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Synchronizability of networks of chaotic systems coupled via a graph with a prescribed degree sequence
Physics Letters A 346 (2005) 281–287 www.elsevier.com/locate/pla
Synchronizability of networks of chaotic systems coupled via a graph with a prescribed degree sequence Chai Wah Wu IBM Research Division, Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598, USA Received 17 May 2005; received in revised form 26 July 2005; accepted 27 July 2005 Available online 18 August 2005 Communicated by C.R. Doering
Abstract Generally, synchronization in a network of chaotic systems depends on the underlying coupling topology. Recently, there have been several studies conducted to determine what features of this topology contribute to the ability to synchronize. A short diameter has been proposed by several authors as a means to facilitate synchronization whereas others point to features such as the homogeneity of the degree sequence. Recently, it has been shown that the degree sequence by itself is not sufficient to determine synchronizability. The purpose of this Letter is to continue this study. For a given degree sequence, we construct two connected graphs with this degree sequence whose synchronizability can be quite different. In particular, we construct a graph with low synchronizability which improves upon previous bounds under certain conditions and we construct a graph which has synchronizability that is asymptotically maximal. On the other hand, we show analytically that for a random network model, homogeneity of the degree sequence is beneficial to synchronization. 2005 Elsevier B.V. All rights reserved. Keywords: Eigenvalue analysis; Nonlinear dynamics; Random graphs; Scale free networks; Synchronization
1. Introduction The object of interest in this Letter is a coupled network of n identical systems with linear coupling:
f (x1 , t) x˙1 . .. − (G ⊗ P )x, x˙ = .. = . x˙n
f (xn , t)
E-mail address: [email protected] (C.W. Wu). 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.07.089
(1)
282
C.W. Wu / Physics Letters A 346 (2005) 281–287
where xi is the state vector of the ith system, x = (x1 , . . . , xn )T and G ⊗ P is the Kronecker product of the matrices G and P . The function f (xi , t) is the nonlinear vector field of the individual systems. The coupling matrix G describes the coupling topology of the network, i.e., Gij = 0 if there is a coupling from system i to system j . If Gij < 0 or Gij > 0, we call such a coupling element cooperative or competitive, respectively. The network in Eq. (1) is said to synchronize globally if xi − xj → 0 as t → ∞ for all i, j . It was shown in [1,2] that for symmetric matrices G and a suitable matrix P , the ability for the network to synchronize depends on the size of the smallest nonzero eigenvalue of G. We will denote this eigenvalue as λ2 (G). An example of such suitable matrices P include the class of positive definite matrices. For other classes of matrices P [3,4] and the discrete time case [5,6], 2 (G) the ratio r(G) = λλmax (G) is important in determining synchronizability, where λmax (G) is the largest eigenvalue of G. In general, the coupled network is a nonlinear dynamical system. For the case when Eq. (1) is a linear system, i.e., f is linear, similar results have been obtained in the context of agreement and consensus problems [7–10]. All these results reduce the study of synchronization properties to properties of eigenvalues of the coupling matrix G. The graph of a matrix G is defined as the weighted graph with an edge of weight Gij from vertex i to vertex j if and only if Gij = 0. We assume that G is symmetric, so the graph of G is undirected.1 The Laplacian matrix of a weighted graph is defined as L = D − A, where D is the diagonal matrix of vertex degrees and A is the adjacency matrix. In this Letter we assume that G is a symmetric zero row sum matrix with nonpositive off-diagonal elements. In this case, G is equal to the Laplacian matrix of the graph of G and λ2 (G) is the algebraic connectivity of the graph [17]. Therefore, we will concentrate in this Letter on the eigenvalues of the Laplacian matrix. There are many results relating various characteristics of the graph of G and the eigenvalues of its Laplacian matrix. The reader is referred to [18–20] and the references therein. In this Letter we will assume that the graphs are unweighted, i.e., the adjacency matrix is a 0–1 matrix. Recently, various models of random graphs have been proposed that mimic more closely man-made and natural networks [21]. These random graphs differ from the classical random graph model in that the degree distribution does not have to be Poisson or binomial [22], but can have other types of distribution such as a power law distribution. Several of these random graph models have two important defining characteristics. First, the graph is generated via a random procedure. Second, the graph has a particular degree distribution. Focusing on the degree distribution alone, while ignoring the random aspect or vice versa can lead to incomplete conclusions. In [23] it was shown that the heterogeneity of the degree distribution can affect λ2 and r and hence the synchronizability. On the other hand, several papers show that the degree sequence by itself is not enough to characterize λ2 and r. In [24–26] it was shown that randomness makes λ2 and r large whereas local coupling makes λ2 and r small. In [27] a graph is constructed for a prescribed degree sequence whose normalized Laplacian matrix has a small nonzero eigenvalue. In this Letter we continue this study by constructing two connected graphs with the same degree sequence whose values for λ2 and r differ significantly. In particular, in some cases the graph with small λ2 has a value for λ2 that is in a sense minimal and improves upon the bound for λ2 in [27] whereas the graph with large λ2 has a value for λ2 which is in a sense maximal. Furthermore, we show analytically that for a model of random graphs, homogeneity of the degree sequence is beneficial for synchronization.
