具有可激励节点网络的随机和确定性动力学
Stochastic and deterministic dynamics in networks with excitable nodes
摘要：
通过对局部树状网络上的一大类可激系统的动力学分析，得出在$λ=1$处发生连续相变的结论，其中$λ$是网络邻接矩阵的最大特征值.本文致力于评估这一要求的一个更一般的情况下，假设的动态传递函数的线性被违反了一个非线性参数$β$之间的随机（$β=0$）和确定性（$β\rightarrow\infty$）动态插值。我们的模型显示了一个丰富的相图与吸收状态和扩展的临界和振荡制度分离的过渡和分叉线，取决于初始状态。我们测试初始状态（$\mathbb{I}$）只有一个初始激发节点，（$\mathbb{II}$）一个固定的分数（$10\%$）的激发节点，所有这些过渡是一阶的$β>0$与滞后效应和间隙功能。对于热力学极限中的情况（$\mathbb{I}$），所有$λ$值和$β>0$的唯一相位中的吸收状态。我们进一步发展了（$\mathbb{I}$）和（$\mathbb{II}$）情形的平均场理论.对于情形（$\mathbb{II}$），我们得到了一个解析的一维映射，它解释了模型的基本性质，包括动力学的滞后图和不动点。
英文摘要：
The analysis of the dynamics of a large class of excitable systems on locally tree-like networks leads to the conclusion that at $λ=1$ a continuous phase transition takes place, where $λ$ is the largest eigenvalue of the adjacency matrix of the network. This paper is devoted to evaluate this claim for a more general case where the assumption of the linearity of the dynamical transfer function is violated with a non-linearity parameter $β$ which interpolates between stochastic ($β=0$) and deterministic ($β\rightarrow\infty$) dynamics. Our model shows a rich phase diagram with an absorbing state and extended critical and oscillatory regimes separated by transition and bifurcation lines which depend on the initial state. We test initial states with ($\mathbb{I}$) only one initial excited node, ($\mathbb{II}$) a fixed fraction ($10\%$) of excited nodes, for all of which the transition is of first order for $β>0$ with a hysteresis effect and a gap function. For the case ($\mathbb{I}$) in the thermodynamic limit the absorbing state in the only phase for all $λ$ values and $β>0$. We further develop mean-field theories for cases ($\mathbb{I}$) and ($\mathbb{II}$). For case ($\mathbb{II}$) we obtain an analytic one-dimensional map which explains the essential properties of the model, including the hysteresis diagrams and fixed points of the dynamics.
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