δ-HYPERBOLICITY IN REAL-WORLD NETWORKS: ALGORITHMIC ANALYSIS AND IMPLICATIONS A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by Hend Mohammed Alrasheed May 2018 Dissertation written by Hend Mohammed Alrasheed B.S., King Saud University, 2004 M.Sc., King Saud University, 2009 Ph.D., Kent State University, 2018 Approved by , Chair, Doctoral Dissertation Committee Dr. Feodor F. Dragan , Members, Doctoral Dissertation Committee Dr. Javed Khan Dr. Hassan Peyravi Dr. Dmitry Ryabogin Accepted by , Chair, Department of Computer Science Dr. Javed Khan , Dean, College of Arts and Sciences Dr. James L. Blank Table of Contents Table of Contents iii List of Figures vii List of Tables ix 1 Introduction 1 1.1 Research contribution . .6 1.1.1 Publications . .8 1.2 Structural properties of graphs . .9 1.3 Gromov δ-hyperbolic spaces . 11 2 Preliminaries and Notations 15 2.1 Basic definitions . 15 2.2 Eccentricity . 21 2.2.1 Eccentricity layering of a graph . 22 2.3 Graph embedding . 23 2.4 Graph centrality measures . 25 3 δ-Hyperbolicity in Graphs 28 iii 3.1 Definitions and basic properties . 28 3.2 Why do we study δ-hyperbolicity in graphs? . 37 3.3 Computational aspect of δ-hyperbolicity . 41 I δ-Hyperbolicity in Graphs - A Structural Point of View 45 4 Core-Periphery Models for Graphs Based on their δ-Hyperbolicity - an Example Using Biological Networks 47 4.1 Introduction . 49 4.2 Background and related work . 52 4.2.1 Core-periphery and network centrality in complex networks . 52 4.2.2 Biological networks and the core-periphery structure . 53 4.3 Datasets . 54 4.4 δ-Hyperbolicity and graph classification . 56 4.4.1 Hyperbolicity of biological networks . 57 4.4.2 Analysis and discussion . 58 4.5 Core-Periphery models based on δ-hyperbolicity . 61 4.5.1 Eccentricity-based bending property of δ-hyperbolic graphs . 62 4.5.2 Core-periphery identification using the eccentricity-based bending property . 70 4.6 Concluding remarks . 78 5 Interplay Between δ-Hyperbolicity and the Core-Periphery Structure in Graphs 82 5.1 Introduction . 84 5.2 Datasets . 86 iv 5.3 δ-Hyperbolicity in graphs . 89 5.3.1 δ-Hyperbolicity and dominated vertices . 89 5.3.2 δ-Hyperbolicity and restricted path lengths . 94 5.4 δ-Hyperbolicity and the core-periphery structure . 97 5.4.1 The minimum-cover-set core . 99 5.4.2 k-core . 101 5.5 Case studies . 103 5.5.1 Biological network . 103 5.5.2 AS-graph . 104 5.6 Concluding remarks . 104 II Eccentricity Approximating Trees 106 6 Eccentricity Approximating Trees in δ-Hyperbolic Graphs 109 6.1 Introduction . 110 6.2 Preliminaries . 114 6.3 Fast approximation of eccentricities . 114 6.3.1 Conditional lower bounds on complexities . 115 6.3.2 Fast additive approximations . 116 6.4 Experimentation on some real-world networks . 123 6.4.1 Estimation of eccentricities . 125 7 Eccentricity Approximating Trees in (α1; ∆)-Metric Graphs 134 7.1 Introduction . 136 7.2 Preliminaries . 138 7.3 Eccentricity function on (α1; ∆)-metric graphs . 141 v 7.4 Eccentricity approximating spanning tree construction . 145 7.4.1 Tree construction for unimodal eccentricity functions . 145 7.4.2 Tree construction for eccentricity functions that are not unimodal . 147 7.5 Experimental results for some real-world networks . 150 7.5.1 Datasets . 152 7.5.2 Analysis of vertex localities and centers . 154 7.5.3 Eccentricity approximating tree construction and analysis . 