'Estimating transmission noise on networks from stationary local order'

14 May 2025

New discovery by international team of researchers published in Europhysics Letters

Christopher R. Kitching, Henri Kauhanen, Jordan Abbott, Deepthi Gopal, Ricardo Bermúdez-Otero & Tobias Galla. 2025. Estimating transmission noise on networks from stationary local order. Europhysics Letters 150(3): 331002.

DOI 10.1209/0295-5075/adc9de

Published on 6 May 2025.

“You see, my son, here time becomes space.” Old Gurnemanz pronounces these resonant words as he leads Parsifal into the hall of the Grail in the first act of Wagner’s famous opera. It turns out, however, that many complex systems in the real world behave in just this way, turning patterns in time into patterns in space: more specifically, traits that are stable in time become clumped in space, and traits that change often as time passes become scattered in space. This discovery, published on 6 May 2025 in the journal Europhysics Letters, is very good news for researchers across a potentially wide range of disciplines in the sciences and the humanities: typically, it is comparatively easy to find out how a system patterns in space now, and much harder to get data on how it evolved in the past, but the article describes a method for recovering information about stability in time from observations about scattering in space.

This result is the product of work by an international team of researchers, with Christopher R. Kitching, a doctoral student at the Department of Physics and Astronomy of the University of Manchester, as the corresponding author. The team’s expertise spans interdisciplinary physics (Christopher R. Kitching, University of Manchester; Jordan Abbott, University of Manchester; Tobias Galla, IFISC, Palma de Mallorca) and linguistics (Henri Kauhanen, University of Konstanz; Deepthi Gopal, Uppsala University; Ricardo Bermúdez-Otero, University of Manchester). The unusual composition of the team reflects the unusual origins of the idea: this insight into the connection between temporal stability and spatial scattering did not originate in pure mathematics or in fundamental physics, but rather in linguistic typology, the discipline that studies how structural traits pattern across the approximately 7000 human languages spoken around the world.

Roughly half a century ago, the founding father of linguistic typology, Joseph Greenberg, made a brilliant guess. He knew that some traits of languages are relatively stable in time, undergoing change only rarely: for example, a language that puts the object before the verb like classical Latin (puer puellam amat ‘boy girl loves’) can change into one that puts the verb before the object like present-day Spanish (el niño ama a la niña ‘the boy loves the girl’), and vice versa; but that happens relatively rarely. Other traits of languages are more unstable, coming and going more frequently: thus, languages that do not have an equivalent of the English definite article the, like Latin, can acquire it relatively easily; those that have it, like Spanish, can lose it relatively easily. Greenberg had the intuition that the more stable traits, like the order of the verb and the object, would be clumped in space: on a linguistic atlas, groups of languages with verb-object order do indeed form large blobs, as do groups of languages with object-verb order. In contrast, more unstable traits, like the presence or absence of a definite article, would be scattered in space: instead of forming large clumps, languages with and without definite articles are interspersed among each other.

As it happens, Greenberg was right, but it took a long time for this to become clear: members of the team proved it in a 2021 article in Science Advances, where they showed that linguistic traits could be ranked by stability (e.g. the order of verb and object is more stable than the presence or absence of a definite article) just by looking at their scattering on a linguistic atlas. Remarkably, the Greenberg-inspired method performed extremely well against benchmarks established by means of far more data-intensive procedures (notably procedures that used information about the genealogy of languages, such as the fact that English is a sister of German, and Hungarian a sister of Finnish).

Yet the very ease with which this method recovered temporal information from spatial data was a surprise. And the mystery only deepened if you looked under the method’s bonnet. To keep the mathematics tractable, the team had made some simplifying assumptions: notably, they assumed that languages were arranged in a regular square lattice. According to this innocent-sounding statement, the Earth should look more like a doughnut covered in a square grid than like the slightly flattened sphere with oceans and continents that we know and love. How could the method work so well under such drastic idealization?

The article just published in Europhysics Letters undertook extensive mathematical analysis to solve the mystery. It turns out that the method inspired by linguistic typology is far more powerful than the team initially suspected. For a large class of complex networks, the relative stability of traits can be inferred from their scattering in space; even if little is known about how the elements of the system actually connect to one another, it is still possible to rank traits in order of stability. Thus, by mathematical necessity rather than mere good luck, the simplifying assumption of the 2021 study (that languages were arranged in a square grid wrapped over a doughnut) had made no difference; the same ranking of linguistic traits by stability would be obtained with a more realistic graph showing the actual connections of languages to their neighbours in geographical space.

The striking robustness of the method suggests a potential for applications across a wide range of domains. Many disciplines produce maps showing how traits of interest are distributed in space: for example, ethnographic atlases show the incidence of specific cultural practices (e.g. property systems, marriage rules) across preindustrial communities across the globe; similarly, ethologists can plot geographical variation in culturally transmitted animal behaviours such as birdsong. In many such cases, it now seems possible to use this easily accessible spatial information to find out if one trait changes more frequently in time than another. That is because all around us, as in the hall of the Grail, patterns in time become patterns in space.