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A new method optimizing the subgraph centrality of large networks
Physica A 444 (2016) 373–387
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Physica A journal homepage: www.elsevier.com/locate/physa
A new method optimizing the subgraph centrality of large networks Xin Yan a,∗ , Chunlin Li a,c , Ling Zhang a , Yaogai Hu b a
Department of Computer Science, Wuhan University of Technology, Wuhan 430070, China
b
School of Electronic Information, Wuhan University, Wuhan 430072, China
c
The State Key Laboratory of Integrated Services Networks, Xidian University, Xi’an 710071, China
highlights • • • • •
We derive a better measure of network connectivity for large networks. A new strategy that can increase/decrease network connectivity the most is derived. We derive two complete functions of spectral density for two types of networks. An optimization algorithm based on spectral density is proposed. Our new findings about spectral density are also concluded.
article
info
Article history: Received 4 April 2015 Received in revised form 16 September 2015 Available online 20 October 2015 Keywords: Spectral density Optimization Network connectivity Structural properties Spectral radius
∗
abstract Since many realistic networks such as wireless sensor/ad hoc networks usually do not agree very well with the basic network models such as small-word and scale-free models, we often need to obtain some expected structural features such as a small average path length and a regular degree distribution while optimizing the connectivity of these networks. Although a minor addition of links for optimizing network connectivity is not likely to change the structural properties of a network, it is necessary to investigate the impact of link addition on network properties as the number of the added links increases. However, to the best of our knowledge, the study of that problem has not been found so far. Furthermore, two closely related questions to that problem, i.e., how to measure and how to improve network connectivity, have not been studied carefully enough yet. To address the three problems above, the authors derive a better measure of network connectivity for large networks and a new strategy that can increase/decrease network connectivity the most, and propose a spectral density algorithm optimizing the connectivity of large networks, which is able to indicate the impact on the structural properties of a network while increasing/decreasing its connectivity, providing us a guided optimization of network connectivity. In other words, our algorithm can optimize not only the connectivity of a large network but also its structural features. Meanwhile, our new findings about spectral density are also concluded in this paper. In addition, we may also apply this algorithm to solve all eigenvalues of an N × N matrix, with a low complexity of O(N 2 ) at most. © 2015 Elsevier B.V. All rights reserved.
Corresponding author. Tel.: +86 13035107916. E-mail addresses: [email protected], [email protected] (X. Yan).
http://dx.doi.org/10.1016/j.physa.2015.10.034 0378-4371/© 2015 Elsevier B.V. All rights reserved.
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1. Introduction Many complex systems can be modeled by networks to capture the possibly inhomogeneous patterns of interactions within complex systems. Due to the diversity of complex systems, the study of complex networks pervades many fields of science, such as mathematics, physics, sociology, computer science, and biology [1,2]. In recent years, there has been a great interest in the study of measuring and optimizing the robustness or resilience of complex networks, as one of the most important topics in complex networks [3,4]. To evaluate the robustness properties of complex networks, various structural and spectral measures have been proposed on the basis of different quantitative properties of underlying networks [5]. Examples are degree distribution, average path length, clustering coefficient, betweenness, global efficiency, largest component (structural measures), as well as algebraic connectivity, spectral radius (spectral measures) etc., which can be employed as the optimization objectives for promoting the robustness of complex networks. Although structural measures represent the topological properties of a network more directly compared to spectral measures, a spectral measure of a network usually contains more abundant characteristic information of this network than a structural measure. For example, a network with a large value of algebraic connectivity, i.e. the second smallest eigenvalue of the Laplacian matrix of this network, has not only high network connectivity but also a low threshold of coupling strength for synchronization [6]. Therefore, in most cases, we prefer to make use of spectral measures while evaluating and optimizing the robustness of complex networks. Network connectivity is a crucial form of network robustness in view of its wealthy robustness implications [3]. As mentioned above, currently algebraic connectivity acts as the prime measure that evaluates the connectivity robustness of complex networks. However, algebraic connectivity has its disadvantages to measure network connectivity while an overall network graph is disconnected, because the value of algebraic connectivity is always equal to zero for any overall disconnected network regardless of the local connectivity of this network. Consequently, in this paper we will introduce subgraph centrality [7] as another spectral measure to evaluate network connectivity and as another optimization objective to enhance network connectivity. To improve the connectivity of an existing large real-world network, instead of substituting its infrastructure for the optimal one that maximizes its connectivity as discussed in Ref. [8], a minor modification on the current network, i.e. adding a small number of links, is usually required due to economic concerns [9]. Nevertheless, here an important question is how we can decide the exact number of the links that need adding in order to increase the connectivity of a network. For an undirected network G(V , E ) consisting of node set V and link set E with N nodes and L links, the number of the added N (N −1) − L to construct a full connected topological graph with the optimal connectivity, if we do not take links should be 2 into account the cost of link additions. Thus, for a large realistic (usually sparse) network, the number of the links that need adding is often bounded by the constraint on the link addition cost. However, if some links are added at an enough low cost that could be ignored, should we add an adequate number of links to a network till it becomes a full connected topology? The answer is NOT in most cases. This reason is that the construction of a full connected topology would extremely likely make the original topological properties of this network changed. For instance, a scale-free network will probably not have some topological features we are expecting any more after some additional links are included. Especially for wireless communication networks, a full connected topology will increase dramatically the MAC collisions due to the lack of its sparseness. In most cases, the optimization objectives for a large realistic network are probably not only to improve its connectivity but also to remain its structural properties unchanged even to promote its certain topological features, e.g., the degree distribution with power-law form, the heterogeneity, and the scalability, etc. Therefore, one of important aims in this paper is to optimize the topological properties of a network besides its connectivity, provided that its cost constraint allows. To optimize the connectivity of a network, another important question is where we should add a link in this network such that its connectivity can be increased the most. The investigation on adding one link to improve network connectivity can guide us how to dynamically add a set of links one by one such that network connectivity is maximally increased. Hence, we propose a new strategy of adding a link, based on the eigenvector components of the adjacency matrix of a network, to improve its connectivity. To the best of our knowledge, these three questions discussed above: (a) how to determine a spectral measure of network connectivity as the optimization objective, (b) how to optimize not only the connectivity of a network but also its topological features if allowed by the cost constraint, and (c) how to add a link to a network such that its connectivity can be increased the most, have not been studied intensively so far. In particular, the study of problem (b) has not been found yet. These three critical problems in network connectivity optimizations will be investigated in this paper. The rest of this paper is organized as follows. Section 2 presents a spectral measure of network connectivity (namely subgraph centrality) and its approximation (namely spectral radius) acting as the optimization objective, instead of the conventional measure of algebraic connectivity. Section 3 presents a new strategy increasing spectral radius based on eigenvector components, and explains why it can be increased the most by using this strategy. In Section 4, we introduce the spectral density measure of network properties as the optimization objective, which is usually treated as the ‘‘fingerprint’’ and ‘‘pulse manifestation’’ of a large network. Section 5 presents the optimization algorithm of network connectivity on the basis of the spectral density of this network. Some optimization examples are given in Section 6. Section 7 presents our conclusions.
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2. Spectral measure of network connectivity Conventionally algebraic connectivity, the second smallest eigenvalue of Laplacian matrix of a network graph, is usually used to measure the connectivity of this network. Fiedler shows that the value of algebraic connectivity reflects how well an overall graph is connected [10]. A graph with a larger numerical value of algebraic connectivity is more difficult to be cut into independent components. However, as mentioned above, algebraic connectivity has its disadvantages to measure network connectivity while an overall network is disconnected, because the value of algebraic connectivity is always equal to zero for any overall disconnected network regardless of the local connectivity of this network. For the sake of avoiding this problem, here it is necessary to introduce another spectral measure of network connectivity, subgraph centrality, as the optimization objective of network connectivity. The topological graph of any network G(V ,E )consisting of node set V and link set E, with N nodes and L links, can be described by an adjacency matrix A(G) = aij N ×N , a non-negative matrix, whose elements aij are positive if there is a link going from node i to node j and zero otherwise, with aii ≡ 0. According to the Perron–Frobenius theorem [11], if A(G) is irreducible, its unique largest eigenvalue is real and positive and the components of the corresponding left and right eigenvectors all are positive. In this paper assuming that the links in G(V , E ) all are undirected, the adjacency matrix A(G) is a symmetric matrix. All eigenvalues are real and A(G) possess an eigen spectral decomposition, A(G) = XDX T , where X = x1 x2 · · · xN is an orthogonal matrix that forms an orthogonal basis, such that XX T = X T X = IN , with the real and normalized eigenvectors x1 , x2 , . . . , xN of A(G) as the columns of matrix X , corresponding to the eigenvalues λ1 > λ2 ≥ · · · ≥ λN −1 ≥ λN in descending order and the diagonal matrix D = diag (λ1 , λ2 , . . . , λN −1 , λN ) [12]. Here note that the left and right eigenvectors of any eigenvalue in a symmetric matrix are the same. 2.1. Approximation of subgraph centrality The closed walks in a network graph are directly associated with its subgraphs. Moreover, a set of closed walks are also able to denote all the alternative routes from a certain node to another in a network. If there are more alternative routes between any pair of nodes in a network, this network may be taken into account connected better. Thus, the number of subgraphs defined by a set of closed walks in a network graph can be used to measure the connectivity of this network. Let nk (i) denote the number of the closed walks with length k starting and ending at the node i in a graph G, which is defined as the ith diagonal entry of the adjacency matrix A(G) with the exponent of k [7]: nk (i) = Ak
ii
.
(1)
Here we take the subgraph centrality (SC in brief) of a network graph G as its connectivity measure, which is defined as the sum of closed walks at all nodes in the network, corresponding to different lengths k [13]. Using Eq. (1), SC can be written as SC =
∞ N
nk (i) =
k=1 i=1
∞ N k
A
ii
.
(2)
k=1 i=1
From A(G) = XDX T and D = diag λk1 , λk2 , . . . , λkN −1 , λkN . Thus,
k λ1 Ak = (x1 , x2 , . . . , xN )
diag (λ1 , λ2 , . . . , λN −1 , λN ), we can obtain that Ak
=
XDk X T , and Dk
=
T x1
λ
k 2
..
. λkN
xT 2 . , .. xTN
and N k
A
ii
= ∥x1 ∥22 λk1 + ∥x2 ∥22 λk2 + · · · + ∥xN ∥22 λkN .
(3)
i =1
Since X = x1
x2
···
xN is a set of orthogonal and normalized real eigenvectors, we obtain ∥x1 ∥22 = ∥x2 ∥22 = · · · =
∥xN ∥22 = 1. Eq. (3) is turned into SC =
∞ N k=1 i=1
λki .
N k i=1
A
ii
=
N
i=1
λki . Meanwhile, Eq. (2) can further be transformed into (4)
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The Perron–Frobenius theorem implies that λ1 > λi for i = 2, . . . , N. When k → ∞ (i.e., the network size is large), we have N
λki
i=1
lim
k→∞
λk1
=1+
N i=2
lim
k→∞
λi λ1
k
= 1,
which leads to N
λki ≈ λk1 .
(5)
i=1
Thus for large k (namely a large network), SC approximates to SC =
∞ N
λki ≈
k=1 i=1
∞
λk1 .
(6)
k=1
For avoiding the possible case of SC → ∞, we may divide the approximation of SC by the factorial of length k, and then take its natural logarithm. Let this result act as the new expression of SC :
∞ λk
.
1
SC = ln
k=1
k!
