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Many physical and biological systems can be studied using complex network theory, a new statistical physics understanding of graph theory. The recent application of complex network theory to the study of functional brain networks has generated great enthusiasm as it allows addressing hitherto non-standard issues in the field, such as efficiency of brain functioning or vulnerability to damage. However, in spite of its high degree of generality, the theory was originally designed to describe systems profoundly different from the brain. We discuss some important caveats in the wholesale application of existing tools and concepts to a field they were not originally designed to describe. At the same time, we argue that complex network theory has not yet been taken full advantage of, as many of its important aspects are yet to make their appearance in the neuroscience literature. Finally, we propose that, rather than simply borrowing from an existing theory, functional neural networks can inspire a fundamental reformulation of complex network theory, to account for its exquisitely complex functioning mode.

Emerging evidence suggests white matter network abnormalities in patients with schizophrenia (SZ) and bipolar disorder (BD), but the alterations in dynamics of the white matter network in patients with SZ and BD are largely unknown. The white matter network of patients with SZ (n = 45) and BD (n = 47) and that of healthy controls (HC, n = 105) were constructed. We used dynamics network control theory to quantify the dynamics metrics of the network, including controllability and synchronizability, to measure the ability to transfer between different states. Experiments show that the patients with SZ and BD showed decreasing modal controllability and synchronizability and increasing average controllability. The correlations between the average controllability and synchronizability of patients were broken, especially for those with SZ. The patients also showed alterations in brain regions with supercontroller roles and their distribution in the cognitive system. Finally, we were able to accurately discriminate and predict patients with SZ and BD. Our findings provide novel dynamic metrics evidence that patients with SZ and BD are characterized by a selective disruption of brain network controllability, potentially leading to reduced brain state transfer capacity, and offer new guidance for the clinical diagnosis of mental illness. We used dynamics network control theory to quantify the dynamics metrics of the network, including controllability and synchronizability of healthy controls, patients with schizophrenia and bipolar disorder. Our findings provide novel dynamic metrics evidence that patients with schizophrenia and bipolar disorder are characterized by a selective disruption of brain network controllability, potentially leading to reduced brain state transfer capacity, and offer new guidance for the clinical diagnosis of mental illness.

Chialvo is one of the two-dimensional map-based neural models. In this paper, a memristor is added to this model to consider the electromagnetic induction’s effects. The memristor is defined based on a hyperbolic tangent function. The dynamical variations are analyzed by obtaining the bifurcation diagrams and Lyapunov spectra. It is shown that the most effective parameters on the dynamics are the magnetic strength and the injected current. The memristive Chialvo can exhibit different neural behaviors. It is also proven that, like the primary Chialvo model, the memristive version has coexisting attractors; an oscillating state coexists with a fixed point. In addition, to understand how memristive neurons behave in a network, two memristive Chialvo models are coupled with electrochemical synapses. By connecting two neurons and calculating the synchronization error, we can determine the system’s synchronizability. It is indicated that the electrical coupling is essential for the occurrence of complete synchronization in the network of memristive Chialvo, and the sole chemical coupling does not lead to synchronization.

We investigate the synchronizability of multilayer star-ring networks. Two types of multilayer networks, including aggregated coupling and divergent coupling, are established based on the connections between the hub node and the leaf nodes in the subnetwork. The eigenvalue spectrum of the two types of multilayer networks is strictly derived, and the correlation between topological parameters and synchronizability is analyzed by the master stability function framework. Moreover, the variable coupling strength has been investigated, revealing that it is significantly related to the synchronizability of the aggregated coupling while having no influence on the divergent coupling. Furthermore, the validity of the synchronizability analysis is obtained by implementing adaptive control on the multilayer star-ring networks previously mentioned. Calculations and comparisons show that the differences caused by the sizes of multilayer networks and interlayer coupling strength are not negligible. Finally, numerical examples are also provided to validate the effectiveness of the theoretical analysis.

We discover a “less-is-more” effect that adding local antagonistic interactions (negative edge weights) can enhance the overall synchronizability of a dynamical network system. To explain this seemingly counterintuitive phenomenon, a condition is established to identify those edges the weight reduction of which improves the synchronizability index of the underlying network. We further reveal that this condition can be interpreted from the perspective of resistance distance and network community structure. The obtained result is also verified via numerical experiments on a 14-node network and a 118-node network. Our finding brings new thoughts and inspirations to the future directions of optimal network design problems.

