Triangle Inequality Violations Research Articles

A Study of Triangle Inequality Violations in Social Network Clustering

A Study of Triangle Inequality Violations in Social Network Clustering

An Adaptive Control Scheme for Data-Driven Traffic Migration Engineering on 5G Network

Adaptive control of traffic engineering (TE) based on 5G network function virtualization (NFV) authorizes the efficient and dynamic network resource allocation, whose utilization is increasingly wide and will become more widespread. In this paper, we first devise an adaptive control scheme for data-driven traffic migration engineering (TME) on the 5G virtual network. The proposed TME technology focuses on a 5G enhancing mobile broadband (eMBB) network application scenario and takes the network operating expenditure (OPEX) as the main research target. Firstly, we predict the network traffic of the virtual network through the constructed traffic predicted mathematical model. Then, based on the triangle inequality violation (TIV) theorem, some local network traffic is adaptively migrated when the predicted link traffic exceeds the peak rate. Consequently, the migrations of logical links in the virtual network layer are completed. Finally, our experiments show that the proposed protocol can effectively improve the key performance indicators (KPIs) of the reconfigured network, such as throughput, delay and energy consumption. Furthermore, the Fridman and Holm statistical hypothesis tests are also used to analyze the simulation data, which proves that the proposed approximate TME algorithm has statistical significance.

All geographical distances are optimal

Triangular inequality is one of the four mathematical properties of distance. Its respect derives from the optimal nature of the measurement of distance. This demonstration (L'Hostis 2016, 2017) reveals key aspects of distances and geographical spaces. We develop this argument by investigating the idea of the optimality of distance through a mathematical and geometric discussion, and by dealing with empirical approaches to applied geography.The first part of the paper explores the consequences of considering that the mathematical property of triangle inequality is always respected. In fact, no violations of triangle inequality are observed in geographical spaces. The study of the optimality of distances in empirical approaches confirms the key role of the property of triangle inequality. The general principle of least-effort applies for most movements and spacings. In addition, however, trajectories with multiple detours, like those of shoppers, runners and nomads, are optimal from a certain point of view. This is also the case for excess travel, i.e. a situation of disjunction between an optimum as perceived by a person in movement and an optimum as perceived by an external observer. Any movement, any spacing within and between cities and in geographical space in general, exhibits a form of optimality, and all geographical distances are optimal.

Learning Feature Interactions with Lorentzian Factorization Machine

Learning representations for feature interactions to model user behaviors is critical for recommendation system and click-trough rate (CTR) predictions. Recent advances in this area are empowered by deep learning methods which could learn sophisticated feature interactions and achieve the state-of-the-art result in an end-to-end manner. These approaches require large number of training parameters integrated with the low-level representations, and thus are memory and computational inefficient. In this paper, we propose a new model named “LorentzFM” that can learn feature interactions embedded in a hyperbolic space in which the violation of triangle inequality for Lorentz distances is available. To this end, the learned representation is benefited by the peculiar geometric properties of hyperbolic triangles, and result in a significant reduction in the number of parameters (20% to 80%) because all the top deep learning layers are not required. With such a lightweight architecture, LorentzFM achieves comparable and even materially better results than the deep learning methods such as DeepFM, xDeepFM and Deep & Cross in both recommendation and CTR prediction tasks.

Network Latency Estimation for Personal Devices: A Matrix Completion Approach

Network latency prediction is important for server selection and quality-of-service estimation in real-time applications on the Internet. Traditional network latency prediction schemes attempt to estimate the latencies between all pairs of nodes in a network based on sampled round-trip times, through either Euclidean embedding or matrix factorization. However, these schemes become less effective in terms of estimating the latencies of personal devices, due to unstable and time-varying network conditions, triangle inequality violation and the unknown ranks of latency matrices. In this paper, we propose a matrix completion approach to network latency estimation. Specifically, we propose a new class of low-rank matrix completion algorithms, which predicts the missing entries in an extracted “network feature matrix” by iteratively minimizing a weighted Schatten- $p$ norm to approximate the rank. Simulations on true low-rank matrices show that our new algorithm achieves better and more robust performance than multiple state-of-the-art matrix completion algorithms in the presence of noise. We further enhance latency estimation based on multiple “frames” of latency matrices measured in the past, and extend the proposed matrix completion scheme to the case of 3-D tensor completion. Extensive performance evaluations driven by real-world latency measurements collected from the Seattle platform show that our proposed approaches significantly outperform various state-of-the-art network latency estimation techniques, especially for networks that contain personal devices.

