the impact of community structures on network robustness.

Findings

Equal degree networks are more tolerant to node failures.

Adding strong community structures increases network vulnerability.

Vulnerability is universal across different degree distributions.

Abstract

Generally, networks are classified into two sides of inequality and equality with respect to the number of links at nodes by the types of degree distributions. One side includes many social, technological, and biological networks which consist of a few nodes with many links, and many nodes with a few links, whereas the other side consists of all nodes with an equal number of links. In comprehensive investigations between them, we have found that, as a more equal network, the tolerance of whole connectivity is stronger without fragmentation against the malfunction of nodes in a wide class of randomized networks. However, we newly find that all networks which include typical well-known network structures between them become extremely vulnerable, if a strong modular (or community) structure is added with commonalities of areas, interests, religions, purpose, and so on. These results will encourage avoiding too dense unions by connecting nodes and taking into account the balanced resource allocation between intra- and inter-links of weak communities. We must reconsider not only efficiency but also tolerance against attacks or disasters, unless no community that is really impossible.

Tables

Table 1. Table 1: Estimated values for the linear relations w = a ​ w ′ + b w=aw^{\prime}+b and Q = α ​ w + β Q=\alpha w+\beta .

		$a$	$b$	$\alpha$	$\beta$
$\nu =1$	(SF)	$0.973\pm 2.21\times {10}^{-3}$	$0.005\pm 1.82\times {10}^{-3}$	$0.911\pm 5.73\times {10}^{-3}$	$0.019\pm 4.63\times {10}^{-3}$
$\nu =1/2$		$0.998\pm 3.47\times {10}^{-4}$	$-0.0008\pm 2.86\times {10}^{-4}$	$0.932\pm 5.71\times {10}^{-3}$	$0.003\pm 4.71\times {10}^{-3}$
$\nu =0$		$0.999\pm 2.00\times {10}^{-4}$	$-0.001\pm 1.65\times {10}^{-4}$	$0.934\pm 5.71\times {10}^{-3}$	$0.001\pm 4.71\times {10}^{-3}$
$\nu =-1$	(Near ER)	$0.999\pm 1.67\times {10}^{-4}$	$-0.001\pm 1.38\times {10}^{-4}$	$0.934\pm 5.71\times {10}^{-3}$	$0.001\pm 4.71\times {10}^{-3}$
$\nu =-3$		$0.999\pm 1.57\times {10}^{-4}$	$-0.001\pm 1.29\times {10}^{-4}$	$0.934\pm 5.70\times {10}^{-3}$	$0.001\pm 4.70\times {10}^{-3}$
$\nu =-5$		$0.999\pm 1.53\times {10}^{-4}$	$-0.001\pm 1.27\times {10}^{-4}$	$0.934\pm 5.71\times {10}^{-3}$	$0.001\pm 4.71\times {10}^{-3}$
$\nu =-100$	(Near Regular)	$0.999\pm 1.52\times {10}^{-4}$	$-0.001\pm 1.26\times {10}^{-4}$	$0.934\pm 5.71\times {10}^{-3}$	$0.001\pm 4.71\times {10}^{-3}$

Table 2. Table S1: The correlation coefficients between μ 2 \mu_{2} and R R against attacks in Fig. 5 for modular networks, whose P ​ ( k ) P(k) are controlled by ν \nu as the attachment probabilities proportional to k i ν k_{i}^{\nu} .

$\nu$		$Q>0.8$
1	0.111	0.822
1/2	0.661	0.977
0	0.555	0.977
-1	0.909	0.975
-3	0.300	0.968
-5	0.097	0.960
-100	-0.095	0.955
Total	0.852	0.951

Table 3. Table S2: The correlation coefficients between μ 2 \mu_{2} and R R against attacks in Fig. S20 for 100 realizations of the networks, whose P ​ ( k ) P(k) are controlled by ν \nu as the attachment probabilities proportional to k i ν k_{i}^{\nu} .

