arXiv subjects

Nils M. Kriege

Publications and source records attributed to Nils M. Kriege.

34 records · Page 2Linked to original sources

arXiv2021

Temporal Graph Kernels for Classifying Dissemination Processes

Many real-world graphs or networks are temporal, e.g., in a social network persons only interact at specific points in time. This information directs dissemination processes on the network, such as the spread of rumors, fake news, or diseases. However, the current state-of-the-art methods for supervised graph classification are designed mainly for static graphs and may not be able to capture temporal information. Hence, they are not powerful enough to distinguish between graphs modeling different dissemination processes. To address this, we introduce a framework to lift standard graph kernels to the temporal domain. Specifically, we explore three different approaches and investigate the trade-offs between loss of temporal information and efficiency. Moreover, to handle large-scale graphs, we propose stochastic variants of our kernels with provable approximation guarantees. We evaluate our methods on a wide range of real-world social networks. Our methods beat static kernels by a large margin in terms of accuracy while still being scalable to large graphs and data sets. Hence, we confirm that taking temporal information into account is crucial for the successful classification of dissemination processes.

cs.SI↗

arXiv2020

Fixed-Parameter Algorithms for the Weighted Max-Cut Problem on Embedded 1-Planar Graphs

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 $O(3^k \cdot n^{3/2} \log n)$.

cs.DS↗

arXiv2020

Maximum Cut Parameterized by Crossing Number

Given an edge-weighted graph $G$ on $n$ nodes, the NP-hard Max-Cut problem asks for a node bipartition such that the sum of edge weights joining the different partitions is maximized. We propose a fixed-parameter tractable algorithm parameterized by the number $k$ of crossings in a given drawing of $G$. Our algorithm achieves a running time of $O(2^k \cdot p(n + k))$, where $p$ is the polynomial running time for planar Max-Cut. The only previously known similar algorithm [8] is restricted to 1-planar graphs (i.e., at most one crossing per edge) and its dependency on $k$ is of order $3^k$ . A direct consequence of our result is that Max-Cut is fixed-parameter tractable w.r.t. the crossing number, even without a given drawing. Moreover, the results naturally carry over to the minor crossing number.

cs.DS↗

arXiv2020

TUDataset: A collection of benchmark datasets for learning with graphs

Recently, there has been an increasing interest in (supervised) learning with graph data, especially using graph neural networks. However, the development of meaningful benchmark datasets and standardized evaluation procedures is lagging, consequently hindering advancements in this area. To address this, we introduce the TUDataset for graph classification and regression. The collection consists of over 120 datasets of varying sizes from a wide range of applications. We provide Python-based data loaders, kernel and graph neural network baseline implementations, and evaluation tools. Here, we give an overview of the datasets, standardized evaluation procedures, and provide baseline experiments. All datasets are available at www.graphlearning.io. The experiments are fully reproducible from the code available at www.github.com/chrsmrrs/tudataset.

cs.LG↗

arXiv2020

A Survey on Graph Kernels

Graph kernels have become an established and widely-used technique for solving classification tasks on graphs. This survey gives a comprehensive overview of techniques for kernel-based graph classification developed in the past 15 years. We describe and categorize graph kernels based on properties inherent to their design, such as the nature of their extracted graph features, their method of computation and their applicability to problems in practice. In an extensive experimental evaluation, we study the classification accuracy of a large suite of graph kernels on established benchmarks as well as new datasets. We compare the performance of popular kernels with several baseline methods and study the effect of applying a Gaussian RBF kernel to the metric induced by a graph kernel. In doing so, we find that simple baselines become competitive after this transformation on some datasets. Moreover, we study the extent to which existing graph kernels agree in their predictions (and prediction errors) and obtain a data-driven categorization of kernels as result. Finally, based on our experimental results, we derive a practitioner's guide to kernel-based graph classification.

cs.LG↗

arXiv2020

Deep Graph Matching Consensus

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. Our source code is available under https://github.com/rusty1s/ deep-graph-matching-consensus.

