Evolution of functional subnetworks in complex systems

Multifunctional reservoir computing

Whereas the power of reservoir computing (RC) in inferring chaotic systems has been well established in the literature, the studies are mostly restricted to monofunctional machines where the training and testing data are acquired from the same attractor. Here, using the strategies of attractor labeling and trajectory separation, we propose a scheme of RC capable of learning multiple attractors generated by entirely different dynamics, namely multifunctional RC. Specifically, we demonstrate that by incorporating a label channel into the standard reservoir computer, a single machine is able to learn from data the dynamics of multiple chaotic attractors, while each attractor can be accurately retrieved by inputting just a scalar in the prediction phase. The dependence of the machine performance on the labeling and separation parameters is investigated, and it is found that the machine performance is optimized when the parameters take intermediate values. The working mechanism of multifunctional RC is analyzed by the method of functional networks in neuroscience, and it is revealed that each attractor is represented by a stable, unique functional network in the reservoir, and the optimal performance arises as a balance between the stability, complexity, and distinguishability of the functional networks.

Enhancing network synchronization by phase modulation

Due to time delays in signal transmission and processing, phase lags are inevitable in realistic complex oscillator networks. Conventional wisdom is that phase lags are detrimental to network synchronization. Here we show that judiciously chosen phase lag modulations can result in significantly enhanced network synchronization. We justify our strategy of phase modulation, demonstrate its power in facilitating and enhancing network synchronization with synthetic and empirical network models, and provide an analytic understanding of the underlying mechanism. Our work provides an alternative approach to synchronization optimization in complex networks, with insights into control of complex nonlinear networks.

Accurate detection of hierarchical communities in complex networks based on nonlinear dynamical evolution

One of the most challenging problems in network science is to accurately detect communities at distinct hierarchical scales. Most existing methods are based on structural analysis and manipulation, which are NP-hard. We articulate an alternative, dynamical evolution-based approach to the problem. The basic principle is to computationally implement a nonlinear dynamical process on all nodes in the network with a general coupling scheme, creating a networked dynamical system. Under a proper system setting and with an adjustable control parameter, the community structure of the network would "come out" or emerge naturally from the dynamical evolution of the system. As the control parameter is systematically varied, the community hierarchies at different scales can be revealed. As a concrete example of this general principle, we exploit clustered synchronization as a dynamical mechanism through which the hierarchical community structure can be uncovered. In particular, for quite arbitrary choices of the nonlinear nodal dynamics and coupling scheme, decreasing the coupling parameter from the global synchronization regime, in which the dynamical states of all nodes are perfectly synchronized, can lead to a weaker type of synchronization organized as clusters. We demonstrate the existence of optimal choices of the coupling parameter for which the synchronization clusters encode accurate information about the hierarchical community structure of the network. We test and validate our method using a standard class of benchmark modular networks with two distinct hierarchies of communities and a number of empirical networks arising from the real world. Our method is computationally extremely efficient, eliminating completely the NP-hard difficulty associated with previous methods. The basic principle of exploiting dynamical evolution to uncover hidden community organizations at different scales represents a "game-change" type of approach to addressing the problem of community detection in complex networks.

Clustering dynamics of complex discrete-time networks and its application in community detection

The clustering phenomenon is common in real world networks. A discrete-time network model is proposed firstly in this paper, and then the phase clustering dynamics of the networks are studied carefully. The proposed model acts as a bridge between the dynamic phenomenon and the topology of a modular network. On one hand, phase clustering phenomenon will occur for a modular network by the proposed model; on the other hand, the communities can be identified from the clustering phenomenon. Beyond the phases' information, it is found that the frequencies of phases can be applied to community detection also with the proposed model. In specific, communities are identified from the information of phases and their frequencies of the nodes. Detailed algorithm for community detection is provided. Experiments show that the performance and efficiency of the dynamics based algorithm are competitive with recent modularity based algorithms in large scale networks.

Dynamics in hybrid complex systems of switches and oscillators

While considerable progress has been made in the analysis of large systems containing a single type of coupled dynamical component (e.g., coupled oscillators or coupled switches), systems containing diverse components (e.g., both oscillators and switches) have received much less attention. We analyze large, hybrid systems of interconnected Kuramoto oscillators and Hopfield switches with positive feedback. In this system, oscillator synchronization promotes switches to turn on. In turn, when switches turn on, they enhance the synchrony of the oscillators to which they are coupled. Depending on the choice of parameters, we find theoretically coexisting stable solutions with either (i) incoherent oscillators and all switches permanently off, (ii) synchronized oscillators and all switches permanently on, or (iii) synchronized oscillators and switches that periodically alternate between the on and off states. Numerical experiments confirm these predictions. We discuss how transitions between these steady state solutions can be onset deterministically through dynamic bifurcations or spontaneously due to finite-size fluctuations.

Onset of synchronization in weighted complex networks: the effect of weight-degree correlation

By numerical simulations, we investigate the onset of synchronization of networked phase oscillators under two different weighting schemes. In scheme-I, the link weights are correlated to the product of the degrees of the connected nodes, so this kind of networks is named as the weight-degree correlated (WDC) network. In scheme-II, the link weights are randomly assigned to each link regardless of the node degrees, so this kind of networks is named as the weight-degree uncorrelated (WDU) network. Interestingly, it is found that by increasing a parameter that governs the weight distribution, the onset of synchronization in WDC network is monotonically enhanced, while in WDU network there is a reverse in the synchronization performance. We investigate this phenomenon from the viewpoint of gradient network, and explain the contrary roles of coupling gradient on network synchronization: gradient promotes synchronization in WDC network, while deteriorates synchronization in WDU network. The findings highlight the fact that, besides the link weight, the correlation between the weight and the node degree is also important to the network dynamics.

Introduction to focus issue: dynamics in systems biology

The methods of nonlinear systems form an extensive toolbox for the study of biology, and systems biology provides a rich source of motivation for the development of new mathematical techniques and the furthering of understanding of dynamical systems. This Focus Issue collects together a large variety of work which highlights the complementary nature of these two fields, showing what each has to offer the other. While a wide range of subjects is covered, the papers often have common themes such as "rhythms and oscillations," "networks and graph theory," and "switches and decision making." There is a particular emphasis on the links between experimental data and modeling and mathematical analysis.