Inferring global network properties from egocentric data with applications to epidemics.

Social networks are often only partly observed, and it is sometimes desirable to infer global properties of the network from 'egocentric' data. In the current paper, we study different types of egocentric data, and show which global network properties are consistent with data. Two global network properties are considered: the size of the largest connected component (the giant) and the size of an epidemic outbreak taking place on the network. The main conclusion is that, in most cases, egocentric data allow for a large range of possible sizes of the giant and the outbreak, implying that egocentric data carry very little information about these global properties. The asymptotic size of the giant and the outbreak is also characterized, assuming the network is selected uniformly among networks with prescribed egocentric data.