$\nu$		$Q>0.8$
1	0.550	0.755
1/2	0.685	0.953
0	0.766	0.963
-1	0.840	0.966
-3	0.861	0.968
-5	0.849	0.963
-100	0.861	0.966
Total	0.775	0.945

Equations

$$Q\stackrel{\mathrm{def}}{=}\frac{1}{2M}\sum _{i,j\in V}\left(A_{ij}-\frac{k_i{k}_j}{2M}\right)\delta (c_i,c_j),$$

$$R\stackrel{\mathrm{def}}{=}\frac{1}{N}\left(\sum _{i=1}^{N-1}\frac{S^{1st}(q)}{N}\right),q=\frac{1}{N},\frac{2}{N},\dots ,\frac{i}{N},\dots ,\frac{N-1}{N},$$

$$\mathbf{x}(t)=\sum _{i=1}^N{c}_i{e}^{-{\mu }_i}{\mathbf{u}}_i\approx c_{1}1+c_2{e}^{-{\mu }_2}{\mathbf{u}}_2,$$

$$O\left(\sqrt{\frac{k_{max}}{{\mu }_2}}\log N\right),$$

$$h(G)\stackrel{\mathrm{def}}{=}\min \left\{\frac{|\partial F|}{|F|}:F\subseteq V,0<|F|<\frac{N}{2}\right\},$$

$$\frac{{\mu }_2}{2}\le h(G)\le \sqrt{2d{\mu }_2},$$

$${\mu }_2\le \frac{N}{N-1}{k}_{min}.$$

Full text

complex networks, module or community, robustness of connectivity, eigenvalue of Laplacian

Introduction

The robustness of connectivity is one of the fundamental issues in network science as an interdisciplinary research area of statistical physics, computer science, and biological or social science, since the disconnections lose the network function for transferring or delivering information or goods. The connectivity is therefore essential even abstracted as independent of what the materials are for linking. Despite persistent efforts [1, 2, 3, 4, 5, 6, 7], it is not elucidated what topological structure is the best or the worst against intentional attacks or large disasters. At least, degree distributions are considered as important. Because it has been found that networks with narrower degree distributions become more robust as the pure effect of degree distributions [8]. However, the robustness may be affected by modules with higher interactions among nodes beyond its degree at each node. In fact at three types of SF networks, ER random graphs [9, 10], and regular networks [11], decreasing the robustness has been numerically shown, when a strong community structure is added artificially [9, 10, 11]. Such vulnerability has also been found in spatially modular networks with local concentrations of nodes connected by short links [12].

On the above background, through an original approach [8, 13], we comprehensively investigate the robustness against intentional attacks in networks with/without modules (or communities as synonym in this paper) under the condition of continuously changing degree distributions which interpolate the above three types of well-known network structures. Moreover, we find a deep relation between the robustness and the algebraic connectivity: the second smallest eigenvalue of Laplacian matrix [14] of graph under the condition of a constant total number of links. Although it has been numerically shown that the eigenvalue increases as the node or link connectivity (defined by the minimum number of nodes or links whose removal causes disconnection) is larger in ER random graphs or SF networks with more (not constant) total links [15], this analysis is also limited for Poisson and power-law degree distributions without considering communities and attacks.

Related Work

So far it has been discovered about twenty years ago that many real networks in social, technological, and biological systems are extremely vulnerable against attacks on hub nodes [1, 2] in commonly existing scale-free (SF) structure whose degree distribution follows a power-law [16, 17]. It was believed about ten years ago that the robustness can be optimized as onion-like by increasing positive degree-degree correlations even from uncorrelated SF networks [3, 4]. Instead, too strong correlations rather weaken the robustness [4], the whole rewiring [5] to enhance correlations is not realistic because of discarding the already constructed relations between nodes. To retain the existing links, incrementally growing methods have been proposed in showing the emergence of onion-like networks [18, 19, 20, 21]. Moreover, it has been pointed out that [21] the enhancing loops may be more crucial than degree-degree correlations for improving the robustness. Just around that time, the equivalence of dismantling and decycling problems has been suggested theoretically in a large class of random networks with light-tailed degree distributions [22]. This means that enhancing loops leads to the strongest tolerance of connectivity for avoiding fragmenting or dismantling as the worst case removal of nodes. Here, the dismantling problem is to find the minimum set of nodes which removal makes a network fragmented into at most a given size, while the decycling problem is to find the minimum set of nodes which are necessary to form loops. They correspond to the critical node detection (CND) [23] and the minimum feedback vertex set (FVS) [24] problems, respectively, known as NP-hard in computer science. Thus, to only measure the size of FVS is intractable, its maximization by enhancing loops becomes more difficult. However, there are several evidences to be solvable somehow as follows.

