Graph Distance Research Articles

LCS graph kernel based on Wasserstein distance in longest common subsequence metric space

For graph learning tasks, many existing methods utilize a message-passing mechanism where vertex features are updated iteratively by aggregation of neighbor information. This strategy provides an efficient means for graph features extraction, but obtained features after many iterations might contain too much information from other vertices, and tend to be similar to each other. This makes their representations less expressive. Learning graphs using paths, on the other hand, can be less adversely affected by this problem because it does not involve all vertex neighbors. However, most of them can only compare paths with the same length, which might engender information loss. To resolve this difficulty, we propose a new Graph Kernel based on a Longest Common Subsequence (LCS) similarity. Moreover, we found that the widely-used R-convolution framework is unsuitable for path-based Graph Kernel because a huge number of comparisons between dissimilar paths might deteriorate graph distances calculation. Therefore, we propose a novel metric space by exploiting the proposed LCS-based similarity, and compute a new Wasserstein-based graph distance in this metric space, which emphasizes more the comparison between similar paths. Furthermore, to reduce the computational cost, we propose an adjacent point merging operation to sparsify point clouds in the metric space.

Chromatic Coloring of Distance Graphs I

The primary aim of this paper is to publicize various problems regarding chromatic coloring of finite, simple and undirected graphs. A simple motivation for this work is that the coloring of graphs gives models for a variety of real world problems such as scheduling. We prove some interesting results related to the computation of chromatic number of certain distance graphs and also discuss some open problems.

Chromatic Coloring of Distance Graphs I

Chromatic Coloring of Distance Graphs I

Graph distances of continuum long-range percolation

We consider a version of continuum long-range percolation on finite boxes of $\mathbb{R}^d$ in which the vertex set is given by the points of a Poisson point process and each pair of two vertices at distance $r$ is connected with probability proportional to $r^{-s}$ for a certain constant $s$. We explore the graph-theoretical distance in this model. The aim of this paper is to show that this random graph model undergoes phase transitions at values $s=d$ and $s=2d$ in analogy to classical long-range percolation on $\mathbb{Z}^d$, by using techniques which are based on an analysis of the underlying Poisson point process.

On hinge domination in graphs

A set Dh of vertices in a graph G = (V,E) is a hinge dset if every vertex u in V −Dh is adjacent to some vertex v in Dh and a vertex w in V −Dh such that (v,w) is not an edge in E. The hinge domination number γh(G) is the minimum size of a hinged dset. In this paper we determine hinge domination number γh(G) for standard graphs and some shadow distance graphs.

Distance complexity measures versus the orbit-entropy measure of dendrimers

In this article, we compute several distance-based topological indices dendrimers. Indeed, based on the distances of related graphs, we investigate the distance-based complexity of these graphs. We also, determine an orbit entropy measure of these structures based on the size of orbits. The analysis results obtained by computing the correlations between given numerical data shows that the correlation between Winer complexity with GP-index, Wiener index, and Wiener polarity is quite low. As a result, we have inferred that the Wiener-complexity can be considered as new measures for quantitative structure-activity relationship/quantitative structure-property relationship (QSAR/QSPR) models of dendrimers. Hence, the W-complexity might be interesting for further investigations in predicting the physicochemical properties of molecules.

Learning graph edit distance by graph neural networks

The emergence of geometric deep learning as a novel framework to deal with graph-based representations has faded away traditional approaches in favor of completely new methodologies. In this paper, we propose a new framework able to combine the advances on deep metric learning with traditional approximations of the graph edit distance. Hence, we propose an efficient graph distance based on the novel field of geometric deep learning. Our method employs a message passing neural network to capture the graph structure, and thus, leveraging this information for its use on a distance computation. The performance of the proposed graph distance is validated on two different scenarios. On the one hand, in a graph retrieval of handwritten words i.e. keyword spotting, showing its superior performance when compared with (approximate) graph edit distance benchmarks. On the other hand, demonstrating competitive results for graph similarity learning when compared with the current state-of-the-art on a recent benchmark dataset.

The distribution of waiting distances in ancestral recombination graphs

The ancestral recombination graph (ARG) contains the full genealogical information of the sample, and many population genetic inference problems can be solved using inferred or sampled ARGs. In particular, the waiting distance between tree changes along the genome can be used to make inference about the distribution and evolution of recombination rates. To this end, we here derive an analytic expression for the distribution of waiting distances between tree changes under the sequentially Markovian coalescent model and obtain an accurate approximation to the distribution of waiting distances for topology changes. We use these results to show that some of the recently proposed methods for inferring sequences of trees along the genome provide strongly biased distributions of waiting distances. In addition, we provide a correction to an undercounting problem facing all available ARG inference methods, thereby facilitating the use of ARG inference methods to estimate temporal changes in the recombination rate.