(7)
According to the Taylor series, ex =
∞ λk
1
SC = ln
k=0
k!
xk k=0 k!
∞
=1+
xk k =1 k ! ,
∞
it yields
− 1 = ln eλ1 − 1 .
(8)
Since ∥A∥2 ≫ 0 in general and ∥A∥2 = λ1 , we arrive at λ1 ≫ 0, eλ1 ≫ 1, and SC = ln eλ1 − 1 ≈ λ1 .
(9)
From the result of Eq. (9), it is observed that the subgraph centrality of a network graph, as another spectral measure of network connectivity, can be represented and figured out approximately by the largest eigenvalue (spectral radius) of its adjacency matrix. 2.2. Comparison of spectral radius to algebraic connectivity In order to demonstrate the advantages of subgraph centrality to measure network connectivity, we respectively plot the values of spectral radius (namely λ1 ) and algebraic connectivity under node random removal (failure) and node targeted removal (attack) with regard to the ratio of the removed nodes to the total nodes, for three types of complex networks: Erdös–Rényi (ER) random model, Watts–Strogats (WS) small-world model, and Barabási–Albert (BA) scale-free model, as shown in Fig. 1. From Fig. 1(a) and (b), it can be observed that the shape and trend of the curves of λ1 are closely similar to those of algebraic connectivity, which is consistent with the accepted conclusions in this scientific field [14]. However, unlike λ1 , as the nodes in a topology are removed, the algebraic connectivity will decline to zero rapidly as soon as any isolated node exists in the topology, whereas at this time the topology still remains the most of its connectivity. Therefore, Algebraic connectivity can only measure the degree of connectivity and indicate whether a topology is connected or not. Without this disadvantage, λ1 can measure not only the degree of connectivity but also the degree of dis-connectivity of a topology, because λ1 is not zero when a topology becomes disconnected. That is why we prefer λ1 as the spectral measure and optimization objective of topological connectivity, rather than algebraic connectivity. 3. Strategy increasing network connectivity In view of the discussed above, we can increase the spectral radius λ1 of a network graph to improve its subgraph centrality or connectivity by dynamically adding a set of links one by one. To reach a certain optimization goal by adding as small number of links as possible, the addition of each of links should make λ1 be increased the most. We will derive a new strategy of link addition, by which λ1 can be increased the most when each of links is added. Suppose that any graph is modified so that its adjacency matrix A becomes A +ζ C , where C is the perturbation matrix and 0 < ζ ≪ 1. Let λ1 > λ2 ≥ · · · ≥ λN be the eigenvalues of A, with x1 , x2 , . . . , xN the corresponding normalized orthogonal
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Fig. 1. The spectral radiuses (a) and the algebraic connectivity (b) under various ratios of the removed nodes to total nodes for three types of network models: ER (square, with 200 nodes and 3875 links), WS (circle, with 200 nodes and 1000 links), and BA (triangle, with 200 nodes and 995 links) respectively. The red dashed curves correspond to node random failure, and the blue full curves corresponding to node targeted attack, i.e. removing nodes according to the descend order of node degrees. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
˜ 1 > λ˜ 2 ≥ · · · ≥ λ˜ N and x˜ 1 , x˜ 2 , . . . , x˜ N denote the eigenvalues and the corresponding eigenvectors. Correspondingly, λ eigenvectors of A + ζ C respectively [12]. For undirected networks, we apply the general perturbation formula [15,16]: λ˜ 1 = λ1 + ζ xT1 Cx1 + ζ 2
T 2 N xk Cx1 + O ζ3 . λ1 − λk k=2
(10)
When using the first-order perturbation from formula (10), the change 1λ1 of the corresponding largest eigenvalue λ1 equals approximately
1λ1 = λ˜ 1 − λ1 ≈ ζ xT1 Cx1 .
(11)
Assume that a perturbation matrix B whose nonzero entries are finite is usually of the same order of magnitude as A. If, however, the modifications are limited to a small number of links in a large network, then ∥B∥2 ≪ ∥A∥2 and the result of Eq. (11) would be valid with ζ C replaced by B, i.e.
1λ1 = λ˜ 1 − λ1 ≈ xT1 Bx1 .
(12)
It can be seen that Eqs. (11) and (12) are of the Rayleigh quotient form [11]. Since the subgraph centrality of a network can be improved by dynamically adding a small number of links one by one, we assume that a certain link is added between node i and node j in this network. At this time (B)ij = (B)ji = w (wherein the constant w is the weight of this undirected link) and the other elements in B are equal to zero, leading to
1λ1 ≈ xT1 Bx1 = 2w(x1 )i (x1 )j .
(13)
According to the Perron–Frobenius theorem, all of the components of principal eigenvector x1 corresponding to λ1 are positive, indicating 1λ1 > 0 (i.e., the value of subgraph centrality continues to be increased as the links are added one by one). From Eq. (13), when we add a link between node i and node j in a network graph, with the largest product of its principal eigenvector components (x1 )i and (x1 )j , λ1 can be increased the most. Thus our strategy increasing network connectivity is to add one undirected link between two distinct nodes in this network every time such that the product of the corresponding two principal eigenvector components is the largest value. 4. Spectral density measure of network features Besides the connectivity of a network, as discussed above, it is also necessary to optimize its topological features. Since the eigenvalue spectrum of adjacency matrix of a network is usually regarded as its ‘‘fingerprint’’ and ‘‘pulse
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manifestation’’ respectively used to identify this network and retrieve some critical structural characteristic information from this network [17], we could employ the eigenvalue spectrum or the spectral density of eigenvalues (for large-sized networks) to measure the features of this network. It is known that the networks with different topological properties have essentially distinct eigenvalue spectra and spectral densities [18]. Thus the spectral density of eigenvalues of a network could be treated as the measure and the optimization objective of its network features. When the size of a network is large, we prefer to employ the spectral density ρ to represent the distribution law (i.e. probability distribution) of adjacency matrix eigenvalues λ of this network. The spectral density of adjacency matrix eigenvalues of a network with N nodes can be written as a sum of Kronecker delta functions [19]:
ρ(λ) =
N 1
N i =1
δ(λ − λi ),
(14)
wherein
δ(λ − λi ) =
1, 0,
λ = λi λ ̸= λi .
When N → ∞, Eq. (14) will converge to a continuous function. In this paper, we select only two representative types of network models, ER and BA models, among a variety of network models such that we can concentrate on our spectral density optimization method. Although the spectral densities of ER and BA models have been mentioned in many Refs. [20–27], it is still short of a complete description about these two spectral densities. Thus we need to derive the detailed mathematical expressions of these two spectral densities for our optimization aims. 4.1. Spectral density of random networks Wigner’s Semicircle Law [20,21] is the fundamental result in the spectral density theory of large random networks. It exhibits a universal property of a class of large, real symmetric matrices with independent random elements. The adjacency matrix of ER random graph satisfies the conditions that are applicable to Wigner’s Semicircle Law. Hence, we can take Wigner’s Semicircle Law as an ideal optimization objective of the structural features of large random networks. Given a finite network size N, the variant of Wigner’s Semicircle Law for the eigenvalue λ distribution of the adjacency matrix of an ER random graph with the link density p is
ρ(λ) ≃
4Np(1 − p) − λ2
√
2π Np(1 − p)
,
(15)
where |λ| ≤ 2 Np(1 − p) [22]. Eq. (15) fits into the bulk density of eigenvalues {λ2 , λ3 , . . . , λN −1 , λN }, ignoring the largest eigenvalue λ1 . According to the Füredi–Komlós theorem [23] and the Perron–Frobenius theorem, the largest eigenvalue with multiplicity one is
λ1 ≃ (N − 2)p + 1.
(16)
For a sufficiently large N, the ‘‘asymptotically equal to’’ in Eqs. (15) and (16) may be regarded as ‘‘equal to’’. In order to represent more directly the semicircle law distribution of adjacency matrix eigenvalues √ of large networks, it is necessary to rescale spectral density ρ (i.e. function) and eigenvalue λ (i.e. variable) in Eq. (15) by Np(1 − p) and √Np(11−p)
√
respectively. Let the rescaled value of λ be R = √Np(λ1−p) , and the rescaled value of ρ be ρ(R) = ρ(λ) Np(1 − p). In terms of Eq. (15), we arrive at
ρ(R) ≃
1 2π
4 − R2 .
(17)
According to the Perron–Frobenius theorem, for the unique largest eigenvalue λ1 , it yields ρ(λ1 ) =
√
ρ(λ1 ) Np(1 − p) =
√
1 N
and ρ(R) =
Np(1 − p)/N. Hence, the complete function of spectral density for large random networks can be
rewritten as
4 − R2 /2π , √ Np(1 − p) ρ(R) = , N 0,
|R| ≤ 2 λ1
R= √ Np(1 − p) otherwise,
(18)
where λ1 = (N − 2)p + 1. To verify Eq. (18) numerically, an experimental result of eigenvalue density is given as shown in Fig. 2. We can observe that the curve (bulk part) and the largest eigenvalue from experiments agree very well with the expressions of Eq. (18) when the network size is large.
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Fig. 2. A spectral density comparison of theoretical curves and experimental results w.r.t. an ER model with N = 500, p = 0.1. The red dashed curve with squares corresponds to the experimental result, and the blue curve is the theoretical curve. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. The diagram of spectral density function of a BA model with N = 1000, m = m0 = 5, where m is the number of added edges at each time-step as the BA model grows, and m0 is the number of initial vertices.
4.2. Spectral density of scale-free networks Although there have been many significant results on the spectral density of scale-free networks, the complete mathematical form of the spectral density has not been found yet. Combining the existing results, we derive the complete function of the spectral density function of BA model. According to the results in Ref. [24], the spectral density function ρ(λ) consists of several well distinguishable parts so long as m > 1, as shown in Fig. 3. Generally speaking, the bulk part of ρ(λ), i.e., the density of eigenvalues {λ2 , λ3 , . . . , λN −1 , λN }, converges to an approximately symmetric continuous function that has a triangle-like shape (see ABD in the center, taking λ = 0 as its axis of symmetry) with two power-law tails (see the curves BC and DE at both edges). Regarding the unique largest eigenvalue λ1 , when ⟨ki ⟩ ≈ 2m ≥ 4, where ⟨ki ⟩ is the average degree of a scale-free network, λ1 will be detached from the bulk part of ρ(λ). Thus, in most cases the spectral density of BA model is made up of three well distinguishable parts: center, tails of bulk, and first eigenvalue. Eigenvalues λ2 and λN in the set of {λ2 , λ3 , . . . , λN −1 , λN } respectively act as the upper bound and lower bound of eigenvalue variable in the bulk part of ρ(λ). Since it is difficult to further analytically compute the values of λ1 , λ2 , and λN [24,25], we can determine these values only by extensive numerical experiments. Being slightly different to the results from Refs. [24,25], our findings are λ1 ∝ N α (α < 0.25), λ2 ∝ N α (α ≈ 0.25), and λN ∝ N α (α ≈ 0.25) rather than λ1 ∝ N 0.25 and λN ∝ N 0.25 . According to the results in Refs. [26,27], for scale-free networks with a degree probability distribution (PDF) P (k) ≈ 2m2 k−r , the tails of the bulk part of ρ(λ) can be described asymptotically as
ρ(λ) ≃ 4m2 |λ|1−2r , (19) √ when |λ| ≥ 1.5 Np(1 − p) [24], where r = 3 for most scale-free networks and p is the link density. We can determine the expressions of eigenvalue density function ρ(λ) only if we evaluate the coordinates of points A, B, C , D, E, F in Fig. 3. For point F (λ1 , ρ(λ1 )), it yields ρ(λ1 ) = N1 in terms of the Perron–Frobenius theorem. From Eq. (19), we
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Fig. 4. The double logarithmic coordinates of λ1 w.r.t. network size N under various m (m0 = m). The straight lines fitting the experimental values have slopes 0.1719, 0.1680, 0.1653, 0.1562, 0.1535, 0.1459, and 0.1594 when m = 2, 3, 4, 5, 6, 7, and 8, respectively.
