Regular Ring Lattice Research Articles

Diffusion-Localization Transition Point of Gravity Type Transport Model on Regular Ring Lattices and Bethe Lattices

Focusing on the diffusion-localization transition, we theoretically analyzed a nonlinear gravity-type transport model defined on networks called regular ring lattices, which have an intermediate structure between the complete graph and the simple ring. Exact eigenvalues were derived around steady states, and the values of the transition points were evaluated for the control parameter characterizing the nonlinearity. We also analyzed the case of the Bethe lattice (or Cayley tree) and found that the transition point is 1/2, which is the lowest value ever reported.

Emergent Behaviors of Thermodynamic Kuramoto Ensemble on a Regular Ring Lattice

The temporal evolution of Kuramoto oscillators influenced by the temperature field often appears in biological oscillator ensembles. In this paper, we propose a generalized Kuramoto type lattice model on a regular ring lattice with the equal spacing assuming that each oscillator has an internal energy (temperature). Our lattice model is derived from the thermodynamical Cucker–Smale model for flocking on the 2D free space under the assumption that the ratio between velocity field and temperature field at each lattice point has a uniform magnitude over lattice points. The proposed model satisfies an entropy principle and exhibits emergent dynamics under some sufficient frameworks formulated in terms of initial data and system parameters. Moreover, the phase-field tends to the Kuramoto phase-field asymptotically.

Numerical Analysis on behavior of coupled maps on regular ring lattice with high degrees

Behavior of coupled dynamical systems strongly depends on network structures, i.e. how dynamical systems connect with each other. We here numerically show what happens when the degree (K) of regular ring lattice increases. In the ring lattice, it is known that the clustering coefficient (C) is 3(K — 2)/4(K — 1). This predicts that C becomes about 3/4, when K is sufficiently large. However, we find that this is not satisfied when K is large, because of the periodic boundary condition of the ring lattice. We further investigate how the change in C of the ring lattice affects behavior of coupled dynamical systems on the ring lattice.

An integrated diversity and fusion framework for binary consensus over fading channels

In this paper, we consider a cooperative network that is trying to reach consensus on the occurrence of an event by communicating over not fully connected and time-invariant network topologies with fading channels. We first discuss the fusion and diversity decision-making strategies over time-invariant network topologies and shed light on the underlying trade-offs. We then propose an integrated diversity and fusion framework. Our approach properly takes advantage of both fusion to enable information flow and diversity to increase robustness to link errors. We mathematically analyze the proposed framework and show how the network achieves accurate consensus asymptotically. To show an example, we then utilize the proposed framework over regular ring lattice networks. Our theoretical and simulation results indicate that the proposed technique improves the consensus performance considerably.

Robustness of random graphs based on graph spectra

It has been recently proposed that the robustness of complex networks can be efficiently characterized through the natural connectivity, a spectral property of the graph which corresponds to the average Estrada index. The natural connectivity corresponds to an average eigenvalue calculated from the graph spectrum and can also be interpreted as the Helmholtz free energy of the network. In this article, we explore the use of this index to characterize the robustness of Erdős-Rényi (ER) random graphs, random regular graphs, and regular ring lattices. We show both analytically and numerically that the natural connectivity of ER random graphs increases linearly with the average degree. It is also shown that ER random graphs are more robust than the corresponding random regular graphs with the same number of vertices and edges. However, the relative robustness of ER random graphs and regular ring lattices depends on the average degree and graph size: there is a critical graph size above which regular ring lattices are more robust than random graphs. We use our analytical results to derive this critical graph size as a function of the average degree.

Evolution of Equity Norms in Small‐World Networks

The topology of interactions has been proved very influential in the results of models based on learning and evolutionary game theory. This paper is aimed at investigating the effect of structures ranging from regular ring lattices to random networks, including small‐world networks, in a model focused on property distribution norms. The model considers a fixed and finite population of agents who play the Nash bargaining game repeatedly. Our results show that regular networks promote the emergence of the equity norm, while less‐structured networks make possible the appearance of fractious regimes. Additionally, our analysis reveals that the speed of adoption can also be affected by the network structure.

Spectral Measure of Structural Robustness in Complex Networks

We introduce the concept of natural connectivity as a measure of structural robustness in complex networks. The natural connectivity characterizes the redundancy of alternative routes in a network by quantifying the weighted number of closed walks of all lengths. This definition leads to a simple mathematical formulation that links the natural connectivity to the spectrum of a network. The natural connectivity can be regarded as an average eigenvalue that changes strictly monotonically with the addition or deletion of edges. We calculate both analytically and numerically the natural connectivity of three typical networks: regular ring lattices, random graphs, and random scale-free networks. We also compare the proposed natural connectivity to other structural robustness measures within a scenario of edge elimination and demonstrate that the natural connectivity provides sensitive discrimination of structural robustness that agrees with our intuition.

Robustness of regular ring lattices based on natural connectivity

It has been recently proposed that natural connectivity can be used to efficiently characterise the robustness of complex networks. The natural connectivity quantifies the redundancy of alternative routes in the network by evaluating the weighted number of closed walks of all lengths and can be seen as an average eigenvalue obtained from the graph spectrum. In this article, we explore both analytically and numerically the natural connectivity of regular ring lattices and regular random graphs obtained through degree-preserving random rewirings from regular ring lattices. We reformulate the natural connectivity of regular ring lattices in terms of generalised Bessel functions and show that the natural connectivity of regular ring lattices is independent of network size and increases with K monotonically. We also show that random regular graphs have lower natural connectivity, and are thus less robust, than regular ring lattices.

EFFECTS OF INDIVIDUAL'S SELF-EXAMINATION ON COOPERATION IN PRISONER'S DILEMMA GAME

We study a spatial evolutionary prisoner's dilemma game on regular network's one-dimensional regular ring and two-dimensional square lattice. The individuals located on the sites of networks can either cooperate with their neighbors or defect. The effects of individual's self-examination are introduced. Using Monte Carlo simulations and pair approximation method, we investigate the average density of cooperators in the stationary state for various values of payoff parameters b and the time interval Δt. The effects of the fraction p of players in the system who are using the self-examination on cooperation are also discussed. It is shown that compared with the case of no individual's self-examination, the persistence of cooperation is inhibited when the payoff parameter b is small and at certain Δt (Δt > 0) or p (p > 0), cooperation is mostly inhibited, while when b is large, the emergence of cooperation can be remarkably enhanced and mostly enhanced at Δt = 0 or p = 1.

A NOVEL SMALL-WORLD NETWORK MODEL: KEEPING CONNECTIVITY WITHOUT ADDING EDGES

Considering the problems of potentially generating a disconnected network in the WS small-world network model [Watts and Strogatz, Nature393, 440 (1998)] and of adding edges in the NW small-world network model [Newman and Watts, Phys. Lett. A263, 341 (1999)], we propose a novel small-world network model. First, generate a regular ring lattice of N vertices. Second, randomly rewire each edge of the lattice with probability p. During the random rewiring procedure, keep the edges between the two nearest neighbor vertices, namely, always keep a connected ring. This model need not add edges and can maintain connectivity of the network at all times in the random rewiring procedure. Simulation results show that the novel model has the typical small-world properties which are small characteristic path length and high clustering coefficient. For large N, the model is approximately equal to the WS model. For large N and small p, the model is approximately equal to the WS model or the NW model.