  In this paper, we study a class of equations representing nonlinear diffusion
on networks. A particular instance of our model can be seen as a network
equivalent of the porous-medium equation. We are interested in studying
perturbations of such a system and describing the consensus dynamics. The
nonlinearity of the equations gives rise to potentially intricate structures of
equilibria that can intersect the consensus space, creating singularities. For
the unperturbed case, we characterise the sets of equilibria by exploiting the
symmetries under group transformations of the nonlinear vector field. Under
small perturbations, we obtain a slow-fast system. Thus, we analyse the
slow-fast dynamics near the singularities on the consensus space. The analysis
at this stage is carried out for complete networks, allowing a detailed
characterisation of the system. We provide a linear approximation of the
intersecting branches of equilibria at the singular points; as a consequence,
we show that, generically, the singularities on the consensus space turn out to
be transcritical. We prove under local assumptions the existence of canard
solutions. For generic graph structures, assuming more strict conditions on the
perturbation, we prove the existence of a maximal canard, which coincides with
the consensus subspace. In addition, we validate by numerical simulations the
principal findings of our main theory, extending the study to non-complete
graphs. Moreover, we show how the delayed loss of stability associated to the
canards induces transient spatio-temporal patterns.
