The nonlinear dynamics of networks is a hot research topic with numerous applications in important areas of science: including neuroscience, gene transcription and regulation, food webs in ecosystems, molecular structure, and digital communications. The paper sets up a general framework for such systems, analogous to the very successful framework of topological dynamics introduced in the 1960s for nonlinear dynamical systems, but taking the network structure into account.

The framework is new, in the sense that no formal setting of this kind seems to have been defined before. Many papers carry out similar analyses for specific networks with specific equations, often making restrictive assumptions on the type of differential equations that is permitted. The formal theory for networks is analogous to the existing theory of dynamical systems with symmetry, but is more general.

A network consists of a number of individual systems, which are "coupled together," meaning that they influence each other. The architecture or topology of the network is a diagram showing these influences, with dots representing the systems and arrows representing couplings.

For example, the dots might correspond to species in a forest, and an arrow from, say, foxes to rabbits means that foxes eat rabbits. Foxes also eat pheasants, giving rise to another arrow from foxes to pheasants, but pheasants don't eat rabbits and rabbits don't eat pheasants so no arrows connect these two species. The way populations of animals change over time as a result of predation can be described by equations that reflect which species eats which.

For another example, the dots might be nerve cells and the arrows connections between these cells, along which signals are sent. Our formal theory provides a general method for associating equations with networks, and in particular it leads to natural ideas of "synchrony," where several systems behave identically at all times—so, for instance, nerve cells might all fire at once. It permits a coherent and logically rigorous analysis of how such networks can change as time passes.

I have been working on dynamics with symmetry for nearly 25 years, in collaboration with Professor Martin Golubitsky of the Department of Mathematics at the University of Houston, using group theory as the basic tool. A few years ago Marty’s postdoc Marcus Pivato came up with a puzzling example of a network that did not have symmetry in the conventional sense, but behaved in a similar manner to systems that do have symmetry.

We eventually traced this curious behavior to the existence of a "quotient network" in which the component systems "clumped together" with the same dynamics. We then realized that this clumping required a new kind of "local symmetry," which mathematically can be set up using the concept of a "groupoid." This idea is reasonably well known in pure mathematics but seldom encountered in applications; a groupoid is like a group but more general. The rest of our work followed naturally once the relevance of groupoids and quotient networks was realized.

Only through applications in the sciences, where, for example, new understanding of the dynamics of nerve cell networks could eventually have medical implications, or where an understanding of ecosystems might help us manage the environment better. At the moment we are following these mathematical ideas to see where they most naturally lead.