In this paper, we apply ensemble methods from statistical physics to analyse fMRI brain networks for Alzheimer's patients. By mapping the nodes in a network to virtual particles in a thermal system, the microcanonical ensemble and the canonical ensemble are analogous to two different fMRI network representations. These representations are obtained by selecting a threshold on the BOLD time series correlations between nodes in different ways. The microcanonical ensemble corresponds to a set of networks with a fixed fraction of edges, while the canonical ensemble corresponds to the set networks with edges obtained with a fixed value of the threshold. In the former case, there is zero variance in the number of edges in each network, while in the latter case the set of networks have a variance in the number of edges. Ensemble methods describe the macroscopic properties of a network by considering the underlying microscopic characterisations which are in turn closely related to the degree configuration and network entropy. Our treatment allows us to specify new partition functions for fMRI brain networks, and to explore a phase transition in the degree distribution. The resulting method turns out to be an effective tool to identify the most salient anatomical brain regions in Alzheimer's disease and provides a tool to distinguish groups of patients in different stages of the disease.

Thermodynamic characterisations or analogies have proved to provide powerful tools for the statistical analysis of network populations or time series, together with the identification of structural anomalies that occur within them. For instance, classical Boltzmann statistics together with the corresponding partition function have been used to apply the tools of statistical physics to the analysis of variations in network structure. However, the physical analogy adopted in this analysis, together with the interpretation of the resulting system of particles is sometimes vague and remains an open question. This, in turn, has implications concerning the definition of quantities such as temperature and energy. In this paper, we take a novel view of the thermal characterisation where we regard the edges in a network as the particles of the thermal system. By considering networks with a fixed number of nodes we obtain a conservation law which applies to the particle occupation configuration. Using this interpretation, we provide a physical meaning for the temperature which is related to the number of network nodes and edges. This provides a fundamental description of a network as a thermal system. If we further interpret the elements of the adjacency matrix as the binary microstates associated with edges, this allows us to further extend the analysis to systems with edge-weights. We thus introduce the concept of the canonical ensemble into the thermal network description and the corresponding partition function and then use this to compute the thermodynamic quantities. Finally, we provide numerical experiments on synthetic and real-world data-sets to evaluate the thermal characterisations for both unweighted and weighted networks.