at least when it comes to complex systems like the power grid, food webs and advanced materials.
For decades, scientists generally assumed that networks function most reliably when their individual components are as similar as possible. But real-world networks are rarely uniform. Generators in a power grid, neurons in a brain, animals in a food web and components in a material all differ in ways that scientists traditionally treated as imperfections.
Now, Northwestern University physicists are overturning that long-held assumption.
In a new study, the scientists developed a mathematical framework that identifies when these differences - a form of variation known as disorder - can actually make a system more stable. The team found that many physical, engineered and biological complex systems can become more stable when their components, or the interactions among them, are different.
Rather than treating variation as a flaw, the findings suggest scientists and engineers could harness deliberately designed differences to build more robust power grids, architected materials and other interconnected systems. The work also could help explain why disorder is so prevalent in natural networks, including neural, biological and ecological systems.
The study was published today (Sept. 17) in the journal Science. The researchers also developed a website that allows users to visualize how the framework works. By changing various parameters, users can watch network components interact with one another, synchronize and organize into patterns.
"Previous studies found a growing number of cases in which disorder (also called heterogeneity, irregularity or asymmetry) across a network's nodes can actually improve stability and desirable behavior," said Northwestern's Adilson Motter, who led the work. "We have seen this in important real-world systems, including power grids, metamaterials and brain computation. But we didn't know how widespread this effect was or which kinds of systems could benefit from it. Our new study answers those questions, explains why these differences can improve stability and even reveals why scientists overlooked this effect for so long."
An expert in complex systems, Motter is the Charles E. and Emma H. Morrison Professor of Physics and Astronomy at Northwestern's Weinberg College of Arts and Sciences and the director of the Center for Network Dynamics. Northwestern postdoctoral researcher Arthur Montanari and graduate student Pietro Zanin, both members of the Motter group, are the study's co-first authors.
The problem with previous models
From flocks and power grids to ecosystems and materials, interconnected systems depend on their ability to withstand disturbances. A gust of wind can scatter a flock, a sudden surge in demand can disrupt a power grid and an impact can deform a material. Scientists study these systems as networks comprising individual components, or nodes, connected by links. In a power grid, for example, generators are the nodes and transmission lines represent the links. And in an ecological network, species are the nodes while their relationships (competition, cooperation and predation) are the links.
Traditionally, network scientists have focused on how these components are connected. They also have relied on simplified mathematical models, such as the widely used Kuramoto model, that describe each node using only one variable. Although these models have provided important insights, they can miss effects that emerge in real systems, where components and interactions have richer dynamics.
"Real systems are rarely uniform," Montanari said. "Birds differ in personalities, neurons vary in shape and even our social relationship can be asymmetric. These differences might appear random, but they can profoundly affect how the whole system behaves."
Previous case studies hinted that these differences could be beneficial. In a 2020 Nature Physics study, Motter's team found that power generators could synchronize more effectively when they operated slightly differently from one another. In a 2025 Nature Communications study, Montanari found similar effects in models of flocking and drone swarms. But researchers did not know whether these findings were isolated examples or evidence of a broader principle.
"Disorder can stabilize networks, but only when the node dynamics are rich enough," Motter said. "Simplified models can inadvertently strip away the very stabilizing effect we want to capture."
Engineering optimal stability
To examine how more realistic network dynamics affect stability, the researchers developed a general mathematical framework. First, they analyzed systems near a stable state. Then, they calculated whether small disturbances faded away - allowing the system to return to its stable state - or grew, pushing the system toward instability. Finally, the team compared networks comprising identical components to networks with varying components and connections. Within this framework, they identified the general conditions under which heterogeneity can outperform uniformity.
Motter and his team tested this framework on models of power grids, neurons, flocks, architected materials and ecological networks. They found two ways disorder can enhance stability: through differences among the network's nodes or among the links connecting them. Overall stability also depends on where the differences occur and how much variation is present. A moderate degree of disorder might enhance stability, for example, while too much could destabilize the same system.
"If you make the system more homogeneous, you lose stability," Montanari said. "But if you increase disorder too much, you also lose stability. Our framework can help pinpoint the level of disorder that helps the system achieve optimal stability."
The researchers also found it wasn't always necessary to design specific differences. In many of the models, even randomly introduced variation improved stability compared to the best completely uniform configuration. An important exception emerged when the disorder occurred in the links rather than among the nodes. In that case, even networks with simple node dynamics could benefit from disorder.
Understanding the role of imperfections
The study's findings could help scientists understand current complex systems, such as ecological networks, and inform the design of new systems, such as architected materials, from scratch.
"Since the 1970s, mathematical models have predicted that large, complex ecosystems should destabilize and collapse," Montanari said. "Yet very large and highly diverse ecosystems persist in nature. Our findings suggest that variation among mutually beneficial interactions, such as those between pollinators and flowers, could help explain this paradox."
Engineers also could put this principle into practice. Architected materials, for example, typically comprise repeated, identical building blocks. Deliberately varying those blocks' shapes, sizes, orientations and physical properties could enable new functions and behaviors across a broad range of applications. The crucial step is to recognize these materials as mechanical networks and build realistic models that preserve the networks' dynamics. Researchers then could use computational methods to search for the most beneficial patterns of disorder.
"When disorder enhances stability, the next challenge is figuring out how best to design it," Motter said.
The study, "Disorder-promoted stability," was supported by the Army Research Office (grant number W911NF-22-2-0109) and the National Science Foundation (grant number DMS-2308341). The study also acknowledges the stimulating research environment provided by the NSF-Simons National Institute for Theory and Mathematics in Biology (NSF grant number DMS-2235451 and Simons Foundation grant number MP-TMPS-00005320).