2. Graphs with a prescribed degree sequence A simple graph is an unweighted graph with no loops nor multiple edges [28]. The degree sequence of a graph is a list of the degrees of the vertices. A list of natural numbers is called graphical if there is a simple graph with this list as its degree sequence. Sufficient and necessary conditions for a list to be graphical are given in [29,30]. The following result gives upper and lower bounds for λ2 and r of the Laplacian matrix of a graph with a given degree sequence. 1 See [11–16] for synchronization results when G is not symmetric and the graph of G is directed.
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Lemma 1. For a connected graph with degree sequence d1 d2 · · · dn , λ2 and r satisfy: 1 − cos πn δ δ π n δ λ2 δ, r , 2 1 − cos n n−1 ∆ ∆ where δ = d1 is the minimum vertex degree and ∆ = dn is the maximum vertex degree. Proof. The first set of inequalities follows from [17]. The second set of inequalities follows from the fact that n n−1 ∆ λmax 2∆, also from [17]. 2 In fact, if the graph is not complete, then λ2 δ [17]. The lower bounds in Lemma 1 are attained or approached asymptotically as n → ∞ for the nearest neighbor path graph with Laplacian matrix 1 −1 −1 2 −1 , L= .. .. .. . . . −1
1
and λ2 = 2(1 − cos(π/n)), r = 1−cos(π/n) 1+cos(π/n) . The upper bounds are attained for the complete graph with Laplacian matrix n − 1 −1 ··· −1 −1 n − 1 −1 , L= .. .. .. . . . −1 and λ2 = n, r = 1.
···
−1 n − 1
2.1. Regular graphs A k-regular graph is a graph where every vertex has degree k. In this section, we consider two types of connected regular graphs whose values of λ2 and r can be quite different. 2.1.1. Construction 1: graph with low λ2 and r For a 2k-regular graph, the degree sequence is (2k, 2k, . . . , 2k). We consider two cases. In the first case, the 1 vertex degree grows as Ω(n 3 − ) for > 0. If we arrange the vertices in a circle, and connect each vertex to its 2k nearest neighbors, then the resulting 2k-regular graph is denoted C2k . The Laplacian matrix L is a circulant matrix: 2k −1 · · · −1 0 · · · 0 −1 · · · −1 −1 2k −1 · · · −1 . .. .. .. .. . . . . −1 · · · −1 0 · · · 0 −1 · · · −1 2k The eigenvalues of L are given by (µ0 , . . . , µn−1 ) with: µ0 = 0,
sin k + 12 2π m 2πml n = 2k + 1 − µm = 2k − 2 cos , n sin πm n l=1 k
A series expansion shows that as n → ∞, 3 4π 2 k + 12 . λ2 µ 1 ≈ 3n2
m = 1, . . . , n − 1.
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n Since λmax 2k n−1 by Lemma 1, 3 2π 2 k + 12 µ1 ≈ . r λmax 3kn2
This means the values λ2 and r decrease as Ω(1/n1+3 ) and Ω(1/n 3 +2 ), respectively. In particular, if k is bounded, i.e., = 1/3, then λ2 , r decrease as Ω(1/n2 ) which is asymptotically the fastest possible by Lemma 1. In the second case, we consider graphs for which k < n2 and use the same approach as in [27]. The main difference is that in [27] the smallest nonzero eigenvalue of the normalized Laplacian matrix L˜ = I − D −1 A is studied.2 For a graph with adjacency matrix A, let e(B, C) = u∈B,v∈C Au,v be the number of edges between B and C. The following lemma is shown in [18]: 4
Lemma 2. λ2 (L)
¯ e(S, S)n λmax (L). |S|(n − |S|)
¯ = 2. By Lemma 2 this means that As in [27], we create a connected k-regular graph with |S| = n2 and e(S, S) n8 , n even 2n 8 λ2 (L) n n = , 8 , n odd n − n1 2 2 n− n1 8(n−1) , n even 8 kn2 . r(L) 8 k(n + 1) , n odd k(n+1)
The reason for studying the two cases is that C2k gives better bounds when k grows slower than n1/3 and the construction in [27] gives better bounds when k grows faster than n1/3 . In either case, λ2 , r → 0 as n → ∞. 2.1.2. Construction 2: graph with high λ2 and r √ √ With high probability, a random 2k-regular graph has eigenvalues λ2 = 2k − O( k ), λmax = 2k + O( k ) as n → ∞ [31]. This means that on a relative scale, λ2 ≈ 2k, and r ≈ 1 as k, n → ∞ which can be considered optimal according to Lemma 1. In conclusion, whereas λ2 and r → 0 as n → ∞ for Construction 1, they remain bounded from below for Construction 2. Lemma 3. If H is a subgraph of G with the same set of vertices,3 then (1)
λ2 (H) + λ2 (G\H) λ2 (G),
(2)
λmax (H) + λmax (G\H) λmax (G), λ2 (H) + λ2 (G\H) min r(H), r(G\H) r(G), λmax (H) + λmax (G\H) λmax (H) λmax (G).
