Ricci Curvature of the Internet Topology

Analysis of Internet topologies has shown that the Internet topology has negative curvature, measured by Gromov's "thin triangle condition", which is tightly related to core congestion and route reliability. In this work we analyze the discrete Ricci curvature of the Internet, defined by Ollivier, Lin, etc. Ricci curvature measures whether local distances diverge or converge. It is a more local measure which allows us to understand the distribution of curvatures in the network. We show by various Internet data sets that the distribution of Ricci cuvature is spread out, suggesting the network topology to be non-homogenous. We also show that the Ricci curvature has interesting connections to both local measures such as node degree and clustering coefficient, global measures such as betweenness centrality and network connectivity, as well as auxilary attributes such as geographical distances. These observations add to the richness of geometric structures in complex network theory.Comment: 9 pages, 16 figures. To be appear on INFOCOM 201

Embedding Directed Graphs in Potential Fields Using FastMap-D

Embedding undirected graphs in a Euclidean space has many computational benefits. FastMap is an efficient embedding algorithm that facilitates a geometric interpretation of problems posed on undirected graphs. However, Euclidean distances are inherently symmetric and, thus, Euclidean embeddings cannot be used for directed graphs. In this paper, we present FastMap-D, an efficient generalization of FastMap to directed graphs. FastMap-D embeds vertices using a potential field to capture the asymmetry between the pairwise distances in directed graphs. FastMap-D learns a potential function to define the potential field using a machine learning module. In experiments on various kinds of directed graphs, we demonstrate the advantage of FastMap-D over other approaches.Comment: 9 pages, Published in Symposium on Combinatorial Search(SoCS-2020). Erratum with updated Result

Embedding Metrics into Ultrametrics and Graphs into Spanning Trees with Constant Average Distortion

This paper addresses the basic question of how well can a tree approximate distances of a metric space or a graph. Given a graph, the problem of constructing a spanning tree in a graph which strongly preserves distances in the graph is a fundamental problem in network design. We present scaling distortion embeddings where the distortion scales as a function of $\epsilon$, with the guarantee that for each $\epsilon$ the distortion of a fraction $1-\epsilon$ of all pairs is bounded accordingly. Such a bound implies, in particular, that the \emph{average distortion} and $\ell_q$-distortions are small. Specifically, our embeddings have \emph{constant} average distortion and $O(\sqrt{\log n})$ $\ell_2$-distortion. This follows from the following results: we prove that any metric space embeds into an ultrametric with scaling distortion $O(\sqrt{1/\epsilon})$. For the graph setting we prove that any weighted graph contains a spanning tree with scaling distortion $O(\sqrt{1/\epsilon})$. These bounds are tight even for embedding in arbitrary trees. For probabilistic embedding into spanning trees we prove a scaling distortion of $\tilde{O}(\log^2 (1/\epsilon))$, which implies \emph{constant} $\ell_q$-distortion for every fixed $q<\infty$.Comment: Extended abstrat apears in SODA 200

A Decentralized Network Coordinate System for Robust Internet Distance

Network distance, measured as round-trip latency be-tween hosts, is important for the performance of many In-ternet applications. For example, nearest server selection and proximity routing in peer-to-peer networks rely on the ability to select nodes based on inter-host latencies. This paper presents PCoord, a decentralized network coordi-nate system for Internet distance prediction. In PCoord, the network is modeled as a D-dimensional geometric space; each host computes its coordinates in this geometric space to characterize its network location based on a small num-ber of peer-to-peer network measurements. The goal is to embed hosts in the geometric space so that the Euclidean distance between two hosts ’ coordinates accurately predicts their actual inter-host network latency. PCoord constructs network coordinates in a fully decentralized fashion. We present several mechanisms in PCoord to stabilize the sys-tem convergence. Our simulation results using real Internet measurements suggest that, even under an extremely chal-lenging flash-crowd scenario where 1740 hosts simultane-ously join the system, PCoord with a 5-dimensional Eu-clidean model is able to converge to 11 % median prediction error in 10 coordinate updates per host on average.

