By Dave DeFusco 

In a groundbreaking study, researchers have revealed crucial insights into how failures spread in interconnected networks, offering new hope for managing systemic risks in various fields, from finance to infrastructure. Their paper,  appears in Physical Review E.

Cascading failures occur when the failure of a single component in a network leads to subsequent failures, potentially causing a system-wide collapse. This phenomenon is especially critical in interdependent networks, where the functioning of one network depends on another. They take place on the internet, where traffic is rerouted to bypass malfunctioning routers unequipped to handle extra traffic. Most failures emerge and evaporate locally, largely unnoticed. A few, however, percolate through dense technological and social networks, hitting users from the most unexpected directions. 

During blackouts, a cascading failure occurs when power lines become overburdened through a surge of electricity, and the grid that distributes the electricity malfunctions in a short period of time. The magnitude of a blackout is rooted in an often-ignored aspect of a globalized world: vulnerability due to interconnectivity.

The study delves into this by examining two interconnected networks where links between them are directional, meaning the relationship flows one way, like a supplier and a customer. The researchers used a mathematical approach, called generating function formalism, to model these cascading failures. This method allowed them to create a set of equations that accurately describe how failures propagate through large networks. The study’s key discovery is that the behavior of these cascading failures depends mainly on the incoming links to each node in the network.

“Through extensive simulations, we found that the type of outgoing links did not significantly impact the overall failure process,” said Bo Tong, lead author of the paper and a Ph.D. student in Mathematics at the Katz School. “This was true for various types of networks, including those with random link distributions, called the Poisson distributions, and those with more complex, uneven distributions, called Pareto. However, networks with certain Pareto distributions experience a more prolonged period of instability before they settle down.”

The researchers found that directional networks can be either more or less vulnerable than their bidirectional counterparts, depending on specific conditions. For practical purposes, they highlighted the importance of predicting whether a network will gradually respond to failures or experience a catastrophic collapse if the initial failure exceeds a certain threshold. Networks with lower average connections were found to be more resilient to sudden, large-scale collapses.

“This research has significant implications for real-world networks,” said David Roth, a co-author of the paper and student in the M.A. in Physics at the Katz School. “For instance, in financial systems, directional links can represent the flow of assets and liabilities between banks. The study’s findings help explain how financial crises, like one that precipitated the Great Recession in 2008, can lead to widespread failures across interconnected institutions.”

By comparing their results to the well-known phase transitions in physics, like liquid to gas, the researchers provided a new way to understand these network collapses. They showed that similar to how water turns into steam at a critical temperature, networks have critical points where small increases in initial failures can lead to dramatic collapses.

In their final analysis, the researchers expanded their model to consider attacks on both interconnected networks simultaneously. They found that their theoretical predictions matched well with simulation results, reinforcing the robustness of their model. This research opens new avenues for further exploration, particularly in developing strategies to strengthen network resilience and mitigate the impact of cascading failures.

“This study marks a significant advancement in understanding cascading failures in interdependent networks,” said Sergey Buldryev, a co-author of the paper and professor of physics at and Boston University. “By focusing on the directionality of links and the specific conditions that lead to network collapse, we have provided valuable tools for predicting and managing risks in complex systems. These insights could prove vital in designing more resilient infrastructures and financial systems, ultimately helping to prevent future crises.”