Misunderstanding geographical distances: two errors and an issue in the interpretation of violations of triangle inequality

We investigate the meaning of the mathematical properties of distances in the fields of geography and economics. The key property of metrics for spatiality is triangle inequality (TI), which is closely related to the distance optimality. We identify three different situations where several authors identify violations of TI. We consider two of them to be errors of interpretation.The first error consists in considering sub-optimal measurements as distances. Yet distances are necessarily optimal if they obey TI.The second set of errors, which is the most widespread, entails a confusion between the Euclidean straight line and the shortest path. The errors lie in treating a detour as a violation of TI, whereas this situation simply corresponds to a non-Euclidean distance.The third problem concerns the additivity of distances. The commonplace situation in geographical space where a break is needed to provide the energy necessary to renew movement, is considered by some authors as another violation of TI. We argue that if these routes are optimal, TI should hold.

Distance outlier detection with a mobile beacon in wireless sensor networks localisation

Range-based localisation algorithms rely on distance measurements between nodes. If outlier distances exist, they make the estimated positions deviate from the true positions. Thus, it is important to detect and remove outlier distance measurements to achieve high localisation accuracy. In this paper, we propose a novel outlier detection approach called middle distance mobile beacon placement MDMP. Different from the traditional triangle-inequality-based approach that can detect only the existence of outlier distances but cannot distinguish which measurements are outliers, MDMP can exactly find which distances are outliers and exclude them from the localisation process. This prevents nodes from getting wrong position estimates caused by these outlier measurements. Simulation results show that MDMP improves the outlier detection accuracy by identifying more outliers than the traditional triangle inequality violation TIV-based method does. The evaluation results also show remarkable improvement in location accuracy by deleting outliers with MDMP.

Conditional clustered matrix factorization based network coordinate system

Network coordinate (NC) systems that use Euclidean distances suffer from the existence of Triangle Inequality Violations (TIVs). Matrix factorization (MF) based NC system is an alternative approach towards better prediction accuracy and can remove TIV. However, the prediction accuracy for short links in these systems still suffers from low prediction accuracy compared with the overall prediction accuracy. Two-layer systems have been proposed to improve the prediction accuracy for short links. They divide the whole space into several location-based clusters and run NC systems on both global layer and local layer. However, these systems do not improve the prediction accuracy for short links in the clusters with a few hosts. In this paper, the Conditional Clustered Network Coordinate (CCNC) system is proposed. It divides the space into a number of clusters in a balanced, dynamic, and decentralized way. In the proposed system, any node can join or disjoin the system without affecting the system accuracy. The performance of the CCNC system is evaluated with the King data set and the PlanetLab data set to be compared against two well known NC systems: Phoenix and Pancake. The simulation results show that CCNC outperforms Phoenix and Pancake significantly in terms of estimation accuracy, expected time to construct the clusters, and the communication overhead. Moreover, CCNC converges very fast and it is simple, scalable, dynamic, and robust under different dimension values.

Network Coordination Algorithm Based on Distributed Probability Matrix Factorization

Network Coordination Systems (NCS) are powerful mechanisms to efficiently predict the network latency of pairwise hosts in Internet without directly network measurement between them. Each host is embedded into a distance space where the latency between nodes is calculated according to the distant function defined by the space with assigned coordination of hosts. In terms of the properties of distant space, most popular NCS can be classified into two categories. One is based on metric space and the other is based on vector space obtained by matrix factorization. Unfortunately, the metric space such as Euclidean space based NCS is known to severely suffer from the weakness of inaccurate formulation of the Triangle Inequality Violation (TIV) and asymmetric latency frequently observed in Internet. To overcome the limitations of metric space, matrix factorization methodology is therefore proposed to represent the latency matrix with missing data by the product of two low-dimension vector matrices. In contrast to the huge amount of work on metric space type of NCS, there are fewer study of matrix factorization NCS up to now. This leads us to consider a probabilistic matrix factorization (PMF) based network distance prediction framework. In this paper we propose a model of distributed probabilistic matrix factorization (DPMF) which provides a probabilistic view of NCS with introduction of latent random variables to the incoming and outgoing vectors of hosts. Simulation on the real word data set shows that DPMF can achieve competitive prediction accuracy compared with two matrix factorization based NCS algorithms. Although the convergence rate of DPMF is relatively slow, the probabilistic viewpoint of delay matrix allows automatic regularization parameter selection in latent vector iteration process which makes the algorithm much practical for various latency data sets. Therefore, DPMF provides a promising generative process of latency value which potentially extends the usage scope of NCS.

