Robustness of Regular Graphs Based on Natural Connectivity
1 Robustness of Regular Graphs Based on Natural Connectivity
Jun Wu , Mauricio Barahona , Yuejin Tan , Hongzhong Deng Abstract : It has been recently proposed that the natural connectivity can be used to characterize efficiently the robustness of complex networks. The natural connectivity quantifies the redundancy of alternative routes in the network by evaluating the weighted number of closed walks of all lengths and can be seen as an average eigenvalue obtained from the graph spectrum. In this paper, we explore both analytically and numerically the natural connectivity of regular ring lattices and regular random graphs obtained through degree-preserving random rewirings from regular ring lattices. We reformulate the natural connectivity of regular ring lattices in terms of generalized Bessel functions and show that the natural connectivity of regular ring lattices is independent of network size and increases with K monotonically. We also show that random regular graphs have lower natural connectivity, and are thus less robust, than regular ring lattices. Keywords : natural connectivity; robustness; regular ring lattices; random regular graphs; generalized Bessel function * Tel.: +86-731-4573593; Fax: +86-731-4573593; E-mail: [email protected] Introduction
Networks are everywhere. Networks with complex topology describe a wide range of systems in nature and society (Albert and Barabási 2002, Boccaletti et al.
Newman et al. et al. et al. et al. et al. et al. et al. NP-complete. This implies that these measures are of no great practical use within the context of complex networks. Another remarkable measure to unfold the robustness of a network is the second smallest (first non-zero) eigenvalue of the Laplacian matrix, also known as the algebraic connectivity. Fiedler (Fiedler 1973) showed that the magnitude of the algebraic connectivity reflects how well connected the overall graph is; the larger the algebraic connectivity is, the more difficult it is to cut a graph into independent components. However, the algebraic connectivity is equal to zero for all disconnected networks. Therefore, it is too coarse a measure for complex networks. An alternative formulation of robustness within the context of complex networks emerged from random graph theory (Bollobás 1985) and was stimulated by the work of Albert et al. (Albert et al. et al. et al. et al. et al. et al. et al. structure (Estrada 2000), bipartivity (Estrada and Rodríguez-Velázquez 2005a), su bgraph centrality (Estrada and Rodríguez-Velázquez 2005b) and expansibility (Estrada 2006a, Estrada 2006b). The natural connectivity has an intuitive physical meaning and a simple mathematical formulation. Physically, it characterizes the redundancy of alternative paths by quantifying the weighted number of closed walks of all lengths leading to a measure that works both in connected and disconnected graphs. Mathematically, the natural connectivity is obtained from the graph spectrum as an average eigenvalue and it increases strictly monotonically with the addition of edges. Rich information about the topology and dynamical processes can be extracted from the spectral analysis of the networks. The natural connectivity sets up a bridge between graph spectra and the robustness of complex networks, allowing a precise quantitative analysis of the network robustness. In this paper, we investigate the natural connectivity of regular graphs, i.e., graphs with constant degree, both of deterministic and of random nature. In particular, we study deterministic ring lattices ( K -cycles or pristine worlds) and regular randomized graphs derived from them through a degree-preserving rewiring (Watts and Strogatz 1998, Barahona and Pecora 2002). Regular graphs are networks of wide interest and constitute examples of expander graphs, which are extremely robust to vertex or edge removal (Sarnak 2004, Hoory et al. Preliminaries
Graphs and Natural Connectivity
A complex network can be formalized in terms of a simple undirected graph ( , )
G V E , where V is the set of vertices and E V V ⊆ × is the set of edges. Let
N V = and M E = be the number of vertices and the number of edges, respectively. The connectivity of the graph G can be represented by the adjacency matrix ( ) ( ) ij N N A G a × = , where ij ji a a = = if vertex i v and j v are adjacent and ij ji a a = = otherwise. It follows immediately that ( ) A G is a real symmetric matrix with real eigenvalues ... N λ λ λ≥ ≥ ≥ , which are usually also called the eigenvalues of the graph G itself. The set { } , ,... N λ λ λ is called the spectrum of G . A walk of length k in a graph G is an alternating sequence of vertices and edges ... k k v e v e e v , where i v V ∈ and ( , ) i i i e v v E − = ∈ . A walk is closed if k v v = . The number of closed walks is an important index for complex networks. Recently, we have proposed that the number of closed walks of all lengths quantify the redundancy of alternative paths in the graph and can therefore serve as a measure of network robustness (Wu et al. k . That is, we define a weighted sum of numbers of closed walks / ! kk S n k ∞= = ∑ , where k n is the number of closed walks of length k . This scaling ensures that the weighted sum does not diverge and it also means that S can be obtained from the powers of the adjacency matrix: ( ) N k k kk i ii n trace A λ λ = = = = ∑ ∑ (1) Using Eq. (1), we can obtain ! ! ! i k kN N Nk i ik k i i k i nS ek k k λ λ λ ∞ ∞ ∞= = = = = = = = = = ∑ ∑ ∑ ∑ ∑ ∑ (2) Hence the proposed weighted sum of closed walks of all lengths can be derived from the graph spectrum. We remark that Eq. (2) corresponds to the Estrada Index of the graph (Estrada 2000) , which has been used in several contexts in graph theory, including su bgraph centrality (Estrada and Rodríguez-Velázquez 2005b) and bipartivity (Estrada and Rodríguez-Velázquez 2005a). The natural connectivity can be defined as an average eigenvalue of the graph as follows. Definition 1 (Wu et al.
