Networks synchronizability, local dynamics and some

Chapter 1

Networks synchronizability, local dynamics and some graph invariants Acilina Caneco, Sara Fernandes, Clara Grácio and J. Leonel Rocha

Abstract The synchronization of a network depends on a number of factors, including the strength of the coupling, the connection topology and the dynamical behaviour of the individual units. In the first part of this work, we fix the network topology and obtain the synchronization interval in terms of the Lyapounov exponents for piecewise linear expanding maps in the nodes. If these piecewise linear maps have the same slope ±s everywhere, we get a relation between synchronizability and the topological entropy. In the second part of this paper we fix the dynamics in the individual nodes and address our work to the study of the effect of clustering and conductance in the amplitude of the synchronization interval.

1.1 Introduction A network with a complex topology is mathematically described by a graph [3]. Classical random graphs were studied by Paul Erd˝os and Alfréd Rényi in the late 1950’s. Examples of such networks include communication and transportation networks, neural and social interaction networks ([1], [13],[20]). Although features of these networks have been studied in the past, it was only recently that massive amount of data are available and computer processing is possible to more easily analyze the behaviour of these networks and verify the applicability of the proposed Acilina Caneco Mathematics Unit, Instituto Superior de Engenharia de Lisboa and CIMA-UE, e-mail: [email protected] Sara Fernandes Department of Mathematics, Universidade de Évora and CIMA-UE, e-mail: [email protected] Clara Grácio Department of Mathematics, Universidade de Évora and CIMA-UE, e-mail: [email protected] J. Leonel Rocha Mathematics Unit, Instituto Superior de Engenharia de Lisboa, e-mail: [email protected]

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Acilina Caneco, Sara Fernandes, Clara Grácio and J. Leonel Rocha

models. In 1998 Watts and Strogatz [20] proposed the new small-world model to describe many of real networks around us and in 1999 Barabási and Albert [1] proposed the new scale-free model based on preferential attachment. These models reflect the natural and man-made networks more accurately than the classical random graph model. This preferential attachment characteristic leads to the formation of clusters, the nodes with more links have a greater probability of getting new ones. One of the most important subjects under investigation is the network synchronizability ([5], [10], [13], [20]). Pecora and Carroll [16] derived the master stability method. Li and Chen [13] derived synchronization and desynchronization values for the coupling parameter in terms of the network topology and the maximum Lyapunov exponent of the individual chaotic nodes. In this work we address the study of network synchronizability in two approaches. One is fixing the connection topology and vary the local dynamics in the nodes and the other is consider the local dynamic fixed and vary the structure of the connections. To the first approach, we study, in Sec. 1.2, the synchronization interval considering fixed the network connection topology, for different kinds of local dynamics. Supposing in the nodes, identical piecewise linear expanding maps, with different slopes in each subinterval, we obtained, in Sec. 1.2.1, the synchronization interval in terms of the Lyapunov exponents of these maps. As a particular case, we derive, in Sec. 1.2.2, the synchronization interval in terms of the topological entropy for piecewise linear maps with slope ±s everywhere and we proved that the synchronizability decreases if the local topological entropy increases. Considering identical chaotic symmetric bimodal maps in the nodes of the network, we express, in Sec. 1.2.3, the synchronization interval in terms of one single critical point of the map. In the second part of this work, we study the network synchronization as a function of the connection topology, fixing the local dynamics. We try to understand the relation of some graph invariants with the spectrum of the Laplacian matrix, ([2], [3]). We can find a great number of formulas relating some graph invariants with the eigenvalues characterizing the synchronization interval, λ2 and λN , but none, as far as we know, for a relation between these eigenvalues and the clustering formation, neither for a relation between the conductance and the clustering. So, in Sec. 1.3, we perform experimental evaluations, that deepens the understanding of the effect of these quantities on the network synchronizability.

1.2 Network synchronizability and local dynamics Mathematically, networks are described by graphs and the theory of dynamical networks is a combination of graph theory and nonlinear dynamics. From the point of view of dynamical systems, we have a global dynamical system emerging from the interactions between the local dynamics of the individual elements and graph theory then analyzes the coupling structure.

