Conditions for the formation of clusters depending on the conductance and the coef cient of clustering Acilina Caneco (1) , Clara Grácio (2) , Sara Fernandes (3) , J.L. Rocha (4) and Carlos Ramos (5) 1 CIMA-UE, Mathematics Unit, ISEL , Portugal,
[email protected] 2 CIMA-UE, Mathematics Department, University of Évora, Portugal,
[email protected] 3 CIMA-UE, Mathematics Department, University of Évora, Portugal,
[email protected] 4 Mathematics Unit, ISEL, Portugal,
[email protected] 5 CIMA-UE, Mathematics Department, University of Évora, Portugal,
[email protected]
Abstract— One of the ultimate goals of researches on complex networks is to understand how the structure of complex networks affects the dynamical process taking place on them, such as traf c ow, epidemic spread, cascading behavior, and so on. In previous works [1] and [2], we have studied the synchronizability of a network in terms of the local dynamics, supposing that the topology of the graph is xed. Now, we are interested in studying the effects of the structure of the network, i.e., the topology of the graph on the network synchronizability. The synchronization interval is given by a formula relating the rst non zero and the largest eigenvalue of the Laplacian matrix of the graph with the maximum Lyapunov exponent of the local nodes. Our goal is to understand under what conditions can ensure the formation of clusters depending on the conductance and the coef cient of clustering
I. P RELIMINARIES 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. A graph is a set G = (V (G); E(G)) where V (G) is a nonempty set of N vertices or nodes (N is called the order of the graph) and E(G) is the set of m edges or links eij that connect two vertices vi and vj (m is called the size of the graph) [5]. If the graph is weighted, for each pair of vertices (vi ; vj ) we set a non negative weight aij such that aij = 0 if the vertices vi and vj are not connected. If the graph is not weighted, aij = 1 if vi and vj are connected and aij = 0 if the vertices vi and vj are not connected. If the graph is not directed, which is the case that we will study, then aij = aji . The matrix A = A(G) = [aij ], where vi , vj 2 V (G); is called the adjacency matrix, That matrix carries an entry 1 at the intersection of the ith row and the jth column if there is a edge from i to j. When there is no edge, the entry will be 0. The degree of a node vi is the number of edges incident on i=N P it and is represented by ki , that is, ki = aij . The degree i=1
distribution is the probability P (k) that a randomly selected node has exactly k edges.
Consider the diagonal matrix D = D(G) = [dij ], 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 fN; 2 g], where is the maximum degree of the vertices. The spectrum of L may be ordered, 1 = 0 2 N . The second eigenvalue 2 is know as the algebraic connectivity or Fiedler value and plays a special role in the graph theory. As much larger 2 is, more dif cult is to separate the graph in disconnected parts. The graph is connected if and only if 2 6= 0 . In fact, the multiplicity of the null eigenvalue 1 is equal to the number of connected components of the graph. As we will see later, as bigger is 2 ; more easily the network synchronizes. II. S YNCHRONIZATION Consider a network of N identical chaotic dynamical oscillators, described by a connected, undirected graph, with no loops and no multiple edges. In each node the dynamics of the oscillators is de ned by x_ i = f (xi ), with f : Rn ! Rn and xi 2 Rn the state variables of the node i. The state equations of this network are x_ i = f (xi ) + b
N X
aij (xj
xi );
with i = 1; 2; :::; N (1)
j=1 j6=i
where b > 0 is the coupling parameter, A = [aij ] is the adjacency matrix and = diag(1; 1; :::1). Equation (1) can be rewritten as N X x_ i = f (xi ) + b lij xj ; with i = 1; 2; :::; N: (2) j=1
where L = (lij ) is the Laplacian matrix or coupling con guration of the network. The network (2) achieves asymptotical synchronization if x1 (t) = x2 (t) = ::: = xN (t) ! e(t); t!1
(3)
where e(t) is a solution of an isolated node (equilibrium point, periodic orbit or chaotic attractor), satisfying e(t) _ = f (e(t)).
Let be hmax the Lyapunov exponent of each individual ndimensional node. If, [7] 1
e hmax 1 + e hmax
