Upper Bounds on the Quantifier Depth for Graph Differentiation in First Order Logic

Abstract

We show that on graphs with n vertices the 2-dimensional Weisfeiler-Leman algorithm requires at most O n 2 / log(n) iterations to reach stabilization. This in particular shows that the previously best, trivial upper bound of O(n 2 ) is asymptotically not tight. In the logic setting this translates to the statement that if two graphs of size n can be distinguished by a formula in first-order logic with counting with 3 variables (i. e., in C 3 ), then they can also be distinguished by a C 3 -formula that has quantifier depth at most O n 2 / log(n) .To prove the result we define a game between two players that enables us to decouple the causal dependencies between the processes happening simultaneously over several iterations of the algorithm. This allows us to treat large color classes and small color classes separately. As part of our proof we show that for graphs with bounded color class size, the number of iterations until stabilization is at most linear in the number of vertices. This also yields a corresponding statement in first-order logic with counting.Similar results can be obtained for the respective logic without counting quantifiers, i. e., for the logic L 3 .

Walk refinement, walk logic, and the iteration number of the Weisfeiler-Leman algorithm

“…We should remark that while the combinatorial techniques from [12] showing the upper bound of O(n 2 / log n) also translate to the setting without counting, our techniques seem to strongly rely on counting, since only the counting itself ensures the correspondence to matrix algebras that we exploit.…”

“…With regard to lower bounds, Fürer [8] proved that there are graphs on which the stabilization number is in Ω(n). In [12], using combinatorial techniques and a case distinction into small and large vertex-color classes, the currently best upper bound of O(n 2 / log n) for the iteration number was proven. In this paper we take an algebraic approach to the problem and show the following upper bound.…”

The Iteration Number of Colour Refinement

“…Similarly as for Colour Refinement, one can consider the number WL k (n) of iterations of k-WL on graphs of order n. Notably, contrasting our results for Colour Refinement, in [21], it was first proved that the trivial upper bound of WL 2 (n) ≤ n 2 − 1 is not even asymptotically tight (see also the journal version [22]). This foundation fostered further work, leading to an astonishingly good new upper bound of O(n log n) for the iteration number of 2-WL [27].…”

The Power of the Weisfeiler-Leman Algorithm for Machine Learning with Graphs

“…Figure 1, which is an important feature in social network analysis. Therefore, it has been generalized to k-tuples leading to a more powerful graph isomorphism heuristic, which has been investigated in depth by the theoretical computer science community [Cai et al, 1992;Kiefer and Schweitzer, 2016;Babai, 2016;Grohe, 2017]. In Shervashidze et al [2011], the 1-WL was first used to obtain a graph kernel, the so-called Weisfeiler-Lehman subtree kernel.…”

“…Kiefer et al, 2015]. Moreover, upper bounds on the running time[Berkholz et al, 2013], and the number of iterations for k = 1[Kiefer and McKay, 2020], and the number of iterations for k = 2[Kiefer and Schweitzer, 2016;Lichter et al, 2019] have been shown. For k = 1 and 2, Arvind et al[2019] studied the abilities of the (folklore) k-WL to detect and count fixed subgraphs, extending the work ofFürer [2017].…”