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Associate Professor of Computer Science Majid Zamani and co-authors have accepted the 2023 IEEE Control Systems Society George S. Axelby Outstanding Paper Award — an honor with a legacy spanning five decades — for the paper entitled "A Lyapunov-based small-gain theorem for infinite networks."

"As a scientist in the field of control theory, this is one of my dream awards. It's the highest recognition I've received in this field. The people who have won this award in the past are pioneers of the field, and I'm thrilled to be acknowledged alongside them," Zamani said. 

The paper takes a transformative view. Previously, scientists had been able to show that stability analysis of large but finite interconnected networks, such as power grids or traffic flow, can be tackled by breaking the networks down into their parts — say, the traffic on a single road — and checking the stability of those parts.  

However, this divide-and-conquer strategy had never been comprehensively theorized for infinite networks, until now. 

Infinite possibilities

This matters because, as our lives become ever more interconnected, we still want to be able to make promises about the systems that we control and model, no matter their complexity and size.

This theorem allows scientists to analyze systems composed of infinitely many complex components. Infinite networks naturally arise as over-approximations of finite yet extremely large networks, potentially encompassing unknown numbers of subsystems.

"We needed to have a fundamental result that works for networks with infinitely many components, and this allows for that,” Zamani said. “You can look at very large-scale networks with possibly unknown numbers of subsystems as infinite networks." 

The theorem has profound impacts across a vast number of fields, from control of quantum systems to consensus in social networks, systems theory and functional analysis. 

"Our result actually generalizes all the existing attempts that have been made to deal with networks of infinite size," Zamani said. 

A safer world 

This theorem is another step toward the guarantees we need for safe, high-performing systems. Complex systems are everywhere around us, from platoons of vehicles to ocean tides and cloud flows and even how glucose flows through the bloodstream. 

However, we must accept that the models we create are imperfect. When controlling a complicated system, there will always be uncertainty because real-life scenarios have too many factors to be properly modeled.

It is crucial that we can analyze the systems we construct for robustness and safety, regardless of their complexity. 

New horizons

When introducing infinite networks, Zamani and his co-authors realized that many of the field's well-established results no longer hold. "Apart from the impact, in terms of a high-level mathematical problem, it was very challenging," Zamani said. 

This leads to an exciting new threshold for fundamental knowledge. 

"Because existing mathematical techniques don't necessarily carry over to infinite networks, you need to resort to different tools, and create new ones," Zamani said.