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. • Developed a nonconservative logistic model on graphs with nontrivial steady states. • Heterogeneous consensus depending on vertex connectivity is reached. • Spatial rendezvous with multiple agent positions can be reached with the model. • The Bass model for the diffusion of innovations can described by the current one. • A Lotka–Volterra model is equivalent to the heterogeneous consensus model.