In 1901, the writer and Nobel laureate Maurice Maeterlinck published “The Life of the Bee”, which popularized the idea that humanity owes our continued survival to the dutiful pollinator. “It is … estimated that more than a hundred thousand varieties of plants would disappear if the bees did not visit them,” Maeterlinck noted, “and possibly even our civilization, for in these mysteries all things intertwine.”

Today, we are getting perilously close to testing Maeterlinck’s hypothesis empirically. Bee colonies, which are responsible for billions of dollars of agriculture, food security, and ecosystem services, are collapsing around the world. It would not take the death of the last hive for life to become grim for at least some of the people, plants and animals who depend on bees, either directly or indirectly.

But precisely how many bee colonies have to collapse before the larger system fails? Are we near the tipping point? Here, we are on much murkier ground.

The same dynamic holds true in our understanding of countless other marquee challenges of our present age, from ocean warming and climate change, to the genetic basis of cancer, the performance of electrical grids, digital networks, and even the stock market. While all of these complex systems have compensatory elements, there is a point when the accumulation of damage becomes unrecoverable, and catastrophic collapse follows. But when?

Last month, network theorists Jianxi Gao, Baruch Barzel and Albert-László Barabási published an important paper in Nature, Universal resilience patterns in complex networks, which takes us a significant step closer to determining these pernicious tipping points. The authors present a universal mathematical framework which allows one to compute a straightforward “resilience function” (or perhaps more aptly, a “collapse threshold”) for any complex system – its point of no return.

Until now, the only way to identify such thresholds was to painstakingly map the interactions of all of the individual elements in the system – a task that was impractical or impossible in most circumstances. (Even our most sophisticated models of the climate, run on supercomputers, model only a fraction of the forces acting upon it.) Gao, Barzel and Barabási have taken much of that complexity out, collapsing it into a single indicator of system health. As such, if borne out, this work will certainly find its way into applied resilience domains – especially disaster risk reduction, climate adaptation and finance.

The above video, which acts as a companion to the paper, does a terrific job of explaining the work (and key concepts of ecological resilience in general).