Diffusion on graphs is a central modeling tool for complex systems, from transport and ecology to social dynamics and neuroscience. In its classical form, graph diffusion is driven by the combinatorial Laplacian and assumes nearest-neighbor interactions and mass conservation. This thesis extends that setting by developing and analyzing families of Laplacian-based operators that incorporate three features frequently required in applications: topology-dependent biases, long-range couplings, and nonconservative gain and loss. The aim is to keep models interpretable and mathematically analyzable, while enlarging the range of dynamical behaviors that can be represented.
First, we introduce hub-biased Laplacians and their generalization to centrality-biased Laplacians, which modify transition rates according to node degree or other centrality measures. These operators provide a principled way to favor or penalize hubs and to tune convergence rates through spectral properties, while their adjoint counterparts generate conservative advection-like dynamics on undirected networks and yield heterogeneous steady states linked to the underlying structure. Second, we study nonlocal diffusion through d-path Laplacians and their transforms, connecting graph-based long-range hopping with ideas from fractional diffusion. Mellin-transformed versions of these operators make it possible to achieve superdiffusive regimes on graphs, with heavy-tailed spreading. Third, we analyse nonconservative diffusion based on the Lerman-Ghosh operators and propose logistic reaction-diffusion models that prevent blow-up and extinction, producing bounded, network-dependent equilibria and heterogeneous consensus states.
These theoretical developments are complemented with applications: advection diffusion modeling of ring-tailed lemur movement in a fragmented landscape, degree-biased effects on explosive synchronization in Kuramoto networks, and laboratory experiments on indirect social influence in innovation adoption. We also apply the non-conservative framework to multi-agent systems and ecological interaction models. Finally, we introduce metaplex models that couple continuous intra-node diffusion with inter-node network transport, and we apply them to brain-inspired simplicial-metaplex diffusion, highlighting differences with higher-order Laplacians and the role of internal geometry.