(3) (4)
Here G\H is a graph with the same set of vertices as G and with edges which are in G but not in H. By a slight abuse of notation, we use λi (H) to denote λi of the Laplacian matrix of the graph H. 2 For k-regular graphs, this difference between L and L ˜ is not important since L = k L˜ and the eigenvalues of L and L˜ differ by a constant
factor k. However, for non-regular graphs, the eigenvalues of L˜ and L will have different properties. 3 By this we mean that A B for all i, j , where A and B are the adjacency matrices of H and G, respectively. ij ij
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Proof. Follows from the facts that λ2 (L) =
min
i xi =0,
x=1
x T Lx,
and L(H) + L(G\H) = L(G).
λmax (L) = max x T Lx x=1
2
2.2. Graphs with prescribed degree sequence For each n, consider a graphical list of degrees 0 < d1 d2 · · · dn . The average vertex degree is defined as k¯ = ( i di )/n. Next, we construct two connected graphs with the same list of degrees but different λ2 and r. 2.2.1. Construction 1: graph with low λ2 and r 1 As in Section 2.1.1 we consider two cases. In the first case, dn grows as Ω(n 3 − ) for > 0. Given a graphical list of degrees 0 < d1 d2 · · · dn , the Havel–Hakimi algorithm [32,33] constructs a connected graph with this degree sequence where the vertices are arranged on a line and each vertex is connected to its nearest neighbors. Let us denote this graph as Ghh with Laplacian matrix Lhh . The graph Ghh is a subgraph of the 2dn -regular graph C2dn defined in Section 2.1.1. Therefore, by Lemma 3, 3 4π 2 dn + 12 1 as n → ∞. = Ω 1+3 λ2 (Lhh ) µ1 ≈ 3n2 n Since λmax (Lhh )
ndn n−1 ,
this means that
3 4π 2 dn + 12 µ1 1 . r(Lhh ) ≈ =Ω 4 λmax 3dn n2 n 3 +2 As in Section 2.1.1, for bounded dn the values of λ2 and r for this construction decrease as Ω(1/n2 ) when n → ∞. By Lemma 1, this is the fastest possible rate. In the second case, assume that the degree sequence is of the form (dmax , . . . , dmax , dmax − 1, . . . , dmax − 1, . . . , 1, . . . , 1) with dmax n/4 and the average degree satisfies k¯ = 2 + dmax n . We further assume that for each 1 i dmax , there are at least 2 vertices with vertex degree i. As in [27], we construct a graph with this degree ¯ = 2. By Lemma 2 this means that sequence such that |S| n2 − dmax and E(S, S) n−4d82 /n , n even 10 23 2n 8 max , λ2 (L) n = n 8 2 /n n , n odd n − 4dmax 2 + dmax 2 − dmax n−4 dmax − 12 2 /n 8(n−1) n even 3 , 10 23 (n − 1)λ2 (L) n2 dmax −4dmax r(L) , 8(n−1) dmax n n odd ndmax 1 2 , ndmax n−4 dmax − 2 /n
where we have used the fact that dmax n/4. Again, in either case r, λ2 → 0 as n → ∞. 2.2.2. Construction 2: graph with high λ2 and r For convenience, we allow loops in the graph, i.e., the graphs are not necessarily simple. Furthermore, rather than constructing graphs with a specific degree sequence, we consider a random graph model which has a prescribed degree sequence in expectation. Given a list of degrees d1 d2 · · · dn satisfying the condition n(dn − d1 )2 (n − d1 ) (dk − d1 ), k
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we construct a random graph as in [25] where an edge between vertex i and vertex j is randomly selected with probability Pij =
d1 (di − d1 )(dj − d1 ) + . n k (dk − d1 )
Since i Pij = dj , the graph has the desired degree sequence in expectation. Let us call this graph model Gpr with Laplacian matrix Lpr . For the special case d1 = d2 = · · · = dn , we choose Pij = d1 /n and denote this graph model as Gr (d1 ). Since Pij d1 /n, this means that Gr (d1 ) is a “subgraph” of Gpr . What we mean by this is that the probability space of Gpr can be partitioned such that for each graph H in Gr (d1 ) with probability p, there exists exactly one event A in Gpr with probability p, such that H is a subgraph of each graph in event A. In particular, consider the edges in Gpr to be of two types. An edge (i, j ) of type 1 occurs with probability d1 /n and an edge (i, j ) of type 2 occurs with probability Pij − dn1 . To each graph H in Gr (d1 ) corresponds a set of graphs A(H) constructed as follows. If (i, j ) is an edge in H, then (i, j ) is an edge of type 1 in graphs in A(H). If (i, j ) is not an edge in H, then either (i, j ) is not an edge or (i, j ) is an edge of type 2 in graphs in A(H). It is clear that these sets A(H) over all graphs H in Gr (d1 ) exactly partition the probability space of Gpr . Furthermore, the probability of A(H) in Gpr is equal to the probability of H in Gr (d1 ) and H is a subgraph of every graph in A(H). By the Courant–Fischer minmax theorem, the eigenvalues of the Laplacian