Topological Characterization of Meta-stable States in Weakly Non-linear Diffusion Processes on Networks

Complex networks are ubiquitous.
Many dynamical processes on networks, including spreading processes such as SIR and SIS, are generally weakly non-linear and the core of the process consists of a diffusion process.
However, while the diffusion may have a trivial fixed point where the probability distributions over all nodes becomes uniform or proportional to their degree, the nonlinearity generally results in the trivial fixed point becoming unstable, yielding non-trivial outcomes.
Especially in real networks with modular and other internal structures, characterizing these outcomes requires numerical simulations.
Here we report discovering a deep connection between these solutions and symmetry groups.
In particular, we consider a multi-dimensional weakly non-linear dynamical process on a network and show that it is equivalent to a force-directed layout problem.
We then show how the topology of the embedding space and its symmetry group can be used to characterize the low-energy solutions, which turn out to be topological in nature.
Often the trivial fixed-point of the dynamics, where the field over all nodes collapses to zero, is unstable due to constraints arising from non-linear interactions among nodes, resulting in topological low-energy states becoming the dominant.