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The voter model has been extensive studied as an opinion dynamic model, and the role of the zealots has only been discussed recently. We introduce the adaptive voter model with zealots and show that the final distribution of the magnetism can be separated into two regions depending on the number of zealots as well as the probability of forming link. When the fraction of zealots is dominated in the population, the probability distribution of magnetism follows a Gaussian-like distribution and the relaxation time is population-size independent. When the population is dominated by the susceptible agents, the relaxation time is proportional to the exponential of the population size. We have found the analytical solution of the relaxation time in the limiting cases and explained the difference of the relaxation time in these two regions based on the approximation method.

We obtain an exact formula for the first-passage time probability distribution for random walks on complex networks using inverse Laplace transform. We write the formula as the summation of finitely many terms with different frequencies corresponding to the poles of Laplace transformed function and separate the short-term and long-term behavior of the first-passage process. We give a formula of the decay rate $\beta$, which is inversely proportional to the characteristic relaxation time $\tau$ of the target node. This exact formula for the first-passage probability between two nodes at a given time can be approximately solved in the mean field approximation by estimation of the characteristic relaxation time $\tau$. Our theoretical results compare well with numerical simulation on artificial as well as real networks., Comment: 8 pages, 6 figures

We design a method to optimize the global mean first-passage time (GMFPT) of multiple random walkers searching in complex networks for a general target, without specifying the property of the target node. According to the Laplace transformed formula of the GMFPT, we can equivalently minimize the overlap between the probability distribution of sites visited by the random walkers. We employ a mutation only genetic algorithm to solve this optimization problem using a population of walkers with different starting positions and a corresponding mutation matrix to modify them. The numerical experiments on two kinds of random networks (WS and BA) show satisfactory results in selecting the origins for the walkers to achieve minimum overlap. Our method thus provides guidance for setting up the search process by multiple random walkers on complex networks., Comment: 5 pages, 2 figures, accepted at ICCS'17

We calculate the topological boundary modes found for quantum wire for a quantum ring. For the symmetric ring we find analytical solutions for two quasi-particles identifiable as the Majorana states and for asymmetric ring, we have also find approximate solution. By turning on a small field at time zero, we find a field induced oscillation of the Majorana states that was initially localized. The range of validity of our solution, the relation between the period of oscillation and the coin parameter and field strength, and the encoding of a message of the on/off of the field using the Majorana states are discussed. Our results suggest ways to reduce decoherence for small systems, thereby providing a good candidate for quantum memory and computing as the Majorana qubit is topologically protected., Comment: arXiv admin note: text overlap with arXiv:1611.00634

The evolution of quantum walk on a finite wire under a small increment of vector potential $\alpha$ can exhibit intrinsic quantum oscillation of the two topologically protected bound states corresponding to the Majorana modes. By tuning an external electric field corresponding to the addition of an $\alpha$ impulse at the end of each intrinsic period, the intrinsic oscillation is enveloped by a beat modulation with a longer period. This beat oscillation is useful in the preparation of decoherence-free qubit in trapped ion chain and may be observed in several experiments.

We generalize a binary majority-vote model on adaptive networks to a plurality-vote counterpart. When opinions are uniformly distributed in the population of voters in the initial state, it is found that having more available opinions in the initial state actually accelerate the time to consensus. In particular, we investigate the three-state plurality-vote model. While time to consensus in two state model scales exponentially with population size $N$, for finite-size system, there is a non-zero probability that either the population reaches the consensus state in a time that is very short and independent of $N$ (in the heterophily regime), or in a time that scales exponentially with $N$ but is still much faster than two-state model.

The original Parrondo game, denoted as AB3, contains two independent games: A and B. The winning or losing of A and B game is defined by the change of one unit of capital. Game A is a losing game if played continuously, with winning probability $p=0.5-\epsilon$, where $\epsilon=0.003$. Game B is also losing and it has two coins: a good coin with winning probability $p_g=0.75-\epsilon$ is used if the player`s capital is not divisible by $3$, otherwise a bad coin with winning probability $p_b=0.1-\epsilon$ is used. Parrondo paradox refers to the situation that the mixture of A and B game in a sequence leads to winning in the long run. The paradox can be resolved using Markov chain analysis. We extend this setting of Parrondo game to involve players with one-step memory. The player can win by switching his choice of A or B game in a Parrondo game sequence. If the player knows the identity of the game he plays and the state of his capital, then the player can win maximally. On the other hand, if the player does not know the nature of the game, then he is playing a (C,D) game, where either (C=A, D=B), or (C=B,D=A). For player with one-step memory playing the AB3 game, he can achieve the highest expected gain with switching probability equal to $3/4$ in the (C,D) game sequence. This result has been found first numerically and then proven analytically. Generalization to AB mod($M$) Parrondo game for other integer $M$ has been made for the general domain of parameters $p_b

The topological phases in one-dimensional quantum walk can be classified by the coin parameters. By solving for the general exact solutions of bound states in one-dimensional quantum walk with boundaries specified by different coin parameters, we show that these bound states are Majorana modes with quasi-energy $E=0,\pi$. These modes are qualitatively different for different boundary conditions used. For two-boundary system with symmetric boundary conditions, the interaction energy between two Majorana bound states can be computed, as in the case of a finite wire. Suggestion of observing these modes are provided., Comment: 6 pages, 3 pages