Global synchronizability of duplex networks induced by three different intra-layer rewiring mechanisms is explored in this paper. The rewiring mechanisms are named as model-preserving rewiring (MPR), simply direct rewiring (SDR), and degree-preserving rewiring (DPR), respectively. It is found that high switching frequencies will certainly enhance global synchronizability for WS-WS duplex networks (i.e., each layer is independently formed by the algorithm proposed by Watts and Strogatz for generating small-world networks), ER-ER duplex networks (i.e., each layer is independently generated by the algorithm proposed by Erdös and Renyi) and BA-BA duplex networks (i.e., each layer is independently formed by the classical BA algorithm). Namely, the faster the intra-layer couplings are reconnected, the faster the duplex networks reach global synchronization. Furthermore, we find that by increasing the intra- or inter-coupling strengths, the WS-WS time-varying network’s global synchronizability is enhanced. Take the WS-WS time-varying network as an example, we find that SDR mechanism has greater impact on global synchronizability than MPR mechanism and DPR mechanism. The related dynamical networks can arrive at synchronization faster by SDR than by MPR or DPR. Thus, we only study the effects of SDR on ER-ER duplex networks and BA-BA duplex networks. In addition, we obtain the fact via numerical simulations that, switching intra-layer coupling topologies under SDR mechanism has the greatest impact on the BA-BA duplex network, followed by the ER-ER network, and has the weakest influence on the WS-WS duplex network in terms of improving the global synchronizability when all the intra-layer networks are sparse and have the same average degree. Finally, the global synchronizability of WS-WS and BA-BA time-varying networks is improved compared with static duplex networks, the reason being that the networks tend to be randomized under SDR according to analysis of the networks’ average clustering coefficients and degree distributions.

This article offers a survey of the recent research advances in pinning control and pinning synchronization on complex dynamical networks. The emphasis is on research ideas and theoretical developments. Some technical details, if deemed necessary for clarity, will be outlined as well.

Many properties of the structure and dynamics of complex networks derive from the characteristics of the spectrum of the associated Laplacian matrix, specifically from the set of its eigenvalues. In this paper, we show that there exist graphs for which the ratio between the length of the spectrum (that is, the difference between the largest and smallest eigenvalues of the Laplacian matrix) and its spread (the difference between the second smallest eigenvalue and the smallest one) is equal to the golden ratio. We call such graphs Golden Laplacian Graphs (GLG). In this paper, we first find all such graphs with a number of nodes n≤10. We then prove several graph-theoretic and algebraic properties that characterize these graphs. These graphs prove to be extremely robust, as they have large vertex and edge connectivity along with a large isoperimetric constant. Finally, we study the synchronization properties of GLGs, showing that they are among the top synchronizable graphs of the same size. Therefore, GLGs represent very good candidates for engineering and communication networks.

This paper investigates the synchronizability of multilayer directed Dutch windmill networks with the help of the master stability function method. Here, we propose three types of multilayer directed networks with different linking patterns, namely, inter-layer directed networks (Networks-A), intra-layer directed networks (Networks-B), and hybrid directed networks (Networks-C), and rigorously derive the analytical expressions of the eigenvalue spectrum on the basis of their supra-Laplacian matrix. It is found that network structure parameters (such as the number of layers and nodes, the intra-layer and the inter-layer coupling strengths) have a significant impact on the synchronizability in the case of the two typical synchronized regions. Finally, in order to confirm that the theoretical conclusions are correct, simulation experiments of multilayer directed network are delivered.

The system model on synchronizability problem of complex networks with multi-layer structure is closer to the real network than the usual single-layer case. Based on the master stability equation (MSF), this paper studies the eigenvalue spectrum of two k-layer variable coupling windmill-type networks. In the case of bounded and unbounded synchronization domain, the relationships between the synchronizability of the layered windmill-type networks and network parameters, such as the numbers of nodes and layers, inter-layers coupling strength, are studied. The simulation of the synchronizability of the layered windmill-type networks are given, and they verify the theoretical results well. Finally, the optimization schemes of the synchronizability are given from the perspective of single-layer and multi-layer networks, and it was found that the synchronizability of the layered windmill-type networks can be improved by changing the parameters appropriately.