The power of stochastic differential equations

A new algorithm developed by Naoki Masuda, with co-athors Kazuyuki Aihara and Neil G. MacLaren, can identify the most predictive data points that a tipping point is near. Published in Nature Communications, this theoretical framework uses the power of stochastic differential equations to observe the fluctuation of data points, or nodes, and then determine which should be used to calculate an early warning signal. The algorithm is unique in that it fully incorporates network science into the process.

Illustration: A demonstration of the study's proposed method for selecting sentinel nodes to construct early warning signals. Left: A network of a freshwater stream food web. Right: A sample of 5,000 different combinations of five nodes selected from the network on the left. Kendall’s τ on the horizontal axis is a measure of how well the early warning signal predicts an on-coming transition. The combination of five nodes maximizing the metric d (shown in green in the network on the left) provides almost the best early warning signal (i.e., almost the largest τ, as shown on the right panel). This figure and caption are provided by Neil G. MacLaren and Naoki Masuda.

Research News

New algorithm cuts through ‘noisy’ data to better predict tipping points

Naoki Masuda, professor of mathematics, is the lead author of a study that identifies the best data points for prediciting tipping points across various systems. Photo: Meredith Forrest Kulwicki/University at Buffalo

UB mathematicians’ theory determines which data points matter most when calculating early warning signals

Release Date: April 26, 2024

Print
Naoki Masuda, with the department of mathematics, poses for a portrait in a common study space in the Mathematics Building in February 2024. Photographer: Meredith Forrest Kulwicki.

Naoki Masuda

Neil MacLaren.

Neil MacLaren

“Every node is somewhat noisy ... but some may change earlier and more drastically than others when a tipping point is near. ”
Naoki Masuda, professor and director of graduate studies
University at Buffalo Department of Mathematics

BUFFALO, N.Y. — Whether you’re trying to predict a climate catastrophe or mental health crisis, mathematics tells us to look for fluctuations. 

Changes in data, from wildlife population to anxiety levels, can be an early warning signal that a system is reaching a critical threshold, known as a tipping point, in which those changes may accelerate or even become irreversible. 

But which data points matter most? And which are simply just noise?

A new algorithm developed by University at Buffalo researchers can identify the most predictive data points that a tipping point is near. Detailed in Nature Communications, this theoretical framework uses the power of stochastic differential equations to observe the fluctuation of data points, or nodes, and then determine which should be used to calculate an early warning signal. 

Simulations confirmed this method was more accurate at predicting theoretical tipping points than randomly selecting nodes.

“Every node is somewhat noisy — in other words, it changes over time — but some may change earlier and more drastically than others when a tipping point is near. Selecting the right set of nodes may improve the quality of the early warning signal, as well as help us avoid wasting resources observing uninformative nodes,” says the study’s lead author, Naoki Masuda, PhD, professor and director of graduate studies in the UB Department of Mathematics, within the College of Arts and Sciences.

The study was co-authored by Neil MacLaren, Phd, a postdoctoral research associate in the Department of Mathematics, and Kazuyuki Aihara, PhD, executive director of the International Research Center for Neurointelligence at the University of Tokyo. 

The work was supported by the National Science Foundation and the Japan Science and Technology Agency.

Warning signals connected via networks

The algorithm is unique in that it fully incorporates network science into the process. While early warning signals have been applied to ecology and psychology for the last two decades, little research has focused on how those signals are connected within a network, Masuda says. 

Consider depression. Recent research has considered it and other mental disorders as a network of symptoms influencing each other by creating feedback loops. A loss of appetite could mean the onset of five other symptoms in the near future, depending on how close those symptoms are on the network.

“As a network scientist, I felt network science could offer a unique or perhaps even improved approach to early warning signals,” Masuda says. 

By thoroughly considering systems as networks, researchers found that simply selecting the nodes with highest fluctuations was not the best strategy. That’s because some selected nodes may be too closely related to other selected nodes.

“Even if we combine two nodes with nice early warning signals, we don’t necessarily get a more accurate signal. Sometimes combining a node with a good signal and another node with a mid-quality signal actually gives us a better signal,” Masuda says. 

While the team validated the algorithm with numerical simulations, they say it can readily be applied to actual data because it does not require information about the network structure itself; it only requires two different states of the networked system to determine an optimal set of nodes. 

“The next steps will be to collaborate with domain experts such as ecologists, climate scientists and medical doctors to further develop and test the algorithm with their empirical data and get insights into their problems,” Masuda says.

Media Contact Information

Tom Dinki
News Content Manager
Physical sciences, economic development
Tel: 716-645-4584
tfdinki@buffalo.edu

Faculty Profile

Naoki Masuda

PhD

Naoki Masuda.

Naoki Masuda

PhD

Naoki Masuda

PhD

Professor, Director of Graduate Studies

Research Interests

Network science; mathematical biology.

Education

PhD, University of Tokyo

Research Summary

Network Science — dynamical processes (e.g. contagion processes, random walks, evolutionary dynamics, early warning signals) on networks, temporal (i.e., time-varying) networks, interdisciplinary applications

Mathematical Biology — gene networks, brain networks, collective behavior and networks of animals, evolutionary graph theory, evolutionary game theory

Selected Publications

Marie Saitou, Naoki Masuda, Omer Gokcumen. Similarity-based analysis of allele frequency distribution among multiple populations identifies adaptive genomic structural variants. Molecular Biology and Evolution, 39, msab313 (2021).

Naoki Masuda, Luis E. C. Rocha. A Gillespie algorithm for non-Markovian stochastic processes. SIAM Review, 60, 95-115 (2018).

Naoki Masuda, Mason A. Porter, Renaud Lambiotte. Random walks and diffusion on networks. Physics Reports, 716-717, 1-58 (2017). 

Naoki Masuda, Renaud Lambiotte. A Guide to Temporal Networks. World Scientific, Singapore (2016; 2020, Second Edition).