The interest in dynamic processes on networks is steadily rising in recent years. In this paper, we consider the $(\alpha,\beta)$-Thresholded Network Dynamics ($(\alpha,\beta)$-Dynamics), where $\alpha\leq \beta$, in which only structural dynamics (dynamics of the network) are allowed, guided by local thresholding rules executed in each node. In particular, in each discrete round $t$, each pair of nodes $u$ and $v$ that are allowed to communicate by the scheduler, computes a value $\mathcal{E}(u,v)$ (the potential of the pair) as a function of the local structure of the network at round $t$ around the two nodes. If $\mathcal{E}(u,v) < \alpha$ then the link (if it exists) between $u$ and $v$ is removed; if $\alpha \leq \mathcal{E}(u,v) < \beta$ then an existing link among $u$ and $v$ is maintained; if $\beta \leq \mathcal{E}(u,v)$ then a link between $u$ and $v$ is established if not already present. The microscopic structure of $(\alpha,\beta)$-Dynamics appears to be simple, so that we are able to rigorously argue about it, but still flexible, so that we are able to design meaningful microscopic local rules that give rise to interesting macroscopic behaviors. Our goals are the following: a) to investigate the properties of the $(\alpha,\beta)$-Thresholded Network Dynamics and b) to show that $(\alpha,\beta)$-Dynamics is expressive enough to solve complex problems on networks. Our contribution in these directions is twofold. We rigorously exhibit the claim about the expressiveness of $(\alpha,\beta)$-Dynamics, both by designing a simple protocol that provably computes the $k$-core of the network as well as by showing that $(\alpha,\beta)$-Dynamics is in fact Turing-Complete. Second and most important, we construct general tools for proving stabilization that work for a subclass of $(\alpha,\beta)$-Dynamics and prove speed of convergence in a restricted setting.

翻译：在最近几年里,对网络动态过程的兴趣正在稳步增加。 特别是, 在每条离散的美元回合中, 每对可以由调度器进行通信的节点美元和美元美元, 将一个值( pha,\beta) 网络动态 ($( pha,\beta) 美元) 视为网络本地结构的函数, 在两个节点周围, 仅允许结构动态( 网络的动力), 在每个节点中执行的本地阈值规则。 特别是, 在每条离散的美元回合中, 每对可由调度器进行通信的节点美元和美元美元, 计算一个价值( mathal) 美元( E) (u, v) 美元(配对网络的潜力) 网络的本地结构的函数, 以两个节点的美元值( 美元( 美元) 向电流向电流转的 。