Quantum Thermodynamics of Time Evolving Networks

In this paper, we present a novel thermodynamic framework for graphs that can be used to analyze time evolving networks, relating the thermodynamics variables to macroscopic changes in network topology, and linking major structural transition to phase changes in the thermodynamic picture. We start from a recent quantum-mechanical characterization of the structure of a network relating the graph Laplacian to a density operator and resulting in a characterization of the network's entropy. Then we adopt a Schrodinger picture of the dynamics of the network, resulting in an estimation of a hidden time-varying Hamiltonian from the data, from which we derive a measure of Energy exchange. From these variables, using the thermodynamic identity, we obtain temperature under the assumption of constant volume of the system. Evaluation of real-world data shows that the thermodynamic variables thus extracted are effective in detecting critical events occurring during network evolution.