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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.
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Complex evolving systems such as the biosphere, ecosystems and societies exhibit sudden collapses, for reasons that are only partially understood. Here we study this phenomenon using a mathematical model of a system that evolves under Darwinian selection and exhibits the spontaneous growth, stasis and collapse of its structure. We find that the typical lifetime of the system increases sharply with the diversity of its components or species. We also find that the prime reason for crashes is a naturally occurring internal fragility of the system. This fragility is captured in the network organizational character and is related to a reduced multiplicity of pathways between its components. This work suggests new parameters for understanding the robustness of evolving molecular networks, ecosystems, societies, and markets.
I analyse a model of an evolving network represented as a directed graph; each node corresponds to one molecular species and the links to catalytic interactions between species. Over short timescales the graph remains fixed while relative populations of the molecular species change according to a set of coupled differential equations. Over long timescales the system is subject to periodic perturbations, each of which adds one new node to the graph, with random links to other nodes, and removes one node with the least relative population. Starting from a sparse random graph, a small autocatalytic set (ACS) inevitably forms and then grows by accreting nodes until it spans the entire graph. The resultant fully autocatalytic graph, whose probability of forming by pure chance is very small, nevertheless forms in this model in an average time that grows only logarithmically with the size of the system. ACSs can also get destroyed, often accompanied by the sudden extinction of a large number of species. I show that the largest of the extinction events in this model are caused by one of three mechanisms, each of which produces a specific discontinuous change in the graphs topology. The model is analytically tractable: two theorems are proved which determine the set of nodes with least relative population in the attractor, for any given graph. This in turn can be used to analytically demonstrate the inevitability of the formation and growth of ACSs and calculate the associated timescales. Finally, I show that the formation and growth of ACSs is robust to the relaxation of many of the idealizations made to enhance the analytical tractability of the model.
Synchronization in networks of coupled oscillators is known to be largely determined by the spectral and symmetry properties of the interaction network. Here we leverage this relation to study a class of networks for which the threshold coupling strength for global synchronization is the lowest among all networks with the same number of nodes and links. These networks, defined as being uniform, complete, and multi-partite (UCM), appear at each of an infinite sequence of network-complement transitions in a larger class of networks characterized by having near-optimal thresholds for global synchronization. We show that the distinct symmetry structure of the UCM networks, which by design are optimized for global synchronizability, often leads to formation of clusters of synchronous oscillators, and that such states can coexist with the state of global synchronization.
Percolation and synchronization are two phase transitions that have been extensively studied since already long ago. A classic result is that, in the vast majority of cases, these transitions are of the second-order type, i.e. continuous and reversible. Recently, however, explosive phenomena have been reported in com- plex networks structure and dynamics, which rather remind first-order (discontinuous and irreversible) transitions. Explosive percolation, which was discovered in 2009, corresponds to an abrupt change in the networks structure, and explosive synchronization (which is concerned, instead, with the abrupt emergence of a collective state in the networks dynamics) was studied as early as the first models of globally coupled phase oscillators were taken into consideration. The two phenomena have stimulated investigations and de- bates, attracting attention in many relevant fields. So far, various substantial contributions and progresses (including experimental verifications) have been made, which have provided insights on what structural and dynamical properties are needed for inducing such abrupt transformations, as well as have greatly enhanced our understanding of phase transitions in networked systems. Our intention is to offer here a monographic review on the main-stream literature, with the twofold aim of summarizing the existing results and pointing out possible directions for future research.
Synchronization has been the subject of intense research during decades mainly focused on determining the structural and dynamical conditions driving a set of interacting units to a coherent state globally stable. However, little attention has been paid to the description of the dynamical development of each individual networked unit in the process towards the synchronization of the whole ensemble. In this paper, we show how in a network of identical dynamical systems, nodes belonging to the same degree class differentiate in the same manner visiting a sequence of states of diverse complexity along the route to synchronization independently on the global network structure. In particular, we observe, just after interaction starts pulling orbits from the initially uncoupled attractor, a general reduction of the complexity of the dynamics of all units being more pronounced in those with higher connectivity. In the weak coupling regime, when synchronization starts to build up, there is an increase in the dynamical complexity whose maximum is achieved, in general, first in the hubs due to their earlier synchronization with the mean field. For very strong coupling, just before complete synchronization, we found a hierarchical dynamical differentiation with lower degree nodes being the ones exhibiting the largest complexity departure. We unveil how this differentiation route holds for several models of nonlinear dynamics including toroidal chaos and how it depends on the coupling function. This study provides new insights to understand better strategies for network identification and control or to devise effective methods for network inference.
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