matrix cannot decrease as more edges are added to the graph. This means that if λ2 (Gr (d1 )) ξ with high probability, then λ2 (Gpr ) ξ with high probability. Let us assume now that d1 = p1 n for some√0 < p1 < 1. In [34] it was shown that for the graph Gr (d1 ), |λ2 − d1 | and |λmax − d1 | are both on the order of o( n ) with high probability as n → ∞. This means that λ2 (Gpr ) √ d1 n d1 , this implies that λ2 ≈ d1 and r 2d for Gpr , i.e., λ2 and r d1 − o( n ) with high probability. Since λ2 n−1 n are bounded away from 0 as n → ∞. The conclusion that λ2 ≈ d1 shows that for the random graph model Gpr , homogeneity of the degree sequence ¯ the highest λ2 can be as enhances synchronizability in terms of λ2 . In particular, for a given average degree k, n n ¯ ¯ i.e., ¯ n → ∞ is approximately k since λ2 n−1 d1 n−1 k. This is achieved for the random graph where d1 = k, the d1 -regular graph which has the most homogeneous degree sequence. This supports the experimental results in [23].
3. Conclusions We present graph constructions that for the same degree sequence generate graphs with low and high synchronizability in networks of coupled systems. In particular, we construct a graph with low synchronizability whose values of λ2 and r decrease to 0 and a graph with high synchronizability whose values of λ2 and r are bounded away from 0 as n → ∞. This further indicates that the degree sequence alone is not sufficient to determine the synchronizability of a network of coupled dynamical systems. Furthermore, the constructions further support the assertion in [25,26] that local coupling tends to have low synchronizability, whereas random coupling tends to have high synchronizability. On the other hand, for the random graph model Gpr in Section 2.2.2, homogeneity in the vertex degree is beneficial for synchronization.
References [1] C.W. Wu, L.O. Chua, IEEE Trans. Circuits Syst. I 42 (8) (1995) 430. [2] C.W. Wu, L.O. Chua, IEEE Trans. Circuits Syst. I 42 (8) (1995) 494.
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[3] L.M. Pecora, T.L. Carroll, Master stability functions for synchronized chaos in arrays of oscillators, in: Proceedings of the IEEE International Symposium on Circuits and Systems, vol. 4, 1998, pp. IV-562–IV-567. [4] M. Barahona, L.M. Pecora, Phys. Rev. Lett. 89 (5) (2002) 054101. [5] C.W. Wu, Global synchronization in coupled map lattices, in: Proceedings of the IEEE International Symposium on Circuits and Systems, vol. 3, 1998, pp. III-302–III-305. [6] C.W. Wu, Synchronization in Coupled Chaotic Circuits and Systems, World Scientific, 2002. [7] A. Jadbabaie, J. Lin, A.S. Morse, IEEE Trans. Automatic Control 48 (6) (2003) 988. [8] L. Mureau, Leaderless coordination via bidirectional and unidirectional time-dependent communication, preprint. [9] R. Olfati-Saber, R.M. Murray, IEEE Trans. Automatic Control 49 (9) (2004) 1520. [10] Y. Hatano, M. Mesbahi, Agreement over random networks, in: Proceedings of the 43rd IEEE Conference on Decision and Control, 2004, pp. 2010–2015. [11] C.W. Wu, IEEE Trans. Circuits Syst. I 50 (2) (2003) 294. [12] C.W. Wu, Linear Multilinear Algebra 53 (3) (2005) 203. [13] C.W. Wu, Linear Algebra Appl. 402 (2005) 207. [14] C.W. Wu, Nonlinearity 18 (2005) 1057. [15] C.W. Wu, Synchronization in an array of chaotic systems coupled via a directed graph, in: Proceedings of the IEEE International Symposium on Circuits and Systems, 2005, pp. 6046–6049. [16] C.W. Wu, IEEE Trans. Circuits Syst. II 53 (5) (2005) 282. [17] M. Fiedler, Czechoslovak Math. J. 23 (98) (1973) 298. [18] B. Mohar, The Laplacian spectrum of graphs, in: Y. Alavi, G. Chartrand, O.R. Oellermann, A.J. Schwenk (Eds.), Graph Theory, Combinatorics, and Applications, vol. 2, Wiley, 1991, pp. 871–898. [19] R. Merris, Linear Algebra Appl. 197–198 (1994) 143. [20] B. Mohar, Some applications of Laplace eigenvalues of graphs, in: G. Hahn, G. Sabidussi (Eds.), Graph Symmetry: Algebraic Methods and Applications, Kluwer Academic Publishers, 1997, pp. 225–275. [21] M.E.J. Newman, SIAM Rev. 45 (2) (2003) 167. [22] B. Bollobás, Random Graphs, second ed., Cambridge Univ. Press, 2001. [23] T. Nishikawa, A.E. Motter, Y.