The Power of the Weisfeiler-Leman Algorithm to Decompose Graphs

The Weisfeiler-Leman procedure is a widely-used approach for graph isomorphism testing that works by iteratively computing an isomorphism-invariant coloring of vertex tuples. Meanwhile, a fundamental tool in structural graph theory, which is often exploited in approaches to tackle the graph isomorphism problem, is the decomposition into 2- and 3-connected components. We prove that the 2-dimensional Weisfeiler-Leman algorithm implicitly computes the decomposition of a graph into its 3-connected components. Thus, the dimension of the algorithm needed to distinguish two given graphs is at most the dimension required to distinguish the corresponding decompositions into 3-connected components (assuming it is at least 2). This result implies that for k >= 2, the k-dimensional algorithm distinguishes k-separators, i.e., k-tuples of vertices that separate the graph, from other vertex k-tuples. As a byproduct, we also obtain insights about the connectivity of constituent graphs of association schemes. In an application of the results, we show the new upper bound of k on the Weisfeiler-Leman dimension of graphs of treewidth at most k. Using a construction by Cai, F\"urer, and Immerman, we also provide a new lower bound that is asymptotically tight up to a factor of 2.Comment: 30 pages, 4 figures, full version of a paper accepted at MFCS 201

Une rencontre entre les noyaux de graphes et la détection d’anomalies dans les réseaux

International audienceLa détection d’anomalies demeure une tâche cruciale pour assurer une gestion efficace et flexible d’un réseau. Récemment, les noyaux de graphes ont connu un grand succès dans de nombreux domaines, notamment en bio-informatique et vision artificielle. Notre travail vise à étudier leur pouvoir de discrimination dans le domaine des réseaux afin de détecter les vulnérabilités et catégoriser le trafic. Dans cet article, nous présentons Nadege, un système d’apprentissage à l’intérieur duquel nous concevons un nouveau noyau de graphe adapté au profilage de réseaux. De surcroît, nousproposons des algorithmes avec des garanties d’approximation théoriques ainsi qu’une politique de détection hybride. Finalement, nous évaluons les performances de Nadege en menant des expériences approfondies sur une variété d’environnements réseaux. Pour différents scénarios, nous montrons son efficacité à empêcher les anomalies de perturber le réseau tout en fournissant une assistance pour la surveillance du trafic

Logarithmic Weisfeiler-Leman Identifies All Planar Graphs

The Weisfeiler-Leman (WL) algorithm is a well-known combinatorial procedure for detecting symmetries in graphs and it is widely used in graph-isomorphism tests. It proceeds by iteratively refining a colouring of vertex tuples. The number of iterations needed to obtain the final output is crucial for the parallelisability of the algorithm. We show that there is a constant k such that every planar graph can be identified (that is, distinguished from every non-isomorphic graph) by the k-dimensional WL algorithm within a logarithmic number of iterations. This generalises a result due to Verbitsky (STACS 2007), who proved the same for 3-connected planar graphs. The number of iterations needed by the k-dimensional WL algorithm to identify a graph corresponds to the quantifier depth of a sentence that defines the graph in the (k+1)-variable fragment C^{k+1} of first-order logic with counting quantifiers. Thus, our result implies that every planar graph is definable with a C^{k+1}-sentence of logarithmic quantifier depth