On the impossibility of localizing multiple rumor sources in a line graph

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.