Recently, by several rewiring methods for heuristically enhancing loops, it has been commonly found that [25] the widths of minimum and maximum degrees become small even with negative degree-degree correlations as counterexamples for onion-like networks [3, 4]. These networks have a large size of FVS with improving the robustness than the conventional optimal [5] against hub attacks. Moreover, the robustness has been comprehensively investigated for continuously changing degree distributions [8] from power-law to exponential [26] further to narrower ones [13] with randomization [27], although the importance of degree distributions has been revealed [1, 2, 6, 7] for mainly power-law and Poisson degree distributions in realistic SF networks and classical Erdös-Rényi (ER) random graphs [28]. Here, because of eliminating degree-degree correlations and higher correlations among degrees, the randomization is necessary for investigating the pure effect of degree distribution on the robustness. From these bases, networks with narrower degree distributions are more robust in randomized networks with continuously changing degree distributions. It has also been numerically shown by using perturbations [29] that random regular networks with zero width of degrees are the optimal as suggested by [30], strictly speaking at least locally, in the state-of-the-art.

On the other hand, there widely exist modules or communities based on local areas, interests, religions, purposes, etc. in social, biological [31], and technological networks [32, 33]. Historically community-like structure has been considered with focusing on internal cohesion among nodes of a subgraph [34]. Such structure is generally characterized by the mixing of dense intra-links between nodes in a community and sparse inter-links between nodes in different communities. Intra- and inter-links correspond to strong-ties or bondings and weak-ties or bridgings, respectively, related to socialcapital [35, 36].

Methods

We explain how various networks are generated by using growing network (GN) [26] and inverse preferential attachment (IPA) [13] models. For growing networks, there is no other construction method to make continuously changing degree distributions $P(k)$ of degree $k$. From an initial configuration (e.g. a complete graph of $m$ nodes), a network is generated in prohibiting self-loops and multi-links by the attachments of $m$ links from a new node to the existing nodes at each time-step until the size $N$ as the total number of nodes. Here, the total number of links is $M\approx m\times N$ except for the initial links, and $1\leq m\ll N$. For selecting the attached node $i$, the attachment probability is proportional to $k_{i}^{\nu}$, where $k_{i}$ denotes the degree of node $i$, $\nu$ is a parameter. Note that a strictly regular network is not created even for $\nu\rightarrow-\infty$ in IPA model, since there are several degrees $m,m+1,\ldots,2m$ [13].

After generating networks with various $P(k)$ as shown later, to investigate the pure effect of degree distribution on the robustness, the randomization eliminates degree-degree correlations and other higher structure such as chain-like [13] by the following rewiring in the configuration model [27]. Off course, the generated network with $N$ nodes and $M$ links is different from an initial one of $m$ nodes. First, all links are cut to free-ends. In the initial stage of randomization, there are $k_{i}$ free-ends of links emanated form each node $i$. Second, two free-ends are randomly chosen and rewired to a link in prohibiting self-loops at a node and multi-links between nodes. Such process is repeated until constructing a connected network of $N$ nodes without free-ends of links. When it is impossible to prohibit self-loops or multi-links, several rewired links are cut to free-ends. Then, the above process is continued again.

