Vectorial Graph Distance

Centrality indices are widely used analytic measures for the importance of nodes in a network. Closeness centrality is very popular among these measures. For a single node v, it takes the sum of the distances of v to all other nodes into account. The currently best algorithms in practical applications for computing the closeness for all nodes exactly in unweighted graphs are based on breadth-first search (BFS) from every node. Thus, even for sparse graphs, these algorithms require quadratic running time in the worst case, which is prohibitive for large networks. In many relevant applications, however, it is unnecessary to compute closeness values for all nodes. Instead, one requires only the k nodes with the highest closeness values in descending order. Thus, we present a new algorithm for computing this top-k ranking in unweighted graphs. Following the rationale of previous work, our algorithm significantly reduces the number of traversed edges. It does so by computing upper bounds on the closeness and stopping the current BFS search when k nodes already have higher closeness than the bounds computed for the other nodes. In our experiments with real-world and synthetic instances of various types, one of these new bounds is good for small-world graphs with low diameter (such as social networks), while the other one excels for graphs with high diameter (such as road networks). Combining them yields an algorithm that is faster than the state of the art for top-k computations for all test instances, by a wide margin for high-diameter graphs.

The Fréchet distance is a distance measure between trajectories in ℝ^d or walks in a graph G. Given constant-time shortest path queries, the Discrete Fréchet distance D_G(P, Q) between two walks P and Q can be computed in O(|P|⋅|Q|) time using a dynamic program. Driemel, van der Hoog, and Rotenberg [SoCG'22] show that for weighted planar graphs this approach is likely tight, as there can be no strongly-subquadratic algorithm to compute a 1.01-approximation of D_G(P, Q) unless the Orthogonal Vector Hypothesis (OVH) fails.<br/>Such quadratic-time conditional lower bounds are common to many Fréchet distance variants. However, they can be circumvented by assuming that the input comes from some well-behaved class: There exist (1+ε)-approximations, both in weighted graphs and in ℝ^d, that take near-linear time for c-packed or κ-straight walks in the graph. In ℝ^d there also exists a near-linear time algorithm to compute the Fréchet distance whenever all input edges are long compared to the distance. <br/>We consider computing the Fréchet distance in unweighted planar graphs. We show that there exist no strongly-subquadratic 1.25-approximations of the discrete Fréchet distance between two disjoint simple paths in an unweighted planar graph in strongly subquadratic time, unless OVH fails. This improves the previous lower bound, both in terms of generality and approximation factor. We subsequently show that adding graph structure circumvents this lower bound: If the graph is a regular tiling with unit-weighted edges, then there exists an Õ((|P|+|Q|)^{1.5})-time algorithm to compute D_G(P, Q). Our result has natural implications in the plane, as it allows us to define a new class of well-behaved curves that facilitate (1+ε)-approximations of their discrete Fréchet distance in subquadratic time.

In this study, we consider the problem of estimating the diameter of a graph, that is, the maximum distance between any two vertices, in linear time. We address a question posed in the literature—whether there exists an interesting graph class with arbitrarily large cycles for which breadth first search (BFS) would always return high‐eccentricity vertices. We answer this question positively, defining a class of graphs that generalizes AT‐free graphs and has no bound on the size of induced cycles, yet BFS always returns a vertex whose eccentricity is within a constant difference from the diameter. In addition, we consider the question—also explicitly stated in the literature—whether some variant of the so‐called multisweep algorithm would always return a high‐eccentricity vertex. We show that the answer is negative by describing a family of graphs for which no variant of multisweep BFS can return a vertex of eccentricity higher than half of the diameter plus a constant.

The idea of the Q-neutrosophic soft set emerges from the neutrosophic soft set by upgrading the membership functions to a two-dimensional entity which indicate uncertainty, indeterminacy and falsity. Hence, it is able to deal with two-dimensional inconsistent, imprecise, and indeterminate information appearing in real life situations. In this study, the tools that measure the similarity, distance and the degree of fuzziness of Q-neutrosophic soft sets are presented. The definitions of distance, similarity and measures of entropy are introduced. Some formulas for Q-neutrosophic soft entropy were presented. The known Hamming, Euclidean and their normalized distances are generalized to make them well matched with the idea of Q-neutrosophic soft set. The distance measure is subsequently used to define the measure of similarity. Lastly, we expound three applications of the measures of Q-neutrosophic soft sets by applying entropy and the similarity measure to a medical diagnosis and decision making problems.

