We study a simple model of epidemics where an infected node transmits the infection to its neighbors independently with probability $p$. This is also known as the independent cascade or Susceptible-Infected-Recovered (SIR) model with fixed recovery time. The size of an outbreak in this model is closely related to that of the giant connected component in ``edge percolation'', where each edge of the graph is kept independently with probability $p$, studied for a large class of networks including configuration model \cite{molloy2011critical} and preferential attachment \cite{bollobas2003,Riordan2005}. Even though these models capture the effects of degree inhomogeneity and the role of super-spreaders in the spread of an epidemic, they only consider graphs that are locally tree like i.e. have a few or no short cycles. Some generalizations of the configuration model were suggested to capture local communities, known as household models \cite{ball2009threshold}, or hierarchical configuration model \cite{Hofstad2015hierarchical}. Here, we ask a different question: what information is needed for general networks to predict the size of an outbreak? Is it possible to make predictions by accessing the distribution of small subgraphs (or motifs)? We answer the question in the affirmative for large-set expanders with local weak limits (also known as Benjamini-Schramm limits). In particular, we show that there is an algorithm which gives a $(1-\epsilon)$ approximation of the probability and the final size of an outbreak by accessing a constant-size neighborhood of a constant number of nodes chosen uniformly at random. We also present corollaries of the theorem for the preferential attachment model, and study generalizations with household (or motif) structure. The latter was only known for the configuration model.
翻译:我们研究一个简单的流行病模型,即一个受感染的节点将感染单独传送给邻居,概率为$p$。这也被称为独立级联或可感知感染-复苏模型(SIR),具有固定恢复时间。这个模型爆发的规模与“隐形透镜”中的巨型连接组件的大小密切相关,该图的每个边缘都以概率独立保存,其概率为$;为一大批网络进行了研究,包括配置模型(cite{mollolo2011zy})和优惠附件{cite{bollobas2003,Riordan2005}。尽管这些模型可以捕捉异异性度和超级扩散者在流行病蔓延中的作用。它们只考虑像“隐形透镜”这样的本地树的图块块。一些配置模型的概略化建议用来捕捉当地社群,称为“我们所知道的直径”的模型(我们所选的直径),或者等级配置模型的模型{cite-cretad-helad-herant-ride-rif-ride-hide-hideal-lide-lifal-listal-listal silstal liverstal listal liverstal lader) strism suder subal subal subal subal subs subs subal subal subs subal suble suble suble suble subil subildslationsal subild subild subilds subsm sub subs subilds subs subs subil suble subs subs sub su sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub sub subs sub subs sub sub sub subsal susususub sub sub sub sub