After the randomization, we artificially add modular structure without overlapping among modules. To simplify the discussion, we assume that the number of nodes in each module is the same integer $N/m_{o}$ as an equal size of module [9, 10], where $m_{o}$ is a divisor of $N$ as the number of modules. This case of equal size is the most meaningful but difficult, since the largest module is dominant, when the sizes are quite different. In that case of different sizes, it is sufficient to easily investigate only the largest one in ignoring the small effects of other modules. One of $1,2,\ldots,m_{o}$ is randomly assigned to each node [10] without any bias. In this stage, the number of intra-links is estimated as $M/m_{o}$ by the probability that one-end node of link has the same module number randomly assigned to the other-end node, while the number of inter-links is estimated as remaining $(1-1/m_{o})M$.

In controlling the strength of modular structure by a given tentative rewiring rate $w^{\prime}$, some inter-links are rewired to intra-links in prohibiting self-loops and multi-links. The rewiring is slightly modified from the previous method [10] to maintain the whole connectivity under a given $P(k)$, and the random process is performed so as not to have any bias. First, $(1-1/m_{o})M\times w^{\prime}$ inter-links are randomly chosen and cut to free-ends. Second, when pairs of free-ends exist in each module, they are randomly chosen and connected to be as many intra-links as possible. Note that the total numbers of free-ends are even both before and after the pairings. Third, if there remain free-ends in different modules, then the free-ends are randomly chosen and connected as returning to inter-links. The return is repeated if possible. Finally, for further remaining two free-ends (denoted by $i,j$) generally belong to different modules, an intra-link is randomly chosen and cut to new free-ends (denoted by $k,l$) in other than those modules. These nodes are connected (as $i$-$k$ and $j$-$l$). Such final process is repeated until there are no free-ends. If a constructed network is fragmented, the sample is discarded. After the rewirings, it is trivial to maintain $P(k)$ because there is nothing to add and remove any links at each node. We remark that the actual rewiring rate $w$ is slightly changeable from a tentative rate $w^{\prime}$, though it is not essential for controlling the strength of modular structure.

For each of the modular networks under the condition of the same $P(k)$ by varying the tentative rewiring rate $w^{\prime}=0.0,0.5,0.7,0.9,0.95$, and $0.98$, we calculate the modularity [37] defined as

[TABLE]

where $\delta(c_{i},c_{j})$ is the Kronecker delta: $1$ if $c_{i}=c_{j}$ or [math] otherwise, $c_{i}$ (or $c_{j}$) is the belonging module number of node $i$ (or $j$). A larger $Q$ represents stronger modular structure.

Results

Various modular networks with continuously changing degree distributions

We set the parameters $\nu=1,1/2,0,-1,-3,-5$, and $-100$ in the attachment probabilities proportional to $k_{i}^{\nu}$, the numbers $m_{o}=5,10,20,50,100$, and $200$ of modules, the number $m=2$ of attached links per time-step in growing, and the size $N=10^{4}$. Figure 1 shows power-law (in SF networks), power-law with exponential cut-off, exponential, near Poisson [38] (corresponded to ER random graphs), and narrower degree distributions [13] (approaching regular networks) for $\nu=1,1/2,0,-1$, and $\nu<-1$, respectively. These function forms [26] are

power-law

$P(k)=Ck^{-\gamma}$,     ($\gamma=3$ for $\nu=1$),

power-law with exponential cut-off

$P(k)\approx C^{\prime}k^{-(1/2-\mu^{2})}\exp(-2\mu\sqrt{k})$,

exponential

$P(k)=C"\exp(-ak)$,

nearly Poisson

$P(k)\approx\frac{\lambda^{k}e^{-\lambda}}{k!}$,

and narrower ones with rapidly decreasing tails of $\exp(-\beta k\log k)$ [13], where $C$, $C^{\prime}$, and $C"$ are normalizing constants, the values of $\mu$, $a$, $\lambda$ and $\beta$ depends on the values of $N$ and $m$ (or $M\approx m\times N$).

