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Recently, some coarse-graining methods based on network synchronization have been proposed to reduce the network size while preserving the synchronizability of the original network. In this paper, we investigate the effects of the coarse graining process on synchronizability over complex networks under different average path lengths and different degrees of distribution. A large amount of experiments demonstrate a close correlation between the average path length, the heterogeneity of the degree distribution and the ability of spectral coarse-grained scheme in preserving the network synchronizability. We find that synchronizability can be well preserved in spectral coarse-grained networks when the considered networks have a longer average path length or a larger degree of variance.

Synchronization, as an emerging phenomenon of a population of dynamically interacting systems, is ubiquitous in nature and plays an important role within various contexts in biology, chemistry, ecology, sociology, and technology [^{11} neurons and more than 10^{15} connections, which brings a big challenge to research on such networks. Especially for the dynamics of large-scale coupling nodes, large number of coupled differential equations result in the trouble of the computation and simulation. Many techniques and methods at the level of mesoscale networks are useless in large networks. Therefore, when using model reduction for simulation and analysis, it is crucial to be able to reduce the network size while keeping most of the relevant properties of the initial networks [

Therefore, some coarse-grained methods are proposed to try to transform large-scale networks into mesoscale networks [

Many articles only studied the maintenance of the topological properties of the initial networks by the coarse-grained method, but few people discussed the influence of the topological properties of the initial network on the effect of the coarse graining method. In this paper, we find that the longer the average path length or the more heterogeneous the degree distribution of the initial network, the spectral coarse-grained algorithm becomes more effective in keeping the synchronization ability.

The rest of the paper is organized as follows. In Section 2, Mathematical basis is introduced. Influence of topological properties of complex networks on the effect of coarse-grained networks are proposed in Section 3. In Section 4, we verify our results through the real world networks. Finally, some conclusions are drawn in Section 5.

Consider a general complex dynamic network with N nodes. The dynamic equation as follow:

x ˙ i = f ( x i ) − c ∑ j = 1 N l i j H ( x j ) , i = 1 , 2 , ⋯ , N (1)

where x i ∈ R n is the n-dimensional state variable of the i_{th} node, c > 0 is the coupling strength, H : R n → R n is the inner coupling function, and the Laplacian matrix L = ( l i j ) N × N describes the coupling topology of the network, in which l i j = − 1 if j connects to i (otherwise 0). The matrix L satisfies the dissipation coupling conditions: ∑ j = 1 N l i j = 0 . If the network is undirected and connected, then L is a symmetric and positive semi-definite matrix with nonnegative eigenvalues satisfying 0 = λ 1 < λ 2 ≤ ⋯ ≤ λ N . If there is s ( t ) ∈ R n and t → ∞ , then x i ( t ) → s ( t ) , i = 1 , 2 , ⋯ , N , the state of all nodes of the Equation (1) is fully (asymptotically) synchronized to s ( t ) and s ( t ) is called a synchronous state, and which ξ i is the variation of the i_{th} node, The variational equation is obtained by:

ξ ˙ = D f ( s ) ξ − c D H ( s ) ξ L T (2)

where D f ( s ) and D H ( s ) are the Jacobian matrices of f ( s ) and H ( s ) with respect to s ( t ) , Diagonalizing Equation (2) yields the following form:

η ˙ = [ D f ( s ) η − c λ k D H ( s ) ] η k , k = 2 , ⋯ , N (3)

where η k is the eigenmode associated with the eigenvalue λ k of L. Generalize Equation (3) to get the main stability equation:

y ˙ = [ D f ( s ) − α D H ( s ) ] y (4)

The largest Lyapunov exponent of this equation is a function of real variables α, It is called the main stable function of network (1) [

In 2008, D. Gfeller et al. proposed a spectral coarse graining method based on merging nodes with similar characteristic components in reference [

First, determine which nodes merge. Let p^{2} denotes the eigenvector for the smallest nonzero eigenvalue λ 2 . Merging nodes that correspond to the same or similar components in p^{2}. Here p max 2 and p min 2 are the largest and the smallest components of p^{2}. Divide the elements in p^{2} evenly between p max 2 and p min 2 into I intervals. The smaller I is, the smaller the size of the coarse-grained network will be, and the more the size of the network is reduced.

Second, update edges and extract the coarse-grained network. Let the N nodes of the initial network be labeled with i = 1 , 2 , ⋯ , N , and the coarse-grained network has N ˜ nodes corresponding to N ˜ groups, labeled with C = 1 , 2 , ⋯ , N ˜ . The edges of the coarse-grained network (corresponding to the new Laplacian matrix L ˜ ) can be updated by the following matrix product

L ˜ = K L Q , (5)

where, K ∈ R N ˜ × N and Q ∈ R N × N ˜ are defined by

K C i = Ψ C , C i / | C | ; Q i C = Ψ C , C i . (6)

Here, | C | is the cardinality of group C; C i is the index of the I’th node group, and Ψ is the Kronecker symbol.

