Random graphs and percolation processes

本研究计划的主题是随机图，概率组合学和极值组合学关于图上的扩散或激活过程。对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真实世界现象所激发的，包括谣言或信息传播以及铁磁性动力学。作为图上的过程，它们提供了自己感兴趣的理论问题。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”，状态更新由局部和均匀规则发生。通常，这些更新被重复，并且如果所有顶点最终都是活动的，则活动顶点的初始集合被称为渗透。对于无限图和大的有限图，中心问题是一个典型的初始激活集的特定密度的行为，以及是否有一个临界阈值的密度随机选择的初始激活集渗透。* 我们对渗流过程的临界阈值有很多了解，一旦激活，顶点就永远保持不变。特别是，这种临界概率的网格图，无限树，一些凯莱图，以及许多模型的随机图和几何图已被广泛研究。大多数这样的结果在很大程度上依赖于过程的单调性和底层图的属性。我建议调查一个更一般的一类问题，无论是过程不受单调性或那些图形包含高度结构化，也随机部分。特别感兴趣的是那些非单调模型的更新规则的配置见证了一个顶点的“失活”的数量比那些见证了激活。在某些情况下，失活的影响是局部的，目的是表明模型的行为，具有很高的概率，在一个“基本上单调”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，我打算讨论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，以最大限度地概括性地研究这些过程，这些过程可以包括那些局部更新规则，其中一些顶点可以被停用，并且可以解释底层图形的随机变化。 在过去，这些类型的图过程的严格阈值结果被发现与实验预测的结果有很大差异，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的深入理解。

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.**