Figure 2 visualizes SF networks with modules by varying the tentative rewiring rate $w^{\prime}$. Similar structural changes are obtained in other networks without power-law $P(k)$ for $\nu\leq 1/2$. For the actual rewiring rate $w$ or the corresponded modularity $Q$, we obtain the linear relations $w=aw^{\prime}+b$ and $Q=\alpha w+\beta$ as estimated in Table 1 by using the least squares method over $100$ realizations of networks with each combination of parameters $\nu$, $w^{\prime}$, and $m_{o}$.

Robustness of connectivity against attacks

The connectivity in a network is usually evaluated by the robustness index [3]

[TABLE]

where $S^{1st}(q)$ denote the size of largest connected component (the 1st maximum or giant component) at the fraction $q$ of attacks or failures. We mainly consider the intentional module-based (MB) attacks [39]. Because they give larger damages to separate the modules by removing inter-links with priority than the conventional initial degree (ID) attacks and initial betweenness (IB) attacks without recalculations of degrees and betweenness centralities (BC) [40] after node removals. Here, the robustness against ID or IB attacks is typically studied and compared to that against MB attacks in considering that end-nodes of inter-links tend to have large BC and to be also chosen by IB attacks. However, it is also possible that end-nodes of intra-links have high BC. In addition, large BC nodes may be hubs, therefore some chosen nodes are coincident in these attacks but others are not as the targets. These differences will appear in the results as shown later (also see Supplement for more details). In the following results, we show the averaged values of $Q$, $S^{1st}(q)/N$, $R$, and the eigenvalue of Laplacian matrix over $100$ realizations. Figure 3 shows that the curves of $S^{1st}(q)/N$ against MB attacks are shifted to left for the larger rewiring rate $w^{\prime}$ with stronger modular structure from red, orange, yellow, green, blue, to light-blue lines in SF (Fig. 3ab) and nearly regular networks (Fig. 3cd) at $\nu=1$ and $-100$, respectively, for $m_{o}=5$ and $200$ modules. Interpolated similar results between these figures are obtained for the intermediate $m_{o}=10,20,50,$ and $100$ modules and in other networks whose degree distributions are exponential, nearly Poisson, and narrower ones for $1/2\geq\nu\geq-5$ (see Supplement Fig. S1 $\sim$ S5). In addition, although there are not many differences in comparing the lines of $S^{1st}(q)/N$ with the same color at a rewiring rate $w^{\prime}$ for SF (Fig. 3ab) and nearly regular networks (Fig. 3cd) against MB attacks, there are remarkable differences in comparing the lines of $S^{1st}(q)/N$ with the same color for SF and nearly regular networks against ID or IB attacks (see Supplement Fig. S6 $\sim$ S15). Such differences are consistent with the previous results [8]. From the definition in Eq. (1), the value of $0\leq R\leq 1/2$ represents the area under the curve of $S^{1st}(q)/N$, a smaller area means more vulnerable against node removals.

A new relation to the robustness

To find a new relation to the robustness, we also investigate the second smallest eigenvalue $\mu_{2}$ of Laplacian matrix $L=A-D$, where $A$ denotes the adjacency matrix whose element is $A_{ij}=1$ if nodes $i$ and $j$ are connected or [math] otherwise, and $D$ denotes the diagonal matrix $\operatorname{diag}(k_{1},k_{2},\ldots,k_{N})$. Note that $\mu_{2}$ is calculated for each of modular networks before attacks. In Fig. 4abcd, we emphasize the relation of $Q$ and $\mu_{2}$ or $R$ against MB attacks in all of the networks denoted by colored lines for $m_{o}=5$ and $200$ modules. Similar results are obtained for the intermediate $m_{o}=10,20,50,$ and $100$ modules. In particular, it is universal that both $\mu_{2}$ and $R$ are rapidly decreased for large $Q\gtrapprox 0.8$ as too strong modules or communities with too dense intra-links in all of these networks whose $P(k)$ are continuously changed from power-law, exponential, near Poisson, to narrower ones by varying $\nu$ denoted as different colored lines (For more details, see Supplement Fig. S16,S17). While, as shown in Fig. 4ef, the decreasing of $R$ against ID attacks for large $Q$ is not remarkable, the results for IB attacks is the intermediate (For more details, see Supplement Fig. S18,S19). However, commonly in Fig. 4abcdef, the decreasing becomes lower than the most gradual parts of red line for almost non-modular SF networks known as the extremely vulnerable [1, 2]. Moreover, we remark that higher values are shown from red to purple lines whose order corresponds to SF networks, ER random graphs, and regular networks.