√
5 −5 2 obtain ρ(λ2 ) = 4m2 λ− , and ρ ±1.5 Np(1 − p) = 2 , ρ(λN ) = 4m |λN |
p=
2m(N −m0 ) N (N −1)
≃
2m . N
Since
1 N
+
λ2 λN
128 2 m 243
5
[Np(1 − p)]− 2 , where the link density
ρ(λ)dλ = 1, by solving this equation we can evaluate the coordinate of point A:
1 4 2 −4 ρ(0) = 1 − + m2 λ− + m λ 1.5 Np(1 − p) + N 2 N
64m2 81
5 .
Np(1 − p)
√
Combining these results above, we arrive at
ρ(0) + kS |λ| , 2 −5 ρ(λ) = 4m |λ| , 1 , N
√
where the slope kS =
|λ| < 1.5 Np(1 − p) |λ|λ∈[λN ,λ2 ] ≥ 1.5 Np(1 − p) λ = λ1 ,
4m2 [1.5 Np(1−p)]
√
(20)
−5
1.5 Np(N −1)
−ρ(0)
.
To√keep a consistent form with the spectral density function of ER model above, it is also necessary to rescale ρ(λ) and Np(1 − p) and √Np(11−p) respectively. Similarly, let the rescaled value of λ be R = √Np(λ1−p) , and the rescaled value of
λ by
√ ρ(λ) be ρ(R) = ρ(λ) Np(1 − p). In terms of Eq. (20), we arrive at Np(1 − p) [ρ(0) + kS |R|] , 2 4m Np(1 − p) |R|−5 , ρ(R) = √ Np(1 − p) , N
|R| < 1.5 |R|R∈[RN ,R2 ] ≥ 1.5 λ1 R= √ , Np(1 − p)
(21)
λ λ where RN = √Np(NN −1) and R2 = √Np(N2 −1) . The values of λ1 , λ2 , and λN are determined according to the experimental results shown in Fig. 4, Fig. 5, and Fig. 6, respectively. The equation of any fitting straight line can be written as
lg λ = k lg N + b, where k indicates the slope of fitting straight line, b is a constant, and λ denotes any of λ1 , λ2 , or λN . The values of both k and b can be determined numerically. Thus, we derive the function expressions of λ1 , λ2 , and λN w.r.t. variable N under various
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381
Fig. 5. The double logarithmic coordinates of λ2 w.r.t. network size N under various m (m0 = m). The straight lines fitting the experimental values have slopes 0.2559, 0.2604, 0.2637, 0.2586, 0.2567, 0.2403, and 0.2418 when m = 2, 3, 4, 5, 6, 7, and 8, respectively.
m:
0.5203 0.1719 10 N , 100.6494 N 0.1680 , 100.7489 N 0.1653 , λ1 = 100.8482 N 0.1562 , 100.9201 N 0.1535 , 0.9937 0.1459 100.9997 N 0.1594 , 10 N , 0.1363 0.2559 10 N , 100.1807 N 0.2604 , 100.2003 N 0.2637 , λ2 = 100.2433 N 0.2586 , 100.2720 N 0.2567 , 0.3349 0.2403 100.3474 N 0.2418 , 10 N ,
m m m m m m m
=2 =3 =4 =5 =6 =7 = 8,
(22)
m m m m m m m
=2 =3 =4 =5 =6 =7 = 8,
(23)
and
0.2231 0.2479 10 N , 0.2533 0.2519 10 N , 100.2538 N 0.2589 , λN = − 100.3168 N 0.2457 , 100.3145 N 0.2528 , 100.3706 N 0.2388 , 100.4131 N 0.2300 ,
m m m m m m m
=2 =3 =4 =5 =6 =7 = 8.
(24)
The experimental results of eigenvalue densities of two scale-free networks with different network sizes are given as shown in Fig. 7(a) and (b) respectively. It is observed that three parts (i.e., center, tails of bulk, and the largest eigenvalue) of each curve from experiments agree very well with the expressions of Eq. (21) when the network size is large. 5. Optimization algorithm based on spectral density A strategy optimizing network connectivity is proposed in Section 3, which can make us reach a certain goal of connectivity optimization by adding the smallest number of links. However, for a large realistic network, as discussed in
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Fig. 6. The double logarithmic coordinates of |λN | w.r.t. network size N under various m (m0 = m). The straight lines fitting the experimental values have slopes 0.2479, 0.2519, 0.2589, 0.2457, 0.2528, 0.2388, and 0.2300 when m = 2, 3, 4, 5, 6, 7, and 8, respectively.
Fig. 7. A spectral density comparison of theoretical curves and experimental results w.r.t. two scale-free networks with N = 500, m = 5 (a) and N = 1000, m = 5 (b) respectively. The red dashed curves with squares correspond to the experimental results, and the blue curves are the theoretical curves. The mean squared error (MSE) between two curves in (a) is 0.0036, and MSE = 0.0024 in (b). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Section 1, in most cases we need to optimize not only its connectivity but also its structural features provided that its cost constraint of link additions allows. Since the structural features of a network may be measured by its spectral density, spectral density should be considered as another optimization objective besides spectral radius. To evaluate the difference in structural properties (or the discrepancy in spectral density) between two networks, in terms of the definition of mean squared error (MSE), we define a measure of spectral density discrepancy:
ˆ 1 − λ1 MD = w1 λ
2
w2 +
2 N λˆ i − λi i=2
,
(N − 1)
(25)
ˆ 1 , λˆ 2 , . . . , λˆ N , in the descend order of eigenvalues, respectively are the eigenvalue spectra where {λ1 , λ2 , . . . , λN } and λ of adjacency matrices of two distinct networks. w1 and w2 in Eq. (25) denote the weights of λ1 and the other eigenvalues, respectively. The smaller value of MD indicates that the structural features of two networks are closer to each other. Because the largest eigenvalue of a network usually has more contributions to its structural properties than the rest of eigenvalue spectrum, we put a larger weight on λ1 than the others, e.g., let w1 = w2 = 0.5 in Eq. (25).
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Fig. 8. The diagram dividing the bulk spectral density curve of ER model.
5.1. Optimal eigenvalue spectrum To minimize the difference in structural properties between the networks that need optimizing and the optimal network model in terms of Eq. (25), it is necessary to determine the eigenvalue spectrum of the optimal network model in advance, i.e., {λ1 , λ2 , . . . , λN } in Eq. (25), based on its spectral density. For example, acting as the optimization goal for random networks, the spectral density of ER model can be transformed into a form of eigenvalue spectrum that is called ‘‘optimal eigenvalue spectrum’’. As shown in Fig. 8, we manage to divide the interval [−2, 2] of variable R in Eq. (18) into r −1 equal subintervals, where r is the number of distinct eigenvalues. The initial value of r may be an arbitrary integer within the range of [2, N ]. Suppose that Ri denotes the midpoint of the ith subinterval, and 1Ri denotes the width of the ith subinterval, where i ∈ {1, 2, . . . , r − 1}. 1R Thus, we can obtain 1Ri = r −4 1 and Ri = 2 − 2 i (2i − 1) (from the right to the left in Fig. 8). Referring to Eq. (18) or Eq. (21),
the probability that variable R lies within the range of Ri −
P Ri −
1Ri 2
< R ≤ Ri +
1Ri 2
1 Ri 2
, Ri +
1 Ri 2
is
= 1Ri ρ (Ri ) , √
when 1Ri → 0. An example is the dark bar shown in Fig. 8. Since R = √Np(λ1−p) and ρ(R) = ρ(λ) Np(1 − p), excluding the largest eigenvalue λ1 , we arrive at
1Ri ρ (Ri ) = 1λi+1 ρ (λi+1 ) 1λi+1 1λi+1 < λ ≤ λi+1 + . = P λi+1 − 2
2
(26)
Therefore, the multiplicity of eigenvalue λi+1 , Ni+1 , can be calculated as
Ni+1 = NP λi+1 −
1λi+1 2
< λ ≤ λi+1 +
1λi+1
2
= ⌊N 1Ri ρ (Ri )⌋ , where ⌊·⌋ denotes the value of multiplicity rounded to its floor integer. Combining the derived results above, it yields
λi+1 = Ri Np(1 − p) Ni+1 = ⌊N 1Ri ρ (Ri )⌋ ,
(27)
where i ∈ {1, 2, . . . , r − 1}. Incorporating the largest eigenvalue λ1 and N1 = 1 with Eq. (27), we obtain eigenvalue spectrum {λ1 , λ2 , . . . , λN } as the optimization goal of topological features for any type of large networks. 5.2. Error analysis of eigenvalue multiplicity The errors of eigenvalue multiplicity in Eq. (27) derive from two aspects: the number of divided subintervals and the rounding of eigenvalue multiplicity.
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Seen from the subinterval divisions in Fig. 8, a larger value of r will lead to a more accurate interpolation to the curve of spectral density. On the other hand, in terms of the definition of spectral density, we obtain 1 N
+ lim
r →N
r −1 1λi+1 1λi+1 < λ ≤ λi+1 + = 1, P λi+1 − i=1
2
2
acting as a constraint on the value of r. To compute the optimal value of r by iteration method, suppose the error bound of interpolation is εr , namely, the iteration precision requirement. Thus, referring to Eq. (26), the end condition of iteration can be written as
r −1 1 1Ri ρ (Ri ) ≤ εr . 1 − − N i =1
(28)
The value of r that satisfies the inequality (28) is the optimal value of r, r ∗ . The error derived from the rounding of eigenvalue multiplicity will directly result in that the sum of eigenvalue r ∗−1 multiplicities is not equal to the total number of eigenvalues N any more, i.e., 1 + ̸= N. If i=1 Ni+1
1 + < N, we in turn add 1 to Ni+1 , namely Ni+1 + 1 ⇒ Ni+1 one by one, in the order of i=1 Ni+1 N⌈(r ∗+1)/2⌉ , N⌈(r ∗+1)/2⌉+1 , N⌈(r ∗+1)/2⌉−1 , N⌈(r ∗+1)/2⌉+2 , N⌈(r ∗+1)/2⌉−2 , . . . , Nr ∗ , N2 , i.e., from the center to the left side and the
r ∗−1
right side of spectral density curve alternately, till 1 +
r ∗−1 i=1
Ni+1 = N. This process may be iterated till 1 +
r ∗−1 i =1
Ni+1 = N.
On the other hand, if 1 + i=1 Ni+1 > N, we in turn subtract 1 from Ni+1 , namely Ni+1 − 1 ⇒ Ni+1 one by one, in the reverse order of the above, i.e., N2 , Nr ∗ , . . . , N⌈(r ∗+1)/2⌉−2 , N⌈(r ∗+1)/2⌉+2 , N⌈(r ∗+1)/2⌉−1 , N⌈(r ∗+1)/2⌉+1 , N⌈(r ∗+1)/2⌉ , till
r ∗−1
1+
r ∗−1 i=1
Ni+1 = N.