H‐plots for displaying nonmetric dissimilarity matrices

AbstractNonmetric pairwise data with violations of symmetry, reflexivity, or triangle inequality appear in fields such as image matching, web mining, or cognitive psychology. When data are inherently nonmetric, we should not enforce metricity as real information could be lost. The multidimensional scaling problem is addressed from a new perspective. I propose a method based on the h‐plot, which naturally handles asymmetric proximity data. Pairwise proximities between the objects are defined, though I do not embed these objects, but rather the variables that give the proximity to or from each object. The method is very simple to implement. The representation goodness can be easily assessed. The methodology is illustrated through several small examples and applied to the analysis of digital images of human corneal endothelia. Comparisons with well‐known methods show its good behavior, especially with nonmetric pairwise data, which motivate my methodology. Other databases and methods are analyzed in the supporting information. © 2013 Wiley Periodicals, Inc. Statistical Analysis and Data Mining 6: 136–143, 2013

Analysis of large scale traceroute datasets in Internet routing overlays by parallel computation

The creation of a routing overlay network on the Internet requires the identification of shorter detour paths between end hosts in comparison to the default path available. These detour paths are typically the edges forming a Triangle Inequality Violation (TIV), an artifact of the Internet delay space where the sum of latencies across an intermediate hop is lesser than the direct latency between the pair of end hosts. These violations are caused mainly due to interdomain routing policies between Autonomous Systems (ASes) and AS peering through Internet eXchange Points (IXPs). Identifying detours for a global overlay network requires large amounts of computational capabilities due to the sheer number of possible paths linking source and destination ASes. In this work, we use parallel programming paradigms to exploit the massively parallel capabilities of analyzing the large network measurement datasets made available to the network research community by CAIDA. We study Internet routes traversing IXPs and measure potential TIVs created by these paths. Large scale analysis of the dataset is carried out by implementing an efficient parallel solution on the CPU and then the general purpose graphics processor unit (GPGPU) as well. Both multicore CPU and GPGPU implementations can be carried out with ease on desktop environments with readily available software. We find both parallel solutions yield high improvements in speedup (2-35x) in comparison to the serial methodologies thereby opening up the possibility of harnessing the power of parallel programming with readily available hardware. The large amount of data analyzed and studied helps draw various inferences for the networking research community in building future scalable Internet routing overlays with greater routing efficiencies.

The effects of triangle inequality on the vehicle routing problem

Routing problems often utilize experimental networks to represent real world scenarios. However most ignore the inclusion of triangle inequality violations, a phenomenon resulting from delays or rounding errors within a network. This work evaluates the effect of both frequency – the number of violations – and severity – the degree of intensity of a violation – of triangle inequality and evaluates both solution quality and solution time based on Simulated Annealing, Ant Colony Optimization and Savings Algorithm methods. Findings indicate that while both frequency and severity degrade solution quality, increased levels of frequency and severity together result in significant adverse affects to solution quality. Solution time, however, is not impacted by the presence of triangle inequality violations within the network. This information should encourage practitioners to identify delays and maintain the presence of triangle inequality violations in a network to ensure accuracy of solution quality.

Selecting Relays for Faster Paths in Embedded Internet Delay Space1

Application relays can be used to reduce latency for end-to-end communication. We propose a novel mechanism called TIVER to discover the latency-reducing relays in a distributed and proactive way. We boil down the relay discovery problem into TIV detours (the edges that are part of relay paths) discovery in the delay space, since the relay path causes the triangle inequality violation (TIV) phenomenon. Network embedding techniques are then introduced to model the delay space and provide a heuristic method to predict TIV detours with low measurement overhead. Subsequently, TIVER mechanism is designed for any given destination to allow nodes to quickly choose appropriate latencyreducing relays. The simulation results show the effects and efficiency of TIVER.

