Complex systems consist of many interacting components, typically exhibiting features such as self-organization, critical transitions, or emergent phenomena.
Network science is particularly useful for studying these kinds of systems, modeling them as graphs, mathematical objects made of nodes (representing the system’s components) and links (representing their interactions).
Beyond graph theory, a more physics-inspired perspective offers additional tools (borrowed, in particular, from statistical physics) for analyzing networks.
Following this approach, we will explore the tight connection between dynamical processes on networks and the topology underneath them.
We will focus in particular on processes governed by the Graph Laplacian, denoted as $\hat{L}$, evolving over time as $\boldsymbol x(\tau)=e^{-\tau \hat{L}}\boldsymbol x(0)$. This equation is valid for diffusion, but also provides a first order description of more general dynamics near the steady state.
We will see the diffusion-based theoretical framework known as the Laplacian Renormalization Group, accounting for the communicability properties of a network and able to unveil its multiscale hierarchical nested structures.
Key tools in this framework, suggesting intriguing physical analogies, include the Graph Density Matrix, which quantifies diffusion streamlines between pairs of nodes, and the Graph Entropy, which measures the heterogeneity of the system’s state after a given diffusion time, and can be considered as an order parameter for second order transitions between diffusion aggregation phases.
We will show that such transitions identify classes of links based on diffusion timescales, corresponding to specific diffusivity modes in the Fourier space and revealing the graph’s topological organization. Finally, we will explain the deep connection between nested modules, Laplacian eigenmodes, and diffusion time scales in hierarchical modular graphs.