Open system quantum thermodynamics of time-varying graphs

In this article, we present a novel analysis of time-evolving networks, based on a thermodynamic representation of graph structure. We show how to characterize the evolution of time-varying complex networks by relating major structural changes to thermodynamic phase transitions. In particular, we derive expressions for a number of different thermodynamic quantities (specifically energy, entropy and temperature), which we use to describe the evolutionary behaviour of the network system over time. Since in the real world no system is truly closed and interactions with the environment are usually strong, we assume an open nature of the system. We adopt the Schrödinger picture as the dynamical representation of the quantum system over time. First, we compute the network entropy using a recent quantum mechanical representation of graph structure, connecting the graph Laplacian to a density operator. Then, we assume the system evolves according to the Schrödinger representation, but we allow for decoherence due to the interaction with the environment in a model akin to Environment-Induced Decoherence. We simplify the model by separating its dynamics into (a) an unknown time-dependent unitary evolution plus (b) an observation/interaction process, and this is the sole cause of the changes in the eigenvalues of the density matrix of the system. This allows us to obtain a measure of energy exchange with the environment through the estimation of the hidden time-varying Hamiltonian responsible for the unitary part of the evolution. Using the thermodynamic relationship between changes in energy, entropy, pressure and volume, we recover the thermodynamic temperature. We assess the utility of the method on real-world time-varying networks representing complex systems in the financial and biological domains. We also compare and contrast the different characterizations provided by the thermodynamic variables (energy, entropy, temperature and pressure). The study shows that the estimation of the time-varying energy operator strongly characterizes different states of a time-evolving system and successfully detects critical events occurring during network evolution.