5.3. Optimization algorithm Given a large network that needs optimizing, we improve its connectivity by link addition, meanwhile, make its topological features approach to those of either ER or BA model as close as possible. Once the optimal eigenvalue spectrum {λ1 , λ2 , . . . , λN } of either random networks or scale-free networks is determined, we start the following optimizing process: (1) If the number of added links does not exceed the upper bound of link addition cost, we continue to add one undirected link to this network according to the link addition strategy based on Eq. (13), which has been proposed in Section 3. Otherwise, the optimizing process ends at this step, and the upper bound of link addition cost is the optimal result which we find out. (2) We compute the eigenvalues of adjacency matrix of thenew network with a new added link, and sort the eigenvalue
ˆ 1 , λˆ 2 , . . . , λˆ N . Afterwards, the value of MD can be calculated spectrum in the descend order of eigenvalues, to acquire λ
according to Eq. (25). (3) The process above iterates till the minimal value of MD is sought out. The optimizing process ends here; and the corresponding number of added links is the optimal result which we search for. The complete metacode of this optimization algorithm is listed in Tables 1 and 2. The 5th line in Table 1 means to add ˆ (N , pˆ ) in terms of a certain link addition strategy Sadd , e.g., Eq. (13) proposed in Section 3. The 6th one undirected link to G ˆ (N , pˆ ) by using the Lanczos algorithm with a computational complexity of O(N 2 ). The 7th line computes the eigenvalues of G
ˆ 1 , λˆ 2 , . . . , λˆ N . EXTRACT_MIN in line 9 is a function line computes the mean squared error between {λ1 , λ2 , . . . , λN } and λ
extracting the smallest element from a queue. The 42nd line in Table 2 assigns r ∗ − 1 distinct eigenvalues λ[i + 1] with their respective multiplicities N [i + 1] to the eigenvalue spectrum {λ2 , λ3 , . . . , λN }. The computational complexity of COMPUTE_EIGENVALUES function is dominated by lines 1–13 in Table 2, which is O(r ∗ N ). When r ∗ = N, the worst case complexity of this function is O(N 2 ). The computational complexities of both line 5 and line 7 in Table 1 are O(N ). In addition, due to the Lanczos algorithm with a complexity of O(N 2 ), the computational complexity of lines 4–8 in Table 1 should be O(Lcos t (N 2 + N )). The complexity of EXTRACT_MIN function is O(Lcos t ). Thus, the computational complexity of our optimization algorithm is O((Lcos t + 1)N 2 + Lcos t (N + 1)), considered as O(Lcos t N 2 + Lcos t N ). 6. Optimization examples In our optimization instances, suppose that the upper bound of link addition cost is infinite, i.e. Lcos t → ∞. Here an ER or BA model with N nodes and a link density p is given as the optimization objective of topological properties. For a BA model, its link density can be calculated by 2m/N and m = m0 . If the present link density of a network that needs optimizing pˆ > p, instead of increasing the connectivity of this network by adding links, we should decrease the number of its links according to a link removal strategy based on Eq. (13), which is an opposite operation to the link addition strategy above, such that its eigenvalue spectrum approaches to the corresponding optimal eigenvalue spectrum as close as possible. Without doubt, in
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Table 1 The metacode of main optimization algorithm.
ˆ (N , pˆ ), G (N, p), Lcos t , εr ) MAIN_ALGORITHM (G ˆ (N , pˆ ) is a given network that needs optimizing with N nodes and the link density pˆ Input: G (usually a sparse large network), and G (N, p) is the objective network with N nodes and the link density p. Lcos t is the upper bound of link addition cost, and εr is the error bound of interpolation. Output: Ladd , the total number of undirected links added to G(N , p). Lanczos λˆ 1 , λˆ 2 , . . . , λˆ N ←−−− Gˆ (N , pˆ ) MSE ˆ 1 , λˆ 2 , . . . , λˆ N MD [Ladd ] ←−−−−−−− λ
Begin
6.
1. Ladd ← 0
7.
2. {λ1 , λ2 , . . . , λN } ← COMPUTE_EIGENVALUES (G (N, p), εr ) 4. for Ladd = 1 to Lcos t
8. end for 9. MDmin [Ladd ] ← EXTRACT_MIN (MD [Ladd ]) 10. return Ladd
5.
Sadd
ˆ (N , pˆ ) ←−− Gˆ (N , pˆ ) G
{λ1 ,λ2 ,...,λN }
End
Table 2 The metacode computing the optimal eigenvalue spectrum. COMPUTE_EIGENVALUES (G (N, p), εr ) Input: G (N, p) is the objective network with N nodes and the link density p, and εr is the iteration precision requirement. Output: {λ1 , λ2 , . . . , λN }, the optimal eigenvalue spectrum. Begin 1. for r = 2 to N 2. sum ← 0 3. for i = 1 to r − 1 4. 1Ri ← 4/(r − 1) (e.g., for ER model) 5. Ri ← 2 − (2i − 1)1Ri /2 6. sum ← sum + 1Ri ρ(Ri ) 7. N [i + 1] ← ⌊N 1Ri ρ(Ri )⌋ 8. end for 9. if |1 − 1/N − sum| ≤ εr then 10. r∗ ← r 11. exit for 12. end if 13. end for 14. for i = 1 to r ∗ − 1 15. sumN ← sumN + N [i + 1] 16. λ[i + 1] ← 2 − (4i − 2)/(r ∗ − 1) 17. end for 18. if sumN + 1 < N then 19. k ← ⌈(r ∗ + 1)/2⌉; j ← 0 20. 21.
22. sumN ← sumN + 1 23. if sumN + 1 < N AND j ̸= 0 then 24. N [k − j ] ← N [k − j ] + 1 25. sumN ← sumN + 1 26. end if 27. j←j+1 28. while (sumN + 1 = N ) 29. else if sumN + 1 > N then 30. j←2 31. do 32. N [j] ← N [j] − 1 33. sumN ← sumN − 1 34. if sumN + 1 > N then 35. N [r ∗ + 2 − j] ← N [r ∗ + 2 − j] − 1 36. sumN ← sumN − 1 37. end if 38. j←j+1 39. while (sumN + 1 = N ) 40. end if 41. λ1 ← (N − 2)P + 1 N [i+1]
42. {λ2 , λ3 , . . . , λN } ←−−− λ[i + 1] 43. return {λ1 , λ2 , . . . , λN } End
do N [k + j] ← N [k + j] + 1
Table 3 The number of added links in random networks with different link density pˆ and network size N. pˆ (or p) N N N N
= 100 = 500 = 1000 = 1500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
15 20 1 1
24 64 96 139
this case, the connectivity of this network declines rather than gets improved. If pˆ < p or pˆ ≈ p, we optimize this network by using our optimization algorithm shown in Tables 1 and 2. Let pˆ = p in the following optimization examples. The optimization results of random networks are listed in Tables 3 and 4, and those of scale-free networks are listed in Tables 5 and 6. The number of added links in random networks and scale-free networks is respectively listed in Tables 3 and 5. The ratio of the added link number to the total link number in random networks and scale-free networks is respectively listed in Tables 4 and 6. The corresponding curves of optimization results in Tables 3–6 are respectively plotted in Fig. 9(a) and (b), as well as Fig. 10(a) and (b). Observing the shapes and trends of these curves in Fig. 9(a) and (b), we can conclude that the spectral density of a random network with a large size N is able to satisfy Wigner’s Semicircle Law quite well when its link density pˆ is less than a certain critical probability value pc , e.g., pc = 0.65 (N = 100), pc = 0.65 (N = 500), pc = 0.65 (N = 1000), and pc = 0.65 (N = 1500). The reason is that the eigenvalue distribution of this network can reach to the optimal result only by adding
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Table 4 The ratio (percent) of the added link number to the total link number in random networks with different pˆ and N. pˆ (or p) N N N N
= 100 = 500 = 1000 = 1500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.2020 0.0080 0.0020 0.0009
0.1010 0.0040 0.0010 0.0004
0.0673 0.0027 0.0007 0.0003
0.0505 0.0020 0.0005 0.0002
0.0404 0.0016 0.0004 0.0002
0.0337 0.0013 0.0003 0.0001
0.1154 0.0011 0.0003 0.0001
0.3788 0.0200 0.0002 0.0001
0.5387 0.0570 0.0214 0.0137
Table 5 The number of added links in scale-free networks with different parameter m (m = m0 ) and network size N. m N N N N
= 100 = 500 = 1000 = 1500
2
3
4
5
6
7
8
5 15 22 28
7 16 24 32
8 17 25 34
8 18 27 36
8 20 28 38
9 21 30 42
9 22 32 46
Table 6 The ratio (percent) of the added link number to the total link number in scale-free networks with different m (m = m0 ) and N. m N N N N
= 100 = 500 = 1000 = 1500
2
3
4
5
6
7
8
2.54 1.50 1.10 0.93
2.38 1.07 0.77 0.71
2.05 0.85 0.63 0.57
1.65 0.72 0.54 0.48
1.38 0.67 0.47 0.42
1.34 0.60 0.43 0.40
1.18 0.55 0.40 0.38
Fig. 9. The number of added links (a) and the ratio (%) of the added link number to the total link number (b) in random networks with different link density pˆ and network size N.
a single undirected link and the corresponding ratio of the added link number to the total link number is also low enough while pˆ ≤ pc . Of course there is an exception when N is small, e.g., N = 100. From Fig. 10(a) and (b), it can be observed that a larger N or m in a scale-free network indicates its structural properties are closer to the optimal result. The optimization results above also verify our optimization algorithm shown in Tables 1 and 2. 7. Conclusions This paper focuses on three critical problems in the connectivity optimizations of large networks: (a) how to determine an appropriate measure of network connectivity, (b) how to add/remove a link to/from a network such that its connectivity can be increased/decreased the most, and (c) how to optimize not only the connectivity of a large network but also its topological features given that its cost constraint allows. First of all, we derive a better measure of network connectivity than algebraic connectivity, spectral radius, when the network size is large. On the basis of this, a new strategy increasing/decreasing network connectivity is derived, by which the spectral radius of a network can be increased/decreased the most. To measure and optimize some network features, we derive two complete functions of spectral density for large random and scalefree networks respectively, acting as the optimization objectives of network properties. At last, the authors propose an optimization algorithm based on spectral density and give some relevant examples to verify it. While our optimization
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Fig. 10. The number of added links (a) and the ratio (%) of the added link number to the total link number (b) in scale-free networks with different m and N.
algorithm is designed for increasing network connectivity, it can be extended to decrease network connectivity, depending on the link density of that network being optimized. In addition, our finding in this paper is that the spectral density of a large random network is in almost total agreement with Wigner’s Semicircle Law when its link density is less than a certain critical probability, and a large scale-free network with a larger growth parameter m has closer properties to the ideal BA model. When deriving the spectral density function of BA model, we also find that the values of three critical eigenvalues w.r.t. network size: λ1 ∝ N α (α < 0.25), λ2 ∝ N α (α ≈ 0.25), and λN ∝ N α (α ≈ 0.25). In a word, our algorithm can optimize some topological features of a large network while improving its network connectivity. It provides a benchmark or an indicator for us while optimizing network connectivity, i.e., a guided optimization. Besides, our algorithm can also be regarded as another algorithm that solves all eigenvalues of an N × N adjacency matrix for large random or scale-free networks, with a low computational complexity of O(N 2 ) at most. Our work in the future is to apply this optimization algorithm to some realistic large networks such as transportation networks. Acknowledgment This work was supported by the National Natural Science Foundation of China under Grant number 61472294. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
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Physica A journal homepage: www.elsevier.com/locate/physa
A new method optimizing the subgraph centrality of large networks Xin Yan a,∗ , Chunlin Li a,c , Ling Zhang a , Yaogai Hu b a
Department of Computer Science, Wuhan University of Technology, Wuhan 430070, China
b
School of Electronic Information, Wuhan University, Wuhan 430072, China
c
The State Key Laboratory of Integrated Services Networks, Xidian University, Xi’an 710071, China
highlights • • • • •
We derive a better measure of network connectivity for large networks. A new strategy that can increase/decrease network connectivity the most is derived. We derive two complete functions of spectral density for two types of networks. An optimization algorithm based on spectral density is proposed. Our new findings about spectral density are also concluded.
