Random graphs and percolation processes

本研究计划的主题是随机图、概率组合和关于图上扩散或激活过程的极值组合。图上这种扩散过程的研究通常是由复杂网络或统计物理中的现实世界现象驱动的，包括谣言或信息传播以及铁磁性动力学。作为图上的过程，它们本身就提供了有趣的理论问题。在这些元胞自动机中，图的顶点处于两种状态之一，“非活动”或“活动”，状态更新通过局部和同构规则发生。通常，这些更新是重复的，如果所有顶点最终都是活动的，那么初始的活动顶点集就被称为渗透。对于无限图和大型有限图，中心问题是具有特定密度的典型初始激活集的行为，以及是否存在随机选择的初始激活集的渗透密度的临界阈值。***对于进程中渗透的临界阈值，一旦激活，顶点将永远保持这种状态，我们已经知道了很多。特别是网格图、无限树、一些Cayley图以及许多随机图和几何图模型的临界概率已经得到了广泛的研究。大多数这样的结果严重依赖于过程的单调性和底层图的性质。我建议研究更一般的一类问题，其中过程不受单调性的约束，或者图中包含高度结构化和随机部分的问题。特别有趣的是那些具有更新规则的非单调模型，其中见证顶点“失活”的配置数量少于见证激活的配置。在某些这样的情况下，失活的影响是局部的，其目的应是表明模型的行为，在高概率下，以一种“本质单调”的方式。为了阐明这些图的结构与激活过程的增长之间的关系，我打算说明底层图中的小扰动如何改变扩散过程及其临界概率。***本研究的计划成果是开发工具和内聚理论，以最大限度地研究这些过程，这些过程可以包含那些局部更新规则，其中一些顶点可以被禁用，并且可以解释底层图的随机变化。在过去，这些类型的图过程的严格阈值结果已经被发现与实验预测的结果有很大的不同，这表明需要有精确的结果来保证对这些激活过程的大规模行为有一个坚实的理解

The topics of this research proposal are random graphs, probabilistic combinatorics, and extremal combinatorics with regard to diffusion or activation processes on graphs. The study of such diffusion processes on graphs is often motivated by real-world phenomena in complex networks or statistical physics including rumour or information-spreading and the dynamics of ferromagnetism. As processes on graphs, they provide theoretical problems of interest in their own right. In these cellular automata, vertices of a graph are in one of two states, 'inactive' or 'active', and state updates occur by a local and homogeneous rule. Generally, these updates are repeated and the initial set of active vertices is said to percolate if all vertices are eventually active. For infinite graphs and for large finite graphs, the central question is the behaviour of a typical initially activated set of a particular density and whether there is a critical threshold for the density of a randomly chosen initially activated set to percolate. ***Much is known about the critical thresholds for percolation in processes where once activated, a vertex remains so forever. In particular, such critical probabilities for grid graphs, infinite trees, some Cayley graphs, and many models of random graphs and geometric graphs have been studied extensively. Most such results rely heavily on the monotone nature of the processes and on the properties of the underlying graph. I propose to investigate a more general class of problems where either the process is not constrained by monotonicity or those where the graph contains both highly structured and also random parts. Of particular interest are those non-monotone models with update rules for which the configurations witnessing the 'deactivation' of a vertex are fewer in number than those witnessing an activation. In some such cases, the effects of deactivation are local and the aim shall be to show that the model behaves, with high probability, in an 'essentially monotone' fashion. To shed light on the relationship between the structure of these graphs and the growth of the activation processes, I intend to address how small perturbations in the underlying graph change a diffusion process and its critical probability.***The planned outcome of this research is the development of tools and of a cohesive theory to study these processes in the greatest generality that can encompass those local update rules in which some vertices can be deactivated and can account for random changes to the underlying graph. In the past, rigorous threshold results for these types of graph processes have been found to differ hugely from those predicted by experiments, indicating a need to have precise results to guarantee a solid understanding of the large-scale behaviour of these activation processes.**