Lower bounds on the clique-chromatic numbers of some distance graphs

Lower bounds on the clique-chromatic numbers of some distance graphs

Density of sets with missing differences and applications

Abstract For a given set M of positive integers, a well-known problem of Motzkin asks to determine the maximal asymptotic density of M-sets, denoted by μ(M), where an M-set is a set of non-negative integers in which no two elements differ by an element in M. In 1973, Cantor and Gordon find μ(M) for |M| ≤ 2. Partial results are known in the case |M| ≥ 3 including some results in the case when M is an infinite set. Motivated by some 3 and 4-element families already discussed by Liu and Zhu in 2004, we study μ(M) for two families namely, M = {a, b,a + b, n(a + b)} and M = {a, b, b − a, n(b − a)}. For both of these families, we find some exact values and some bounds on μ(M). This number theory problem is also related to various types of coloring problems of the distance graphs generated by M. So, as an application, we also study these coloring parameters associated with these families.

Netrd: A library for network reconstruction and graph distances

Over the last two decades, alongside the increased availability of large network datasets, we have witnessed the rapid rise of network science. For many systems, however, the data we have access to is not a direct description of the underlying network. More and more, we see the drive to study networks that have been inferred or reconstructed from non-network data---in particular, using time series data from the nodes in a system to infer likely connections between them. Selecting the most appropriate technique for this task is a challenging problem in network science. Different reconstruction techniques usually have different assumptions, and their performance varies from system to system in the real world. One way around this problem could be to use several different reconstruction techniques and compare the resulting networks. However, network comparison is also not an easy problem, as it is not obvious how best to quantify the differences between two networks, in part because of the diversity of tools for doing so. The netrd Python package seeks to address these two parallel problems in network science by providing, to our knowledge, the most extensive collection of both network reconstruction techniques and network comparison techniques (often referred to as graph distances) in a single library (https://github.com/netsiphd/netrd). In this article, we detail the two main functionalities of the netrd package. Along the way, we describe some of its other useful features. This package builds on commonly used Python packages and is already a widely used resource for network scientists and other multidisciplinary researchers. With ongoing open-source development, we see this as a tool that will continue to be used by all sorts of researchers to come.

Graph Classification Method Based on Wasserstein Distance

Graph classification is a challenging problem, which attracts more and more attention. The key to solving this problem is based on what metric to compare graphs, that is, how to define graph similarity. Common graph classification methods include graph kernel, graph editing distance, graph embedding and so on. We introduce a new graph similarity metric, namely GRD (Geometric gRaph Distance). Our model GRD is composed of three sub-modules, which capture the differences between the graph structures from different aspects. Finally, the graph distances defined by the three modules are fused to define the similarity between graphs. Experiments show that GRD is superior to the baseline methods on the benckmark datasets.

General degree distance of graphs

We generalize several topological indices and introduce the general degree distance of a connected graph $G$. For $a, b \in \mathbb{R}$, the general degree distance $DD_{a,b} (G) = \sum_{ v \in V(G)} [deg_{G}(v)]^a S^b_{G} (v)$, where $V(G)$ is the vertex set of $G$, $deg_G (v)$ is the degree of a vertex $v$, $S^b_{G} (v) = \sum_{ w \in V(G) \setminus \{ v \} } [d_{G} (v,w) ]^{b}$ and $d_{G} (v,w)$ is the distance between $v$ and $w$ in $G$. We present some sharp bounds on the general degree distance for multipartite graphs and trees of given order, graphs of given order and chromatic number, graphs of given order and vertex connectivity, and graphs of given order and number of pendant vertices.

NASGEM: Neural Architecture Search via Graph Embedding Method

Neural Architecture Search (NAS) automates and prospers the design of neural networks. Estimator-based NAS has been proposed recently to model the relationship between architectures and their performance to enable scalable and flexible search. However, existing estimator-based methods encode the architecture into a latent space without considering graph similarity. Ignoring graph similarity in node-based search space may induce a large inconsistency between similar graphs and their distance in the continuous encoding space, leading to inaccurate encoding representation and/or reduced representation capacity that can yield sub-optimal search results. To preserve graph correlation information in encoding, we propose NASGEM which stands for Neural Architecture Search via Graph Embedding Method. NASGEM is driven by a novel graph embedding method equipped with similarity measures to capture the graph topology information. By precisely estimating the graph distance and using an auxiliary Weisfeiler-Lehman kernel to guide the encoding, NASGEM can utilize additional structural information to get more accurate graph representation to improve the search efficiency. GEMNet, a set of networks discovered by NASGEM, consistently outperforms networks crafted by existing search methods in classification tasks, i.e., with 0.4%-3.6% higher accuracy while having 11%- 21% fewer Multiply-Accumulates. We further transfer GEMNet for COCO object detection. In both one-stage and twostage detectors, our GEMNet surpasses its manually-crafted and automatically-searched counterparts.