1 Synchronizability, local dynamics and some graph invariants

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A graph is a set G = (V (G), E(G)) where V (G) is a nonempty set of N vertices or nodes and E(G) is the set of m edges or links ei j that connect two vertices vi and v j , [3]. If the graph is weighted, for each pair of vertices (vi , v j ) we set a non negative weight ai j such that ai j = 0 if the vertices vi and v j are not connected. If the graph is not weighted, ai j = 1 if vi and v j are connected and ai j = 0 if the vertices vi and v j are not connected. If the graph is not directed, which is the case that we will study, ai j = a ji . The matrix A = A(G) = [ai j ], where vi , v j ∈ V (G), is called the adjacency matrix. The degree of a node vi is the number of edges incident on it and i=N

is represented by ki , that is, ki =

∑ ai j . The degree distribution is the probability

i=1

P(k) that a randomly selected node has exactly k edges. Consider the diagonal matrix D = D(G) = [di j ], where dii = ki . We call Laplacian matrix to L = D − A. The matrix L acts in `2 (V ) and sometimes is called Kirchhoff matrix of the graph, due to its role in the Kirchhoff Matrix-Tree Theorem. The eigenvalues of L are all real and non negatives and are contained in the interval [0, min {N, 2∆ }], where ∆ is the maximum degree of the vertices. The spectrum of L may be ordered, λ1 = 0 ≤ λ2 ≤ · · · ≤ λN . The second eigenvalue λ2 is know as the algebraic connectivity or Fiedler value and plays a special role in the graph theory. As much larger λ2 is, more difficult is to separate the graph in disconnected parts. The graph is connected if and only if λ2 6= 0 . In fact, the multiplicity of the null eigenvalue λ1 is equal to the number of connected components of the graph. As we will see later, as bigger is λ2 , more easily the network synchronizes. Consider a network of N identical chaotic dynamical oscillators, described by a connected, unoriented graph, with no loops and no multiple edges. In each node the dynamics of the oscillators is defined by x˙i = f (xi ), with f : Rn → Rn and xi ∈ Rn the state variables of the node i. The state equations of this network are N

x˙i = f (xi ) + c ∑ ai j Γ (x j − xi ),

with i = 1, 2, ..., N

(1.1)

j=1 j6=i

where c > 0 is the coupling parameter, A = [ai j ] is the adjacency matrix and Γ = diag(1, 1, ...1). Equation (1.1) can be rewritten as N

x˙i = f (xi ) + c ∑ li j x j ,

with i = 1, 2, ..., N.

(1.2)

j=1

where L = (li j ) = D − A is the Laplacian matrix or coupling configuration of the network. The network (1.2) achieves asymptotical synchronization if x1 (t) = x2 (t) = ... = xN (t) → e(t), where e(t) is a solution of an isolate node (equilibrium point, t→∞

periodic orbit or chaotic attractor), satisfying e(t) ˙ = f (e(t)).

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1.2.1 Synchronization interval for piecewise linear maps with different slopes Consider the network (1.2) with identical chaotic nodes. In this work we will consider the network in the discretized form N

xi (k + 1) = f (xi (k)) + c ∑ li j f (x j (k)),

with i = 1, 2, ..., N.

(1.3)

