Jean-Jil Duchamps

ANR project: Graphes Aléatoires pour les Réseaux Phylogénétiques

ANR project ID: ANR-24-CE40-7154

The ANR-funded GARP project (graphes aléatoires pour les réseaux phylogénétiques, i.e. random graphs for phylogenetic network) is active for the 2025-2028 period. It is a starting grant (JCJC) that I obtained with François Bienvenu, with a main focus on studying new stochastic models of phylogenetic networks. More generally, we are interested in understanding the geometries of networks generated by models exhibiting both branching and coalescence mechanisms.

We have obtained the recruitment of a postdoc in Laboratoire de mathématiques de Besançon, and more precisely within the (informal) research group that is focused on probabilistic modeling of ecology and evolutionary biology. The two-year position was obtained by Frederic Alberti, who started in October 2025.

Works and events supported by the project

We co-organize and participate in funding the events organized in Besançon around stochastic modeling in ecology and evolution, including a 3-day workshop in September 2026.

Publications and preprints

• A Parameterized Family of Balance Indices for Phylogenetic Networks (2026).
  → with F Bienvenu and H Maffioli – arXiv. HAL.
  ShowHide abstract

  We introduce a new family of balance indices for phylogenetic networks: the $H_{\alpha }$ indices, where $\alpha$ is a positive real number. This family includes the $B_2$ index as a special case ($\alpha =1$) and provides a natural extension of the Sackin index to phylogenetic networks. We show that the $H_{\alpha }$ indices share many structural properties with the $B_2$ index, most notably a "grafting property" that makes it possible to express the $H_{\alpha }$ index of a network in terms of the $H_{\alpha }$ indices of its biconnected components. These properties allow us to identify networks that minimize / maximize $H_{\alpha }$ for various classes of phylogenetic networks, and to study its distribution for several models of random trees and networks (in particular, Galton-Watson trees and binary Markov branching trees, with a focus on the Yule and PDA models). Finally, we show how local limits can be used to analyze the asymptotic behavior of $H_{\alpha }$ for large trees and networks, and we obtain general results for the moments of $H_{\alpha }$ for a broad class of random phylogenetic networks known as blowups of Galton-Watson trees.

• Sharp $L\log L$ Condition for Supercritical Galton-Watson Processes with Countable Types (2025).
  → with M André – arXiv. HAL.
  ShowHide abstract

  We investigate Kesten-Stigum-like results for multi-type Galton-Watson processes with a countable number of types in a general setting, allowing us in particular to consider processes with an infinite total population at each generation. Specifically, a sharp $L\log L$ condition is found under the only assumption that the mean reproduction matrix is positive recurrent in the sense of Vere-Jones (1967). The type distribution is shown to always converge in probability in the recurrent case, and under conditions covering many cases it is shown to converge almost surely.

• The $B_2$ Index of Galled Trees (2024).
  → with F Bienvenu, M Fuchs and TC Yu – arXiv. HAL.
  ShowHide abstract

  In recent years, there has been an effort to extend the classical notion of phylogenetic balance, originally defined in the context of trees, to networks. One of the most natural ways to do this is with the so-called $B_2$ index. In this paper, we study the $B_2$ index for a prominent class of phylogenetic networks: galled trees. We show that the $B_2$ index of a uniform leaf-labeled galled tree converges in distribution as the network becomes large. We characterize the corresponding limiting distribution, and show that its expected value is 2.707911858984... This is the first time that a balance index has been studied to this level of detail for a random phylogenetic network.
  One specificity of this work is that we use two different and independent approaches, each with its advantages: analytic combinatorics, and local limits. The analytic combinatorics approach is more direct, as it relies on standard tools; but it involves slightly more complex calculations. Because it has not previously been used to study such questions, the local limit approach requires developing an extensive framework beforehand; however, this framework is interesting in itself and can be used to tackle other similar problems.