How complex a dynamical network can be?

Positive Lyapunov exponents measure the asymptotic exponential divergence of nearby trajectories of a dynamical system. Not only they quantify how chaotic a dynamical system is, but since their sum is an upper bound for the rate of information production, they also provide a convenient way to quantify the complexity of a dynamical network. We conjecture based on numerical evidences that for a large class of dynamical networks composed by equal nodes, the sum of the positive Lyapunov exponents is bounded by the sum of all the positive Lyapunov exponents of both the synchronization manifold and its transversal directions, the last quantity being in principle easier to compute than the latter. As applications of our conjecture we: (i) show that a dynamical network composed of equal nodes and whose nodes are fully linearly connected produces more information than similar networks but whose nodes are connected with any other possible connecting topology; (ii) show how one can calculate upper bounds for the information production of realistic networks whose nodes have parameter mismatches, randomly chosen; (iii) discuss how to predict the behavior of a large dynamical network by knowing the information provided by a system composed of only two coupled nodes.     

非一致通信时滞动力学网络上的接连滞后同步

在动力学网络中,节点与节点之间的通信通常存在时滞,并且不同节点之间的通信时滞往往是不同的(即非一致通信时滞),研究非一致通信时滞动力学网络上的接连滞后同步,更具现实意义.为此,本文首先构建含有非一致通信时滞的动力学网络模型.其次分别设计线性反馈控制和自适应反馈控制,利用Lyapunov函数方法,重点分析了该网络的接连滞后同步的稳定性,得到了同步稳定的充分条件.最后,选取蔡氏电路作为局部动力学,又分别选取了链式网络和星型网络这两种拓扑结构来验证理论结果的正确性和有效性.     

Exponential synchronization of complex networks with nonidentical time-delayed dynamical nodes

In this Letter, exponential synchronization of a complex network with nonidentical time-delayed dynamical nodes is considered. Two effective control schemes are proposed to drive the network to synchronize globally exponentially onto any smooth goal dynamics. By applying open-loop control to all nodes and adding some intermittent controllers to partial nodes, some simple criteria for exponential synchronization of such network are established. Meanwhile, a pinning scheme deciding which nodes need to be pinned and a simply approximate formula for estimating the least number of pinned nodes are also provided. By introducing impulsive effects to the open-loop controlled network, another synchronization scheme is developed for the network with nonidentical time-delayed dynamical nodes, and an estimate of the upper bound of impulsive intervals ensuring global exponential stability of the synchronization process is also given. Numerical simulations are presented finally to demonstrate the effectiveness of the theoretical results.     

Characterizing the dynamical importance of network nodes and links

The largest eigenvalue of the adjacency matrix of networks is a key quantity determining several important dynamical processes on complex networks. Based on this fact, we present a quantitative, objective characterization of the dynamical importance of network nodes and links in terms of their effect on the largest eigenvalue. We show how our characterization of the dynamical importance of nodes can be affected by degree-degree correlations and network community structure. We discuss how our characterization can be used to optimize techniques for controlling certain network dynamical processes and apply our results to real networks.     

Spectral coarse graining and synchronization in oscillator networks

Coarse graining techniques offer a promising alternative to large-scale simulations of complex dynamical systems, as long as the coarse-grained system is truly representative of the initial one. Here, we investigate how the dynamical properties of oscillator networks are affected when some nodes are merged together to form a coarse-grained network. Moreover, we show that there exists a way of grouping nodes preserving as much as possible some crucial aspects of the network dynamics. This coarse graining approach provides a useful method to simplify complex oscillator networks, and more generally, networks whose dynamics involves a Laplacian matrix.     

Synchronization-based topology identification of weighted general complex dynamical networks with time-varying coupling delay

Many existing papers investigated the geometric features, control and synchronization of complex dynamical networks provided with certain topology. However, the exact topology of a network is sometimes unknown or uncertain. Based on LaSalle’s invariance principle, we propose an adaptive feedback technique to identify the exact topology of a weighted general complex dynamical network model with time-varying coupling delay. By receiving the network nodes evolution, the topology of such a kind of network with identical or different nodes, or even with varying topology can be monitored. In comparison with previous methods, time delay is taken into account in this simple, analytical and systematic synchronization-based technique. Particularly, the weight configuration matrix is not necessarily symmetric or irreducible, and the inner-coupling matrix need not be symmetric. Illustrative simulations are provided to verify the correctness and effectiveness of the proposed scheme.     

Estimating Topology of Discrete Dynamical Networks

In this paper, by applying Lasalle's invariance principle and some results about the trace of a matrix, we propose a method for estimating the topological structure of a discrete dynamical network based on the dynamical evolution of the network. The network concerned can be directed or undirected, weighted or unweighted, and the local dynamics of each node can be nonidentical. The connections among the nodes can be all unknown orpartially known. Finally, two examples, including a Hénon map and a central network, are illustrated to verify the theoretical results.     

