Many empirical complex systems, from biochemical to social ones, exhibit a network structure with heterogeneous connectivity patterns, their topology being characterized by mesoscale and/or hierarchical organization. However, such systems are identified not only by their structure but also from the dynamical processes on its top: the interplay between topology and dynamics often leads to a rich spectrum of phenomena, from localization to phase transitions and collective behavior. Recently, we have introduced a novel framework for the analysis of complex networks in terms of a macroscopic description which accounts, simultaneously, for the full microscopic knowledge of the system – i.e., its adjacency matrix W – and its interplay with diffusive dynamics – such as continuous diffusion and random walks [1]. There are two main novelties coming from this framework, which is based on a new fundamental operator – i.e., the density matrix ρβ – which encodes the network state in terms of a Gibbsian operator ρβ = e −βH(W)/Z – where H(·) is a function encoding dynamics, e.g. H = L is the combinatorial Laplacian governing continuous diffusion, and Z is the partition function. Remarkably, this framework i) allows to calculate macroscopic descriptors such as information entropy without limiting one’s focus to a specific subset of network features (e.g., degree distribution, which provide only partial information about the structure); ii) by using dynamical processes, their evolution across time is used to probe the system at different temporal scales which, in turn, provide multiscale information about distinct topological scales. In this talk we will briefly introduce the theoretical foundations of this framework while pointing to the broad range of its successful applications to practical problems: the dimensionality reduction of multilayer biological and transportation networks [2], the functional reducibility of multilayer social and transportation systems to enhance their transport properties [3], the quantification of the unit-system entanglement and its effect on the disintegration of social and biological networks [4], the multiscale analysis of virus-host interactions with a special focus on the SARS-CoV2–human interactome [5].
