Structural properties of the synchronized cluster on complex networks

Percolation critical exponents in cluster kinetics of pulse-coupled oscillators

Transient dynamics leading to the synchrony of a type of pulse-coupled oscillators, so-called scrambler oscillators, has previously been studied as an aggregation process of synchronous clusters, and a rate equation for the cluster size distribution has been proposed. However, the evolution of the cluster size distribution for general cluster sizes has not been fully understood yet. In this paper, we study the evolution of the cluster size distribution from the perspective of a percolation model by regarding the number of aggregations as the number of attached bonds. Specifically, we derive the scaling form of the cluster size distribution with specific values of the critical exponents using the property that the characteristic cluster size diverges as the percolation threshold is approached from below. Through simulation, it is confirmed that the scaling form well explains the evolution of the cluster size distribution. Based on the distribution behavior, we find that a giant cluster of all oscillators is formed discontinuously at the threshold and also that further aggregation does not occur like in a one-dimensional bond percolation model. Finally, we discuss the origin of the discontinuous formation of the giant cluster from the perspective of global suppression in explosive percolation models. For this, we approximate the aggregation process as a cluster-cluster aggregation with a given collision kernel. We believe that the theoretical approach presented in this paper can be used to understand the transient dynamics of a broad range of synchronizations.

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.

Community structure revealed by phase locking

Community structure can naturally emerge in paths to synchronization, and scratching it from the paths is a tough issue that accounts for the diverse dynamics of synchronization. In this paper, with assumption that the synchronization on complex networks is made up of local and collective processes, we proposed a scheme to lock the local synchronization (phase locking) at a stable state, meanwhile, suppress the collective synchronization based on Kuramoto model. Through this scheme, the network dynamics only contains the local synchronization, which suggests that the nodes in the same community synchronize together and these synchronization clusters well reveal the community structure of network. Furthermore, by analyzing the paths to synchronization, the relations or overlaps among different communities are also obtained. Thus, the community detection based on the scheme is performed on five real networks and the observed community structures are much more apparent than modularity-based fast algorithm. Our results not only provide a deep insight to understand the synchronization dynamics on complex network but also enlarge the research scope of community detection.

Classification of transport backbones of complex networks

Transport properties in random and scale-free (SF) networks are studied by analyzing the betweenness centrality (BC) distribution P(B) in the minimum spanning trees (MSTs) and infinite incipient percolation clusters (IIPCs) of the networks. It is found that P(B) in MSTs scales as P(B)∼B(-δ). The obtained values of δ are classified into two different categories, δ≃1.6 and δ≃2.0. Using the mapping between BC and the branch size of tree structures, it is proved that δ in MSTs which are close to critical trees is 1.6. In contrast, δ in MSTs which are supercritical trees is shown to be 2.0. We also find δ=1.5 in IIPCs, which is a natural result because IIPC is physically critical. Based on the results in MSTs, a physical reason why δ≥2 in the original networks is suggested.

Optimal topology for parallel discrete-event simulations

The effect of shortcuts on the task completion landscape in parallel discrete-event simulation (PDES) is investigated. The morphology of the task completion landscape in PDES is known to be described well by the Langevin-type equation for nonequillibrium interface growth phenomena, such as the Kardar-Parisi-Zhang equation. From the numerical simulations, we find that the root-mean-squared fluctuation of task completion landscape, W(t,N), scales as W(t→∞,N)~N when the number of shortcuts, ℓ, is finite. Here N is the number of nodes. This behavior can be understood from the mean-field type argument with effective defects when ℓ is finite. We also study the behavior of W(t,N) when ℓ increases as N increases and provide a criterion to design an optimal topology to achieve a better synchronizability in PDES.

Evolution of microscopic and mesoscopic synchronized patterns in complex networks

Previous studies about synchronization of Kuramoto oscillators in complex networks have shown how local patterns of synchronization emerge differently in homogeneous and heterogeneous topologies. The main difference between the paths to synchronization in both topologies is rooted in the growth of the largest connected component of synchronized nodes when increasing the coupling between the oscillators. Nevertheless, a recent study focusing on this same phenomenon has claimed the contrary, stating that the statistical distribution of synchronized clusters for both types of networks is similar. Here we provide extensive numerical evidences that confirm the original claims, namely, that the microscopic and mesoscopic dynamics of the synchronized patterns indeed follow different routes.