Let ( )
A G be the adjacency matrix of G and let ... N λ λ λ≥ ≥ ≥ be the eigenvalues of ( ) A G , the natural connectivity or natural eigenvalue of G is defined by ln / i Ni e N λ λ = ⎛ ⎞= ⎜ ⎟⎝ ⎠ ∑ (3) It is evident from Eq. (3) that N λ λ λ≥ ≥ . Moreover, the natural connectivity changes strictly monotonically with the addition or deletion of edges (Estrada and Rodríguez-Velázquez 2005a, Wu et al. et al. Generalized Bessel Functions
The Bessel functions of the first kind ( )
J x α are defined as the solutions to the Bessel differential equation
22 2 22 ( ) 0 d y dyx x x ydxdx α+ + − = (4) which are nonsingular at the origin (Abramowitz and Stegun 1972). Another definition of the Bessel function for integer values of n is possible using an integral representation (Abramowitz and Stegun 1972)
1( ) cos( sin ) n J x n x d π τ τ τ π = − ∫ (5) The generalized Bessel functions of the first kind for integer values of n are defined by (Dattoli et al.
1( , ,... ) cos( sin sin 2 ... sin ) n M M
J x x x n x x x M d π τ τ τ τ τπ= − − − − ∫ (6) A related class of functions are the modified Bessel functions of the first kind (Abramowitz and Stegun 1972) ( ) ( ) I x i J ix αα α − = (7) which are the solutions to the modified Bessel differential equation
22 2 22 ( ) 0 d y dyx x x ydxdx α+ − + = (8) The corresponding modified generalized Bessel functions are defined by (Dattoli et al.
1( , ,... ) cos( ) exp( cos )
Mn M ss
I x x x n x s d π τ τ τπ = = ∑∫ (9) The following are some important properties of modified generalized Bessel functions (Dattoli and Torre 1996, Dattoli et al. ( , ,... ) ( , ,... ) n M n M I x x x I x x x − = (10) ( , ,... ) 0 as n M I x x x n → → ∞ (11) e ( , ,... ) exp cos( )
Min n M sn s
I x x x x s τ τ ∞=−∞ = ⎡ ⎤= ⎢ ⎥⎣ ⎦ ∑ ∑ (12) ( , ,... ) ( , ,... ) ( ) n M n Ml M l Ml I x x x I x x x I x ∞ − −=−∞ = ∑ (13) These will be used in the derivation of the natural connectivity of regular ring lattices in the next section. Natural Connectivity of Regular Ring Lattices
A regular ring lattice , N K R is a K − regular graph with N vertices in a ring in which each vertex is connected to its K neighbors ( K on either side). They have also been called K -cycles and pristine worlds (Barahona and Pecora 2002) as they are the starting point for the small-world construction (Watts and Strogatz 1998). The adjacency matrix A of , N K R is a circulant matrix in the form ......... ... ... ...... NN N c c cc c cA c c c −− − ⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦ (14) where k c = if mod k N K ± > or k = , otherwise k c = . We show an example of a regular ring lattice in Fig. 1. Fig. 1 Regular ring lattice , N K R with N = and K = . The eigenvectors of a circulant matrix are the columns of the discrete Fourier transform matrix of the same size. Consequently, the eigenvalues of a circulant matrix can be obtained by the Fast Fourier transform (Gray 2006, Davis 1979) as Nj kk ik jc j NN πλ −= −⎛ ⎞= − =⎜ ⎟⎝ ⎠ ∑ (15) where i = − . For the regular ring lattices , N K R , the spectrum is then given by
11 1
K Nj k k N KKk ik j ik jN Nk jN π πλ π −= = −= − −⎛ ⎞ ⎛ ⎞= − + −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠−⎛ ⎞= ⎜ ⎟⎝ ⎠ ∑ ∑∑ (16) which can be substituted into Eq. (3) to obtain the expression of the natural connectivity ,2 N K
N KR j k k jN N πλ = = ⎡ ⎤⎛ − ⎞⎛ ⎞= ⎢ ⎥⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎣ ⎦ ∑ ∑ (17) Using Eq.(12), we can rewrite Eq. (17) as ,2 N K