-C. Lai, F.C. Hoppensteadt, Phys. Rev. Lett. 91 (1) (2003) 014101. [24] C.W. Wu, Synchronization in arrays of coupled nonlinear systems: passivity, circle criterion and observer design, in: Proceedings of IEEE International Symposium on Circuits and Systems, vol. 3, 2001, pp. III-692–III-695. [25] C.W. Wu, Phys. Lett. A 319 (2003) 495. [26] C.W. Wu, Synchronization in systems coupled via complex networks, in: Proceedings of the IEEE International Symposium on Circuits and Systems, vol. 4, 2004, pp. IV-724–IV-727. [27] F.M. Atay, T. Biyiko˘glu, J. Jost, On the synchronization of networks with prescribed degree distributions, Technical report, Max Planck Institute for Mathematics in the Sciences, nlin/0407024. [28] B. Bollobás, Modern Graph Theory, Graduate Texts in Mathematics, vol. 184, Springer, 1998. [29] P. Erdös, T. Gallai, Mat. Lapok 11 (1960) 264. [30] G. Sierksma, H. Hoogeveen, J. Graph Theory 15 (2) (1991) 223. [31] J. Friedman, J. Kahn, E. Szemerédi, On the second eigenvalue in random regular graphs, in: Proceedings of the 21st Annual ACM Symposium on Theory of Computing, 1989, pp. 587–598. [32] V. Havel, Casopis Pest. Mat. 80 (1955) 477. [33] S.L. Hakimi, SIAM J. Appl. Math. 10 (1962) 496. [34] F. Juhász, Discrete Math. 96 (1991) 59.
Synchronizability of networks of chaotic systems coupled via a graph with a prescribed degree sequence Chai Wah Wu IBM Research Division, Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598, USA Received 17 May 2005; received in revised form 26 July 2005; accepted 27 July 2005 Available online 18 August 2005 Communicated by C.R. Doering
Abstract Generally, synchronization in a network of chaotic systems depends on the underlying coupling topology. Recently, there have been several studies conducted to determine what features of this topology contribute to the ability to synchronize. A short diameter has been proposed by several authors as a means to facilitate synchronization whereas others point to features such as the homogeneity of the degree sequence. Recently, it has been shown that the degree sequence by itself is not sufficient to determine synchronizability. The purpose of this Letter is to continue this study. For a given degree sequence, we construct two connected graphs with this degree sequence whose synchronizability can be quite different. In particular, we construct a graph with low synchronizability which improves upon previous bounds under certain conditions and we construct a graph which has synchronizability that is asymptotically maximal. On the other hand, we show analytically that for a random network model, homogeneity of the degree sequence is beneficial to synchronization. 2005 Elsevier B.V. All rights reserved. Keywords: Eigenvalue analysis; Nonlinear dynamics; Random graphs; Scale free networks; Synchronization
1. Introduction The object of interest in this Letter is a coupled network of n identical systems with linear coupling:
f (x1 , t) x˙1 . .. − (G ⊗ P )x, x˙ = .. = . x˙n
f (xn , t)
E-mail address: [email protected] (C.W. Wu). 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.07.089
(1)
282
C.W. Wu / Physics Letters A 346 (2005) 281–287
where xi is the state vector of the ith system, x = (x1 , . . . , xn )T and G ⊗ P is the Kronecker product of the matrices G and P . The function f (xi , t) is the nonlinear vector field of the individual systems. The coupling matrix G describes the coupling topology of the network, i.e., Gij = 0 if there is a coupling from system i to system j . If Gij < 0 or Gij > 0, we call such a coupling element cooperative or competitive, respectively. The network in Eq. (1) is said to synchronize globally if xi − xj → 0 as t → ∞ for all i, j . It was shown in [1,2] that for symmetric matrices G and a suitable matrix P , the ability for the network to synchronize depends on the size of the smallest nonzero eigenvalue of G. We will denote this eigenvalue as λ2 (G). An example of such suitable matrices P include the class of positive definite matrices. For other classes of matrices P [3,4] and the discrete time case [5,6], 2 (G) the ratio r(G) = λλmax (G) is important in determining synchronizability, where λmax (G) is the largest eigenvalue of G. In general, the coupled network is a nonlinear dynamical system. For the case when Eq. (1) is a linear system, i.e., f is linear, similar results have been obtained in the context of agreement and consensus problems [7–10]. All these results reduce the study of synchronization properties to properties of eigenvalues of the coupling matrix G. The graph of a matrix G is defined as the weighted graph with an edge of weight Gij from vertex i to vertex j if and only if Gij = 0. We assume that G is symmetric, so the graph of G is undirected.1 The Laplacian matrix of a weighted graph is defined as L = D − A, where D is the diagonal matrix of vertex degrees and A is the adjacency matrix. In this Letter we assume that G is a symmetric zero row sum matrix with nonpositive