Unsupervised RGB-T saliency detection by node classification distance and sparse constrained graph learning

Unsupervised RGB-T saliency detection by node classification distance and sparse constrained graph learning

Density theorems for anisotropic point configurations

AbstractSeveral results in the existing literature establish Euclidean density theorems of the following strong type. These results claim that every set of positive upper Banach density in the Euclidean space of an appropriate dimension contains isometric copies of all sufficiently large elements of a prescribed family of finite point configurations. So far, all results of this type discussed linear isotropic dilates of a fixed point configuration. In this paper, we initiate the study of analogous density theorems for families of point configurations generated by anisotropic dilations, i.e., families with power-type dependence on a single parameter interpreted as their size. More specifically, we prove nonisotropic power-type generalizations of a result by Bourgain on vertices of a simplex, a result by Lyall and Magyar on vertices of a rectangular box, and a result on distance trees, which is a particular case of the treatise of distance graphs by Lyall and Magyar. Another source of motivation for this paper is providing additional evidence for the versatility of the approach stemming from the work of Cook, Magyar, and Pramanik and its modification used recently by Durcik and the present author. Finally, yet another purpose of this paper is to single out anisotropic multilinear singular integral operators associated with the above combinatorial problems, as they are interesting on their own.

Chromatic Numbers of Distance Graphs without Short Odd Cycles in Rational Spaces

Chromatic Numbers of Distance Graphs without Short Odd Cycles in Rational Spaces

External diffusion-limited aggregation on a spanning-tree-weighted random planar map

Let M be the infinite spanning-tree-weighted random planar map, which is the local limit of finite random planar maps sampled with probability proportional to the number of spanning trees they admit. We show that a.s. the M-graph-distance diameter of the external diffusion-limited aggregation (DLA) cluster on M run for m steps is of order m2/d+om(1), where d is the metric ball volume growth exponent for M (which was shown to exist by Ding and Gwynne (Comm. Math. Phys. 374 (2020) 1877–1934). By known bounds for d, one has 0.55051⋯≤2/d≤0.563315…. Along the way, we also prove that loop-erased random walk (LERW) on M typically travels graph distance m2/d+om(1) in m units of time and that the graph-distance diameter of a finite spanning-tree-weighted random planar map with n edges, with or without boundary, is of order n1/d+on(1) except on an event with probability decaying faster than any negative power of n. Our proofs are based on a special relationship between DLA and LERW on spanning-tree-weighted random planar maps as well as estimates for distances in such maps which come from the theory of Liouville quantum gravity.

Formula omitted]-packing colorings of distance graphs [formula omitted

Given a graph G and a non-decreasing sequence S=(a1,a2,…) of positive integers, the mapping f:V(G)→{1,…,k} is an S-packing k-coloring of G if for any distinct vertices u,v∈V(G) with f(u)=f(v)=i the distance between u and v in G is greater than ai. The smallest k such that G has an S-packing k-coloring is the S-packing chromatic number, χS(G), of G. In this paper, we continue the study of S-packing colorings of distance graphs, initiated by Togni (2014) and Ekstein et al. (2012). We focus on the distance graphs G(Z,{2,t}), where t>1 is an odd integer, which has Z as its vertex set, and i,j∈Z are adjacent if |i−j|∈{2,t}. We determine the S-packing chromatic numbers of the graphs G(Z,{2,t}), where S is any sequence with ai∈{1,2} for all i. In addition, we give lower and upper bounds for the d-distance chromatic numbers of the distance graphs G(Z,{2,t}), which in the cases d≥t−3 give the exact values. Implications for the corresponding S-packing chromatic numbers of the circulant graphs are also discussed.

Study of Circular Distance in Graphs

Circular distance between vertices of a graph has a significant role, which is defined as summation of detour distance and geodesic distance. Attention is paid, this is metric on the set of all vertices of graph and it plays an important role in graph theory. Some bounds have been carried out for circular distance in terms of pendent vertices of graph . Some results and properties have been found for circular distance for some classes of graphs and applied this distance to Cartesian product of graphs〖 P〗_2×C_n. Including 〖 P〗_2×C_n, some graphs acted as a circular self-centered. Using this circular distance there exists some relations between various radii and diameters in path graphs. The possible applications were briefly discussed.