j=1

Let 0 = λ1 < λ2 ≤ ... ≤ λN be the eigenvalues of the coupling matrix L and let hmax be the Lyapunov exponent of each individual n-dimensional node. If c > h|λmax| , 2 then the synchronized states are exponentially stable [13]. We may fix f , the local dynamic in each node and vary the connection topology, L, or fix L and vary f . In a previous work [4] we have considered, in each node of the network, piecewise linear maps with slope ±s, motivated by the fact that every m-modal map f : I = [a, b] ⊂ R → I, with growth rate s and positive topological entropy htop ( f ) (log s = htop ( f )) is, by theorem 7.4 from Milnor and Thurston [15] and Parry, topologically semi-conjugated to a p + 1 piecewise linear map T , with p ≤ m, defined on the interval J = [0, 1], with slope ±s everywhere and htop (T ) = htop ( f ) = log s. As a generalization, we will consider now a network having in each node a piecewise linear map with different slopes in each subinterval [19]. Let I ⊂ R be a compact interval and f : I → I, f = ( f1 , ..., fn ), a piecewise linear expanding map. The set of n laps of f defines a partition PI = {I1 , ..., In } of the interval I. Let ai , with i = 1, ..., n + 1, be the discontinuity points and the turning points of the map f . Considering the orbits of these points, we define a Markov partition PI0 of I. The orbit of each point ai is defined by n o (i) (i) o (ai ) = xk : xk = f k (ai ) , k ∈ N0 . To simplify the presentation, we consider the points a1 and an+1 as fixed points of the map f . Let {b1 , ..., bm+1 } = {o (ai ) : i = 1, ..., n + 1} be the set of the points correspondent to the orbits of the discontinuity points and the turning points, ordered on the interval I. This set allows us to define a subpartition PI0 of PI . The subpartition PI0 = {J1 , ..., Jm } with m ≥ n determines a Markov partition of the interval I. Note that f determines PI0 uniquely, but the converse is not true. The piecewise linear expanding map f induces a subshift of finite type whose m × m transition matrix A = [ai j ] is defined by ½ 1 if f (int J j ) ⊇ int Ji ai j = 0 otherwise. We denote this subshift by (ΣA , σ ), where σ is the shift map on ΣmN defined by σ (x1 x2 ...) = x2 x3 ..., where Σm = {1, ..., m} correspondent to the m states of the subshift. The topological entropy of (ΣA , σ ) is log λA , where λA is the spectral radius of the transition matrix A. In [18], using the signal of f 0 , is defined a weighted


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matrix which describes the transitions between the points b1 , ..., bm+1 . The relation between the transition matrix and the weighted matrix is established in [18]. This result allows us to compute the topological entropy of a subshift of finite type by a different method. See [18] and [19], for the relation between the kneading data associated to f and the topological entropy. To the subshift (ΣA , σ ) and the Markov partition PI0 , we associated a Lipschitz function φ : I → R, [19], defined by

φ = {φi : Ji → R, 1 ≤ i ≤ m} where

¯ ¯ φi (x) = −β ϕi (x) and ϕi (x) = log ¯ fi0 (x)¯ , with β ∈ R.

This function is a weight for the dynamical system associated to (ΣA , σ ) depending on the parameter β . Let L 1 (I) be the set of all Lebesgue integrable functions on I. The transfer operator Lφ : L 1 (I) → L 1 (I), associated with f and PI0 , ¡

¢ Lφ j g (x) =

m

∑ exp φ j

j=1

³

´ ³ ´ f j−1 (x) g f j−1 (x) χ f (int I j )

where χI j is the characteristic function of I j . In this section we consider a class of one-dimensional transformations that are piecewise linear Markov transformations. Consequently, the transfer operator has the following matrix representation. Let C be the class of all functions that are piecewise constant on the partition PI0 . The transfer operator has the following matrix representation Lφ g = Qβ πg with g ∈ C , where C is the class of all functions that are piecewise constants, on the partition PI0 and πg = (π1 , ..., πm )T . If Dβ is the diagonal matrix defined by Dβ = (exp φ1 , ..., exp φm ) and A is the transition matrix, then the matrix Qβ is the m × m weighted transition matrix defined by ai j Qβ = A Dβ = [qi j ] where qi j = ¯ ¯β . ¯ 0¯ ¯ f j¯ By the Ruelle-Perron-Frobenius Theorem there exist λβ > 0 and vβ ∈ C , with vβ (Ji ) > 0 for all 1 ≤ i ≤ m, such that vβ is the eigenvector of Qβ with largest eigenvalue λβ , i.e., Qβ vβ = λβ vβ . This eigenvector of Qβ is used to construct a transition probability matrix, as follows. Let µ be a measure with support in PI0 , then we denote the adjoint operator of Lφ by Lφ∗ , which is defined by a bounded linear map on measures, i.e., ¡ ∗ ¢ ¡ ¢ Lφ µ (g) = µ Lφ g .