A lot of effort has been put into researching image interpretation, but there is still no universally accepted approach to map low-level feature into high level image semantic interpretation. In this paper, a method for continuous low-level features vector quantization is presented so as to define appropriate values for descriptive variables. The similarity among different concepts of the domain is examined and compared by using the measure of similarity which is based on the probabilistic model and the measure of distance. Also, an abstract image description vector suitable for image analysis is given.

The species conservation technique described here, in which the population of a genetic algorithm is divided into several groups according to their similarity, is inspired by ecology. Each group with similar characteristics is called a species and is centred on a dominating individual, called the species seed. A genetic algorithm based on this species conservation technique, called the species-conserving genetic algorithm (SCGA), was established and has been proved to be effective in finding multiple solutions of multimodal optimization problems. In this article, the SCGA is used to solve engineering design optimization problems. Different distance measures (measures of similarity) are investigated to analyse the performance of the SCGA. It is shown that the Euclidean distance is not the only possible basis for defining a species and sometimes may not make sense in engineering applications. Two structural design problems are used to demonstrate how the choice of a meaningful measure of similarity will help the exploration for significant designs.

The methodology of semantic clustering analysis of customer's text-opinions collection is developed. The author's version of the mathematical models of formalization and practical realization of short textual messages semantic clustering procedure is proposed, based on the customer's text-opinions collection Latent Semantic Analysis knowledge extracting method. An algorithm for semantic clustering of the text-opinions is developed, the distinctive characteristics of which is the introduction of concepts and methods of identification point of reference in the scale of text-opinions collection closeness determination, instrument of the documents' closeness degree identification, measure of similarity between pairs of documents. The version of quantitative evaluation of the clustering results is developed. The concepts of resolving power of the method of semantic clustering and level of the clustering procedure quality are proposed. Analysis of the specific features and the effectiveness level of various distance measures is conducted.

Both supervised and unsupervised machine learning techniques require selection of a measure of distance between, or similarity among, the objects to be classified or clustered. Different measures of distance or similarity will lead to different machine learning performance. The appropriateness of a distance measure will typically depend on the types of features being used in the learning process.

Various metrics and functions of metrics are examined. A number of results concerning the identification of similarities and distinctions between objects are presented for different measurement techniques.

Meditation with mobile apps has been shown to improve mental and physical health. However, regular, long-term meditation app use is needed to maintain these health benefits, and many people have a difficult time maintaining engagement with meditation apps over time. Our goal was to determine the length of the timeframe over which usage data must be collected before future app abandonment can be predicted accurately in order to better target additional behavioral support to those who are most likely to stop using the app. Data were collected from a randomly drawn sample of 2600 new subscribers to a 1-year membership of the mobile app Calm, who started using the app between July and November of 2018. App usage data contained the duration and start time of all meditation sessions with the app over 365 days. We used these data to construct the following predictive model features: total daily sessions, total daily duration, and a measure of temporal similarity between consecutive days based on the dynamic time warping (DTW) distance measure. We then fit random forest models using increasingly longer periods of data after users subscribed to Calm to predict whether they performed any meditation sessions over 2-week intervals in the future. Model fit was assessed using the area under the receiver operator characteristic curve (AUC), and an exponential growth model was used to determine the minimal amount of data needed to reach an accurate prediction (95% of max AUC) of future engagement. After first subscribing to Calm, 83.1% of the sample used the Calm app on at least 1 more day. However, by day 350 after subscribing, 58.0% of users abandoned their use of the app. For the persistent users, the average number of daily sessions was 0.33 (SD = 0.02), the average daily duration of meditating was 3.93 minutes (SD = 0.25), and the average DTW distance to the previous day was 1.50 (SD = 0.17). The exponential growth models revealed that an average of 64 days of observations after subscribing to Calm are needed to reach an accurate prediction of future app engagement. Our results are consistent with existing estimates of the time required to develop a new habit. Additionally, this research demonstrates how to use app usage data to quickly and accurately predict the likelihood of users' future app abandonment. This research allows future researchers to better target just-in-time interventions towards users at risk of abandonment.