cs.LG↗

arXiv2019

Computing Optimal Assignments in Linear Time for Approximate Graph Matching

Finding an optimal assignment between two sets of objects is a fundamental problem arising in many applications, including the matching of `bag-of-words' representations in natural language processing and computer vision. Solving the assignment problem typically requires cubic time and its pairwise computation is expensive on large datasets. In this paper, we develop an algorithm which can find an optimal assignment in linear time when the cost function between objects is represented by a tree distance. We employ the method to approximate the edit distance between two graphs by matching their vertices in linear time. To this end, we propose two tree distances, the first of which reflects discrete and structural differences between vertices, and the second of which can be used to compare continuous labels. We verify the effectiveness and efficiency of our methods using synthetic and real-world datasets.

cs.LG↗

arXiv2019

A Unifying View of Explicit and Implicit Feature Maps of Graph Kernels

Non-linear kernel methods can be approximated by fast linear ones using suitable explicit feature maps allowing their application to large scale problems. We investigate how convolution kernels for structured data are composed from base kernels and construct corresponding feature maps. On this basis we propose exact and approximative feature maps for widely used graph kernels based on the kernel trick. We analyze for which kernels and graph properties computation by explicit feature maps is feasible and actually more efficient. In particular, we derive approximative, explicit feature maps for state-of-the-art kernels supporting real-valued attributes including the GraphHopper and graph invariant kernels. In extensive experiments we show that our approaches often achieve a classification accuracy close to the exact methods based on the kernel trick, but require only a fraction of their running time. Moreover, we propose and analyze algorithms for computing random walk, shortest-path and subgraph matching kernels by explicit and implicit feature maps. Our theoretical results are confirmed experimentally by observing a phase transition when comparing running time with respect to label diversity, walk lengths and subgraph size, respectively.

cs.LG↗

arXiv2019

Deep Weisfeiler-Lehman Assignment Kernels via Multiple Kernel Learning

Kernels for structured data are commonly obtained by decomposing objects into their parts and adding up the similarities between all pairs of parts measured by a base kernel. Assignment kernels are based on an optimal bijection between the parts and have proven to be an effective alternative to the established convolution kernels. We explore how the base kernel can be learned as part of the classification problem. We build on the theory of valid assignment kernels derived from hierarchies defined on the parts. We show that the weights of this hierarchy can be optimized via multiple kernel learning. We apply this result to learn vertex similarities for the Weisfeiler-Lehman optimal assignment kernel for graph classification. We present first experimental results which demonstrate the feasibility and effectiveness of the approach.

cs.LG↗

arXiv2018

A note on block-and-bridge preserving maximum common subgraph algorithms for outerplanar graphs

Schietgat, Ramon and Bruynooghe proposed a polynomial-time algorithm for computing a maximum common subgraph under the block-and-bridge preserving subgraph isomorphism (BBP-MCS) for outerplanar graphs. We show that the article contains the following errors: (i) The running time of the presented approach is claimed to be $\mathcal{O}(n^{2.5})$ for two graphs of order $n$. We show that the algorithm of the authors allows no better bound than $\mathcal{O}(n^4)$ when using state-of-the-art general purpose methods to solve the matching instances arising as subproblems. This is even true for the special case, where both input graphs are trees. (ii) The article suggests that the dissimilarity measure derived from BBP-MCS is a metric. We show that the triangle inequality is not always satisfied and, hence, it is not a metric. Therefore, the dissimilarity measure should not be used in combination with techniques that rely on or exploit the triangle inequality in any way. Where possible, we give hints on techniques that are suitable to improve the algorithm.

cs.DS↗

arXiv2018

A Fixed-Parameter Algorithm for the Max-Cut Problem on Embedded 1-Planar Graphs

We propose a fixed-parameter tractable algorithm for the \textsc{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 algorithm recursively reduces a 1-planar graph to at most $3^k$ planar graphs, using edge removal and node contraction. The \textsc{Max-Cut} problem is then solved on the planar graphs using established polynomial-time algorithms. We show that a maximum cut in the given 1-planar graph can be derived from the solutions for the planar graphs. Our algorithm computes a maximum cut in an embedded 1-planar graph with $n$ nodes and $k$ edge crossings in time $\mathcal{O}(3^k \cdot n^{3/2} \log n)$.

