An old result of M\"uller and R\"odl states that a countable graph $G$ has a subgraph whose vertices all have infinite degree if and only if for any vertex labeling of $G$ by positive integers, an infinite increasing path can be found. They asked whether an analogous equivalence holds for edge labelings, which Reiterman answered in the affirmative. Recently, Arman, Elliott, and R\"odl extended this problem to linear $k$-uniform hypergraphs $H$ and generalized the original equivalence for vertex labelings. They asked whether Reiterman's result for edge labelings can similarly be extended. We confirm this for the case where $H$ admits finitely many $\beta$-cycles involving any fixed vertex.
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The approximate uniform sampling of graph realizations with a given degree sequence is an everyday task in several social science, computer science, engineering etc. projects. One approach is using Markov chains. The best available current result about the well-studied switch Markov chain is that it is rapidly mixing on P-stable degree sequences (see DOI:10.1016/j.ejc.2021.103421). The switch Markov chain does not change any degree sequence. However, there are cases where degree intervals are specified rather than a single degree sequence. (A natural scenario where this problem arises is in hypothesis testing on social networks that are only partially observed.) Rechner, Strowick, and M\"uller-Hannemann introduced in 2018 the notion of degree interval Markov chain which uses three (separately well-studied) local operations (switch, hinge-flip and toggle), and employing on degree sequence realizations where any two sequences under scrutiny have very small coordinate-wise distance. Recently Amanatidis and Kleer published a beautiful paper (arXiv:2110.09068), showing that the degree interval Markov chain is rapidly mixing if the sequences are coming from a system of very thin intervals which are centered not far from a regular degree sequence. In this paper we extend substantially their result, showing that the degree interval Markov chain is rapidly mixing if the intervals are centred at P-stable degree sequences.
One fundamental problem in temporal graph analysis is to count the occurrences of small connected subgraph patterns (i.e., motifs), which benefits a broad range of real-world applications, such as anomaly detection, structure prediction, and network representation learning. However, existing works focused on exacting temporal motif are not scalable to large-scale temporal graph data, due to their heavy computational costs or inherent inadequacy of parallelism. In this work, we propose a scalable parallel framework for exactly counting temporal motifs in large-scale temporal graphs. We first categorize the temporal motifs based on their distinct properties, and then design customized algorithms that offer efficient strategies to exactly count the motif instances of each category. Moreover, our compact data structures, namely triple and quadruple counters, enable our algorithms to directly identify the temporal motif instances of each category, according to edge information and the relationship between edges, therefore significantly improving the counting efficiency. Based on the proposed counting algorithms, we design a hierarchical parallel framework that features both inter- and intra-node parallel strategies, and fully leverages the multi-threading capacity of modern CPU to concurrently count all temporal motifs. Extensive experiments on sixteen real-world temporal graph datasets demonstrate the superiority and capability of our proposed framework for temporal motif counting, achieving up to 538* speedup compared to the state-of-the-art methods. The source code of our method is available at: //github.com/steven-ccq/FAST-temporal-motif.
A partial orientation $\vec{H}$ of a graph $G$ is a weak $r$-guidance system if for any two vertices at distance at most $r$ in $G$, there exists a shortest path $P$ between them such that $\vec{H}$ directs all but one edge in $P$ towards this edge. In case $\vec{H}$ has bounded maximum outdegree, this gives an efficient representation of shortest paths of length at most $r$ in $G$. We show that graphs from many natural graph classes admit such weak guidance systems, and study the algorithmic aspects of this notion.
In this paper, we propose a depth-first search (DFS) algorithm for searching maximum matchings in general graphs. Unlike blossom shrinking algorithms, which store all possible alternative alternating paths in the super-vertices shrunk from blossoms, the newly proposed algorithm does not involve blossom shrinking. The basic idea is to deflect the alternating path when facing blossoms. The algorithm maintains detour information in an auxiliary stack to minimize the redundant data structures. A benefit of our technique is to avoid spending time on shrinking and expanding blossoms. This DFS algorithm can determine a maximum matching of a general graph with $m$ edges and $n$ vertices in $O(mn)$ time with space complexity $O(n)$.
