text: We establish a thorough treatment of reduced-state synchronization for qubit networks with the aim of driving the qubits’ reduced states to a common trajectory. The evolution of the quantum network’s state is described by a master equation, where the network Hamiltonian is either a direct sum or a tensor product of identical qubit Hamiltonians, and the coupling terms are given by a set of permutation operators over the network. The permutations introduce naturally quantum directed interactions. We show that reduced-state synchronization is achieved if and only if the quantum interaction graphs corresponding to the permutation operators form a strongly connected union graph. The proof is based on an algebraic analysis making use of the Perron-Frobenius theorem for non-negative matrices. The convergence rate and the limiting orbit are explicitly characterized. Numerical examples are provided illustrating the obtained results. Further, we investigate the missing symmetry in the reduced-state synchronization from a graphical point of view. The information-flow hierarchy in quantum permutation operators is characterized by different layers of information-induced graphs. We show that the quantum synchronization equation is by nature equivalent to several parallel cut-balanced consensus processes, and a necessary and sufficient condition is obtained for quantum reduced-state synchronization under switching interactions applying recent work of Hendrickx and Tsitsiklis.
  @cite_30: 
  @cite_36: In a recent Physical Review Letters article, propose a simple but compelling discrete-time model of n autonomous agents (i.e., points or particles) all moving in the plane with the same speed but with different headings. Each agent's heading is updated using a local rule based on the average of its own heading plus the headings of its "neighbors." In their paper, provide simulation results which demonstrate that the nearest neighbor rule they are studying can cause all agents to eventually move in the same direction despite the absence of centralized coordination and despite the fact that each agent's set of nearest neighbors change with time as the system evolves. This paper provides a theoretical explanation for this observed behavior. In addition, convergence results are derived for several other similarly inspired models. The Vicsek model proves to be a graphic example of a switched linear system which is stable, but for which there does not exist a common quadratic Lyapunov function.
  @cite_28: In this paper, we discuss consensus problems for networks of dynamic agents with fixed and switching topologies. We analyze three cases: 1) directed networks with fixed topology; 2) directed networks with switching topology; and 3) undirected networks with communication time-delays and fixed topology. We introduce two consensus protocols for networks with and without time-delays and provide a convergence analysis in all three cases. We establish a direct connection between the algebraic connectivity (or Fiedler eigenvalue) of the network and the performance (or negotiation speed) of a linear consensus protocol. This required the generalization of the notion of algebraic connectivity of undirected graphs to digraphs. It turns out that balanced digraphs play a key role in addressing average-consensus problems. We introduce disagreement functions for convergence analysis of consensus protocols. A disagreement function is a Lyapunov function for the disagreement network dynamics. We proposed a simple disagreement function that is a common Lyapunov function for the disagreement dynamics of a directed network with switching topology. A distinctive feature of this work is to address consensus problems for networks with directed information flow. We provide analytical tools that rely on algebraic graph theory, matrix theory, and control theory. Simulations are provided that demonstrate the effectiveness of our theoretical results.
  @cite_32: The clock synchronization problem is to determine the time difference @math between two spatially separated clocks. When message delivery times between the two clocks are uncertain, @math classical messages must be exchanged between the clocks to determine @math digits of @math . On the other hand, as we show, there exists a quantum algorithm to obtain @math digits of @math while communicating only @math quantum messages.
  @cite_24: This paper gives sufficient conditions for an array of linearly coupled systems to synchronize. A typical result states that the array will synchronize if the nonzero eigenvalues of the coupling matrix have real parts that are negative enough. In particular, we show that the intuitive idea that strong enough mutual diffusive coupling will synchronize an array of identical cells is true in general. Sufficient conditions for synchronization for several coupling configurations will be considered. For coupling that leaves the array decoupled at the synchronized state, the cells each follow their natural uncoupled dynamics at the synchronized state. We illustrate this with an array of chaotic oscillators. Extensions of these results to general coupling are discussed. >
  @cite_0: Power-grid networks must be synchronized in order to function. A condition for the stability of the synchronous state enables identification of network parameters that enhance spontaneous synchronization—heralding the possibility of smart grids that operate optimally in real-world systems.
  @cite_23: Abstract : This thesis investigates certain fundamental problems in decentralized decision making and computation. We study the problem of whether a set of decision makers (or processors) with different (but related) information may make compatible decisions without communication and we characterize the computational complexity of this problem. We also analyze the complexity of other basic problems of decentralized decision making, such as decentralized detection (hypothesis testing). We then consider a scheme whereby a set of decision makers (processors) exchange and update tentative decisions which minimize a common cost function, given information they possess; we show that they are guaranteed to converge to consensus. Finally, we consider a broad class of asynchronous distributed (deterministic and stochastic) iterative optimization algorithms tolerating communication delays. We associate communication requirements with such algorithms and show that they converge appropriately under certain conditions which are no more severe than those required for their centralized counterparts. Several applications in human organizations, parallel computation and distributed signal processing are indicated. Additional keywords: decision theory. (Author).
  @cite_31: We provide an analysis of the classic Kuramoto model of coupled nonlinear oscillators that goes beyond the existing results for all-to-all networks of identical oscillators. Our work is applicable to oscillator networks of arbitrary interconnection topology with uncertain natural frequencies. Using tools from spectral graph theory and control theory, we prove that for couplings above a critical value all the oscillators synchronize, resulting in convergence of all phase differences to a constant value, both in the case of identical natural frequencies as well as uncertain ones. We also provide a series of bounds for the critical values of the coupling strength.
  @cite_16: 
summary: Consensus and synchronization of node states in a network of coupled dynamics have been extensively studied in the past decades @cite_24 @cite_36 @cite_31 . The basic idea is that @cite_23 nodes in a network with suitable connectivity can reach a common state or trajectory where all nodes' initial values are encoded with distributed node interactions, and the strong involvement of graph-theoretic approaches has reshaped the study of networked control systems @cite_16 . Excellent results have been derived towards the understandings of how bidirectional or directed, fixed or switching, deterministic or random, node interactions influence the consensus value and the speed of consensus, e.g., @cite_28 @cite_30 @cite_0 @cite_32 .