cs.DS↗

arXiv2018

Largest Weight Common Subtree Embeddings with Distance Penalties

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 $\mathcal O(|T|\,|T'|Δ)$ for two rooted or unrooted trees $T$ and $T'$, where $Δ=\min\{Δ(T),Δ(T')\}$ with $Δ(X)$ the maximum degree of $X$. If the weights are integral and at most $C$, we obtain a running time of $\mathcal O(|T|\,|T'|\sqrtΔ\log (C\min\{|T|,|T'|\}))$ for rooted trees.

cs.DS↗

arXiv2018

Recognizing Cuneiform Signs Using Graph Based Methods

The cuneiform script constitutes one of the earliest systems of writing and is realized by wedge-shaped marks on clay tablets. A tremendous number of cuneiform tablets have already been discovered and are incrementally digitalized and made available to automated processing. As reading cuneiform script is still a manual task, we address the real-world application of recognizing cuneiform signs by two graph based methods with complementary runtime characteristics. We present a graph model for cuneiform signs together with a tailored distance measure based on the concept of the graph edit distance. We propose efficient heuristics for its computation and demonstrate its effectiveness in classification tasks experimentally. To this end, the distance measure is used to implement a nearest neighbor classifier leading to a high computational cost for the prediction phase with increasing training set size. In order to overcome this issue, we propose to use CNNs adapted to graphs as an alternative approach shifting the computational cost to the training phase. We demonstrate the practicability of both approaches in an extensive experimental comparison regarding runtime and prediction accuracy. Although currently available annotated real-world data is still limited, we obtain a high accuracy using CNNs, in particular, when the training set is enriched by augmented examples.

cs.CV↗

arXiv2017

On Valid Optimal Assignment Kernels and Applications to Graph Classification

The success of kernel methods has initiated the design of novel positive semidefinite functions, in particular for structured data. A leading design paradigm for this is the convolution kernel, which decomposes structured objects into their parts and sums over all pairs of parts. Assignment kernels, in contrast, are obtained from an optimal bijection between parts, which can provide a more valid notion of similarity. In general however, optimal assignments yield indefinite functions, which complicates their use in kernel methods. We characterize a class of base kernels used to compare parts that guarantees positive semidefinite optimal assignment kernels. These base kernels give rise to hierarchies from which the optimal assignment kernels are computed in linear time by histogram intersection. We apply these results by developing the Weisfeiler-Lehman optimal assignment kernel for graphs. It provides high classification accuracy on widely-used benchmark data sets improving over the original Weisfeiler-Lehman kernel.

cs.LG↗

arXiv2016

Faster Kernels for Graphs with Continuous Attributes via Hashing

While state-of-the-art kernels for graphs with discrete labels scale well to graphs with thousands of nodes, the few existing kernels for graphs with continuous attributes, unfortunately, do not scale well. To overcome this limitation, we present hash graph kernels, a general framework to derive kernels for graphs with continuous attributes from discrete ones. The idea is to iteratively turn continuous attributes into discrete labels using randomized hash functions. We illustrate hash graph kernels for the Weisfeiler-Lehman subtree kernel and for the shortest-path kernel. The resulting novel graph kernels are shown to be, both, able to handle graphs with continuous attributes and scalable to large graphs and data sets. This is supported by our theoretical analysis and demonstrated by an extensive experimental evaluation.

cs.LG↗

arXiv2016

Faster Algorithms for the Maximum Common Subtree Isomorphism Problem

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 ${\sf NP}$-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 $Δ$ our algorithm achieves a running time of $\mathcal{O}(n^2Δ)$ 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 $\mathcal{O}(n^6+Tn^2)$ to $\mathcal{O}(n^3+TnΔ)$, where $n$ is the order of the larger tree, $T$ is the number of different solutions, and $Δ$ 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.

cs.DS↗