Inferring Edges from Weights in the Debye Model

The measurement of information on the connections between pairs of nodes is an important part of studying network structures. Real-world network datasets contain fine features on their edges, which are usually represented as edge weights, to reflect the strength of the connection. However, to remove spurious connections and understand the topological structure, network edges are usually represented as binary states, being either connected or not. This is still a controversial issue about how to infer binary adjacency matrices from edge weight distribution. Usually, it is achieved by simply thresholding to distinguish true links and to obtain a set of sparse connections. Previously, tools developed in statistical mechanics have provided effective ways to find the optimal threshold so as to maintain the statistical properties in the network structure. Thermodynamic analogies together with statistical ensembles have been proved to be useful in analysing edge-weighted networks. To extend this work, in this paper, we use Debye's solid model to describe the probability distribution of edge weights. This models the distribution of edge weights using the mixed Gamma distribution. We treat the derived edge-weight distribution as the combination of two Gamma functions and then apply the Expectation-Maximization algorithm to estimate the corresponding parameters. This gives the optimal threshold to convert weighted networks to binary connections. Numerical analysis shows that Debye's solid model provides a new way to describe the edge weight probability. Moreover, there exists a phase transition in the low-temperature region, corresponding to a structural transition caused by applying the threshold. Experimental results on real-world weighted networks reveal an improved threshold performance for inferring edge connections from edge weights.