article
info
Article history: Received 4 April 2015 Received in revised form 16 September 2015 Available online 20 October 2015 Keywords: Spectral density Optimization Network connectivity Structural properties Spectral radius
∗
abstract Since many realistic networks such as wireless sensor/ad hoc networks usually do not agree very well with the basic network models such as small-word and scale-free models, we often need to obtain some expected structural features such as a small average path length and a regular degree distribution while optimizing the connectivity of these networks. Although a minor addition of links for optimizing network connectivity is not likely to change the structural properties of a network, it is necessary to investigate the impact of link addition on network properties as the number of the added links increases. However, to the best of our knowledge, the study of that problem has not been found so far. Furthermore, two closely related questions to that problem, i.e., how to measure and how to improve network connectivity, have not been studied carefully enough yet. To address the three problems above, the authors derive a better measure of network connectivity for large networks and a new strategy that can increase/decrease network connectivity the most, and propose a spectral density algorithm optimizing the connectivity of large networks, which is able to indicate the impact on the structural properties of a network while increasing/decreasing its connectivity, providing us a guided optimization of network connectivity. In other words, our algorithm can optimize not only the connectivity of a large network but also its structural features. Meanwhile, our new findings about spectral density are also concluded in this paper. In addition, we may also apply this algorithm to solve all eigenvalues of an N × N matrix, with a low complexity of O(N 2 ) at most. © 2015 Elsevier B.V. All rights reserved.
Corresponding author. Tel.: +86 13035107916. E-mail addresses: [email protected], [email protected] (X. Yan).
http://dx.doi.org/10.1016/j.physa.2015.10.034 0378-4371/© 2015 Elsevier B.V. All rights reserved.
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1. Introduction Many complex systems can be modeled by networks to capture the possibly inhomogeneous patterns of interactions within complex systems. Due to the diversity of complex systems, the study of complex networks pervades many fields of science, such as mathematics, physics, sociology, computer science, and biology [1,2]. In recent years, there has been a great interest in the study of measuring and optimizing the robustness or resilience of complex networks, as one of the most important topics in complex networks [3,4]. To evaluate the robustness properties of complex networks, various structural and spectral measures have been proposed on the basis of different quantitative properties of underlying networks [5]. Examples are degree distribution, average path length, clustering coefficient, betweenness, global efficiency, largest component (structural measures), as well as algebraic connectivity, spectral radius (spectral measures) etc., which can be employed as the optimization objectives for promoting the robustness of complex networks. Although structural measures represent the topological properties of a network more directly compared to spectral measures, a spectral measure of a network usually contains more abundant characteristic information of this network than a structural measure. For example, a network with a large value of algebraic connectivity, i.e. the second smallest eigenvalue of the Laplacian matrix of this network, has not only high network connectivity but also a low threshold of coupling strength for synchronization [6]. Therefore, in most cases, we prefer to make use of spectral measures while evaluating and optimizing the robustness of complex networks. Network connectivity is a crucial form of network robustness in view of its wealthy robustness implications [3]. As mentioned above, currently algebraic connectivity acts as the prime measure that evaluates the connectivity robustness of complex networks. However, algebraic connectivity has its disadvantages to measure network connectivity while an overall network graph is disconnected, because the value of algebraic connectivity is always equal to zero for any overall disconnected network regardless of the local connectivity of this network. Consequently, in this paper we will introduce subgraph centrality [7] as another spectral measure to evaluate network connectivity and as another optimization objective to enhance network connectivity. To improve the connectivity of an existing large real-world network, instead of substituting its infrastructure for the optimal one that maximizes its connectivity as discussed in Ref. [8], a minor modification on the current network, i.e. adding a small number of links, is usually required due to economic concerns [9]. Nevertheless, here an important question is how we can decide the exact number of the links that need adding in order to increase the connectivity of a network. For an undirected network G(V , E ) consisting of node set V and link set E with N nodes and L links, the number of the added N (N −1) − L to construct a full connected topological graph with the optimal connectivity, if we do not take links should be 2 into account the cost of link additions. Thus, for a large realistic (usually sparse) network, the number of the links that need adding is often bounded by the constraint on the link addition cost. However, if some links are added at an enough low cost that could be ignored, should we add an adequate number of links to a network till it becomes a full connected topology? The answer is NOT in most cases. This reason is that the construction of a full connected topology would extremely likely make the original topological properties of this network changed. For instance, a scale-free network will probably not have some topological features we are expecting any more after some additional links are included. Especially for wireless communication networks, a full connected topology will increase dramatically the MAC collisions due to the lack of its sparseness. In most cases, the optimization objectives for a large realistic network are probably not only to improve its connectivity but also to remain its structural properties unchanged even to promote its certain topological features, e.g., the degree distribution with power-law form, the heterogeneity, and the scalability, etc. Therefore, one of important aims in this paper is to optimize the topological properties of a network besides its connectivity, provided that its cost constraint allows. To optimize the connectivity of a network, another important question is where we should add a link in this network such that its connectivity can be increased the most. The investigation on adding one link to improve network connectivity can guide us how to dynamically add a set of links one by one such that network connectivity is maximally increased. Hence, we propose a new strategy of adding a link, based on the eigenvector components of the adjacency matrix of a network, to improve its connectivity. To the best of our knowledge, these three questions discussed above: (a) how to determine a spectral measure of network connectivity as the optimization objective, (b) how to optimize not only the connectivity of a network but also its topological features if allowed by the cost constraint, and (c) how to add a link to a network such that its connectivity can be increased the most, have not been studied intensively so far. In particular, the study of problem (b) has not been found yet. These three critical problems in network connectivity optimizations will be investigated in this paper. The rest of this paper is organized as follows. Section 2 presents a spectral measure of network connectivity (namely subgraph centrality) and its approximation (namely spectral radius) acting as the optimization objective, instead of the conventional measure of algebraic connectivity. Section 3 presents a new strategy increasing spectral radius based on eigenvector components, and explains why it can be increased the most by using this strategy. In Section 4, we introduce the spectral density measure of network properties as the optimization objective, which is usually treated as the ‘‘fingerprint’’ and ‘‘pulse manifestation’’ of a large network. Section 5 presents the optimization algorithm of network connectivity on the basis of the spectral density of this network. Some optimization examples are given in Section 6. Section 7 presents our conclusions.
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2. Spectral measure of network connectivity Conventionally algebraic connectivity, the second smallest eigenvalue of Laplacian matrix of a network graph, is usually used to measure the connectivity of this network. Fiedler shows that the value of algebraic connectivity reflects how well an overall graph is connected [10]. A graph with a larger numerical value of algebraic connectivity is more difficult to be cut into independent components. However, as mentioned above, algebraic connectivity has its disadvantages to measure network connectivity while an overall network is disconnected, because the value of algebraic connectivity is always equal to zero for any overall disconnected network regardless of the local connectivity of this network. For the sake of avoiding this problem, here it is necessary to introduce another spectral measure of network connectivity, subgraph centrality, as the optimization objective of network connectivity. The topological graph of any network G(V ,E )consisting of node set V and link set E, with N nodes and L links, can be described by an adjacency matrix A(G) = aij N ×N , a non-negative matrix, whose elements aij are positive if there is a link going from node i to node j and zero otherwise, with aii ≡ 0. According to the Perron–Frobenius theorem [11], if A(G) is irreducible, its unique largest eigenvalue is real and positive and the components of the corresponding left and right eigenvectors all are positive. In this paper assuming that the links in G(V , E ) all are undirected, the adjacency matrix A(G) is a symmetric matrix. All eigenvalues are real and A(G) possess an eigen spectral decomposition, A(G) = XDX T , where X = x1 x2 · · · xN is an orthogonal matrix that forms an orthogonal basis, such that XX T = X T X = IN , with the real and normalized eigenvectors x1 , x2 , . . . , xN of A(G) as the columns of matrix X , corresponding to the eigenvalues λ1 > λ2 ≥ · · · ≥ λN −1 ≥ λN in descending order and the diagonal matrix D = diag (λ1 , λ2 , . . . , λN −1 , λN ) [12]. Here note that the left and right eigenvectors of any eigenvalue in a symmetric matrix are the same. 2.1. Approximation of subgraph centrality The closed walks in a network graph are directly associated with its subgraphs. Moreover, a set of closed walks are also able to denote all the alternative routes from a certain node to another in a network. If there are more alternative routes between any pair of nodes in a network, this network may be taken into account connected better. Thus, the number of subgraphs defined by a set of closed walks in a network graph can be used to measure the connectivity of this network. Let nk (i) denote the number of the closed walks with length k starting and ending at the node i in a graph G, which is defined as the ith diagonal entry of the adjacency matrix A(G) with the exponent of k [7]: nk (i) = Ak
ii
.
(1)
Here we take the subgraph centrality (SC in brief) of a network graph G as its connectivity measure, which is defined as the sum of closed walks at all nodes in the network, corresponding to different lengths k [13]. Using Eq. (1), SC can be written as SC =
∞ N
nk (i) =
k=1 i=1
∞ N k
A
ii
.
(2)
k=1 i=1
From A(G) = XDX T and D = diag λk1 , λk2 , . . . , λkN −1 , λkN . Thus,
k λ1 Ak = (x1 , x2 , . . . , xN )
diag (λ1 , λ2 , . . . , λN −1 , λN ), we can obtain that Ak
=
XDk X T , and Dk
=
T x1
λ
k 2
..
. λkN
xT 2 . , .. xTN
and N k
A
ii
= ∥x1 ∥22 λk1 + ∥x2 ∥22 λk2 + · · · + ∥xN ∥22 λkN .
(3)
i =1
Since X = x1
x2
···
xN is a set of orthogonal and normalized real eigenvectors, we obtain ∥x1 ∥22 = ∥x2 ∥22 = · · · =
∥xN ∥22 = 1. Eq. (3) is turned into SC =
∞ N k=1 i=1
λki .
N k i=1
A
ii
=
N
i=1
λki . Meanwhile, Eq. (2) can further be transformed into (4)
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The Perron–Frobenius theorem implies that λ1 > λi for i = 2, . . . , N. When k → ∞ (i.e., the network size is large), we have N
λki
i=1
lim
k→∞
λk1
=1+
N i=2
lim
k→∞
λi λ1
k
= 1,
which leads to N
λki ≈ λk1 .
(5)
i=1
Thus for large k (namely a large network), SC approximates to SC =
∞ N
λki ≈
k=1 i=1
∞
λk1 .
(6)
k=1
For avoiding the possible case of SC → ∞, we may divide the approximation of SC by the factorial of length k, and then take its natural logarithm. Let this result act as the new expression of SC :
∞ λk
.
1
SC = ln
k=1
k!
(7)
According to the Taylor series, ex =
∞ λk
1
SC = ln
k=0
k!
xk k=0 k!
