Graph coarsening: from scientific computing to machine learning

The general method of graph coarsening or graph reduction has been a remarkably useful and ubiquitous tool in scientific computing and it is now just starting to have a similar impact in machine learning. The goal of this paper is to take a broad look into coarsening techniques that have been successfully deployed in scientific computing and see how similar principles are finding their way in more recent applications related to machine learning. In scientific computing, coarsening plays a central role in algebraic multigrid methods as well as the related class of multilevel incomplete LU factorizations. In machine learning, graph coarsening goes under various names, e.g., graph downsampling or graph reduction. Its goal in most cases is to replace some original graph by one which has fewer nodes, but whose structure and characteristics are similar to those of the original graph. As will be seen, a common strategy in these methods is to rely on spectral properties to define the coarse graph.

Inertia Sets of Semicliqued Graphs

In this paper, we investigate inertia sets of simple connected undirected graphs. The main focus is on the shape of their corresponding inertia tables, in particular whether or not they are trapezoidal. This paper introduces a special family of graphs created from any given graph, $G$, coined semicliqued graphs and denoted $\widetilde{K}G$. We establish the minimum rank and inertia sets of some $\widetilde{K}G$ in relation to the original graph $G$. For special classes of graphs, $G$, it can be shown that the inertia set of $G$ is a subset of the inertia set of $\widetilde{K}G$. We provide the inertia sets for semicliqued cycles, paths, stars, complete graphs, and for a class of trees. In addition, we establish an inertia set bound for semicliqued complete bipartite graphs.

Metaknowledge Enhanced Open Domain Question Answering with Wiki Documents

The commonly-used large-scale knowledge bases have been facing challenges in open domain question answering tasks which are caused by the loose knowledge association and weak structural logic of triplet-based knowledge. To find a way out of this dilemma, this work proposes a novel metaknowledge enhanced approach for open domain question answering. We design an automatic approach to extract metaknowledge and build a metaknowledge network from Wiki documents. For the purpose of representing the directional weighted graph with hierarchical and semantic features, we present an original graph encoder GE4MK to model the metaknowledge network. Then, a metaknowledge enhanced graph reasoning model MEGr-Net is proposed for question answering, which aggregates both relational and neighboring interactions comparing with R-GCN and GAT. Experiments have proved the improvement of metaknowledge over main-stream triplet-based knowledge. We have found that the graph reasoning models and pre-trained language models also have influences on the metaknowledge enhanced question answering approaches.

Publishing Triangle Counting Histogram in Social Networks Based on Differential Privacy

The continuous expansion of the number and scale of social networking sites has led to an explosive growth of social network data. Mining and analyzing social network data can bring huge economic value and social benefits, but it will result in privacy leakage and other issues. The research focus of social network data publishing is to publish available data while ensuring privacy. Aiming at the problem of low data availability of social network node triangle counting publishing under differential privacy, this paper proposes a privacy protection method of edge triangle counting. First, an edge-removal projection algorithm TSER based on edge triangle count sorting is proposed to obtain the upper bound of sensitivity. Then, two edge triangle count histogram publishing methods satisfying edge difference privacy are given based on the TSER algorithm. Finally, experimental results show that compared with the existing algorithms, the TSER algorithm can retain more triangles in the original graph, reduce the error between the published data and the original data, and improve the published data availability.

Hybrid pointer networks for traveling salesman problems optimization

In this work, we proposed a hybrid pointer network (HPN), an end-to-end deep reinforcement learning architecture is provided to tackle the travelling salesman problem (TSP). HPN builds upon graph pointer networks, an extension of pointer networks with an additional graph embedding layer. HPN combines the graph embedding layer with the transformer’s encoder to produce multiple embeddings for the feature context. We conducted extensive experimental work to compare HPN and Graph pointer network (GPN). For the sack of fairness, we used the same setting as proposed in GPN paper. The experimental results show that our network significantly outperforms the original graph pointer network for small and large-scale problems. For example, it reduced the cost for travelling salesman problems with 50 cities/nodes (TSP50) from 5.959 to 5.706 without utilizing 2opt. Moreover, we solved benchmark instances of variable sizes using HPN and GPN. The cost of the solutions and the testing times are compared using Linear mixed effect models. We found that our model yields statistically significant better solutions in terms of the total trip cost. We make our data, models, and code publicly available https://github.com/AhmedStohy/Hybrid-Pointer-Networks.

