Inhomogeneity and generalised bootstrap percolation in stochastic networks - 27785

自举逾渗过程是由物理学家Chalupa、Leath和赖希在1979年为了描述某些磁现象而提出的，从那时起，概率学家、数学物理学家和组合学家就对它进行了广泛的研究。这是一类基于以下简单规则的图上的“感染”过程：如果至少有一定数量的邻居被感染，则节点被感染。特别是在过去的10到15年里，该领域已经看到了巨大的增长，特别是在这些过程的概率和极端方面。在这个项目中，我们将研究更复杂的版本的自举渗流过程，这是基于多数规则。也就是说，如果一个节点的邻居中至少有一部分被感染，那么该节点就会被感染。这些过程有相当数量的应用主题，从思想或趋势在社会中的传播到进化博弈论。尽管它们具有基本性质，但对其进行严格的研究仍然是一个重大挑战。该项目的主要目标是发展这种过程的随机网络，表现出各种形式的不均匀性的理论。这个一般概念捕捉了在现实世界的网络中观察到的许多属性，例如社交或生物网络。事实上，不均匀性是真实世界网络的一个关键特征，它反映了群体内部的多样性。该项目将探索各种形式的不均匀性对这些过程演变方式的影响，从而更深入地了解在“均匀”结构中观察不到的现象。其结果将为许多社会和经济进程的基础机制提供新的见解。在过去的10年里，对这些过程的研究变得越来越重要，主要是由于互联网平台上的经济活动的巨大增长以及社交媒体的影响力越来越大。

Bootstrap percolation processes were introduced by the physicists Chalupa, Leath and Reich in 1979 in order to describe certain magnetic phenomena and have been studied extensively since then by probabilists, mathematical physicists and combinatorialists. This is a class of "infection" processes on graphs that are based on the following simple rule: a node becomes infected if at least a certain number of its neighbours have become infected. Especially during the last 10 to 15 years, the field has seen a huge amount of growth, in particular, regarding probabilistic and extremal aspects of these processes. In this project, we will study more sophisticated versions of bootstrap percolation processes, which are based on majority rules. That is, a node becomes infected if at least a certain fraction of its neighbours have become infected. These processes have a considerable number of applications in themes ranging from the spread of ideas or trends in a society to evolutionary game theory. Despite their basic nature, their rigorous study remains a major challenge. The main goal of the project is to develop the theory of such processes on stochastic networks that exhibit various forms of inhomogeneity. This general notion captures a number of properties that have been observed in real-world networks, such as social or biological networks. In fact, inhomogeneity is a key feature of real-world networks, which reflects the diversity within a population. The project will explore the effects that various forms of inhomogeneity have on the way these processes evolve, leading to a deeper understanding of phenomena that are not observed in "homogeneous" structures. Its results will give new insight on mechanisms that underpin many societal and economic processes. The study of these processes has become increasingly important during the last 10 years, mainly due to the enormous growth of economic activity over internet platforms as well as the increasing impact of social media.