Small noise may diversify collective motion in Vicsek model

Natural systems are inextricably affected by noise. Within recent decades, the manner in which noise affects the collective behavior of self-organized systems, specifically, has garnered considerable interest from researchers and developers in various fields. To describe the collective motion of multiple interacting particles, Vicsek et al. proposed a well-known self-propelled particle (SPP) system, which exhibits a second-order phase transition from disordered to ordered motion in simulation; due to its non-equilibrium, randomness, and strong coupling nonlinear dynamics, however, there has been no rigorous analysis of such a system to date. To decouple systems consisting of deterministic laws and randomness, we propose a general method which transfers the analysis of these systems to the design of cooperative control algorithms. In this study, we rigorously analyzed the original Vicsek model under both open and periodic boundary conditions for the first time, and developed extensions to heterogeneous SPP systems (including leaderfollower models) using the proposed method. Theoretical results show that SPP systems switch an infinite number of times between ordered and disordered states for any noise intensity and population density, which implies that the phase transition indeed takes a nontraditional form. We also investigated the robust consensus and connectivity of these systems. Moreover, the findings presented in this paper suggest that our method can be used to predict possible configurations during the evolution of complex systems, including turn, vortex, bifurcation and flock merger phenomena as they appear in SPP systems

Recovery of signals by a weighted ℓ2/ℓ1\ell_2/\ell_1 minimization under arbitrary prior support information

In this paper, we introduce a weighted $\ell_2/\ell_1$ minimization to recover block sparse signals with arbitrary prior support information. When partial prior support information is available, a sufficient condition based on the high order block RIP is derived to guarantee stable and robust recovery of block sparse signals via the weighted $\ell_2/\ell_1$ minimization. We then show if the accuracy of arbitrary prior block support estimate is at least $50\%$, the sufficient recovery condition by the weighted $\ell_2/\ell_{1}$ minimization is weaker than that by the $\ell_2/\ell_{1}$ minimization, and the weighted $\ell_2/\ell_{1}$ minimization provides better upper bounds on the recovery error in terms of the measurement noise and the compressibility of the signal. Moreover, we illustrate the advantages of the weighted $\ell_2/\ell_1$ minimization approach in the recovery performance of block sparse signals under uniform and non-uniform prior information by extensive numerical experiments. The significance of the results lies in the facts that making explicit use of block sparsity and partial support information of block sparse signals can achieve better recovery performance than handling the signals as being in the conventional sense, thereby ignoring the additional structure and prior support information in the problem

A sharp recovery condition for sparse signals with partial support information via orthogonal matching pursuit

This paper considers the exact recovery of $k$-sparse signals in the noiseless setting and support recovery in the noisy case when some prior information on the support of the signals is available. This prior support consists of two parts. One part is a subset of the true support and another part is outside of the true support. For $k$-sparse signals $\mathbf{x}$ with the prior support which is composed of $g$ true indices and $b$ wrong indices, we show that if the restricted isometry constant (RIC) $\delta_{k+b+1}$ of the sensing matrix $\mathbf{A}$ satisfies \begin{eqnarray*} \delta_{k+b+1}<\frac{1}{\sqrt{k-g+1}}, \end{eqnarray*} then orthogonal matching pursuit (OMP) algorithm can perfectly recover the signals $\mathbf{x}$ from $\mathbf{y}=\mathbf{Ax}$ in $k-g$ iterations. Moreover, we show the above sufficient condition on the RIC is sharp. In the noisy case, we achieve the exact recovery of the remainder support (the part of the true support outside of the prior support) for the $k$-sparse signals $\mathbf{x}$ from $\mathbf{y}=\mathbf{Ax}+\mathbf{v}$ under appropriate conditions. For the remainder support recovery, we also obtain a necessary condition based on the minimum magnitude of partial nonzero elements of the signals $\mathbf{x}$

A sharp recovery condition for block sparse signals by block orthogonal multi-matching pursuit

We consider the block orthogonal multi-matching pursuit (BOMMP) algorithm for the recovery of block sparse signals. A sharp bound is obtained for the exact reconstruction of block $K$-sparse signals via the BOMMP algorithm in the noiseless case, based on the block restricted isometry constant (block-RIC). Moreover, we show that the sharp bound combining with an extra condition on the minimum $\ell_2$ norm of nonzero blocks of block $K-$sparse signals is sufficient to recover the true support of block $K$-sparse signals by the BOMMP in the noise case. The significance of the results we obtain in this paper lies in the fact that making explicit use of block sparsity of block sparse signals can achieve better recovery performance than ignoring the additional structure in the problem as being in the conventional sense