∞
=1+
xk k =1 k ! ,
∞
it yields
− 1 = ln eλ1 − 1 .
(8)
Since ∥A∥2 ≫ 0 in general and ∥A∥2 = λ1 , we arrive at λ1 ≫ 0, eλ1 ≫ 1, and SC = ln eλ1 − 1 ≈ λ1 .
(9)
From the result of Eq. (9), it is observed that the subgraph centrality of a network graph, as another spectral measure of network connectivity, can be represented and figured out approximately by the largest eigenvalue (spectral radius) of its adjacency matrix. 2.2. Comparison of spectral radius to algebraic connectivity In order to demonstrate the advantages of subgraph centrality to measure network connectivity, we respectively plot the values of spectral radius (namely λ1 ) and algebraic connectivity under node random removal (failure) and node targeted removal (attack) with regard to the ratio of the removed nodes to the total nodes, for three types of complex networks: Erdös–Rényi (ER) random model, Watts–Strogats (WS) small-world model, and Barabási–Albert (BA) scale-free model, as shown in Fig. 1. From Fig. 1(a) and (b), it can be observed that the shape and trend of the curves of λ1 are closely similar to those of algebraic connectivity, which is consistent with the accepted conclusions in this scientific field [14]. However, unlike λ1 , as the nodes in a topology are removed, the algebraic connectivity will decline to zero rapidly as soon as any isolated node exists in the topology, whereas at this time the topology still remains the most of its connectivity. Therefore, Algebraic connectivity can only measure the degree of connectivity and indicate whether a topology is connected or not. Without this disadvantage, λ1 can measure not only the degree of connectivity but also the degree of dis-connectivity of a topology, because λ1 is not zero when a topology becomes disconnected. That is why we prefer λ1 as the spectral measure and optimization objective of topological connectivity, rather than algebraic connectivity. 3. Strategy increasing network connectivity In view of the discussed above, we can increase the spectral radius λ1 of a network graph to improve its subgraph centrality or connectivity by dynamically adding a set of links one by one. To reach a certain optimization goal by adding as small number of links as possible, the addition of each of links should make λ1 be increased the most. We will derive a new strategy of link addition, by which λ1 can be increased the most when each of links is added. Suppose that any graph is modified so that its adjacency matrix A becomes A +ζ C , where C is the perturbation matrix and 0 < ζ ≪ 1. Let λ1 > λ2 ≥ · · · ≥ λN be the eigenvalues of A, with x1 , x2 , . . . , xN the corresponding normalized orthogonal
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Fig. 1. The spectral radiuses (a) and the algebraic connectivity (b) under various ratios of the removed nodes to total nodes for three types of network models: ER (square, with 200 nodes and 3875 links), WS (circle, with 200 nodes and 1000 links), and BA (triangle, with 200 nodes and 995 links) respectively. The red dashed curves correspond to node random failure, and the blue full curves corresponding to node targeted attack, i.e. removing nodes according to the descend order of node degrees. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
˜ 1 > λ˜ 2 ≥ · · · ≥ λ˜ N and x˜ 1 , x˜ 2 , . . . , x˜ N denote the eigenvalues and the corresponding eigenvectors. Correspondingly, λ eigenvectors of A + ζ C respectively [12]. For undirected networks, we apply the general perturbation formula [15,16]: λ˜ 1 = λ1 + ζ xT1 Cx1 + ζ 2
T 2 N xk Cx1 + O ζ3 . λ1 − λk k=2
(10)
When using the first-order perturbation from formula (10), the change 1λ1 of the corresponding largest eigenvalue λ1 equals approximately
1λ1 = λ˜ 1 − λ1 ≈ ζ xT1 Cx1 .
(11)
Assume that a perturbation matrix B whose nonzero entries are finite is usually of the same order of magnitude as A. If, however, the modifications are limited to a small number of links in a large network, then ∥B∥2 ≪ ∥A∥2 and the result of Eq. (11) would be valid with ζ C replaced by B, i.e.
1λ1 = λ˜ 1 − λ1 ≈ xT1 Bx1 .
(12)
It can be seen that Eqs. (11) and (12) are of the Rayleigh quotient form [11]. Since the subgraph centrality of a network can be improved by dynamically adding a small number of links one by one, we assume that a certain link is added between node i and node j in this network. At this time (B)ij = (B)ji = w (wherein the constant w is the weight of this undirected link) and the other elements in B are equal to zero, leading to
1λ1 ≈ xT1 Bx1 = 2w(x1 )i (x1 )j .
(13)
According to the Perron–Frobenius theorem, all of the components of principal eigenvector x1 corresponding to λ1 are positive, indicating 1λ1 > 0 (i.e., the value of subgraph centrality continues to be increased as the links are added one by one). From Eq. (13), when we add a link between node i and node j in a network graph, with the largest product of its principal eigenvector components (x1 )i and (x1 )j , λ1 can be increased the most. Thus our strategy increasing network connectivity is to add one undirected link between two distinct nodes in this network every time such that the product of the corresponding two principal eigenvector components is the largest value. 4. Spectral density measure of network features Besides the connectivity of a network, as discussed above, it is also necessary to optimize its topological features. Since the eigenvalue spectrum of adjacency matrix of a network is usually regarded as its ‘‘fingerprint’’ and ‘‘pulse
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manifestation’’ respectively used to identify this network and retrieve some critical structural characteristic information from this network [17], we could employ the eigenvalue spectrum or the spectral density of eigenvalues (for large-sized networks) to measure the features of this network. It is known that the networks with different topological properties have essentially distinct eigenvalue spectra and spectral densities [18]. Thus the spectral density of eigenvalues of a network could be treated as the measure and the optimization objective of its network features. When the size of a network is large, we prefer to employ the spectral density ρ to represent the distribution law (i.e. probability distribution) of adjacency matrix eigenvalues λ of this network. The spectral density of adjacency matrix eigenvalues of a network with N nodes can be written as a sum of Kronecker delta functions [19]:
ρ(λ) =
N 1
N i =1
δ(λ − λi ),
(14)
wherein
δ(λ − λi ) =
1, 0,
λ = λi λ ̸= λi .
When N → ∞, Eq. (14) will converge to a continuous function. In this paper, we select only two representative types of network models, ER and BA models, among a variety of network models such that we can concentrate on our spectral density optimization method. Although the spectral densities of ER and BA models have been mentioned in many Refs. [20–27], it is still short of a complete description about these two spectral densities. Thus we need to derive the detailed mathematical expressions of these two spectral densities for our optimization aims. 4.1. Spectral density of random networks Wigner’s Semicircle Law [20,21] is the fundamental result in the spectral density theory of large random networks. It exhibits a universal property of a class of large, real symmetric matrices with independent random elements. The adjacency matrix of ER random graph satisfies the conditions that are applicable to Wigner’s Semicircle Law. Hence, we can take Wigner’s Semicircle Law as an ideal optimization objective of the structural features of large random networks. Given a finite network size N, the variant of Wigner’s Semicircle Law for the eigenvalue λ distribution of the adjacency matrix of an ER random graph with the link density p is
ρ(λ) ≃
4Np(1 − p) − λ2
√
2π Np(1 − p)
,
(15)
where |λ| ≤ 2 Np(1 − p) [22]. Eq. (15) fits into the bulk density of eigenvalues {λ2 , λ3 , . . . , λN −1 , λN }, ignoring the largest eigenvalue λ1 . According to the Füredi–Komlós theorem [23] and the Perron–Frobenius theorem, the largest eigenvalue with multiplicity one is
λ1 ≃ (N − 2)p + 1.
(16)
For a sufficiently large N, the ‘‘asymptotically equal to’’ in Eqs. (15) and (16) may be regarded as ‘‘equal to’’. In order to represent more directly the semicircle law distribution of adjacency matrix eigenvalues √ of large networks, it is necessary to rescale spectral density ρ (i.e. function) and eigenvalue λ (i.e. variable) in Eq. (15) by Np(1 − p) and √Np(11−p)
√
respectively. Let the rescaled value of λ be R = √Np(λ1−p) , and the rescaled value of ρ be ρ(R) = ρ(λ) Np(1 − p). In terms of Eq. (15), we arrive at
ρ(R) ≃
1 2π
4 − R2 .
(17)
According to the Perron–Frobenius theorem, for the unique largest eigenvalue λ1 , it yields ρ(λ1 ) =
√
ρ(λ1 ) Np(1 − p) =
√
1 N
and ρ(R) =
Np(1 − p)/N. Hence, the complete function of spectral density for large random networks can be
rewritten as
4 − R2 /2π , √ Np(1 − p) ρ(R) = , N 0,
|R| ≤ 2 λ1
R= √ Np(1 − p) otherwise,
(18)
where λ1 = (N − 2)p + 1. To verify Eq. (18) numerically, an experimental result of eigenvalue density is given as shown in Fig. 2. We can observe that the curve (bulk part) and the largest eigenvalue from experiments agree very well with the expressions of Eq. (18) when the network size is large.
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Fig. 2. A spectral density comparison of theoretical curves and experimental results w.r.t. an ER model with N = 500, p = 0.1. The red dashed curve with squares corresponds to the experimental result, and the blue curve is the theoretical curve. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. The diagram of spectral density function of a BA model with N = 1000, m = m0 = 5, where m is the number of added edges at each time-step as the BA model grows, and m0 is the number of initial vertices.
4.2. Spectral density of scale-free networks Although there have been many significant results on the spectral density of scale-free networks, the complete mathematical form of the spectral density has not been found yet. Combining the existing results, we derive the complete function of the spectral density function of BA model. According to the results in Ref. [24], the spectral density function ρ(λ) consists of several well distinguishable parts so long as m > 1, as shown in Fig. 3. Generally speaking, the bulk part of ρ(λ), i.e., the density of eigenvalues {λ2 , λ3 , . . . , λN −1 , λN }, converges to an approximately symmetric continuous function that has a triangle-like shape (see ABD in the center, taking λ = 0 as its axis of symmetry) with two power-law tails (see the curves BC and DE at both edges). Regarding the unique largest eigenvalue λ1 , when ⟨ki ⟩ ≈ 2m ≥ 4, where ⟨ki ⟩ is the average degree of a scale-free network, λ1 will be detached from the bulk part of ρ(λ). Thus, in most cases the spectral density of BA model is made up of three well distinguishable parts: center, tails of bulk, and first eigenvalue. Eigenvalues λ2 and λN in the set of {λ2 , λ3 , . . . , λN −1 , λN } respectively act as the upper bound and lower bound of eigenvalue variable in the bulk part of ρ(λ). Since it is difficult to further analytically compute the values of λ1 , λ2 , and λN [24,25], we can determine these values only by extensive numerical experiments. Being slightly different to the results from Refs. [24,25], our findings are λ1 ∝ N α (α < 0.25), λ2 ∝ N α (α ≈ 0.25), and λN ∝ N α (α ≈ 0.25) rather than λ1 ∝ N 0.25 and λN ∝ N 0.25 . According to the results in Refs. [26,27], for scale-free networks with a degree probability distribution (PDF) P (k) ≈ 2m2 k−r , the tails of the bulk part of ρ(λ) can be described asymptotically as
ρ(λ) ≃ 4m2 |λ|1−2r , (19) √ when |λ| ≥ 1.5 Np(1 − p) [24], where r = 3 for most scale-free networks and p is the link density. We can determine the expressions of eigenvalue density function ρ(λ) only if we evaluate the coordinates of points A, B, C , D, E, F in Fig. 3. For point F (λ1 , ρ(λ1 )), it yields ρ(λ1 ) = N1 in terms of the Perron–Frobenius theorem. From Eq. (19), we
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Fig. 4. The double logarithmic coordinates of λ1 w.r.t. network size N under various m (m0 = m). The straight lines fitting the experimental values have slopes 0.1719, 0.1680, 0.1653, 0.1562, 0.1535, 0.1459, and 0.1594 when m = 2, 3, 4, 5, 6, 7, and 8, respectively.