Graph Connectivity in Log Steps Using Label Propagation

The fastest deterministic algorithms for connected components take logarithmic time and perform superlinear work on a Parallel Random Access Machine (PRAM). These algorithms maintain a spanning forest by merging and compressing trees, which requires pointer-chasing operations that increase memory access latency and are limited to shared-memory systems. Many of these PRAM algorithms are also very complicated to implement. Another popular method is “leader-contraction” where the challenge is to select a constant fraction of leaders that are adjacent to a constant fraction of non-leaders with high probability, but this can require adding more edges than were in the original graph. Instead we investigate label propagation because it is deterministic, easy to implement, and does not rely on pointer-chasing. Label propagation exchanges representative labels within a component using simple graph traversal, but it is inherently difficult to complete in a sublinear number of steps. We are able to overcome the problems with label propagation for graph connectivity. We introduce a surprisingly simple framework for deterministic, undirected graph connectivity using label propagation that is easily adaptable to many computational models. It achieves logarithmic convergence independently of the number of processors and without increasing the edge count. We employ a novel method of propagating directed edges in alternating direction while performing minimum reduction on vertex labels. We present new algorithms in PRAM, Stream, and MapReduce. Given a simple, undirected graph [Formula: see text] with [Formula: see text] vertices, [Formula: see text] edges, our approach takes O(m) work each step, but we can only prove logarithmic convergence on a path graph. It was conjectured by Liu and Tarjan (2019) to take [Formula: see text] steps or possibly [Formula: see text] steps. Our experiments on a range of difficult graphs also suggest logarithmic convergence. We leave the proof of convergence as an open problem.

The diameter vulnerability of two-dimensional optimal circulant networks

This paper studies the effect of changing the diameter of the network in circulant networks of dimension two with unreliable elements (nodes, links). The well-known (Δ,D,D′, s)-problem is to find (Δ, D)-graphs with maximum degree Δ and diameter D such that the subgraphs obtained from the original graph by deleting any set of up to s vertices (edges) have diameter at most D′. For a family of optimal circulants of degree four we found the ranges of the orders of the graphs that preserve the diameter of the graph for one (two) vertex or edge failures. It is proved that in the investigated circulant networks in case of failure of one or two edges (vertices), the diameter can increase by no more than one, and in case of failure of three elements by no more than two (edge failures) or three (vertex failures). It is shown that two-dimensional optimal circulants, in comparison with two-dimensional tori, have a better diameter in case of element failures.

Better Distance Preservers and Additive Spanners

We study two popular ways to sketch the shortest path distances of an input graph. The first is distance preservers , which are sparse subgraphs that agree with the distances of the original graph on a given set of demand pairs. Prior work on distance preservers has exploited only a simple structural property of shortest paths, called consistency , stating that one can break shortest path ties such that no two paths intersect, split apart, and then intersect again later. We prove that consistency alone is not enough to understand distance preservers, by showing both a lower bound on the power of consistency and a new general upper bound that polynomially surpasses it. Specifically, our new upper bound is that any p demand pairs in an n -node undirected unweighted graph have a distance preserver on O( n 2/3 p 2/3 + np 1/3 edges. We leave a conjecture that the right bound is O ( n 2/3 p 2/3 + n ) or better. The second part of this paper leverages these distance preservers in a new construction of additive spanners , which are subgraphs that preserve all pairwise distances up to an additive error function. We give improved error bounds for spanners with relatively few edges; for example, we prove that all graphs have spanners on O(n) edges with + O ( n 3/7 + ε ) error. Our construction can be viewed as an extension of the popular path-buying framework to clusters of larger radii.

Metaknowledge Enhanced Open Domain Question Answering with Wiki Documents

The commonly-used large-scale knowledge bases have been facing challenges in open domain question answering tasks which are caused by the loose knowledge association and weak structural logic of triplet-based knowledge. To find a way out of this dilemma, this work proposes a novel metaknowledge enhanced approach for open domain question answering. We design an automatic approach to extract metaknowledge and build metaknowledge network from Wiki documents. For the purpose of representing the directional weighted graph with hierarchical and semantic features, we present an original graph encoder GE4MK to model the metaknowledge network. Then a metaknowledge enhanced graph reasoning model MEGr-Net is proposed for question answering, which aggregates both relational and neighboring interactions comparing with R-GCN and GAT. Experiments have proved the improvement of metaknowledge over main-stream triplet-based knowledge. We have found that the graph reasoning models and pre-trained language models also have influences on the metaknowledge enhanced question answering approaches.

Omega Index of Line and Total Graphs

A derived graph is a graph obtained from a given graph according to some predetermined rules. Two of the most frequently used derived graphs are the line graph and the total graph. Calculating some properties of a derived graph helps to calculate the same properties of the original graph. For this reason, the relations between a graph and its derived graphs are always welcomed. A recently introduced graph index which also acts as a graph invariant called omega is used to obtain such relations for line and total graphs. As an illustrative exercise, omega values and the number of faces of the line and total graphs of some frequently used graph classes are calculated.