The shape of a network has an unexpected but predictable effect on its dynamics, mathematician Davide Sclosa shows in new research. This provides new insights into how networks function and could contribute to a better understanding of social dynamics, information distribution and even technological infrastructures.
Dynamical systems on a graph
A group of people exchanging opinions. An artificial neural network making predictions. A virus spreading through a population. A server distributing tasks across multiple computers. A network evolving randomly with connections switching on and off. Each of these phenomena represents a dynamical system on a graph: a mathematical structure that illustrates the interconnections between different points. Sclosa focused his research on these different types of networks, such as social networks, artificial neural networks, electricity networks, and even viruses spreading through a population.
Influence of network structure on dynamics
It turns out that the structure of a network has a major influence on how information, opinions or even electricity spread. For example, social networks with many recurring connections – loops – ensure that information frequently returns to its starting point. This ensures that multiple opinions can coexist stably, making the network more democratic. ''For example, imagine a group of people standing in a line, where each person only speaks to their immediate neighbours. The opinion models suggest that after a while, a consensus is reached among them, for example that everyone becomes politically moderate. In contrast, for a group of people arranged in a circle, a different stable configuration can emerge, with a spectrum of different opinions coexisting. For example, from far left to far right, and looping back again'', Sclosa explains.
Practical applications
The findings of this study are not limited to social networks. As the findings are mathematical theorems about general networks, they can be applied to a wide range of systems. Whether it concerns a neural network, an electricity grid or a server distributing tasks across computers, if the network meets the study's hypotheses, its dynamics can be predicted. For instance, network analysis can determine which connections should be reinforced or severed in a power grid to prevent blackouts. Similarly, it can identify which roads should be widened or closed to improve traffic flow.
Theoretical and computer-based methods
The research was largely conducted using theoretical mathematical methods. However, in one specific case the predicted dynamics turned out to be so surprising that a practical test was needed. Python code was used to simulate a network where an infinite number of stable equilibria were possible. This illustrates how mathematics and computer science can go hand in hand to explain complex phenomena.