Experimental observations of quantum walks in one dimension have provided many exciting applications in quantum computing, while recent theoretical investigation of single phase defect in these system points towards interesting phenomena associated with bounds states. Here we obtain analytical solutions of quantum walk with a general quantum coin in one dimension with multiple defects, with new prediction on the condition for zero reflectance for scattering state, and the existence of an analogy to the Ramsauer effect for multiple defects. We also show the transition from the zero reflectance state to the bound state can provide a method for preparing the quantum walk in a bound state. Applications to systems similar to thin film optics are suggested., Comment: 5 pages, 5 figures. Submitted to PRA

Reliability is one of the important measures of how well the system meets its design objective, and mathematically is the probability that a system will perform satisfactorily for at least a given period of time. When the system is described by a connected network of N components (nodes) and their L connection (links), the reliability of the system becomes a difficult network design problem which solutions are of great practical interest in science and engineering. This paper discusses the numerical method of finding the most reliable network for a given N and L using genetic algorithm. For a given topology of the network, the reliability is numerically computed using adjacency matrix. For a search in the space of all possible topologies of the connected network with N nodes and L links, genetic operators such as mutation and crossover are applied to the adjacency matrix through a string representation. In the context of graphs, the mutation of strings in genetic algorithm corresponds to the rewiring of graphs, while crossover corresponds to the interchange of the sub-graphs. For small networks where the most reliable network can be found by exhaustive search, genetic algorithm is very efficient. For larger networks, our results not only demonstrate the efficiency of our algorithm, but also suggest that the most reliable network will have high symmetry., Comment: 9 pages, 10 figures, 3 tables

Inspired by the flashing ratchet, Parrondo's game presents an apparently paradoxical situation. Parrondo's game consists of two individual games, game A and game B. Game A is a slightly losing coin-tossing game. Game B has two coins, with an integer parameter $M$. If the current cumulative capital (in discrete unit) is a multiple of $M$, an unfavorable coin $p_b$ is used, otherwise a favorable $p_g$ coin is used. Paradoxically, combination of game A and game B could lead to a winning game, which is the Parrondo effect. We extend the original Parrondo's game to include the possibility of $M$ being either $M_1$ or $M_2$. Also, we distinguish between strong Parrondo effect, i.e. two losing games combine to form a winning game, and weak Parrondo effect, i.e. two games combine to form a better-performing game. We find that when $M_2$ is not a multiple of $M_1$, the combination of $B(M_1)$ and $B(M_2)$ has strong and weak Parrondo effect for some subsets in the parameter space $(p_b,p_g)$, while there is neither strong nor weak effect when $M_2$ is a multiple of $M_1$. Furthermore, when $M_2$ is not a multiple of $M_1$, stochastic mixture of game A may cancel the strong and weak Parrondo effect. Following a discretization scheme in the literature of Parrondo's game, we establish a link between our extended Parrondo's game with the analysis of discrete Brownian ratchet. We find a relation between the Parrondo effect of our extended model to the macroscopic bias in a discrete ratchet. The slope of a ratchet potential can be mapped to the fair game condition in the extended model, so that under some conditions, the macroscopic bias in a discrete ratchet can provide a good predictor for the game performance of the extended model. On the other hand, our extended model suggests a design of a ratchet in which the potential is a mixture of two periodic potentials.

We provide a method for the ranking of the reliability of two networks with the same connectance. Our method is based on the Cheeger constant linking the topological property of a network with its spectrum. We first analyze a set of twisted rings with the same connectance and degree distribution, and obtain the ranking of their reliability using their eigenvalue gaps. The results are generalized to general networks using the method of rewiring. The success of our ranking method is verified numerically for the IEEE57, the Erd\H{o}s-R\'enyi, and the Small-World networks., Comment: 7 pages, 3 figures

The Gauss's law, in an abstract sense, is a theorem that relates quantities on the boundary (flux) to the interior (charge) of a surface. An identity for soap froths were proved with the same boundary-interior relation. In this article, we try to construct a definition of flux for other graphs, such that a similar boundary-interior relation can be satisfied., Comment: 6 pages

The organizational structure of a network is investigated with a simulated precipitation model which does not make use of prior knowledge about the community structure of the network. The result is presented as a structure profile through which various definitions of communities can be applied for specific applications. The simulated precipitation model performs the grouping of nodes so that nodes belonging to the same 'community' automatically aggregate, thereby revealing regions of the adjacency matrix with denser interconnections. The process is analogous to massive particles precipitating towards the lower potential layer. Without loss of the infrastructure information, a community structure profile of a network can be obtained as the ground state of the Hamiltonian. The method is also applicable to directed and weighted networks.

The first passage time (FPT) distribution for random walk in complex networks is calculated through an asymptotic analysis. For network with size $N$ and short relaxation time $\tau\ll N$, the computed mean first passage time (MFPT), which is inverse of the decay rate of FPT distribution, is inversely proportional to the degree of the destination. These results are verified numerically for the paradigmatic networks with excellent agreement. We show that the range of validity of the analytical results covers networks that have short relaxation time and high mean degree, which turn out to be valid to many real networks., Comment: 6 pages, 4 figures, 1 table