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Synchronization of an evolving complex hyper-network

In this paper, the synchronization in a hyper-network of coupled dynamical systems is investigated for the first time. An evolving hyper-network model is proposed for better describing some complex systems. A concept of joint degree is introduced, and the evolving mechanism of hyper-network is given with respect to the joint degree. The hyper-degree distribution of the proposed evolving hyper-network is derived based on a rate equation method and obeys a power law distribution. Furthermore, the synchronization in a hyper-network of coupled dynamical systems is investigated for the first time. By calculating the joint degree matrix, several simple yet useful synchronization criteria are obtained and illustrated by several numerical examples.


Introduction
Complex networks [1,2,3,4,5] have been used to model the real complex systems with a large number of interacting individuals, such as Internet [6], world trade web [7], metabolic networks [8], coauthor networks [9], and so on. The individuals in complex systems are denoted by nodes in complex networks, and the interactions between individuals are denoted by edges between nodes. For example, in a collaboration network, the scientists can be regarded as nodes and their joint papers can be treated as edges. From this collaboration network, we can find the collaborative relationship between any two scientists. However, many papers have more than two coauthors. Therefore, the collaborative relationship among any three or more scientists can not be described in the above simple collaboration network. Then, what kind of networks can solve this problem is an interesting issue, and the complex hyper-networks corresponding to hyper-graphs emerge in this context.
In complex networks, an edge connects only two nodes, however, an hyper-edge of complex hypernetworks can contain more than two nodes, which can describe the multifaceted collaborative relationship appropriately. Though a hyper-edge can connect an arbitrary number of nodes, it is often useful to study hyper-networks where each hyper-edge connects the same number of nodes: a k-uniform hyper-complex is a hyper-network in which each hyper-edge connects exactly k nodes.
There has been a growing interest in the research of hyper-networks [10][11][12][13][14] recently. For example, in virtual enterprises, to respond to market changes rapidly and exploit unexpected business opportunities efficiently, the virtual breeding environment (VBE) actors collaborate, and share competencies, skills as well as resources [11]. In order to describe the relationship among VBE actors, a hyper-graph (a hyper-network) is proposed as a meaningful logical structure, in which a hyper-path (i.e., a hyper-edge) represents the structure underlying a minimal cluster of enterprises. By the hyper-graph model, the formation process of a virtual enterprise can be presented and the pre-identified business opportunity may be caught.
Recently, an evolving hyper-network model is introduced to describe real-life systems with the hypernetwork characteristics [14]. Two evolving mechanisms with respect to a hyper-degree, namely hyper-edge growth and preferential attachment, are proposed to construct the hyper-network mode. The hyper-degree is defined as the number of the hyper-edge attached to that node. In this paper, a different evolving kuniform hyper-network model is introduced, whose evolving mechanism involves the joint degree of k − 1 nodes, which is defined as the number of the hyper-edges attached to the k − 1 nodes.
On the other hand, synchronization as a typical collective dynamical behavior of complex networks has drawn considerable attentions recently [15][16][17][18][19][20][21][22][23][24][25][26][27]. However, less attention has been paid to synchronization of complex hyper-networks. Thus in this paper, synchronization of the 3-uniform hyper-network will be investigated for the first time. The rest of this paper is organized as follows. In Section 2, an evolving hypernetwork model is introduced and several topological characteristics are studied. In Section 3, synchronization of a 3-uniform hyper-network is investigated and several synchronization criteria are obtained. In Section 4, several numerical examples are provided to verify the effectiveness of derived results. Conclusions are drawn in Section 5.

An evolving hyper-network model
An evolving mechanism with respect to joint degree to construct the k-uniform hyper-network model is proposed in this section. In the scale-free network model put forward by Barabási and Albert [2], two simple evolving mechanisms, i.e., growth and preferential attachment, are proposed to construct the network model. Inspired by [2], the following generation algorithm are proposed to construct the k-uniform hyper-network model: (1) Growth: the hyper-network starting with m 0 (m 0 ≥ k) nodes, at every time step, we add one node and m hyper-edges to the existing hyper-network, where (2) Preferential attachment: the probability Π i1i2...i k−1 of a new hyper-edge contains the new node and selected k − 1 nodes in the existing hyper-network depends on the joint degree of the k − 1 nodes at time t. Let d i1i2...i k−1 denotes the joint degree of nodes i 1 , i 2 , · · · , i k−1 , which is defined as the number of hyper-edges containing nodes i 1 , i 2 , · · · , i k−1 , then Then, after t time steps, this constructs a k-uniform hyper-network with m 0 +t nodes and mt hyper-edges. Define the hyper-degree d i h of node i as the number of the hyper-edges containing node i and node-node distance as the number of the hyper-edges in the shortest paths between two nodes. According to the generation algorithm and the relation between hyper-degree and joint degree, when a new node enter into the hyper-network, the probability of node i with hyper-degree d i h acquires a hyper-edge is Let i be the new node added to the hyper-network at time t i , and p(d h , t i , t) be the probability that node i has hyper-degree d i h when it is being picked up at time t (t ≥ t i ). Suppose that d i h is a continuous variable, where a is a constant and denotes the rate of change of d i h . Since the new node i brings in m new hyperedges, which has hyper-degree m at time t i , so the change of hyper-degree at t i is a = m. Solving equation Assuming that the time variables t i have a uniform distribution with P (t i ) = 1/t, i.e., the new nodes are being added at equal time intervals, then one has which is the same as that of BA scale-free network when k = 2. Fig. 1 shows the hyper-degree distribution of the 3-uniform hyper-network models with m 0 = 4, m = 1, 3, 6 and t = 196, from which one can find that the 3-uniform hyper-network has power distribution property, i.e., scale-free property. Table I shows the average path length of the 3-uniform hyper-network models with m 0 = 4 and different pair of t and m, which exhibits small world property.