Condensation and evolution of a space–time network

In this work, we try to propose in a novel way, using the Bose and Fermi quantum network approach, a framework studying condensation and evolution of a space–time network described by the Loop quantum gravity. Considering quantum network connectivity features in Loop quantum gravity, we introduce a link operator, and through extending the dynamical equation for the evolution of the quantum network posed by Ginestra Bianconi to an operator equation, we get the solution of the link operator. This solution is relevant to the Hamiltonian of the network, and then is related to the energy distribution of network nodes. Showing that tremendous energy distribution induces a huge curved space–time network may indicate space time condensation in high-energy nodes. For example, in the case of black holes, quantum energy distribution is related to the area, thus the eigenvalues of the link operator of the nodes can be related to the quantum number of the area, and the eigenvectors are just the spin network states. This reveals that the degree distribution of nodes for the space–time network is quantized, which can form space–time network condensation. The black hole is a sort of result of space–time network condensation, however there may be more extensive space–time network condensations, such as the universe singularity (big bang).     

Condensation and evolution of a space-time network

In this work, we try to propose in a novel way, using the Bose and Fermi quantum network approach, a framework studying condensation and evolution of a space-time network described by the Loop quantum gravity. Considering quantum network connectivity features in Loop quantum gravity, we introduce a link operator, and through extending the dynamical equation for the evolution of the quantum network posed by Ginestra Bianconi to an operator equation, we get the solution of the link operator. This solution is relevant to the Hamiltonian of the network, and then is related to the energy distribution of network nodes. Showing that tremendous energy distribution induces a huge curved space-time network may indicate space time condensation in high-energy nodes. For example, in the case of black holes, quantum energy distribution is related to the area, thus the eigenvalues of the link operator of the nodes can be related to the quantum number of the area, and the eigenvectors are just the spin network states. This reveals that the degree distribution of nodes for the space-time network is quantized, which can form space-time network condensation. The black hole is a sort of result of space-time network condensation, however there may be more extensive space-time network condensations, such as the universe singularity (big bang).      

Smoluchowski Equation for Networks: Merger Induced Intermittent Giant Node Formation and Degree Gap

The dynamical phase diagram of a network undergoing annihilation, creation, and coagulation of nodes is found to exhibit two regimes controlled by the combined effect of preferential attachment for initiator and target nodes during coagulation and for link assignment to new nodes. The first regime exhibits smooth dynamics and power law degree distributions. In the second regime, giant degree nodes and gaps in the degree distribution are formed intermittently. Data for the Japanese firm network in 1994 and 2014 suggests that this network is moving towards the intermittent switching region.     

Node importance for dynamical process on networks: a multiscale characterization

Defining the importance of nodes in a complex network has been a fundamental problem in analyzing the structural organization of a network, as well as the dynamical processes on it. Traditionally, the measures of node importance usually depend either on the local neighborhood or global properties of a network. Many real-world networks, however, demonstrate finely detailed structure at various organization levels, such as hierarchy and modularity. In this paper, we propose a multiscale node-importance measure that can characterize the importance of the nodes at varying topological scale. This is achieved by introducing a kernel function whose bandwidth dictates the ranges of interaction, and meanwhile, by taking into account the interactions from all the paths a node is involved. We demonstrate that the scale here is closely related to the physical parameters of the dynamical processes on networks, and that our node-importance measure can characterize more precisely the node influence under different physical parameters of the dynamical process. We use epidemic spreading on networks as an example to show that our multiscale node-importance measure is more effective than other measures.     

Pinning control of weighted general complex dynamical networks with time delay

Time delays commonly exist in the real world. In the present work we consider weighted general complex dynamical networks with time delay, which are undirected and connected. Control of such networks, by applying local feedback injections to a fraction of network nodes, is investigated for both continuous-time and discrete-time cases. Both delay-independent and delay-dependent asymptotical stability criteria for network stabilization are derived. It is also shown that the whole network can be stabilized by controlling only one node. The efficiency of the derived results was illustrated by numerical examples.     

Topology identification of weighted complex dynamical networks

Recently, various papers investigated the geometry features, synchronization and control of complex network provided with certain topology. While, the exact topology of a network is sometimes unknown or uncertain. Using Lyapunov theory, we propose an adaptive feedback controlling method to identify the exact topology of a rather general weighted complex dynamical network model. By receiving the network nodes evolution, the topology of such kind of network with identical or different nodes, or even with switching topology can be monitored. Experiments show that the methods presented in this paper are of high accuracy with good performance.     

