Nils M. Kriege

AAAI Conference 2026 Conference Paper

Spectral Basis Learning for Expressive Graph Neural Networks in Link Prediction

• Niloofar Azizi
• Nils M. Kriege
• Nicholas J. A. Harvey
• Horst Bischof

Graph Neural Networks (GNNs) excel in handling graph-structured data but often underperform in link prediction tasks compared to classical methods, mainly due to the limitations of the commonly used message-passing principle. Notably, their ability to distinguish non-isomorphic graphs is limited by the 1-dimensional Weisfeiler-Lehman test (WL). Our study presents a novel method to enhance the expressivity of GNNs by embedding induced subgraphs into the eigenbasis of the graph Laplacian. We introduce a Learnable Lanczos algorithm with Linear Constraints (LLwLC), proposing two novel subgraph extraction strategies: encoding vertex-deleted subgraphs and applying Neumann eigenvalue constraints. For the former, we demonstrate the ability to distinguish graphs that are indistinguishable by 2-WL, while maintaining efficiency. The latter focuses on link representations enabling differentiation between k-regular graphs and node automorphism, a vital aspect for link prediction tasks. Our approach results in a lightweight architecture, reducing the need for extensive training datasets. Empirically, our method improves performance in challenging link prediction tasks across benchmark datasets, establishing its practical utility and supporting our theoretical findings. Notably, LLwLC achieves 20x and 10x speedups by requiring only 5% and 10% of the data from the PubMed and OGBL-Vessel datasets, while comparing to the state-of-the-art.

ICML Conference 2025 Conference Paper

Weisfeiler and Leman Go Gambling: Why Expressive Lottery Tickets Win

• Lorenz Kummer
• Samir Moustafa
• Anatol Ehrlich
• Franka Bause
• Nikolaus Süss
• Wilfried N. Gansterer
• Nils M. Kriege

The lottery ticket hypothesis (LTH) is well-studied for convolutional neural networks but has been validated only empirically for graph neural networks (GNNs), for which theoretical findings are largely lacking. In this paper, we identify the expressivity of sparse subnetworks, i. e. their ability to distinguish non-isomorphic graphs, as crucial for finding winning tickets that preserve the predictive performance. We establish conditions under which the expressivity of a sparsely initialized GNN matches that of the full network, particularly when compared to the Weisfeiler-Leman test, and in that context put forward and prove a Strong Expressive Lottery Ticket Hypothesis. We subsequently show that an increased expressivity in the initialization potentially accelerates model convergence and improves generalization. Our findings establish novel theoretical foundations for both LTH and GNN research, highlighting the importance of maintaining expressivity in sparsely initialized GNNs. We illustrate our results using examples from drug discovery.

JMLR Journal 2023 Journal Article

Weisfeiler and Leman go Machine Learning: The Story so far

• Christopher Morris
• Yaron Lipman
• Haggai Maron
• Bastian Rieck
• Nils M. Kriege
• Martin Grohe
• Matthias Fey
• Karsten Borgwardt

In recent years, algorithms and neural architectures based on the Weisfeiler–Leman algorithm, a well-known heuristic for the graph isomorphism problem, have emerged as a powerful tool for machine learning with graphs and relational data. Here, we give a comprehensive overview of the algorithm’s use in a machine-learning setting, focusing on the supervised regime. We discuss the theoretical background, show how to use it for supervised graph and node representation learning, discuss recent extensions, and outline the algorithm’s connection to (permutation-)equivariant neural architectures. Moreover, we give an overview of current applications and future directions to stimulate further research. [abs] [ pdf ][ bib ] &copy JMLR 2023. ( edit, beta )

NeurIPS Conference 2022 Conference Paper

Weisfeiler and Leman Go Walking: Random Walk Kernels Revisited

• Nils M. Kriege

Random walk kernels have been introduced in seminal work on graph learning and were later largely superseded by kernels based on the Weisfeiler-Leman test for graph isomorphism. We give a unified view on both classes of graph kernels. We study walk-based node refinement methods and formally relate them to several widely-used techniques, including Morgan's algorithm for molecule canonization and the Weisfeiler-Leman test. We define corresponding walk-based kernels on nodes that allow fine-grained parameterized neighborhood comparison, reach Weisfeiler-Leman expressiveness, and are computed using the kernel trick. From this we show that classical random walk kernels with only minor modifications regarding definition and computation are as expressive as the widely-used Weisfeiler-Leman subtree kernel but support non-strict neighborhood comparison. We verify experimentally that walk-based kernels reach or even surpass the accuracy of Weisfeiler-Leman kernels in real-world classification tasks.

TCS Journal 2021 Journal Article

Fixed-parameter algorithms for the weighted Max-Cut problem on embedded 1-planar graphs

• Christine Dahn
• Nils M. Kriege
• Petra Mutzel
• Julian Schilling

We propose two fixed-parameter tractable algorithms for the weighted Max-Cut problem on embedded 1-planar graphs parameterized by the crossing number k of the given embedding. A graph is called 1-planar if it can be drawn in the plane with at most one crossing per edge. Our algorithms recursively reduce a 1-planar graph to at most 3 k planar graphs, using edge removal and node contraction. Our main algorithm then solves the Max-Cut problem for the planar graphs using the FCE-MaxCut introduced by Liers and Pardella [23]. In the case of non-negative edge weights, we suggest a variant that allows to solve the planar instances with any planar Max-Cut algorithm. We show that a maximum cut in the given 1-planar graph can be derived from the solutions for the planar graphs. Our algorithms compute a maximum cut in an embedded weighted 1-planar graph with n nodes and k edge crossings in time O ( 3 k ⋅ n 3 / 2 log ⁡ n ).