Figure 5 shows the correlations between $\mu_{2}$ and $R$ against attacks in these modular networks (see Supplement Fig. S20 as the detail results for 100 realizations of the networks by each parameter $\nu$ or $w^{\prime}$). Several superimposed points with a color represent the results in networks for the different number $m_{o}$ of modules and varying the rewiring rate $w^{\prime}$ under the condition of the same $P(k)$ at each value of $\nu$ which corresponds to power-law (as SF networks), exponential, near Poisson (as ER random graphs), or narrower ones (in approaching regular networks). It is common that the plots are located from down-left to top-right as decreasing both $\nu$ (from red to purple points) and $w^{\prime}$ or its corresponding $Q$ to weaken the modular structure, especially, the networks become stronger as $P(k)$ becomes narrower for smaller $\nu$. The correlation coefficients are $0.951$ for $Q>0.8$ and $0.852$ for $Q<0.8$ at the lower and the upper dashed-oval parts in Fig. 5a, respectively. In other words, against MB attacks, small $\mu_{2}$ and $R$ appear clearly with higher correlations as the modularity $Q$ is larger. In contrast, Fig. 5bc shows that the correlation coefficients are $0.690$ for $Q>0.8$ and $0.967$ for $Q<0.8$ against IB attacks, and $0.198$ for $Q>0.8$ and $0.963$ for $Q<0.8$ against ID attacks. Rather, these higher correlations in the upper dashed-oval parts represent the pure effect of $P(k)$ on the robustness against IB or ID attacks, because of weak modular structure in $Q<0.8$ (For more detail correlations by each value of $\nu$, see Supplement Tables S1 and S2).

Discussion

We briefly discusses what the maximization of $\mu_{2}$ means from the explanations to mathematical theories. For a network with the maximum and the minimum degrees $k_{max}$ and $k_{min}$, $\mu_{2}$ is dominant to be fast epidemic or information delivery, since the solution of diffusion equation $d\bm{x}/dt=-L\bm{x}$ is given by

[TABLE]

where $\bm{u}_{i}$ is the $i$-th eigenvector of Laplacian matrix $L$, $\bm{u}_{1}=\bm{1}\stackrel{{\scriptstyle\rm def}}{{=}}(1,1,\ldots,1)$, and the coefficients $c_{1},c_{2},\ldots,c_{N}$ are determined by the initial state $\bm{x}(0)=\sum_{i=1}^{N}c_{i}\bm{u}_{i}$. Note that $e^{-\mu_{i}}$ is more quickly converged to zero than $e^{-\mu_{2}}$ for $i\geq 3$ because of $0=\mu_{1}<\mu_{2}\leq\mu_{3}\leq\ldots\leq\mu_{N}$. In other words, a larger $\mu_{2}$ gives a more compact network with fast diffusion. This property is related to the following theoretical results. First, the diameter $\rho$ (as the maximum length of the shortest path between nodes) and the average length $\langle l_{ij}\rangle$ of the shortest paths between nodes are upper bounded by

[TABLE]

for a connected network [41, 14], where the length of the shortest path is counted by hops between nodes. Thus, these smaller values of $\rho$ and $\langle l_{ij}\rangle$ give a compact network as $\mu_{2}$ is larger. We remark that the construction of SF networks is based on efficiency and therefore compact with small $\langle l_{ij}\rangle$ even for $M=O(N)$ links [42, 43] but vulnerable against hub attacks [1, 2], however the coexisting of compactness (or small-worldness) and strong robustness is possible in other networks especially as random regular.

