Heterogeneous consensus dynamics through reaction–diffusion models on graphs/networks

Consensus protocols (linear and nonlinear) are extensively used for multi-agent systems to
perform a wide variety of tasks. Its connections with diffusive models make a natural bridge between
engineering and social/natural complex systems. These protocols give rise to conservative
processes on the graphs representing the systems, and reach homogeneous (consensus) steady
states. Here we develop a model that extends the concept of global consensus (diffusion) to
dynamical processes which are nonconservative on the graphs, and reach bounded steady states,
which are not necessarily a global consensus. That is, they allow heterogeneous steady states in
which different subsets of agents can reach a consensus among them, which may be different
from the ones reached by other groups. We prove some mathematical results indicating how
the structure of the graphs representing the system determines the final states of the dynamics.
Finally, we illustrate the applications of the model by considering a heterogeneous rendezvous
of a group of agents, as well as the connections of the model with the Bass model for diffusion
of innovation and the Lotka–Volterra equations to model the evolution of groups of species that
interact in different ways.