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We study a simple model of epidemics where an infected node transmits theinfection to its neighbors independently with probability $p$. This is alsoknown as the independent cascade or Susceptible-Infected-Recovered (SIR) modelwith fixed recovery time. The size of an outbreak in this model is closelyrelated to that of the giant connected component in ``edge percolation'', whereeach edge of the graph is kept independently with probability $p$, studied fora large class of networks including configuration model\cite{molloy2011critical} and preferential attachment\cite{bollobas2003,Riordan2005}. Even though these models capture the effectsof degree inhomogeneity and the role of super-spreaders in the spread of anepidemic, they only consider graphs that are locally tree like i.e. have a fewor no short cycles. Some generalizations of the configuration model weresuggested 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 generalnetworks to predict the size of an outbreak? Is it possible to make predictionsby accessing the distribution of small subgraphs (or motifs)? We answer thequestion in the affirmative for large-set expanders with local weak limits(also known as Benjamini-Schramm limits). In particular, we show that there isan algorithm which gives a $(1-\epsilon)$ approximation of the probability andthe final size of an outbreak by accessing a constant-size neighborhood of aconstant number of nodes chosen uniformly at random. We also presentcorollaries of the theorem for the preferential attachment model, and studygeneralizations with household (or motif) structure. The latter was only knownfor the configuration model.

Algorithms Using Local Graph Features to Predict Epidemics