Second, the expanding constant

[TABLE]

is also bounded as

[TABLE]

for a finite connected $d$-regular graph $G$ [44], where $V$ denotes a set of nodes with the size $|V|=N$, $\partial F$ denotes a set of links emanated from nodes in $F$ to other nodes in $V-F$, $\mu_{2}$ equals to the spectral gap $\lambda_{N}-\lambda_{N-1}$, $d=\lambda_{N}\geq\lambda_{N-1}\geq\ldots\geq\lambda_{1}$ as the eigenvalues of adjacency matrix $A$. Thus, $h(G)$ can be larger as the order of $\mu_{2}$ from the lower bound in the left-hand of Eq. (3), which means the strong connectivity with many inter-links in $\partial F$ against a bisection into $F$ and $V-F$ by these link removals [14]. Note that $h(G)$ is also limited by the order of $\sqrt{\mu_{2}}$ from the upper bound in the right-hand of Eq. (3). In addition, for a network, $\mu_{2}$ can be larger, when $k_{min}$ is large from [14]

[TABLE]

These Eqs. (3) and (4) become the bases for supporting our obtained result: Weak modular networks with narrower degree distributions (larger $k_{min}$) have stronger connectivity against attacks, at the extreme case the nearly regular networks have the largest $R$ and $\mu_{2}$ in Figs. 3, 4, and 5. From Eqs. (2) and (4), smaller $k_{max}$ and larger $k_{min}$ are also desirable.

Third, in a $d$-regular graph, a larger $\mu_{2}$ corresponds to a smaller $\lambda_{N-1}$. However, $\lambda_{N-1}$ is greater than or equal to $2\sqrt{d-1}$ [45]. The smallest case $\lambda_{N-1}=2\sqrt{d-1}$ is the condition whether or not a $d$-regular graph becomes a Ramanujan graph known as good with small diameter $\rho$ and large expanding constant $h(G)$. Strictly, a Ramanujan graph has the constraints for the limited numbers $N$ and $M=d\times N/2$ of nodes and links, e.g. $N=p^{r}$, $r\geq 2$, $q\geq 2\sqrt{p}$, $d=(p^{r}-1)/(p-1)$, where $p$ and $q$ are odd primes in an algebraic construction [44, 45]. Therefore it belongs to a very small subclass of $d$-regular graphs. However, from the asymptotic property [46], the strong robustness of weak modular $d$-regular networks may be corresponded to the goodness of Ramanujan graphs. In more detail, it has been derived that [46], for any integer $d\geq 4$ and $\epsilon>0$, a random $d$-regular graph with sufficiently large $N$ asymptotically approach a Ramanujan graph in the meaning that the eigenvalues $\lambda_{i}$ of adjacency matrix except for $\lambda_{N}=d$ satisfy $|\lambda_{i}|\leq 2\sqrt{d-1}+\epsilon$, $i=1,\ldots,N-1$, as $\epsilon\rightarrow 0$ for $N\rightarrow\infty$. Note that, for $d=4$, the maximum spectral gap is $d-2\sqrt{d-1}\approx 0.53$ as nearly as vertical axis intercepts of purple lines in Fig. 4ab. In addition, since a Ramanujan graph has links determined by operations on the finite group of projective general linear group $PGL_{2}(q)$ or projective special linear group $PSL_{2}(q)$ using an equivalence class based on modulo [44, 45], neighbor nodes of two end-nodes of a link seem to be far away on the paths except for the link, when $N$ is large. This is also supported by that ”the frequency of times spent by simple random walk in a nontrivial cycle is almost surely [math] on every infinite Ramanujan graph”, where ”nontrivial cycle is not purely a backtracking cycle” [47]. Note that short cycles exist in the local module structure. Thus, Ramanujan graphs are probably non-modular $d$-regular graphs, and close to random ones for a large $N$. As further studies with deeper insights, there remain theoretical issues or more careful numerical checks, which may also be useful from an application points of view.

