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Irreversible k-threshold processes: Graph-theoretical threshold models of the spread of disease and of opinion

机译：k不可逆阈值过程：疾病传播和观点传播的图论阈值模型

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We will consider models of the spread of disease or opinion through social networks, represented as graphs. In our models, vertices will be in one of two states, I ("infected") or 0 ("uninfected") and change of state will take place at discrete times. After describing several such models, we will concentrate on the model, called an irreversible k-threshold process, where a vertex enters state 1 if at least k of its neighbors are in state 1, and where a vertex never leaves state 1 once it is in it. We will seek sets of vertices with the property that, if they are in state 1 at the beginning, then eventually all vertices end up in state 1. Such vertex sets correspond to vertices that can be infected with a disease or opinion so as to guarantee saturation of the population with the disease or opinion. We will also discuss ways to "defend" against such saturating sets, for example by "vaccination" or designing network topologies.
机译：我们将考虑通过社交网络传播疾病或意见的模型，以图形表示。在我们的模型中，顶点将处于两种状态之一，即I（“感染”）或0（“未感染”），并且状态变化将在不连续的时间发生。在描述了几个这样的模型之后，我们将集中于称为不可逆k阈值过程的模型，如果至少一个邻居的k处于状态1，则该顶点进入状态1，而一旦顶点处于状态1，则该顶点永远不会离开状态1。在里面。我们将寻求具有以下属性的顶点集：如果它们在开始时处于状态1，那么最终所有顶点最终都将处于状态1。此类顶点集对应于可以感染疾病或观点的顶点，以确保疾病或意见使人口饱和。我们还将讨论“防御”此类饱和集的方法，例如通过“接种疫苗”或设计网络拓扑。

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