√
5 −5 2 obtain ρ(λ2 ) = 4m2 λ− , and ρ ±1.5 Np(1 − p) = 2 , ρ(λN ) = 4m |λN |
p=
2m(N −m0 ) N (N −1)
≃
2m . N
Since
1 N
+
λ2 λN
128 2 m 243
5
[Np(1 − p)]− 2 , where the link density
ρ(λ)dλ = 1, by solving this equation we can evaluate the coordinate of point A:
1 4 2 −4 ρ(0) = 1 − + m2 λ− + m λ 1.5 Np(1 − p) + N 2 N
64m2 81
5 .
Np(1 − p)
√
Combining these results above, we arrive at
ρ(0) + kS |λ| , 2 −5 ρ(λ) = 4m |λ| , 1 , N
√
where the slope kS =
|λ| < 1.5 Np(1 − p) |λ|λ∈[λN ,λ2 ] ≥ 1.5 Np(1 − p) λ = λ1 ,
4m2 [1.5 Np(1−p)]
√
(20)
−5
1.5 Np(N −1)
−ρ(0)
.
To√keep a consistent form with the spectral density function of ER model above, it is also necessary to rescale ρ(λ) and Np(1 − p) and √Np(11−p) respectively. Similarly, let the rescaled value of λ be R = √Np(λ1−p) , and the rescaled value of
λ by
√ ρ(λ) be ρ(R) = ρ(λ) Np(1 − p). In terms of Eq. (20), we arrive at Np(1 − p) [ρ(0) + kS |R|] , 2 4m Np(1 − p) |R|−5 , ρ(R) = √ Np(1 − p) , N
|R| < 1.5 |R|R∈[RN ,R2 ] ≥ 1.5 λ1 R= √ , Np(1 − p)
(21)
λ λ where RN = √Np(NN −1) and R2 = √Np(N2 −1) . The values of λ1 , λ2 , and λN are determined according to the experimental results shown in Fig. 4, Fig. 5, and Fig. 6, respectively. The equation of any fitting straight line can be written as
lg λ = k lg N + b, where k indicates the slope of fitting straight line, b is a constant, and λ denotes any of λ1 , λ2 , or λN . The values of both k and b can be determined numerically. Thus, we derive the function expressions of λ1 , λ2 , and λN w.r.t. variable N under various
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381
Fig. 5. The double logarithmic coordinates of λ2 w.r.t. network size N under various m (m0 = m). The straight lines fitting the experimental values have slopes 0.2559, 0.2604, 0.2637, 0.2586, 0.2567, 0.2403, and 0.2418 when m = 2, 3, 4, 5, 6, 7, and 8, respectively.
m:
0.5203 0.1719 10 N , 100.6494 N 0.1680 , 100.7489 N 0.1653 , λ1 = 100.8482 N 0.1562 , 100.9201 N 0.1535 , 0.9937 0.1459 100.9997 N 0.1594 , 10 N , 0.1363 0.2559 10 N , 100.1807 N 0.2604 , 100.2003 N 0.2637 , λ2 = 100.2433 N 0.2586 , 100.2720 N 0.2567 , 0.3349 0.2403 100.3474 N 0.2418 , 10 N ,
m m m m m m m
=2 =3 =4 =5 =6 =7 = 8,
(22)
m m m m m m m
=2 =3 =4 =5 =6 =7 = 8,
(23)
and
0.2231 0.2479 10 N , 0.2533 0.2519 10 N , 100.2538 N 0.2589 , λN = − 100.3168 N 0.2457 , 100.3145 N 0.2528 , 100.3706 N 0.2388 , 100.4131 N 0.2300 ,
m m m m m m m
=2 =3 =4 =5 =6 =7 = 8.
(24)
The experimental results of eigenvalue densities of two scale-free networks with different network sizes are given as shown in Fig. 7(a) and (b) respectively. It is observed that three parts (i.e., center, tails of bulk, and the largest eigenvalue) of each curve from experiments agree very well with the expressions of Eq. (21) when the network size is large. 5. Optimization algorithm based on spectral density A strategy optimizing network connectivity is proposed in Section 3, which can make us reach a certain goal of connectivity optimization by adding the smallest number of links. However, for a large realistic network, as discussed in
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Fig. 6. The double logarithmic coordinates of |λN | w.r.t. network size N under various m (m0 = m). The straight lines fitting the experimental values have slopes 0.2479, 0.2519, 0.2589, 0.2457, 0.2528, 0.2388, and 0.2300 when m = 2, 3, 4, 5, 6, 7, and 8, respectively.
Fig. 7. A spectral density comparison of theoretical curves and experimental results w.r.t. two scale-free networks with N = 500, m = 5 (a) and N = 1000, m = 5 (b) respectively. The red dashed curves with squares correspond to the experimental results, and the blue curves are the theoretical curves. The mean squared error (MSE) between two curves in (a) is 0.0036, and MSE = 0.0024 in (b). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Section 1, in most cases we need to optimize not only its connectivity but also its structural features provided that its cost constraint of link additions allows. Since the structural features of a network may be measured by its spectral density, spectral density should be considered as another optimization objective besides spectral radius. To evaluate the difference in structural properties (or the discrepancy in spectral density) between two networks, in terms of the definition of mean squared error (MSE), we define a measure of spectral density discrepancy:
ˆ 1 − λ1 MD = w1 λ
2
w2 +
2 N λˆ i − λi i=2
,
(N − 1)
(25)
ˆ 1 , λˆ 2 , . . . , λˆ N , in the descend order of eigenvalues, respectively are the eigenvalue spectra where {λ1 , λ2 , . . . , λN } and λ of adjacency matrices of two distinct networks. w1 and w2 in Eq. (25) denote the weights of λ1 and the other eigenvalues, respectively. The smaller value of MD indicates that the structural features of two networks are closer to each other. Because the largest eigenvalue of a network usually has more contributions to its structural properties than the rest of eigenvalue spectrum, we put a larger weight on λ1 than the others, e.g., let w1 = w2 = 0.5 in Eq. (25).
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Fig. 8. The diagram dividing the bulk spectral density curve of ER model.
5.1. Optimal eigenvalue spectrum To minimize the difference in structural properties between the networks that need optimizing and the optimal network model in terms of Eq. (25), it is necessary to determine the eigenvalue spectrum of the optimal network model in advance, i.e., {λ1 , λ2 , . . . , λN } in Eq. (25), based on its spectral density. For example, acting as the optimization goal for random networks, the spectral density of ER model can be transformed into a form of eigenvalue spectrum that is called ‘‘optimal eigenvalue spectrum’’. As shown in Fig. 8, we manage to divide the interval [−2, 2] of variable R in Eq. (18) into r −1 equal subintervals, where r is the number of distinct eigenvalues. The initial value of r may be an arbitrary integer within the range of [2, N ]. Suppose that Ri denotes the midpoint of the ith subinterval, and 1Ri denotes the width of the ith subinterval, where i ∈ {1, 2, . . . , r − 1}. 1R Thus, we can obtain 1Ri = r −4 1 and Ri = 2 − 2 i (2i − 1) (from the right to the left in Fig. 8). Referring to Eq. (18) or Eq. (21),
the probability that variable R lies within the range of Ri −
P Ri −
1Ri 2
< R ≤ Ri +
1Ri 2
1 Ri 2
, Ri +
1 Ri 2
is
= 1Ri ρ (Ri ) , √
when 1Ri → 0. An example is the dark bar shown in Fig. 8. Since R = √Np(λ1−p) and ρ(R) = ρ(λ) Np(1 − p), excluding the largest eigenvalue λ1 , we arrive at
1Ri ρ (Ri ) = 1λi+1 ρ (λi+1 ) 1λi+1 1λi+1 < λ ≤ λi+1 + . = P λi+1 − 2
2
(26)
Therefore, the multiplicity of eigenvalue λi+1 , Ni+1 , can be calculated as
Ni+1 = NP λi+1 −
1λi+1 2
< λ ≤ λi+1 +
1λi+1
2
= ⌊N 1Ri ρ (Ri )⌋ , where ⌊·⌋ denotes the value of multiplicity rounded to its floor integer. Combining the derived results above, it yields
λi+1 = Ri Np(1 − p) Ni+1 = ⌊N 1Ri ρ (Ri )⌋ ,
(27)
where i ∈ {1, 2, . . . , r − 1}. Incorporating the largest eigenvalue λ1 and N1 = 1 with Eq. (27), we obtain eigenvalue spectrum {λ1 , λ2 , . . . , λN } as the optimization goal of topological features for any type of large networks. 5.2. Error analysis of eigenvalue multiplicity The errors of eigenvalue multiplicity in Eq. (27) derive from two aspects: the number of divided subintervals and the rounding of eigenvalue multiplicity.
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Seen from the subinterval divisions in Fig. 8, a larger value of r will lead to a more accurate interpolation to the curve of spectral density. On the other hand, in terms of the definition of spectral density, we obtain 1 N
+ lim
r →N
r −1 1λi+1 1λi+1 < λ ≤ λi+1 + = 1, P λi+1 − i=1
2
2
acting as a constraint on the value of r. To compute the optimal value of r by iteration method, suppose the error bound of interpolation is εr , namely, the iteration precision requirement. Thus, referring to Eq. (26), the end condition of iteration can be written as
r −1 1 1Ri ρ (Ri ) ≤ εr . 1 − − N i =1
(28)
The value of r that satisfies the inequality (28) is the optimal value of r, r ∗ . The error derived from the rounding of eigenvalue multiplicity will directly result in that the sum of eigenvalue r ∗−1 multiplicities is not equal to the total number of eigenvalues N any more, i.e., 1 + ̸= N. If i=1 Ni+1
1 + < N, we in turn add 1 to Ni+1 , namely Ni+1 + 1 ⇒ Ni+1 one by one, in the order of i=1 Ni+1 N⌈(r ∗+1)/2⌉ , N⌈(r ∗+1)/2⌉+1 , N⌈(r ∗+1)/2⌉−1 , N⌈(r ∗+1)/2⌉+2 , N⌈(r ∗+1)/2⌉−2 , . . . , Nr ∗ , N2 , i.e., from the center to the left side and the
r ∗−1
right side of spectral density curve alternately, till 1 +
r ∗−1 i=1
Ni+1 = N. This process may be iterated till 1 +
r ∗−1 i =1
Ni+1 = N.
On the other hand, if 1 + i=1 Ni+1 > N, we in turn subtract 1 from Ni+1 , namely Ni+1 − 1 ⇒ Ni+1 one by one, in the reverse order of the above, i.e., N2 , Nr ∗ , . . . , N⌈(r ∗+1)/2⌉−2 , N⌈(r ∗+1)/2⌉+2 , N⌈(r ∗+1)/2⌉−1 , N⌈(r ∗+1)/2⌉+1 , N⌈(r ∗+1)/2⌉ , till
r ∗−1
1+
r ∗−1 i=1
Ni+1 = N.