Conclusion

We have comprehensively investigated the robustness of connectivity in modular networks with continuously changing degree distributions, which include power-law (SF networks), nearly Poisson (ER random graphs), and nearly unimodal (regular networks) for interpolating the previous analyses [9, 10, 11] at only three types of typical networks. Although tremendous efforts [1, 2, 3, 4, 5, 6, 7] has been devoted to slow progress research on the robustness in complex networks, we have additionally revealed the universal property that strong modular networks, even with any degree distributions, are extremely vulnerable against intentional attacks. Moreover, between the statistical robustness of connectivity [3, 4] and the algebraic connectivity [14, 44, 41], we have found a new relation that both the robustness index $R$ and the second smallest eigenvalue $\mu_{2}$ of Laplacian matrix become smaller as the modularity $Q$ is larger. In particular, the correlations between $R$ and $\mu_{2}$ are high, when MB attacks are given in a strong modular structure with $Q>0.8$, while ID or IB attacks are given in a weak modular structure with $Q<0.8$. The later case as the pure effect of degree distributions is also supported by that the robustness becomes stronger as networks approach regular from SF [8], and that $\mu_{2}$ can become larger in the upper and lower bounds [41, 14, 44] as $k_{min}$ is larger and $k_{max}$ is smaller with narrower degree distributions. Therefore too dense modules or communities should be avoided for maintaining the essential network function of connectivity against intentional attacks or disasters. In other words, the excessive numbers of links to neighbor nodes counted by hops are unsuitable for not only technological infrastructures or biochemical reactions but also social meanings in increasing contemporary threats of military conflicts or large disasters, though dense unions as connecting nodes seem to give a positive impression. In more details, future studies are remained to elucidate the mechanism in slightly different behavior for varying the number $m_{o}$ of modules and the robustness against other more destructive attacks [48], e.g. attacks on collective influencers [49] or candidates of FVS [50], in taking into account that the worst node removals is CND problem [23] as NP-hard.

Data availability

The data analyzed in this study are available from the corresponding author on reasonable request.

Abbreviations

SF: scale-free, CND: critical node detection, FVS: feedback vertex set, NP: non-deterministic polynomial time, ER: Erdös-Rényi, GN: growing network, IPA: inverse preferential attachment, MB: module-based, ID: initial degree, IB: initial betweenness, BC: betweenness centralities.

Author Information

Affiliation

Graduate School of Advanced Institute of Science and Technology, Japan Advanced Institute of Science and Technology, Ishikawa 923-1292 Japan

Yukio Hayashi, and Taishi Ogawa

Contributions

Y.H. conceived and designed the research, analyzed data and discuss results, and wrote the manuscript. T.O. implemented and performed numerical experiments.

Competing Interests

The authors declare no competing interests.

Corresponding Author

Yukio Hayashi

Supplementary Materials

We set the parameters $\nu=1,1/2,0,-1,-3,-5$, and $-100$ in the attachment probabilities proportional to $k_{i}^{\nu}$, the numbers $m_{o}=5,10,20,50.100$, and $200$ of modules, the number $m=2$ of attached links per time-step in growing, and the size $N=10^{4}$. In the following results, we show the averaged values over $100$ realizations.

In Figs. S1 $\sim$ S15, it is common that the curves of $S(q)/N$ against attacks are shifted to left for larger rewiring rate $w^{\prime}$ with stronger modular structure colored from red, orange, yellow, green, blue, to light-blue. However, in comparing these figures, the areas under the curves are larger as $\nu$ is smaller in approaching regular from SF networks.

In Figs. S16 $\sim$ S19, it is common that higher values for each of the vertical axes are shown from red to purple lines, which correspond to $P(k)$ changed from power-law (as SF networks), power-law with exponential cut-off, exponential, nearly Poisson (as ER random graphs), and narrower ones (approaching regular networks) by varying $\nu$, although the intervals of lines and decreasing behavior are slightly different in comparing these figures.