Phoenix: A Weight-Based Network Coordinate System Using Matrix Factorization

Network coordinate (NC) systems provide a lightweight and scalable way for predicting the distances, i.e., round-trip latencies among Internet hosts. Most existing NC systems embed hosts into a low dimensional Euclidean space. Unfortunately, the persistent occurrence of Triangle Inequality Violation (TIV) on the Internet largely limits the distance prediction accuracy of those NC systems. Some alternative systems aim at handling the persistent TIV, however, they only achieve comparable prediction accuracy with Euclidean distance based NC systems. In this paper, we propose an NC system, so-called Phoenix, which is based on the matrix factorization model. Phoenix introduces a weight to each reference NC and trusts the NCs with higher weight values more than the others. The weight-based mechanism can substantially reduce the impact of the error propagation. Using the representative aggregate data sets and the newly measured dynamic data set collected from the Internet, our simulations show that Phoenix achieves significantly higher prediction accuracy than other NC systems. We also show that Phoenix quickly converges to steady state, performs well under host churn, handles the drift of the NCs successfully by using regularization, and is robust against measurement anomalies. Phoenix achieves a scalable yet accurate end-to-end distances monitoring. In addition, we study how well an NC system can characterize the TIV property on the Internet by introducing two new quantitative metrics, so-called RE <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RPL</sub> and AE <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RPL</sub> . We show that Phoenix is able to characterize TIV better than other existing NC systems.

On Suitability of Euclidean Embedding for Host-Based Network Coordinate Systems

In this paper, we investigate the suitability of embedding Internet hosts into a Euclidean space given their pairwise distances (as measured by round-trip time). Using the classical scaling and matrix perturbation theories, we first establish the (sum of the) magnitude of negative eigenvalues of the (doubly centered, squared) distance matrix as a measure of suitability of Euclidean embedding. We then show that the distance matrix among Internet hosts contains negative eigenvalues of large magnitude, implying that embedding the Internet hosts in a Euclidean space would incur relatively large errors. Motivated by earlier studies, we demonstrate that the inaccuracy of Euclidean embedding is caused by a large degree of triangle inequality violation (TIV) in the Internet distances, which leads to negative eigenvalues of large magnitude. Moreover, we show that the TIVs are likely to occur locally; hence the distances among these close-by hosts cannot be estimated accurately using a global Euclidean embedding. In addition, increasing the dimension of embedding does not reduce the embedding errors. Based on these insights, we propose a new hybrid model for embedding the network nodes using only a two-dimensional Euclidean coordinate system and small error adjustment terms. We show that the accuracy of the proposed embedding technique is as good as, if not better than, that of a seven-dimensional Euclidean embedding.

Dissimilarity cumulation as a procedure correcting for violations of triangle inequality

Abstract Dissimilarity on a finite set is a function that assigns to every pair of points (stimuli) a nonnengative number vanishing if and only if the two points are identical. For any two points in a finite set, the minimum of the length of all finite chains connecting these points is a quasimetric (i.e., it satisfies all metric axioms except for symmetry). The computation of the quasimetric from a dissimilarity can be viewed as a data-analytic procedure of “correcting” dissimilarities into (quasi)distances, an alternative to nonmetric Multidimensional Scaling: the procedure simply replaces the dissimilarity between any two points with the shortest length across all chains of points connecting them. It is shown in this paper that this procedure can be equivalently described as a series of recursive corrections for violations of the triangle inequality across all triples of points considered in an arbitrary order.

On the information and representation of non-Euclidean pairwise data

Two common data representations are mostly used in intelligent data analysis, namely the vectorial and the pairwise representation. Pairwise data which satisfy the restrictive conditions of Euclidean spaces can be faithfully translated into a Euclidean vectorial representation by embedding. Non-metric pairwise data with violations of symmetry, reflexivity or triangle inequality pose a substantial conceptual problem for pattern recognition since the amount of predictive structural information beyond what can be measured by embeddings is unclear. We show by systematic modeling of non-Euclidean pairwise data that there exists metric violations which can carry valuable problem specific information. Furthermore, Euclidean and non-metric data can be unified on the level of structural information contained in the data. Stable component analysis selects linear subspaces which are particularly insensitive to data fluctuations. Experimental results from different domains support our pattern recognition strategy.