Structural graph parameters, such as treewidth, pathwidth, and clique-width, are a central topic of study in parameterized complexity. A main aim of research in this area is to understand the "price of generality" of these widths: as we transition from more restrictive to more general notions, which are the problems that see their complexity status deteriorate from fixed-parameter tractable to intractable? This type of question is by now very well-studied, but, somewhat strikingly, the algorithmic frontier between the two (arguably) most central width notions, treewidth and pathwidth, is still not understood: currently, no natural graph problem is known to be W-hard for one but FPT for the other. Indeed, a surprising development of the last few years has been the observation that for many of the most paradigmatic problems, their complexities for the two parameters actually coincide exactly, despite the fact that treewidth is a much more general parameter. It would thus appear that the extra generality of treewidth over pathwidth often comes "for free". Our main contribution in this paper is to uncover the first natural example where this generality comes with a high price. We consider Grundy Coloring, a variation of coloring where one seeks to calculate the worst possible coloring that could be assigned to a graph by a greedy First-Fit algorithm. We show that this well-studied problem is FPT parameterized by pathwidth; however, it becomes significantly harder (W[1]-hard) when parameterized by treewidth. Furthermore, we show that Grundy Coloring makes a second complexity jump for more general widths, as it becomes para-NP-hard for clique-width. Hence, Grundy Coloring nicely captures the complexity trade-offs between the three most well-studied parameters. Completing the picture, we show that Grundy Coloring is FPT parameterized by modular-width.
In this short note, we show that for any $\epsilon >0$ and $k<n^{0.5-\epsilon}$ the choice number of the Kneser graph $KG_{n,k}$ is $\Theta (n\log n)$.
For any small positive real $\varepsilon$ and integer $t > \frac{1}{\varepsilon}$, we build a graph with a vertex deletion set of size $t$ to a tree, and twin-width greater than $2^{(1-\varepsilon) t}$. In particular, this shows that the twin-width is sometimes exponential in the treewidth, in the so-called oriented twin-width and grid number, and that adding an apex may multiply the twin-width by at least $2-\varepsilon$. Except for the one in oriented twin-width, these lower bounds are essentially tight.
Graph Convolutional Networks (GCNs) have received increasing attention in recent machine learning. How to effectively leverage the rich structural information in complex graphs, such as knowledge graphs with heterogeneous types of entities and relations, is a primary open challenge in the field. Most GCN methods are either restricted to graphs with a homogeneous type of edges (e.g., citation links only), or focusing on representation learning for nodes only instead of jointly optimizing the embeddings of both nodes and edges for target-driven objectives. This paper addresses these limitations by proposing a novel framework, namely the GEneralized Multi-relational Graph Convolutional Networks (GEM-GCN), which combines the power of GCNs in graph-based belief propagation and the strengths of advanced knowledge-base embedding methods, and goes beyond. Our theoretical analysis shows that GEM-GCN offers an elegant unification of several well-known GCN methods as specific cases, with a new perspective of graph convolution. Experimental results on benchmark datasets show the advantageous performance of GEM-GCN over strong baseline methods in the tasks of knowledge graph alignment and entity classification.
InteractE: Improving Convolution-based Knowledge Graph Embeddings by Increasing Feature Interactions
Most existing knowledge graphs suffer from incompleteness, which can be alleviated by inferring missing links based on known facts. One popular way to accomplish this is to generate low-dimensional embeddings of entities and relations, and use these to make inferences. ConvE, a recently proposed approach, applies convolutional filters on 2D reshapings of entity and relation embeddings in order to capture rich interactions between their components. However, the number of interactions that ConvE can capture is limited. In this paper, we analyze how increasing the number of these interactions affects link prediction performance, and utilize our observations to propose InteractE. InteractE is based on three key ideas -- feature permutation, a novel feature reshaping, and circular convolution. Through extensive experiments, we find that InteractE outperforms state-of-the-art convolutional link prediction baselines on FB15k-237. Further, InteractE achieves an MRR score that is 9%, 7.5%, and 23% better than ConvE on the FB15k-237, WN18RR and YAGO3-10 datasets respectively. The results validate our central hypothesis -- that increasing feature interaction is beneficial to link prediction performance. We make the source code of InteractE available to encourage reproducible research.
Graph-based semi-supervised learning (SSL) is an important learning problem where the goal is to assign labels to initially unlabeled nodes in a graph. Graph Convolutional Networks (GCNs) have recently been shown to be effective for graph-based SSL problems. GCNs inherently assume existence of pairwise relationships in the graph-structured data. However, in many real-world problems, relationships go beyond pairwise connections and hence are more complex. Hypergraphs provide a natural modeling tool to capture such complex relationships. In this work, we explore the use of GCNs for hypergraph-based SSL. In particular, we propose HyperGCN, an SSL method which uses a layer-wise propagation rule for convolutional neural networks operating directly on hypergraphs. To the best of our knowledge, this is the first principled adaptation of GCNs to hypergraphs. HyperGCN is able to encode both the hypergraph structure and hypernode features in an effective manner. Through detailed experimentation, we demonstrate HyperGCN's effectiveness at hypergraph-based SSL.
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