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Note that the adjoint operator Lφ∗ is represented by the matrix QTβ . The eigenvalues of the matrices Qβ and QβT are the same. For the m-dimensional vector space PI0 , we consider two bases © ª B = {e1 , ..., em } and B 0 = e01 , ..., e0m . The set of vectors in B are defined by the column vector e j = (0, ..., 0, 1, 0, ..., 0)T where 1 is in the jth -position. These vectors correspond to the intervals of the Markov partition. On the other hand, the set of vectors in B 0 are defined by e0j = (0, ..., 0, v j , 0, ..., 0)T , which correspond to the coordinates of the vector vβ . If Mβ is the matrix which describes the change from the basis B 0 to the basis B, then we define a new matrix, the m × m matrix Rβ = Mβ−1 Qβ Mβ = [ri j ] where ri j = qi j

vj with ri j ≥ 0. vi

The matrix Rβ is the matrix representation of Lφ , with respect to the basis B 0 . As the matrices Qβ and Rβ are similar, the largest eigenvalue λβ of these matrices is the same. Define a m × m stochastic matrix Sβ = [si j ] where si j =

ri j with si j ≥ 0 and λβ

m

∑ si j = 1.

j=1

The transpose matrix SβT corresponds to a modified or normalized transfer operator, with respect to the basis B 0 . Let uβ0 = (u01 , ..., u0m ) be the left eigenvector and vβ0 = (v01 , ..., v0m ) be the strictly positive right eigenvector of the matrix Rβ . The probability vector pβ = (p1 , ..., pm ) is defined by pi =

u0i v0i

m

, such that

∑ u0i v0i

m

m

i=1

i=1

∑ pi si j = p j and ∑ pi = 1.

i=1

This vector defines the unique f -invariant equilibrium state for φ = −β log | f 0 (x)|. Note that, if we consider µ ∗ = (u1 v1 , ..., um vm ), up to a multiplicative constant, then µ ∗ = pβ , see [19] and references therein. The stochastic matrix Sβ and the probability vector pβ allow us to define an invariant probability measure µβ on the repeller, depending on the parameter β . Let ΣA and Σm be as above. Define µβ on the semi-algebra of measurable intervals by

µβ

¡© ª¢ (xi )i∈N ∈ ΣA : xq = a1 , ..., xq+k−1 = ak , with ak ∈ Σm and k ∈ N

= pa1 sa1 a2 sa2 a3 ...sak−1 ak . We call this ¡ measure ¢ the weighted Markov measure, associated to the weighted one-sided pβ , Sβ -Markov shift, supported by the repeller. This invariant measure gives nonvanishing probabilities only for the trajectories staying in the repeller.


¡

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¢

Lemma 1. The weighted one-sided pβ , Sβ -Markov shift has Lyapunov exponent χµβ ( f ) with respect to the measure µβ , given by m ¡¯ ¯¢ χµβ ( f ) = ∑ pi log ¯ fi0 ¯

(1.4)

i=1

where the derivative fi0 is evaluated on the interval Ji of the partition PI0 . See [19] for the proof. Attending that, there exist a unique invariant probability measure µ1 , (β = 1) for the map f , generated by the absolutely continuous conditionally invariant measure µ (see Proposition 2 of [19]), we may express the network synchronizability interval in terms of the Lyapunov exponent χµ1 ( f ). Theorem 1. Consider the network (1.3), having a connection topology given by some coupling matrix L with eigenvalues 0 = λ1 < λ2 ≤ ... ≤ λN and in each node identical piecewise linear expanding maps f with Lyapunov exponent χµ1 ( f ) given by (1.4). Then, the network synchronizes if the coupling parameter c verifies 1 − e−χµ1 ( f ) 1 + e−χµ1 ( f )