Small patterns, big impact: Predicting how networks behave by looking at their parts

What is it about?

Networks are often used to describe the interaction between individual units. We can use networks to describe a multitude of different systems, like food webs, power grids, or even a global supply chain. One of the big questions is: When something disturbs that network, how does the system react? Does it stay calm, can it deal with the disturbance and return to its normal state, or does it spiral into chaos? Scientists traditionally try to understand these systems by looking at the "big picture", but it is reasonable to ask if this is always necessary. Our work is looking deeper into this question and searching for small, specific groups of just two or three members that can be used as predictors for wider system dynamics. We explain why those structures are rare to find when trying to explain long-term responses to disturbances, but we can also show that the most violent initial reactions can indeed be explained by only using these small graphs, which we call functional motifs. In short, big systems don't usually fail because the whole network is weak; they fail because tiny, influential clusters within them are prone to overreacting. By finding and monitoring these small groups, we can better predict and prevent systemic failures before they start.

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Photo by JJ Ying on Unsplash

Photo by JJ Ying on Unsplash

Why is it important?

This work addresses one of the biggest challenges in science today: predicting how a massive, complex system will react when it is suddenly "pushed" by a disturbance. Even though these systems are large, it is usually the normal procedure to analyze them as a whole, with every single interaction, to understand them. We provide a mathematical "shortcut" by showing that we can identify risky subgroups that cause the most trouble. This is an especially interesting finding when data on the rest of the system is missing. This works because we shift the focus away from a traditional concept of long-term behavior called "stability" and toward the immediate response called "reactivity". The Problem with Stability When searching for smaller structures that could help to explain why certain systems are more stable than others, the first look went to a system's stability. This is a reasonable approach, especially for systems like food webs that developed naturally through a long process of trial and error until they ended up in the networks we see today. Some smaller substructures in these networks have been shown to be overrepresented compared to totally random networks. One can imagine that there has to be a reason why these substructures are favored. However, we prove that such groups are mathematically rare and hard to find because long-term stability usually depends on the entire network working together. The Breakthrough with Reactivity Reactivity is the immediate shock response a system shows right after it is disturbed. For a system that is able to return to a stable equilibrium state after a disturbance, this way back to normal can happen in two ways. Either it does not respond to the shock and returns right back to where it started. More important is the second way of response: Right after a disturbing event, the system can react by amplifying the shock and push further away from a stable state. This reaction is ground for concern since the "push" can be enough to cross a tipping point. We discovered that "every small group is a functional motif for reactivity". This means that the violent reaction of a giant network is actually driven by tiny, localized hotspots. In fact, our models showed that groups of just two or three members can account for over 99% of a system's total overreaction. We offer a way to simplify how we monitor critical systems. Although we explain our work using food webs as an example system, the same theory applies to all kinds of networks, like power grids, epidemics, or global supply chains. Ultimately, this work allows us to simplify complex problems. By focusing on these tiny, high-risk groups, we can better predict and manage failures in networks.