Will a realistic complex system be stable?

Dr Paul Kirk from the MRC Biostatistics Unit (BSU) and colleagues from Imperial College London — including joint first author, Delphine Rolando — have just published a paper in the New Journal of Physics investigating the stability of complex systems. This new research has implications for a variety of fields, including ecology, molecular biology, disease transmission, and even the global financial network.

Below Dr Kirk explains some of the fascinating ideas investigated in the paper, which are also illustrated in an animated video abstract. To check out the video, the first video abstract BSU have ever been part of click here.

The world is a complex place. All around us are people, animals, technology, and many other things that encounter and interact with one another. To try to make sense of these complicated webs of interactions, scientists often represent complex systems as networks. Social media website Facebook provides a good example: we can represent people as nodes in a network, and draw edges between people who are friends. Networks are common in ecology too. Ecologists often deal with food webs, which are networks that describe what eats what.

Kirk story image

We often associate positive and negative numbers with the edges in networks to indicate the “strength” and “type” of each interaction. For example, when a fox encounters a rabbit, it is good news for the fox, so the edge from the fox to rabbit will have a small positive number associated with it. On the other hand, it is bad news for the rabbit, so the edge from the rabbit to fox will have a larger negative number.

Scientists are frequently interested in the long-term behaviour of complex systems. In our ecological example, we would like to know if the fox and rabbit populations will eventually settle down, so that the number of foxes and rabbits remains approximately constant over time. Or will both populations just grow larger and larger, until foxes and rabbits overrun the Earth?

The answers to questions like these depend on the properties of the network, such as the number of nodes, number of edges, and strengths and types of interactions. To investigate how these network properties affect the long-term behaviour of complex systems, researchers have made use of results from a branch of mathematics called Random Matrix Theory (RMT). This involves using a mathematical model for the network to assess the effects of changing the network's properties (such as the number of nodes or edges). Traditionally, scientists have used quite a specific type of simple model, which allows them to use famous results from RMT in order to work out how network properties affect long-term behaviour.

In the paper Conditional random matrix ensembles and the stability of dynamical systems (Kirk*, Rolando* et al.), Dr Kirk and colleagues demonstrate that traditional simple models can actually give misleading results. Instead, they propose to use models that better reflect our current understanding of the world around us.

Dr Kirk says: “Although this means we can no longer make use of elegant results from RMT, we can instead use computer simulations to provide us with more realistic answers. As a general rule, results obtained using mathematical models will only be as sensible as the choice of model, and we should always be wary of putting too much faith in conclusions drawn from over-simplified models. “