off-diagonal elements. In this case, G is equal to the Laplacian matrix of the graph of G and λ2 (G) is the algebraic connectivity of the graph [17]. Therefore, we will concentrate in this Letter on the eigenvalues of the Laplacian matrix. There are many results relating various characteristics of the graph of G and the eigenvalues of its Laplacian matrix. The reader is referred to [18–20] and the references therein. In this Letter we will assume that the graphs are unweighted, i.e., the adjacency matrix is a 0–1 matrix. Recently, various models of random graphs have been proposed that mimic more closely man-made and natural networks [21]. These random graphs differ from the classical random graph model in that the degree distribution does not have to be Poisson or binomial [22], but can have other types of distribution such as a power law distribution. Several of these random graph models have two important defining characteristics. First, the graph is generated via a random procedure. Second, the graph has a particular degree distribution. Focusing on the degree distribution alone, while ignoring the random aspect or vice versa can lead to incomplete conclusions. In [23] it was shown that the heterogeneity of the degree distribution can affect λ2 and r and hence the synchronizability. On the other hand, several papers show that the degree sequence by itself is not enough to characterize λ2 and r. In [24–26] it was shown that randomness makes λ2 and r large whereas local coupling makes λ2 and r small. In [27] a graph is constructed for a prescribed degree sequence whose normalized Laplacian matrix has a small nonzero eigenvalue. In this Letter we continue this study by constructing two connected graphs with the same degree sequence whose values for λ2 and r differ significantly. In particular, in some cases the graph with small λ2 has a value for λ2 that is in a sense minimal and improves upon the bound for λ2 in [27] whereas the graph with large λ2 has a value for λ2 which is in a sense maximal. Furthermore, we show analytically that for a model of random graphs, homogeneity of the degree sequence is beneficial for synchronization.
2. Graphs with a prescribed degree sequence A simple graph is an unweighted graph with no loops nor multiple edges [28]. The degree sequence of a graph is a list of the degrees of the vertices. A list of natural numbers is called graphical if there is a simple graph with this list as its degree sequence. Sufficient and necessary conditions for a list to be graphical are given in [29,30]. The following result gives upper and lower bounds for λ2 and r of the Laplacian matrix of a graph with a given degree sequence. 1 See [11–16] for synchronization results when G is not symmetric and the graph of G is directed.
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Lemma 1. For a connected graph with degree sequence d1 d2 · · · dn , λ2 and r satisfy: 1 − cos πn δ δ π n δ λ2 δ, r , 2 1 − cos n n−1 ∆ ∆ where δ = d1 is the minimum vertex degree and ∆ = dn is the maximum vertex degree. Proof. The first set of inequalities follows from [17]. The second set of inequalities follows from the fact that n n−1 ∆ λmax 2∆, also from [17]. 2 In fact, if the graph is not complete, then λ2 δ [17]. The lower bounds in Lemma 1 are attained or approached asymptotically as n → ∞ for the nearest neighbor path graph with Laplacian matrix 1 −1 −1 2 −1 , L= .. .. .. . . . −1
1
and λ2 = 2(1 − cos(π/n)), r = 1−cos(π/n) 1+cos(π/n) . The upper bounds are attained for the complete graph with Laplacian matrix n − 1 −1 ··· −1 −1 n − 1 −1 , L= .. .. .. . . . −1 and λ2 = n, r = 1.
···
−1 n − 1
2.1. Regular graphs A k-regular graph is a graph where every vertex has degree k. In this section, we consider two types of connected regular graphs whose values of λ2 and r can be quite different. 2.1.1. Construction 1: graph with low λ2 and r For a 2k-regular graph, the degree sequence is (2k, 2k, . . . , 2k). We consider two cases. In the first case, the 1 vertex degree grows as Ω(n 3 − ) for > 0. If we arrange the vertices in a circle, and connect each vertex to its 2k nearest neighbors, then the resulting 2k-regular graph is denoted C2k . The Laplacian matrix L is a circulant matrix: 2k −1 · · · −1 0 · · · 0 −1 · · · −1 −1 2k −1 · · · −1 . .. .. .. .. . . . . −1 · · · −1 0 · · · 0 −1 · · · −1 2k The eigenvalues of L are given by (µ0 , . . . , µn−1 ) with: µ0 = 0,
sin k + 12 2π m 2πml n = 2k + 1 − µm = 2k − 2 cos , n sin πm n l=1 k
A series expansion shows that as n → ∞, 3 4π 2 k + 12 . λ2 µ 1 ≈ 3n2
m = 1, . . . , n − 1.
