Arthur Queiroz Moura | Searcharxiv

arXiv2026

Message-passing graph persistence

Message-passing graph neural networks (GNNs) are limited in expressive power by the one-dimensional Weisfeiler-Leman test. Persistent homology (PH) provides topological descriptors that can complement GNN representations, but existing approaches typically compute persistence separately from the representation learning process. In this paper, we introduce Message-Passing Graph Persistence (MGP), a family of topological descriptors that incorporates message passing into graph filtrations. At each filtration step, node embeddings are updated using the current subgraph and associated with persistence events. These embeddings also determine the representatives of connected components during persistence computation. We prove that MGP is invariant under graph isomorphism and that it is strictly more expressive than the PH-based descriptors it extends. We evaluate MGP on an expressivity benchmark and on several graph classification and regression datasets. Experiments show that MGP outperforms other PH-based descriptors on most of these datasets.

cs.LG↗

arXiv2026

Group-product rigidity and Higman-Thompson reassociation groups

Fix an arity $r\ge 2$, a group $G$, and a full ordered $r$-ary tree $T$ with at least two internal vertices. For an arbitrary operation $ω:G^r\to G$, let $ω_T$ denote the operation obtained by iterating $ω$ according to $T$. We classify all $ω$ for which there exists a bijection $F_T:G\to G$ such that $ω_T(x_1,\ldots,x_n)=F_T(x_1\cdots x_n)$. We prove that $ω$ must have one of the forms $ax_1\cdots x_r$, $x_1\cdots x_rb$, $d\,ψ(x_1\cdots x_r)$, where $a,b\in G$, $d\in Z(G)$, and $ψ\in\operatorname{Aut}(G)$, with explicit conditions on the parameters determined by $T$. We next reverse the problem. Fix an operation $ω$ of one of these three forms and determine every full ordered $r$-ary tree $T$ for which there exists a bijection $F_T:G\to G$ satisfying $ω_T(x_1,\ldots,x_n)=F_T(x_1\cdots x_n)$. The answer is governed by the order of $aZ(G)$ or $bZ(G)$ in $G/Z(G)$, or by the order of $ψ$ in $\operatorname{Aut}(G)$. We also ask, for each of the three solution forms above, how much associativity remains. More precisely, for two full ordered $r$-ary trees $S$ and $T$ with the same number of leaves, we determine exactly when $ω_S=ω_T$. Pairs of $r$-ary trees encode changes of parenthesization, and modulo simultaneous expansion they represent elements of the Higman--Thompson group $F_r$. The changes of parenthesization that preserve the iterated operation form a subgroup of $F_r$, which we determine explicitly.

math.GR↗