KNR nj n K Nnn j jin IN N jI inN N πλ π ∞= =−∞∞=−∞ = ⎡ ⎤−⎛ ⎞⎢ ⎥= ⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦⎡ ⎤−⎛ ⎞⎢ ⎥= ⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦ ∑ ∑∑ ∑ (cid:13)(cid:11)(cid:14)(cid:11)(cid:15)(cid:13)(cid:11)(cid:14)(cid:11)(cid:15) (18) Note that
N N Nj j jn Nt n j n j n ji iN N N N N N π π πδ = = = − − −⎛ ⎞ ⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠= ∑ ∑ ∑ (19) where , , n Nt t δ ∈ Z is the Kronecker delta, i.e., , n Nt δ = if n Nt = , and , n Nt δ = otherwise. Then Eq. (18) simplifies to , , 01 ln (2, 2,...2) ln 2 (2, 2,...2) (2, 2,...2) N K
K K KR n n Nt Ntn t
I I I λ δ ∞ ∞=−∞ = ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟= ⋅ = +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ∑ ∑ (cid:13)(cid:11)(cid:14)(cid:11)(cid:15) (cid:13)(cid:11)(cid:14)(cid:11)(cid:15) (cid:13)(cid:11)(cid:14)(cid:11)(cid:15) (20) The asymptotic convergence shown in Eq. (11) implies that (2, 2,...2) 0
KNt I → (cid:13)(cid:11)(cid:14)(cid:11)(cid:15) as N → ∞ , which leads to the following asymptotic result. Theorem 1
Let , N K R be a regular ring lattice, then the natural connectivity of , N K R is , ln (2, 2,...2)+ (1) N K KR I λ ο⎛ ⎞⎜ ⎟= ⎜ ⎟⎝ ⎠ (cid:13)(cid:11)(cid:14)(cid:11)(cid:15) (21) where (1) 0 as N ο → → ∞ . Using the properties shown in Eq. (13) and Eq. (10), we then obtain that (2, 2,...2) (2, 2,...2) (2) (2, 2,...2) (2) 2 (2, 2,...2) (2) K K K KKl l Kl ll l
I I I I I I I − − −∞ ∞=−∞ = = = + ∑ ∑ (cid:13)(cid:11)(cid:14)(cid:11)(cid:15) (cid:13)(cid:11)(cid:14)(cid:11)(cid:15) (cid:13)(cid:11)(cid:14)(cid:11)(cid:15) (cid:13)(cid:11)(cid:14)(cid:11)(cid:15) (22) Note that (2, 2,...2) (2) 0 KKl ll
I I −∞= > ∑ (cid:13)(cid:11)(cid:14)(cid:11)(cid:15) and (2) 2.2796>1 I = whence it follows that
10 0 (2, 2,...2) (2, 2,...2)
K K
I I − > (cid:13)(cid:11)(cid:14)(cid:11)(cid:15) (cid:13)(cid:11)(cid:14)(cid:11)(cid:15) (23) Thus , , 1 N K N K
R R λ λ − > , i.e., the natural connectivity of regular ring lattices , N K R increases monotonically with K . The implication is that as we increase the number of neighbours in regular ring lattices the robustness of the graph increases. Note that a cycle graph N C , which consists of a single cycle, is a special regular ring lattice with K = , i.e., ,1 N N
C R = , we obtain the following corollary: Corollary 1
Let N C be a cycle graph, then the natural connectivity of N C is ( ) ln (2)+ (1) N C I λ ο = (24) where (1) 0 as N ο → → ∞ . Using the recurrence property shown in Eq. (13) , we can devise a recursive analytical procedure to calculate (2, 2,...2) K I (cid:13)(cid:11)(cid:14)(cid:11)(cid:15) in terms of standard Bessel functions ( ) n I x . In Fig. 2, we show both the numerical and analytical results and find that the natural connectivity of regular ring lattices increases with K monotonically, and that the natural connectivity is independent of the network size when N is large. This agrees well with the analytical results. Fig. 2. Natural connectivity of regular ring lattices, , N K R : (a) λ vs. N with
