Weighted Network Analysis Using the Debye Model

Statistical mechanics provides effective means for complex network analysis, and in particular the classical Boltzmann partition function has been extensively used to explore network structure. One of the shortcomings of this model is that it is couched in terms of unweighted edges. To overcome this problem and to extend the utility of this type of analysis, in this paper, we explore how the Debye solid model can be used to describe the probability density function for particles in such a system. According to our analogy the distribution of node degree and edge-weight in the network can be derived from the distribution of molecular energy in the Debye model. This allows us to derive a probability density function for nodes, and thus is identical to the degree distribution for the case of uniformly weighted edges. We also consider the case where the edge weights follow a distribution (non-uniformly weighted edges). The corresponding network energy is the cumulative distribution function for the node degree. This distribution reveals a phase transition for the temperature dependence. The Debye model thus provides a new way to describe the node degree distribution in both unweighted and weighted networks.