Identification of unstable individuals in dynamic networks

To identify the unstable individuals of networks is of great importance for information mining and security management. Exploring a broad range of steady-state dynamical processes including biochemical dynamics, epidemic processes,birth–death processes and regulatory dynamics, we propose a new index from the microscopic perspective to measure the stability of network nodes based on the local correlation matrix. The proposed index describes the stability of each node based on the activity change of the node after its neighbor is disturbed. Simulation and comparison results show our index can identify the most unstable nodes in the network with various dynamical behaviors, which would actually create a richer way and a novel insight of exploring the problem of network controlling and optimization.     

Synchronization in Complex Networks with Multiple Connections

In this paper a class of networks with multiple connections are discussed. The multiple connections include two different types of links between nodes in complex networks. For this new model, we give a simple generating procedure. Furthermore, we investigate dynamical synchronization behavior in a delayed two-layer network, giving corresponding theoretical analysis and numerical examples.     

Transition from the self-organized to the driven dynamical clusters

We study the mechanism of formation of synchronized clusters in coupled maps on networks with various connection architectures. The nodes in a cluster are self-synchronized or driven-synchronized, based on the coupling strength and underlying network structures. A smaller coupling strength region shows driven clusters independent of the network rewiring strategies, whereas a larger coupling strength region shows the transition from the self-organized cluster to the driven cluster as network connections are rewired to the bi-partite type. Lyapunov function analysis is performed to understand the dynamical origin of cluster formation. The results provide insights into the relationship between the topological clusters which are based on the direct connections between the nodes, and the dynamical clusters which are based on the functional behavior of these nodes.     

Information transfer dynamics in fixed-pathways networks

Most complex technological networks are defined in such a way that their global properties are manifested at a dynamical level. An example of this is when internal dynamical processes are constrained to predefined pathways, without the possibility of alternate routes. For instance, large corporation software networks, where several flow processes take place, are typically routed along specific paths. In this work, we propose a model to describe the global characteristics of this kind of processes, where the dynamics depends on the state of the nodes, represented by two possibilities: responsive or blocked. We present numerical simulations that show rich global behavior with unexpected emerging properties. In particular, we show that two different regimes appear as a function of the total network load. Each regime is characterized by developing either a unimodal or a bimodal distribution for the density of responsive nodes, directly related to global efficiency. We provide a detailed explanation for the main characteristics of our results as well as an analysis of the implications for real technological systems.     

Disturbance rejection and H∞ pinning control of linear complex dynamical networks

This paper concerns the disturbance rejection problem of a linear complex dynamical network subject to external disturbances.A dynamical network is said to be robust to disturbance,if the H ∞ norm of its transfer function matrix from the disturbance to the performance variable is satisfactorily small.It is shown that the disturbance rejection problem of a dynamical network can be solved by analysing the H ∞ control problem of a set of independent systems whose dimensions are equal to that of a single node.A counter-intuitive result is that the disturbance rejection level of the whole network with a diffusive coupling will never be better than that of an isolated node.To improve this,local feedback injections are applied to a small fraction of the nodes in the network.Some criteria for possible performance improvement are derived in terms of linear matrix inequalities.It is further demonstrated via a simulation example that one can indeed improve the disturbance rejection level of the network by pinning the nodes with higher degrees than pinning those with lower degrees.     

Periodically intermittent controlling complex dynamical networks with time-varying delays to a desired orbit

This Letter investigates the problem of synchronization in complex dynamical networks with time-varying delays. A periodically intermittent control scheme is proposed to achieve global exponential synchronization for a general complex network with both time-varying delays dynamical nodes and time-varying delays coupling. It is shown that the sates of the general complex network with both time-varying delays dynamical nodes and time-varying delays coupling can globally exponentially synchronize with a desired orbit under the designed intermittent controllers. Moreover, a typical network consisting of the time-delayed Chua oscillator with nearest-neighbor unidirectional time-varying delays coupling is given as an example to verify the effectiveness of the proposed control methodology.     

A general fractional-order dynamical network: synchronization behavior and state tuning

A general fractional-order dynamical network model for synchronization behavior is proposed. Different from previous integer-order dynamical networks, the model is made up of coupled units described by fractional differential equations, where the connections between individual units are nondiffusive and nonlinear. We show that the synchronous behavior of such a network cannot only occur, but also be dramatically different from the behavior of its constituent units. In particular, we find that simple behavior can emerge as synchronized dynamics although the isolated units evolve chaotically. Conversely, individually simple units can display chaotic attractors when the network synchronizes. We also present an easily checked criterion for synchronization depending only on the eigenvalues distribution of a decomposition matrix and the fractional orders. The analytic results are complemented with numerical simulations for two networks whose nodes are governed by fractional-order Lorenz dynamics and fractional-order Ro?ssler dynamics, respectively.