Matching two geometric objects in two-dimensional (2D) and three-dimensional (3D) spaces is a central problem in computer vision, pattern recognition, and protein structure prediction. In particular, the problem of aligning two polygonal chains under translation and rotation to minimize their distance has been studied using various distance measures. It is well known that the Hausdorff distance is useful for matching two point sets, and that the Fréchet distance is a superior measure for matching two polygonal chains. The discrete Fréchet distance closely approximates the (continuous) Fréchet distance, and is a natural measure for the geometric similarity of the folded 3D structures of biomolecules such as proteins. In this paper, we present new algorithms for matching two polygonal chains in two dimensions to minimize their discrete Fréchet distance under translation and rotation, and an effective heuristic for matching two polygonal chains in three dimensions. We also describe our empirical results on the application of the discrete Fréchet distance to protein structure-structure alignment.

Measures of sequence similarity and distance based more or less explicitly on subword composition are attracting an increasing interest driven by intensive applications such as massive document classification and genome-wide molecular taxonomy. A uniform character of such measures is in some underlying notion of relative compressibility, whereby two similar sequences are expected to share a larger number of common substrings than two distant ones. This paper reviews some of the approaches to sequence comparison based on subword composition and suggests that their common denominator may ultimately reside in special classes of subwords, the nature of which resonates in interesting ways with the structure of popular subword trees and graphs.

The uses of biochemical genetics as an aid to taxonomy are discussed briefly. A survey is made of published data to establish expected levels of genetic similarity or identity between conspecific populations, between congeneric species and between species of different genera. Existing measures of genetic similarity, identity or distance are discussed and some of the difficulties involved in their use in taxonomy are outlined. A new measure of genetic similarity is proposed specifically designed for use in making interspecific comparisons for taxonomic purposes. This measure offers considerable advantages in ease of calculation and also greatly simplifies the estimation of sampling errors. An empirical comparison is made, using published data, of the measures and confidence limits obtained by the proposed memod against published measures of genetic variation between species.

Attribute theories propose that a cognitive system constructs representations with constituent elements that reflect the attributes, features, properties, or characteristics of the world and the things in it. These theories are flexible; attributes may be used to describe tangible physical things in one’s environment but also intangible things such as plans and concepts. In the domain of memory, attribute theories describe how representations of one’s experience are stored, retrieved, and manipulated by the cognitive system. The mathematical language of linear algebra is central to these theories. Vector operations are used to define various measures of representational distance and similarity. Matrices are used to define associative structures such as synaptic weights in neural network models and sets of memories in instance theory models. This chapter examines some foundational attribute-based approaches to memory theory, including early attempts to define psychological similarity, and the influential stimulus-sampling theory. It reviews important empirical phenomena involving memory attributes, including proactive interference, encoding specificity, and source memory. Finally, the chapter considers how attribute-based approaches have influenced modern cognitive neuroscientific investigations and theory.

Ontologies, taxonomies, and other semantic hierarchies are increasingly necessary for organizing large quantities of data. We continue our development of knowledge discovery techniques based on combinatorial algorithms rooted in order theory by aiming to supplement the pseudo-distances previously developed as structural measures of vertical height in poset-based ontologies with quantitative measures of vertical distance based on additional statistical information. In this way, we seek to accommodate weighting of different portions of the underlying ontology according to this external information source. We also wish to improve on the deficiencies of existing such measures, in particular Resnik's measure of semantic similarity in lexical databases such as Wordnet. We begin by recalling and developing some basic concepts for ordered data objects, including our pseudo-distances and the operation of probability distributions as weights on posets. We then discuss and critique Resnik's measure before introducing our own sense of links weights and weighted normalized pseudo-distances among comparable nodes.