5.3. Optimization algorithm Given a large network that needs optimizing, we improve its connectivity by link addition, meanwhile, make its topological features approach to those of either ER or BA model as close as possible. Once the optimal eigenvalue spectrum {λ1 , λ2 , . . . , λN } of either random networks or scale-free networks is determined, we start the following optimizing process: (1) If the number of added links does not exceed the upper bound of link addition cost, we continue to add one undirected link to this network according to the link addition strategy based on Eq. (13), which has been proposed in Section 3. Otherwise, the optimizing process ends at this step, and the upper bound of link addition cost is the optimal result which we find out. (2) We compute the eigenvalues of adjacency matrix of thenew network with a new added link, and sort the eigenvalue
ˆ 1 , λˆ 2 , . . . , λˆ N . Afterwards, the value of MD can be calculated spectrum in the descend order of eigenvalues, to acquire λ
according to Eq. (25). (3) The process above iterates till the minimal value of MD is sought out. The optimizing process ends here; and the corresponding number of added links is the optimal result which we search for. The complete metacode of this optimization algorithm is listed in Tables 1 and 2. The 5th line in Table 1 means to add ˆ (N , pˆ ) in terms of a certain link addition strategy Sadd , e.g., Eq. (13) proposed in Section 3. The 6th one undirected link to G ˆ (N , pˆ ) by using the Lanczos algorithm with a computational complexity of O(N 2 ). The 7th line computes the eigenvalues of G
ˆ 1 , λˆ 2 , . . . , λˆ N . EXTRACT_MIN in line 9 is a function line computes the mean squared error between {λ1 , λ2 , . . . , λN } and λ
extracting the smallest element from a queue. The 42nd line in Table 2 assigns r ∗ − 1 distinct eigenvalues λ[i + 1] with their respective multiplicities N [i + 1] to the eigenvalue spectrum {λ2 , λ3 , . . . , λN }. The computational complexity of COMPUTE_EIGENVALUES function is dominated by lines 1–13 in Table 2, which is O(r ∗ N ). When r ∗ = N, the worst case complexity of this function is O(N 2 ). The computational complexities of both line 5 and line 7 in Table 1 are O(N ). In addition, due to the Lanczos algorithm with a complexity of O(N 2 ), the computational complexity of lines 4–8 in Table 1 should be O(Lcos t (N 2 + N )). The complexity of EXTRACT_MIN function is O(Lcos t ). Thus, the computational complexity of our optimization algorithm is O((Lcos t + 1)N 2 + Lcos t (N + 1)), considered as O(Lcos t N 2 + Lcos t N ). 6. Optimization examples In our optimization instances, suppose that the upper bound of link addition cost is infinite, i.e. Lcos t → ∞. Here an ER or BA model with N nodes and a link density p is given as the optimization objective of topological properties. For a BA model, its link density can be calculated by 2m/N and m = m0 . If the present link density of a network that needs optimizing pˆ > p, instead of increasing the connectivity of this network by adding links, we should decrease the number of its links according to a link removal strategy based on Eq. (13), which is an opposite operation to the link addition strategy above, such that its eigenvalue spectrum approaches to the corresponding optimal eigenvalue spectrum as close as possible. Without doubt, in
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385
Table 1 The metacode of main optimization algorithm.
ˆ (N , pˆ ), G (N, p), Lcos t , εr ) MAIN_ALGORITHM (G ˆ (N , pˆ ) is a given network that needs optimizing with N nodes and the link density pˆ Input: G (usually a sparse large network), and G (N, p) is the objective network with N nodes and the link density p. Lcos t is the upper bound of link addition cost, and εr is the error bound of interpolation. Output: Ladd , the total number of undirected links added to G(N , p). Lanczos λˆ 1 , λˆ 2 , . . . , λˆ N ←−−− Gˆ (N , pˆ ) MSE ˆ 1 , λˆ 2 , . . . , λˆ N MD [Ladd ] ←−−−−−−− λ
Begin
6.
1. Ladd ← 0
7.
2. {λ1 , λ2 , . . . , λN } ← COMPUTE_EIGENVALUES (G (N, p), εr ) 4. for Ladd = 1 to Lcos t
8. end for 9. MDmin [Ladd ] ← EXTRACT_MIN (MD [Ladd ]) 10. return Ladd
5.
Sadd
ˆ (N , pˆ ) ←−− Gˆ (N , pˆ ) G
{λ1 ,λ2 ,...,λN }
End
Table 2 The metacode computing the optimal eigenvalue spectrum. COMPUTE_EIGENVALUES (G (N, p), εr ) Input: G (N, p) is the objective network with N nodes and the link density p, and εr is the iteration precision requirement. Output: {λ1 , λ2 , . . . , λN }, the optimal eigenvalue spectrum. Begin 1. for r = 2 to N 2. sum ← 0 3. for i = 1 to r − 1 4. 1Ri ← 4/(r − 1) (e.g., for ER model) 5. Ri ← 2 − (2i − 1)1Ri /2 6. sum ← sum + 1Ri ρ(Ri ) 7. N [i + 1] ← ⌊N 1Ri ρ(Ri )⌋ 8. end for 9. if |1 − 1/N − sum| ≤ εr then 10. r∗ ← r 11. exit for 12. end if 13. end for 14. for i = 1 to r ∗ − 1 15. sumN ← sumN + N [i + 1] 16. λ[i + 1] ← 2 − (4i − 2)/(r ∗ − 1) 17. end for 18. if sumN + 1 < N then 19. k ← ⌈(r ∗ + 1)/2⌉; j ← 0 20. 21.
22. sumN ← sumN + 1 23. if sumN + 1 < N AND j ̸= 0 then 24. N [k − j ] ← N [k − j ] + 1 25. sumN ← sumN + 1 26. end if 27. j←j+1 28. while (sumN + 1 = N ) 29. else if sumN + 1 > N then 30. j←2 31. do 32. N [j] ← N [j] − 1 33. sumN ← sumN − 1 34. if sumN + 1 > N then 35. N [r ∗ + 2 − j] ← N [r ∗ + 2 − j] − 1 36. sumN ← sumN − 1 37. end if 38. j←j+1 39. while (sumN + 1 = N ) 40. end if 41. λ1 ← (N − 2)P + 1 N [i+1]
42. {λ2 , λ3 , . . . , λN } ←−−− λ[i + 1] 43. return {λ1 , λ2 , . . . , λN } End
do N [k + j] ← N [k + j] + 1
Table 3 The number of added links in random networks with different link density pˆ and network size N. pˆ (or p) N N N N
= 100 = 500 = 1000 = 1500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
15 20 1 1
24 64 96 139
this case, the connectivity of this network declines rather than gets improved. If pˆ < p or pˆ ≈ p, we optimize this network by using our optimization algorithm shown in Tables 1 and 2. Let pˆ = p in the following optimization examples. The optimization results of random networks are listed in Tables 3 and 4, and those of scale-free networks are listed in Tables 5 and 6. The number of added links in random networks and scale-free networks is respectively listed in Tables 3 and 5. The ratio of the added link number to the total link number in random networks and scale-free networks is respectively listed in Tables 4 and 6. The corresponding curves of optimization results in Tables 3–6 are respectively plotted in Fig. 9(a) and (b), as well as Fig. 10(a) and (b). Observing the shapes and trends of these curves in Fig. 9(a) and (b), we can conclude that the spectral density of a random network with a large size N is able to satisfy Wigner’s Semicircle Law quite well when its link density pˆ is less than a certain critical probability value pc , e.g., pc = 0.65 (N = 100), pc = 0.65 (N = 500), pc = 0.65 (N = 1000), and pc = 0.65 (N = 1500). The reason is that the eigenvalue distribution of this network can reach to the optimal result only by adding
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Table 4 The ratio (percent) of the added link number to the total link number in random networks with different pˆ and N. pˆ (or p) N N N N
= 100 = 500 = 1000 = 1500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.2020 0.0080 0.0020 0.0009
0.1010 0.0040 0.0010 0.0004
0.0673 0.0027 0.0007 0.0003
0.0505 0.0020 0.0005 0.0002
0.0404 0.0016 0.0004 0.0002
0.0337 0.0013 0.0003 0.0001
0.1154 0.0011 0.0003 0.0001
0.3788 0.0200 0.0002 0.0001
0.5387 0.0570 0.0214 0.0137
Table 5 The number of added links in scale-free networks with different parameter m (m = m0 ) and network size N. m N N N N
= 100 = 500 = 1000 = 1500
2
3
4
5
6
7
8
5 15 22 28
7 16 24 32
8 17 25 34
8 18 27 36
8 20 28 38
9 21 30 42
9 22 32 46
Table 6 The ratio (percent) of the added link number to the total link number in scale-free networks with different m (m = m0 ) and N. m N N N N
= 100 = 500 = 1000 = 1500
2
3
4
5
6
7
8
2.54 1.50 1.10 0.93
2.38 1.07 0.77 0.71
2.05 0.85 0.63 0.57
1.65 0.72 0.54 0.48
1.38 0.67 0.47 0.42
1.34 0.60 0.43 0.40
1.18 0.55 0.40 0.38
Fig. 9. The number of added links (a) and the ratio (%) of the added link number to the total link number (b) in random networks with different link density pˆ and network size N.
a single undirected link and the corresponding ratio of the added link number to the total link number is also low enough while pˆ ≤ pc . Of course there is an exception when N is small, e.g., N = 100. From Fig. 10(a) and (b), it can be observed that a larger N or m in a scale-free network indicates its structural properties are closer to the optimal result. The optimization results above also verify our optimization algorithm shown in Tables 1 and 2. 7. Conclusions This paper focuses on three critical problems in the connectivity optimizations of large networks: (a) how to determine an appropriate measure of network connectivity, (b) how to add/remove a link to/from a network such that its connectivity can be increased/decreased the most, and (c) how to optimize not only the connectivity of a large network but also its topological features given that its cost constraint allows. First of all, we derive a better measure of network connectivity than algebraic connectivity, spectral radius, when the network size is large. On the basis of this, a new strategy increasing/decreasing network connectivity is derived, by which the spectral radius of a network can be increased/decreased the most. To measure and optimize some network features, we derive two complete functions of spectral density for large random and scalefree networks respectively, acting as the optimization objectives of network properties. At last, the authors propose an optimization algorithm based on spectral density and give some relevant examples to verify it. While our optimization
X. Yan et al. / Physica A 444 (2016) 373–387
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Fig. 10. The number of added links (a) and the ratio (%) of the added link number to the total link number (b) in scale-free networks with different m and N.
algorithm is designed for increasing network connectivity, it can be extended to decrease network connectivity, depending on the link density of that network being optimized. In addition, our finding in this paper is that the spectral density of a large random network is in almost total agreement with Wigner’s Semicircle Law when its link density is less than a certain critical probability, and a large scale-free network with a larger growth parameter m has closer properties to the ideal BA model. When deriving the spectral density function of BA model, we also find that the values of three critical eigenvalues w.r.t. network size: λ1 ∝ N α (α < 0.25), λ2 ∝ N α (α ≈ 0.25), and λN ∝ N α (α ≈ 0.25). In a word, our algorithm can optimize some topological features of a large network while improving its network connectivity. It provides a benchmark or an indicator for us while optimizing network connectivity, i.e., a guided optimization. Besides, our algorithm can also be regarded as another algorithm that solves all eigenvalues of an N × N adjacency matrix for large random or scale-free networks, with a low computational complexity of O(N 2 ) at most. Our work in the future is to apply this optimization algorithm to some realistic large networks such as transportation networks. Acknowledgment This work was supported by the National Natural Science Foundation of China under Grant number 61472294. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]
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