article published yes David A. Brewster author Jakub Svoboda author 130759D2-D7DD-11E9-87D2-DE0DE66974250000-0002-1419-3267 Dylan Roscow author Krishnendu Chatterjee author 2E5DCA20-F248-11E8-B48F-1D18A9856A870000-0002-4561-241X Josef Tkadlec author 3F24CCC8-F248-11E8-B48F-1D18A9856A870000-0002-1097-9684 Martin A. Nowak author KrCh department Formal Methods for Stochastic Models: Algorithms and Applications project We examine population structures for their ability to maintain diversity in neutral evolution. We use the general framework of evolutionary graph theory and consider birth–death (bd) and death–birth (db) updating. The population is of size N. Initially all individuals represent different types. The basic question is: what is the time TN until one type takes over the population? This time is known as consensus time in computer science and as total coalescent time in evolutionary biology. For the complete graph, it is known that TN is quadratic in N for db and bd. For the cycle, we prove that TN is cubic in N for db and bd. For the star, we prove that TN is cubic for bd and quasilinear (N log N) for db. For the double star, we show that TN is quartic for bd. We derive upper and lower bounds for all undirected graphs for bd and db. We also show the Pareto front of graphs (of size N = 8) that maintain diversity the longest for bd and db. Further, we show that some graphs that quickly homogenize can maintain high levels of diversity longer than graphs that slowly homogenize. For directed graphs, we give simple contracting star-like structures that have superexponential time scales for maintaining diversity. https://research-explorer.ista.ac.at/download/20254/20280/2025_PNASNexus_Brewster.pdf application/pdfno https://creativecommons.org/licenses/by/4.0/ Oxford University Press2025 eng 2752-6542 2503.0984110.1093/pnasnexus/pgaf252 48 Brewster, David A., Jakub Svoboda, Dylan Roscow, Krishnendu Chatterjee, Josef Tkadlec, and Martin A. Nowak. “Maintaining Diversity in Structured Populations.” <i>PNAS Nexus</i>. Oxford University Press, 2025. <a href="https://doi.org/10.1093/pnasnexus/pgaf252">https://doi.org/10.1093/pnasnexus/pgaf252</a>. D. A. Brewster, J. Svoboda, D. Roscow, K. Chatterjee, J. Tkadlec, and M. A. Nowak, “Maintaining diversity in structured populations,” <i>PNAS Nexus</i>, vol. 4, no. 8. Oxford University Press, 2025. Brewster, D. A., Svoboda, J., Roscow, D., Chatterjee, K., Tkadlec, J., &#38; Nowak, M. A. (2025). Maintaining diversity in structured populations. <i>PNAS Nexus</i>. Oxford University Press. <a href="https://doi.org/10.1093/pnasnexus/pgaf252">https://doi.org/10.1093/pnasnexus/pgaf252</a> Brewster DA, Svoboda J, Roscow D, Chatterjee K, Tkadlec J, Nowak MA. Maintaining diversity in structured populations. <i>PNAS Nexus</i>. 2025;4(8). doi:<a href="https://doi.org/10.1093/pnasnexus/pgaf252">10.1093/pnasnexus/pgaf252</a> Brewster DA, Svoboda J, Roscow D, Chatterjee K, Tkadlec J, Nowak MA. 2025. Maintaining diversity in structured populations. PNAS Nexus. 4(8), pgaf252. Brewster, David A., et al. “Maintaining Diversity in Structured Populations.” <i>PNAS Nexus</i>, vol. 4, no. 8, pgaf252, Oxford University Press, 2025, doi:<a href="https://doi.org/10.1093/pnasnexus/pgaf252">10.1093/pnasnexus/pgaf252</a>. D.A. Brewster, J. Svoboda, D. Roscow, K. Chatterjee, J. Tkadlec, M.A. Nowak, PNAS Nexus 4 (2025). 202542025-08-31T22:01:32Z2026-06-11T09:11:17Z