Synchronization in hyper-network of coupled dynamical systems
Complex network consisting of N identical linearly and diffusively coupled nodes can be described bẏ where x i (t) = (x i1 (t), x i2 (t), · · · , x in (t)) T ∈ R n is the state variable of node i, f : R n → R n represents the local dynamics of an isolated node, ε > 0 is coupling strength, Γ ∈ R n×n is inner coupling matrix. Matrix A = (a ij ) N ×N is outer coupling matrix, which denotes the network topology and is defined as, if there is a connection from node i to node j (i = j), then a ij = 0, otherwise, a ij = 0. According to the definition of the coupling matrix A, the network (2) covers a variety of models ranging from simple weightless undirected networks to more complicated weighted directed networks. Similar to the network (2), a k-uniform hyper-network of N coupled systems can be described bẏ where i l (l = 1, 2, · · · , k) are different each other and a i1i2···i k is defined as: if there exists a hyper-edge contains nodes i 1 , i 2 , · · · , i k , then a i1i2···i k = 1, otherwise, a i1i2···i k = 0. For k = 3, the hyper-network (3) can be simplified asẋ In the following, consider synchronization of the 3-uniform hyper-network (4), which can be rewritten aṡ Let d ij be the joint degree of nodes i and j (j = i), which is defined as the number of hyper-edges containing nodes i and j. Then one has and rewrite (5) aṡ Thus,ẋ Define the diagonal elements of D as follows Then, equation (6) givesẋ In the subsequent studies, the hyper-network considered is always assumed to be connected, i.e., any pair of nodes is reachable along hyper-edge. It is easy to verify that the joint degree matrix D = (d ij ) is irreducible and its eigenvalues are 0 = λ 1 > λ 2 ≥ · · · ≥ λ N . Our objective here is to synchronize the network (7) with a given orbit s(t), i.e., where s(t) is a solution of an isolated node, namely,ṡ(t) = f (s(t)). Here, s(t) can be an equilibrium point, a periodic orbit, or even a chaotic attractor. Let x i (t) = s(t) + η i (t) and linearize (7) about s(t). This leads tȯ where Df (s(t)) is is the Jacobian of f on s(t). Then refer to the proofs of lemmas 1 and 2 in [27], one has the following theorems.
Proof. The proof will be given in the Appendix.
Proof. The proof will be given in the Appendix. It is clear that the inequality (12) is equivalent to Therefore, synchronizability of the 3-uniform hyper-network (5) with respect to a given coupling matrix can be characterized by the second-largest eigenvalue λ 2 of the joint degree matrix. Remark 1. Similar to the above discussions, synchronization of general k-uniform hyper-network can be investigated as well. In fact, refer to the definition of joint degree, hyper-network (3) can be simplified aṡ
Secondly, consider the synchronizability of 3-uniform hyper-network with different pair of m and N . Fig.  4 shows the second-largest eigenvalues of the joint degree matrix with different pair of N and m, where λ 2 (N, m) is obtained by averaging the results of 10 runs. It is clear that λ 2 (N, 1) > λ 2 (N, 2) > · · · > λ 2 (N, 6), which means the hyper-network with larger m has stronger synchronizability. Fig. 5 shows the synchronization errors with N = 100 and different coupling strength ε = 0.3, ε = 0.5 and ε = 5, from which one can easily find that the hyper-network with m = 6 has stronger synchronizability.

Conclusions
An evolving mechanism with respect to joint degree is proposed to construct a k-uniform hyper-network model. Based on a rate equation method, the hyper-degree distribution of the hyper-network is derived,  which obeys a power law distribution. Moreover, a complex hyper-network coupled with dynamical systems is introduced. By defining the joint degree of two nodes, the hyper-network can be simplified and its synchronization is investigated for the first time. Further, synchronizability of 3-uniform hyper-network with different pair of m and N is considered by calculating the second-largest eigenvalue λ 2 of the joint degree matrix.
Refer to the proof of Lemma 1 in Ref. [27], λ 1 = 0 corresponds to the synchronization of the system states. Then if the N − 1 n-dimensional linear time-varying systems (10) are exponentially stable, the synchronized states (8) is exponentially stable.