ICLR Conference 2020 Conference Paper

Deep Graph Matching Consensus

• Matthias Fey
• Jan Eric Lenssen
• Christopher Morris 0001
• Jonathan Masci
• Nils M. Kriege

This work presents a two-stage neural architecture for learning and refining structural correspondences between graphs. First, we use localized node embeddings computed by a graph neural network to obtain an initial ranking of soft correspondences between nodes. Secondly, we employ synchronous message passing networks to iteratively re-rank the soft correspondences to reach a matching consensus in local neighborhoods between graphs. We show, theoretically and empirically, that our message passing scheme computes a well-founded measure of consensus for corresponding neighborhoods, which is then used to guide the iterative re-ranking process. Our purely local and sparsity-aware architecture scales well to large, real-world inputs while still being able to recover global correspondences consistently. We demonstrate the practical effectiveness of our method on real-world tasks from the fields of computer vision and entity alignment between knowledge graphs, on which we improve upon the current state-of-the-art.

IJCAI Conference 2018 Conference Paper

A Property Testing Framework for the Theoretical Expressivity of Graph Kernels

• Nils M. Kriege
• Christopher Morris
• Anja Rey
• Christian Sohler

Graph kernels are applied heavily for the classification of structured data. However, their expressivity is assessed almost exclusively from experimental studies and there is no theoretical justification why one kernel is in general preferable over another. We introduce a theoretical framework for investigating the expressive power of graph kernels, which is inspired by concepts from the area of property testing. We introduce the notion of distinguishability of a graph property by a graph kernel. For several established graph kernels we show that they cannot distinguish essential graph properties. In order to overcome this, we consider a kernel based on k-disc frequencies. We show that this efficiently computable kernel can distinguish fundamental graph properties. Finally, we obtain learning guarantees for nearest neighbor classifiers in our framework.

MFCS Conference 2018 Conference Paper

Largest Weight Common Subtree Embeddings with Distance Penalties

• Andre Droschinsky
• Nils M. Kriege
• Petra Mutzel

The largest common embeddable subtree problem asks for the largest possible tree embeddable into two input trees and generalizes the classical maximum common subtree problem. Several variants of the problem in labeled and unlabeled rooted trees have been studied, e. g. , for the comparison of evolutionary trees. We consider a generalization, where the sought embedding is maximal with regard to a weight function on pairs of labels. We support rooted and unrooted trees with vertex and edge labels as well as distance penalties for skipping vertices. This variant is important for many applications such as the comparison of chemical structures and evolutionary trees. Our algorithm computes the solution from a series of bipartite matching instances, which are solved efficiently by exploiting their structural relation and imbalance. Our analysis shows that our approach improves or matches the running time of the formally best algorithms for several problem variants. Specifically, we obtain a running time of O(|T| |T'|Delta) for two rooted or unrooted trees T and T', where Delta=min{Delta(T), Delta(T')} with Delta(X) the maximum degree of X. If the weights are integral and at most C, we obtain a running time of O(|T| |T'|sqrt Delta log (C min{|T|, |T'|})) for rooted trees.

MFCS Conference 2016 Conference Paper

Faster Algorithms for the Maximum Common Subtree Isomorphism Problem

• Andre Droschinsky
• Nils M. Kriege
• Petra Mutzel

The maximum common subtree isomorphism problem asks for the largest possible isomorphism between subtrees of two given input trees. This problem is a natural restriction of the maximum common subgraph problem, which is NP-hard in general graphs. Confining to trees renders polynomial time algorithms possible and is of fundamental importance for approaches on more general graph classes. Various variants of this problem in trees have been intensively studied. We consider the general case, where trees are neither rooted nor ordered and the isomorphism is maximum w. r. t. a weight function on the mapped vertices and edges. For trees of order n and maximum degree Delta our algorithm achieves a running time of O(n^2*Delta) by exploiting the structure of the matching instances arising as subproblems. Thus our algorithm outperforms the best previously known approaches. No faster algorithm is possible for trees of bounded degree and for trees of unbounded degree we show that a further reduction of the running time would directly improve the best known approach to the assignment problem. Combining a polynomial-delay algorithm for the enumeration of all maximum common subtree isomorphisms with central ideas of our new algorithm leads to an improvement of its running time from O(n^6+T*n^2) to O(n^3+T*n*Delta), where n is the order of the larger tree, T is the number of different solutions, and Delta is the minimum of the maximum degrees of the input trees. Our theoretical results are supplemented by an experimental evaluation on synthetic and real-world instances.

ICML Conference 2012 Conference Paper

Subgraph Matching Kernels for Attributed Graphs

• Nils M. Kriege
• Petra Mutzel