SEIR (Susceptible-Exposed-Infected-Recovered) is a general and widely-used diffusion model that can model the diffusion in different contexts such as idea spreading and disease propagation. Here, we tackle the problem of inferring graph edges if we can only observe a SEIR diffusion process spreading over the nodes of a graph. This problem is of importance in the common case where node states can be estimated with less cost than the edges can be found. Some applications include inferring a contact network from disease spread data, inferring a reference network from idea spreading, or estimating influenza diffusion rates between U.S. states. We improve upon the existing approaches for this problem in three ways: (1) we assume we are provided only with the probabilistic information about the state of each node which may also be undersampled or incomplete; (2) we present a more general framework that better uses trace data to model edge non-existence under SEIR model; (3) we can infer the network at both micro and macro scales.
Experiments on both real and synthetic data show that our method is accurate under these challenging cases at multiple scales, and it performs consistently better than the existing methods. For instance, we can infer a high school human contact network at microscale by tracking influenza diffusion almost 10\% better than the existing methods as well as the estimated networks closely mimick the full range of properties of the true network. We also estimated the strength of the influenza diffusion between and inside the U.S. states from Google Flu Trends data at macroscale. Estimated rates are correlated with the human transportation rates between the states to a certain degree, and we gain interesting insight into the influenza diffusion in U.S. such as the importance of the less populous states in epidemics as well as the asymmetric influenza diffusion between U.S. states.