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n Since λmax 2k n−1 by Lemma 1, 3 2π 2 k + 12 µ1 ≈ . r λmax 3kn2
This means the values λ2 and r decrease as Ω(1/n1+3 ) and Ω(1/n 3 +2 ), respectively. In particular, if k is bounded, i.e., = 1/3, then λ2 , r decrease as Ω(1/n2 ) which is asymptotically the fastest possible by Lemma 1. In the second case, we consider graphs for which k < n2 and use the same approach as in [27]. The main difference is that in [27] the smallest nonzero eigenvalue of the normalized Laplacian matrix L˜ = I − D −1 A is studied.2 For a graph with adjacency matrix A, let e(B, C) = u∈B,v∈C Au,v be the number of edges between B and C. The following lemma is shown in [18]: 4
Lemma 2. λ2 (L)
¯ e(S, S)n λmax (L). |S|(n − |S|)
¯ = 2. By Lemma 2 this means that As in [27], we create a connected k-regular graph with |S| = n2 and e(S, S) n8 , n even 2n 8 λ2 (L) n n = , 8 , n odd n − n1 2 2 n− n1 8(n−1) , n even 8 kn2 . r(L) 8 k(n + 1) , n odd k(n+1)
The reason for studying the two cases is that C2k gives better bounds when k grows slower than n1/3 and the construction in [27] gives better bounds when k grows faster than n1/3 . In either case, λ2 , r → 0 as n → ∞. 2.1.2. Construction 2: graph with high λ2 and r √ √ With high probability, a random 2k-regular graph has eigenvalues λ2 = 2k − O( k ), λmax = 2k + O( k ) as n → ∞ [31]. This means that on a relative scale, λ2 ≈ 2k, and r ≈ 1 as k, n → ∞ which can be considered optimal according to Lemma 1. In conclusion, whereas λ2 and r → 0 as n → ∞ for Construction 1, they remain bounded from below for Construction 2. Lemma 3. If H is a subgraph of G with the same set of vertices,3 then (1)
λ2 (H) + λ2 (G\H) λ2 (G),
(2)
λmax (H) + λmax (G\H) λmax (G), λ2 (H) + λ2 (G\H) min r(H), r(G\H) r(G), λmax (H) + λmax (G\H) λmax (H) λmax (G).
(3) (4)
Here G\H is a graph with the same set of vertices as G and with edges which are in G but not in H. By a slight abuse of notation, we use λi (H) to denote λi of the Laplacian matrix of the graph H. 2 For k-regular graphs, this difference between L and L ˜ is not important since L = k L˜ and the eigenvalues of L and L˜ differ by a constant
factor k. However, for non-regular graphs, the eigenvalues of L˜ and L will have different properties. 3 By this we mean that A B for all i, j , where A and B are the adjacency matrices of H and G, respectively. ij ij
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Proof. Follows from the facts that λ2 (L) =
min
i xi =0,
x=1
x T Lx,
and L(H) + L(G\H) = L(G).
