Using covariance of node states to design early warning signals for network dynamics
Shilong Yu, Neil G. MacLaren, Naoki Masuda · Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences · 2026
Real-life systems often experience regime shifts. An early warning signal (EWS) is a quantity that attempts to anticipate such a regime shift. Because complex systems of practical interest showing regime shifts are often dynamics on networks, a research interest is to design EWSs for networks, including determining sentinel nodes that are useful for constructing high-quality EWSs. Previous work has shown that the sample variance is a viable EWS, including in the case of networks. We explore the use of the sample covariance of two nodes, or sentinel node pairs, for improving EWSs for networks. We perform analytical calculations in four-node networks and numerical simulations in larger networks to find that the sample covariance and its combination over node pairs is inferior to the sample variance and its combination over nodes; the latter are previously proposed EWSs based on sentinel node selection. The present results support the predominant use of diagonal entries of the covariance matrix (i.e. variance) as opposed to off-diagonal entries in EWS construction. This article is part of the theme issue 'Critical transitions and intelligent control in complex systems'.