Critical Connectivity and Fastest Convergence Rates of Distributed Consensus with Switching Topologies and Additive Noises

Consensus conditions and convergence speeds are crucial for distributed consensus algorithms of networked systems. Based on a basic first-order average-consensus protocol with time-varying topologies and additive noises, this paper first investigates its critical consensus condition on network topology by stochastic approximation frameworks. A new joint-connectivity condition called extensible joint-connectivity that contains a parameter $\delta$ (termed the extensible exponent) is proposed. With this and a balanced topology condition, we show that a critical value of $\delta$ for consensus is $1/2$. Optimization on convergence rate of this protocol is further investigated. It is proved that the fastest convergence rate, which is the theoretic optimal rate among all controls, is of the order $1/t$ for the best topologies, and is of the order $1/t^{1-2\delta}$ for the worst topologies which are balanced and satisfy the extensible joint-connectivity condition. For practical implementation, certain open-loop control strategies are introduced to achieve consensus with a convergence rate of the same order as the fastest convergence rate. Furthermore, a consensus condition is derived for non stationary and strongly correlated random topologies. The algorithms and consensus conditions are applied to distributed consensus computation of mobile ad-hoc networks; and their related critical exponents are derived from relative velocities of mobile agents for guaranteeing consensus.Comment: 36 pages, 0 figur

Noise-induced synchronization of Hegselmann-Krause dynamics in full space

The Hegselmann-Krause (HK) model is a typical self-organizing system with local rule dynamics. In spite of its widespread use and numerous extensions, the underlying theory of its synchronization induced by noise still needs to be developed. In its original formulation, as a model first proposed to address opinion dynamics, its state-space was assumed to be bounded, and the theoretical analysis of noise-induced synchronization for this particular situation has been well established. However, when system states are allowed to exist in an unbounded space, mathematical difficulties arise whose theoretical analysis becomes non-trivial and is as such still lacking. In this paper, we completely resolve this problem by exploring the topological properties of HK dynamics and by employing the theory of independent stopping time. The associated result in full statespace provides a solid interpretation of the randomness-induced synchronization of self-organizing system

Robust Fragmentation Modeling of Hegselmann-Krause-Type Dynamics

In opinion dynamics, how to model the enduring fragmentation phenomenon (disagreement, cleavage, and polarization) of social opinions has long possessed a central position. It is widely known that the confidence-based opinion dynamics provide an acceptant mechanism to produce fragmentation phenomenon. In this study, taking the famous confidence-based Hegselmann-Krause (HK) model, we examine the robustness of the fragmentation coming from HK dynamics and its variations with prejudiced and stubborn agents against random noise. Prior to possible insightful explanations, the theoretical results in this paper explicitly reveal that the well-appearing fragmentation of HK dynamics and its homogeneous variations finally vanishes in the presence of arbitrarily tiny noise, while only the HK model with heterogenous prejudices displays a solid cleavage in noisy environment

Regional boundary controllability of time fractional diffusion processes

In this paper, we are concerned with the regional boundary controllability of the Riemann-Liouville time fractional diffusion systems of order $\alpha\in (0,1]$. The characterizations of strategic actuators are established when the systems studied are regionally boundary controllable. The determination of control to achieve regional boundary controllability with minimum energy is explored. We also show a connection between the regional internal controllability and regional boundary controllability. Several useful results for the optimal control from an implementation point of view are presented in the end.Comment: 19 pages 2figures in IMA J. Math. Control Info. 201

Revisiting the tunneling spectrum and information recovery of a general charged and rotating black hole

In this paper we revisit the tunneling spectrum of a charged and rotating black hole--Kerr-Newman black hole by using Parikh and Wilczek's tunneling method and get the most general result compared with the works [9, 10]. We find an ambiguity in Parikh and Wilczek's tunneling method, and give a reasonable description. We use this general spectrum to discuss the information recovery based on the Refs. [11-13]. For the tunneling spectrum we obtained, there exit correlations between sequential Hawking radiations, information can be carried out by such correlations, and the entropy is conserved during the radiation process. So we resolve the information loss paradox based on the methods [11-13] in the most general case

Entropy spectrum of the apparent horizon of Vaidya black holes via adiabatic invariance

The spectroscopy of the apparent horizon of Vaidya black holes is investigated via adiabatic invariance. We obtain an equally spaced entropy spectrum with its quantum equal to the one given by Bekenstein. We demonstrate that the quantization of entropy and area is a generic property of horizon, not only for stationary black holes, and the results also exit in a dynamical black hole. Our work also shows that the quantization of black hole is closely related to Hawking temperature, which is an interesting thing