λmax (L) = max x T Lx x=1
2
2.2. Graphs with prescribed degree sequence For each n, consider a graphical list of degrees 0 < d1 d2 · · · dn . The average vertex degree is defined as k¯ = ( i di )/n. Next, we construct two connected graphs with the same list of degrees but different λ2 and r. 2.2.1. Construction 1: graph with low λ2 and r 1 As in Section 2.1.1 we consider two cases. In the first case, dn grows as Ω(n 3 − ) for > 0. Given a graphical list of degrees 0 < d1 d2 · · · dn , the Havel–Hakimi algorithm [32,33] constructs a connected graph with this degree sequence where the vertices are arranged on a line and each vertex is connected to its nearest neighbors. Let us denote this graph as Ghh with Laplacian matrix Lhh . The graph Ghh is a subgraph of the 2dn -regular graph C2dn defined in Section 2.1.1. Therefore, by Lemma 3, 3 4π 2 dn + 12 1 as n → ∞. = Ω 1+3 λ2 (Lhh ) µ1 ≈ 3n2 n Since λmax (Lhh )
ndn n−1 ,
this means that
3 4π 2 dn + 12 µ1 1 . r(Lhh ) ≈ =Ω 4 λmax 3dn n2 n 3 +2 As in Section 2.1.1, for bounded dn the values of λ2 and r for this construction decrease as Ω(1/n2 ) when n → ∞. By Lemma 1, this is the fastest possible rate. In the second case, assume that the degree sequence is of the form (dmax , . . . , dmax , dmax − 1, . . . , dmax − 1, . . . , 1, . . . , 1) with dmax n/4 and the average degree satisfies k¯ = 2 + dmax n . We further assume that for each 1 i dmax , there are at least 2 vertices with vertex degree i. As in [27], we construct a graph with this degree ¯ = 2. By Lemma 2 this means that sequence such that |S| n2 − dmax and E(S, S) n−4d82 /n , n even 10 23 2n 8 max , λ2 (L) n = n 8 2 /n n , n odd n − 4dmax 2 + dmax 2 − dmax n−4 dmax − 12 2 /n 8(n−1) n even 3 , 10 23 (n − 1)λ2 (L) n2 dmax −4dmax r(L) , 8(n−1) dmax n n odd ndmax 1 2 , ndmax n−4 dmax − 2 /n
where we have used the fact that dmax n/4. Again, in either case r, λ2 → 0 as n → ∞. 2.2.2. Construction 2: graph with high λ2 and r For convenience, we allow loops in the graph, i.e., the graphs are not necessarily simple. Furthermore, rather than constructing graphs with a specific degree sequence, we consider a random graph model which has a prescribed degree sequence in expectation. Given a list of degrees d1 d2 · · · dn satisfying the condition n(dn − d1 )2 (n − d1 ) (dk − d1 ), k
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we construct a random graph as in [25] where an edge between vertex i and vertex j is randomly selected with probability Pij =
d1 (di − d1 )(dj − d1 ) + . n k (dk − d1 )
Since i Pij = dj , the graph has the desired degree sequence in expectation. Let us call this graph model Gpr with Laplacian matrix Lpr . For the special case d1 = d2 = · · · = dn , we choose Pij = d1 /n and denote this graph model as Gr (d1 ). Since Pij d1 /n, this means that Gr (d1 ) is a “subgraph” of Gpr . What we mean by this is that the probability space of Gpr can be partitioned such that for each graph H in Gr (d1 ) with probability p, there exists exactly one event A in Gpr with probability p, such that H is a subgraph of each graph in event A. In particular, consider the edges in Gpr to be of two types. An edge (i, j ) of type 1 occurs with probability d1 /n and an edge (i, j ) of type 2 occurs with probability Pij − dn1 . To each graph H in Gr (d1 ) corresponds a set of graphs A(H) constructed as follows. If (i, j ) is an edge in H, then (i, j ) is an edge of type 1 in graphs in A(H). If (i, j ) is not an edge in H, then either (i, j ) is not an edge or (i, j ) is an edge of type 2 in graphs in A(H). It is clear that these sets A(H) over all graphs H in Gr (d1 ) exactly partition the probability space of Gpr . Furthermore, the probability of A(H) in Gpr is equal to the probability of H in Gr (d1 ) and H is a subgraph of every graph in A(H). By the Courant–Fischer minmax theorem, the eigenvalues of the Laplacian matrix cannot decrease as more edges are added to the graph. This means that if λ2 (Gr (d1 )) ξ with high probability, then λ2 (Gpr ) ξ with high probability. Let us assume now that d1 = p1 n for some√0 < p1 < 1. In [34] it was shown that for the graph Gr (d1 ), |λ2 − d1 | and |λmax − d1 | are both on the order of o( n ) with high probability as n → ∞. This means that λ2 (Gpr ) √ d1 n d1 , this implies that λ2 ≈ d1 and r 2d for Gpr , i.e., λ2 and r d1 − o( n ) with high probability. Since λ2 n−1 n are bounded away from 0 as n → ∞. The conclusion that λ2 ≈ d1 shows that for the random graph model Gpr , homogeneity of the degree sequence ¯ the highest λ2 can be as enhances synchronizability in terms of λ2 . In particular, for a given average degree k, n n ¯ ¯ i.e., ¯ n → ∞ is approximately k since λ2 n−1 d1 n−1 k. This is achieved for the random graph where d1 = k, the d1 -regular graph which has the most homogeneous degree sequence. This supports the experimental results in [23].
3. Conclusions We present graph constructions that for the same degree sequence generate graphs with low and high synchronizability in networks of coupled systems. In particular, we construct a graph with low synchronizability whose values of λ2 and r decrease to 0 and a graph with high synchronizability whose values of λ2 and r are bounded away from 0 as n → ∞. This further indicates that the degree sequence alone is not sufficient to determine the synchronizability of a network of coupled dynamical systems. Furthermore, the constructions further support the assertion in [25,26] that local coupling tends to have low synchronizability, whereas random coupling tends to have high synchronizability. On the other hand, for the random graph model Gpr in Section 2.2.2, homogeneity in the vertex degree is beneficial for synchronization.
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