To investigate the structural effects on network synchronizability, we use random interchanging algorithm [

1) Randomly pick two existing edges e 1 = x 1 x 2 and e 2 = x 3 x 4 , such that x 1 ≠ x 2 ≠ x 3 ≠ x 4 and there is no edge between x 1 and x 4 as well as x 2 and x 3 .

2) Cross reconnect these four nodes., that is, connect x 1 and x 4 as well as x 2 and x 3 , and remove the edges e 1 and e 2 .

3) Ensure that the network is connected and calculate whether this interchange increases/decreases the network average path length. If it does, accept the new configuration, else restore the original network structure.

4) Repeat step 1) unless the desired average path length is achieved.

Because the algorithm is only reconnected, it does not change the degree of any node. So the degree distribution and degree sequence are fixed.

We consider a small-world network with average degree 〈 d e g 〉 = 4 , 1000 nodes. By random interchanging, we obtain four small-world networks with the same degree distribution and different average path lengths, respectively 5.5219, 5.5518, 5.9301 and 6.6772. Similarly, we consider the scale-free network with

power exponent 2.05, 1000 nodes. Through random interchanging, we obtain four scale-free networks with the same degree distribution and different average path lengths, and the average path lengths are 4.1000, 4.2530, 4.3048, 4.4591.

Based on the spectral coarse-graining method, The initial network of N = 1000 nodes is merged into N ˜ groups, and λ ˜ 2 and λ ˜ N denote the smallest non-zero eigenvalue and the largest non-zero eigenvalue of the new Laplacian matrix, respectively. The relationship between N ˜ , λ ˜ N and λ ˜ 2 / λ ˜ N for different values of average path lengths are shown in

network. On the other hand,

In summary, regardless of the small-world networks or scale-free networks, as the average path length increases, the eigenvalue λ ˜ 2 and the eigenratio λ ˜ 2 / λ ˜ N of coarse-grained networks become closer to the original values of the initial networks, which is to say that, with the increase of the average path length, the spectral coarse-grained method becomes more effective in keeping the network synchronization.

In

In order to measure the heterogeneity of the degree distribution of the network, we use the degree variance σ to represent the heterogeneity of the degree distribution. The greater the degree variance, the more inhomogeneous the network.

Similarly, we first obtain small-world networks with different four-degree variances by reconnecting probabilities p = 0.1, 0.2, 0.3, and 0.4. We use the random interchanging algorithm to equalize the average path lengths of these four networks. Then we get four networks with the same average path length and different degree distribution. The degree variance is 0.3657, 0.7640, 1.0840, and 1.3200, respectively. Equally, we consider the four scale-free networks with 1000 nodes, uniform average path length and different degree distributions. The degree variances are 14.2060, 15.2419, 18.6693, and 24.4806, respectively.

The simulation results show, with the increase of degree variance, the distribution of network degree becomes more heterogeneous in both small-world networks and scale-free networks. The λ ˜ 2 and λ ˜ 2 / λ ˜ N of coarse-grained network are more close to the corresponding values of the initial networks, that is to say, the effect of coarse-grained network will be better with the increase of degree variance.

To demonstrate our conclusions, we consider two real world networks-scientific cooperation network ( N = 379 ) [

interchanging algorithm. Moreover, we get three protein-protein interaction networks with average path lengths of 4.0122, 4.2111, and 4.3763 through the random interchanging algorithm. The results are shown in

This paper studies the influence of the average path length and the heterogeneity of the degree distribution on the ability of spectral coarse-graining method in keeping the network synchronizability. According to the large number of simulation experiments, the average path length and the heterogeneity of degree distribution are closely related to keep the λ ˜ 2 and λ ˜ 2 / λ ˜ N of the initial networks when applying spectral coarse-graining method to reduce the network size. The longer the average path length, the more inhomogeneous the degree distribution, which can better maintain the network synchronization ability in the coarse-grained process.

This project is supported by National Natural Science Foundation of China (Nos.61563013, 61663006) and the Natural Science Foundation of Guangxi (No.2018GXNSFAA138095).

Zeng, L., Jia, Z. and Wang, Y.Y. (2018) Influence of Topological Properties of Complex Networks on the Effect of Spectral Coarse-Grained Network. Communications and Network, 10, 93-104. https://doi.org/10.4236/cn.2018.103008