Here we examine the problem of rumor source identification in line graphs. We assume the SI model for rumor propagation with exponential waiting times. We consider the case where a rumor originates from two sources simultaneously, and evaluate the likelihood function for the given observations given those sources. As the size of the infected region grows arbitrarily large, we show that unlike the single source case, where the likelihood function concentrates near the midpoint of the infected region, the support of the likelihood function in this case remains widely distributed over the middle half of the infected region. This makes the rumor sources impossible to localize with high probability on any scale smaller than that of the infection size itself.
Previous work has shown that for contagion processes on extended star networks (trees with exactly one node of degree > 2), there is a simple, closed-form expression for a highly accurate approximation to the maximum likelihood infection source. Here, we generalize that result to a class of hypertrees which, although somewhat structurally analogous, provides a much richer representation space. In particular, this approach can be used to estimate patient zero sources, even when the infection has been propagated via large group gatherings rather than person-to-person spread, and when it is spreading through interrelated social bubbles with varying degrees of overlap. In contact tracing contexts, this estimator may be used to identify the source of a local outbreak, which can then be used for forward tracing or for further backward tracing (by similar or other means) to an upstream source.
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