3, 4,5 K = from bottom to top; (b) λ vs. K with N = . The symbols represent numerical results and lines represent the asymptotic analytical results as N → ∞ according to Eq. (21). For the sake of comparison, we show in Fig. 3 the algebraic connectivity a of regular ring lattices. The algebraic connectivity is the first non zero eigenvalue of the Laplacian matrix of the graph and has been widely used as a measure of the connectedness of the network and the resistance to edge deletion. Our results show that the algebraic connectivity of regular ring lattices increases with K monotonically, but decreases with N monotonically. In fact, the algebraic connectivity approaches zero when N is large. Our intuitive understanding of robustness suggests that the robustness of a regular ring lattice with given K should not depend on the size of the graph N , when N is large. Therefore the natural connectivity as a measure of robustness agrees with our intuition, while the algebraic connectivity does not. Fig. 3. Algebraic connectivity a of regular ring lattices, , N K R : (a) a vs. N with
3, 4,5 K = from bottom to top; (b) a vs. K with N = . Natural Connectivity of Random Regular Graphs
To explore the natural connectivity of regular ring lattices in depth, we randomize regular ring lattices and generate random regular graphs through a degree-preserving, random rewiring procedure. It is difficult to obtain the eigenvalues of random regular graphs analytically. In this paper, we present numerical results obtained from Eq. (3). The rewiring algorithm is as follows: (1) Choose at random two edges denoted by ( , ) v w and ( , ) v w ; (2)
Replace the two edges with two new ones ( , ) v v and ( , ) w w . Note that the above process could lead to multi-edges or self-loops. We avoid that by rejecting any rewirings that lead to non-simple graphs. In Fig. 4 we show the natural connectivity as a function of the number of rewirings starting from regular ring lattices with different N and K . We find that the natural connectivity decreases with the process of random degree-preserving rewirings and then achieves a steady value. This means that random regular graphs are less robust than regular ring lattices reaching an asymptotic value for a totally randomized regular graph. Fig. 4 Natural connectivity as a function of the number of random degree-preserving rewiring starting from regular ring lattices with
3, 4,5 K = where N = . Each quantity is an averaged over 100 realizations. We show in Fig. 5 the steady value of natural connectivity of random regular graphs as a function of N and K respectively. We find that the natural connectivity of random regular graphs decreases with N and increases with K . For the sake of comparison, we also show the natural connectivity of regular ring lattices with the same number of vertices and edges. It is clear that regular ring lattices are more robust than random regular graphs. Fig. 5 Natural connectivity of random regular graphs: (a) λ vs. N with
3, 4,5 K = . The solid lines represent the natural connectivity of regular ring lattices; (b) λ vs. K with N = . Each quantity is an average over 100 realizations. The solid line here represents the asymptotic behavior as N → ∞ of regular ring lattices. Conclusions
We have investigated the natural connectivity of regular graphs in this paper. We have shown that the natural connectivity of regular ring lattices is asymptotically independent of network size and it increases monotonically with the degree. We have generated random regular graphs by degree-preserving, random rewirings from regular ring lattices and have demonstrated that the natural connectivity of random regular graphs is smallet than that of regular ring lattices. For these regular graphs, randomness decreases the robustness of the network. Acknowledgment
This work is in part supported by the National Science Foundation of China under Grant No. 60904065, No. 70501032 and No. 70771111. References
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