Quantitative metric profiles capture three-dimensional temporospatial architecture to discriminate cellular functional states

<p>Abstract</p> <p>Background</p> <p>Computational analysis of tissue structure reveals sub-visual differences in tissue functional states by extracting quantitative signature features that establish a diagnostic profile. Incomplete and/or inaccurate profiles contribute to misdiagnosis.</p> <p>Methods</p> <p>In order to create more complete tissue structure profiles, we adapted our cell-graph method for extracting quantitative features from histopathology images to now capture temporospatial traits of three-dimensional collagen hydrogel cell cultures. Cell-graphs were proposed to characterize the spatial organization between the cells in tissues by exploiting graph theory wherein the nuclei of the cells constitute the <it>nodes </it>and the approximate adjacency of cells are represented with <it>edges</it>. We chose 11 different cell types representing non-tumorigenic, pre-cancerous, and malignant states from multiple tissue origins.</p> <p>Results</p> <p>We built cell-graphs from the cellular hydrogel images and computed a large set of features describing the structural characteristics captured by the graphs over time. Using three-mode tensor analysis, we identified the five most significant features (metrics) that capture the compactness, clustering, and spatial uniformity of the 3D architectural changes for each cell type throughout the time course. Importantly, four of these metrics are also the discriminative features for our histopathology data from our previous studies.</p> <p>Conclusions</p> <p>Together, these descriptive metrics provide rigorous quantitative representations of image information that other image analysis methods do not. Examining the changes in these five metrics allowed us to easily discriminate between all 11 cell types, whereas differences from visual examination of the images are not as apparent. These results demonstrate that application of the cell-graph technique to 3D image data yields discriminative metrics that have the potential to improve the accuracy of image-based tissue profiles, and thus improve the detection and diagnosis of disease.</p

Metric Embedding via Shortest Path Decompositions

We study the problem of embedding shortest-path metrics of weighted graphs into $\ell_p$ spaces. We introduce a new embedding technique based on low-depth decompositions of a graph via shortest paths. The notion of Shortest Path Decomposition depth is inductively defined: A (weighed) path graph has shortest path decomposition (SPD) depth $1$. General graph has an SPD of depth $k$ if it contains a shortest path whose deletion leads to a graph, each of whose components has SPD depth at most $k-1$. In this paper we give an $O(k^{\min\{\frac{1}{p},\frac{1}{2}\}})$-distortion embedding for graphs of SPD depth at most $k$. This result is asymptotically tight for any fixed $p>1$, while for $p=1$ it is tight up to second order terms. As a corollary of this result, we show that graphs having pathwidth $k$ embed into $\ell_p$ with distortion $O(k^{\min\{\frac{1}{p},\frac{1}{2}\}})$. For $p=1$, this improves over the best previous bound of Lee and Sidiropoulos that was exponential in $k$; moreover, for other values of $p$ it gives the first embeddings whose distortion is independent of the graph size $n$. Furthermore, we use the fact that planar graphs have SPD depth $O(\log n)$ to give a new proof that any planar graph embeds into $\ell_1$ with distortion $O(\sqrt{\log n})$. Our approach also gives new results for graphs with bounded treewidth, and for graphs excluding a fixed minor

An Overview of Internet Measurements:Fundamentals, Techniques, and Trends

The Internet presents great challenges to the characterization of its structure and behavior. Different reasons contribute to this situation, including a huge user community, a large range of applications, equipment heterogeneity, distributed administration, vast geographic coverage, and the dynamism that are typical of the current Internet. In order to deal with these challenges, several measurement-based approaches have been recently proposed to estimate and better understand the behavior, dynamics, and properties of the Internet. The set of these measurement-based techniques composes the Internet Measurements area of research. This overview paper covers the Internet Measurements area by presenting measurement-based tools and methods that directly influence other conventional areas, such as network design and planning, traffic engineering, quality of service, and network management

Impact of the Inaccuracy of Distance Prediction Algorithms on Internet Applications--an Analytical and Comparative Study

Distance prediction algorithms use O(N) Round Trip Time (RTT) measurements to predict the N2 RTTs among N nodes. Distance prediction can be applied to improve the performance of a wide variety of Internet applications: for instance, to guide the selection of a download server from multiple replicas, or to guide the construction of overlay networks or multicast trees. Although the accuracy of existing prediction algorithms has been extensively compared using the relative prediction error metric, their impact on applications has not been systematically studied. In this paper, we consider distance prediction algorithms from an application\u27s perspective to answer the following questions: (1) Are existing prediction algorithms adequate for the applications? (2) Is there a significant performance difference between the different prediction algorithms, and which is the best from the application perspective? (3) How does the prediction error propagate to affect the user perceived application performance? (4) How can we address the fundamental limitation (i.e., inaccuracy) of distance prediction algorithms? We systematically experiment with three types of representative applications (overlay multicast, server selection, and overlay construction), three distance prediction algorithms (GNP, IDES, and the triangulated heuristic), and three real-world distance datasets (King, PlanetLab, and AMP). We find that, although using prediction can improve the performance of these applications, the achieved performance can be dramatically worse than the optimal case where the real distances are known. We formulate statistical models to explain this performance gap. In addition, we explore various techniques to improve the prediction accuracy and the performance of prediction-based applications. We find that selectively conducting a small number of measurements based on prediction-based screening is most effective