157 7.6 Concluding remarks . 162 8 Eccentricity k-Approximating Tree Algorithms - Comparison 164 8.1 The algorithms . 164 8.2 Comparison . 166 9 Conclusions and Future Directions 168 Bibliography 171 vi List of Figures 1.1 Graph and vertex eccentricity. .3 1.2 A visualization of the hyperbolic plane using the Poincar´edisc model . .5 1.3 Parallel lines in (a) Euclidean space and (b) hyperbolic space. 13 1.4 Triangles in (a) Euclidean space and (b) hyperbolic space. 13 1.5 An example of a hyperbolic space . 14 2.1 Diameter, radius, and center in a graph. 17 2.2 Some common graph types. 20 2.3 Eccentricity Layering of a graph . 22 2.4 Graph eccentricity layering example . 23 3.1 The embedding of a complete graph into a weighted tree. 29 3.2 δ-Hyperbolicity examples. 30 3.3 Geodesic triangles, Gromov product, and tripods. 32 3.4 Four-point-condition definition for δ-hyperbolicity. 33 3.5 Slim triangles definition for δ-hyperbolicity. 34 3.6 Thin triangles definition for δ-hyperbolicity. 35 3.7 A graph with two bi-connected components. 43 vii 4.1 Distribution of quadruples over different values of δ-hyperbolicity. 57 4.2 Distribution of quadruples over different values of δ-hyperbolicity (three datasets). 58 4.3 Classification graphs based on their δ-hyperbolicity. 60 4.4 Graph eccentricity layering. 63 4.5 Distribution of vertices over different layers of the graph's eccentricity layering. 64 4.6 A case in which shortest paths between a diamteral pair do not bend towards the graph's center. 68 4.7 µk values for each network in the graph datasets. 71 4.8 A simplified illustration of the eccentricity layering of a graph and the Maximum-Peak model. 72 4.9 The percentage of the uncovered vertex pairs after the orderly addition of vertices to the core set. 78 5.1 A graph with dominated vertices. 90 5.2 Illustration of Lemma 9. 95 5.3 p-δ-Hyperbolicity of real-world and artificial networks. 97 5.4 k-core example. 102 7.1 The relationships between the different graph types used in this chapter. 138 ++ 7.2 Forbidden isometric subgraph W6 ....................... 139 7.3 An α1-metric graph. 143 7.4 Distribution of vertices with loc( ) > 1 over different layers of the eccentricity · layering in each graph. 156 viii List of Tables 2.1 Summary of notations. 16 4.1 Graph datasets and their parameters. 55 4.2 The hyperbolicity of each graph in the datasets. 56 4.3 The ratio of the δ-hyperbolicity to logarithm of the size and logarithm of the diameter for each network in the datasets. 62 4.4 Distribution of vertices over different layers with respect to the graph's ec- centricity layering. 65 4.5 The effect of the distance between vertex pairs on the bending property. 67 4.6 The absolute and the effective curvity for our graph datasets. 69 4.7 The cores of the graph datasets based on the Maximum-Peak model. 74 4.8 The cores of the graph datasets based on the Minimum Cover Set model. 77 4.9 Summary of the graph datasets' parameters and cores. 80 5.1 Graph datasets and their parameters. 87 5.2 Statistics of dominated vertices. 91 5.3 p-δ-Hyperbolicity. 93 5.4 p-δ-hyperbolicity and the center of the graph. 98 5.5 δ-Hyperbolicity and the graph's minimum-cover-set core. 99 ix 5.6 δ-hyperbolicity and the graph's k-core. 102 6.1 Graph datasets and their parameters. 124 6.2 Eccentricity estimation based on Proposition 8. 126 6.3 Distribution of distortion values listed in Table 6.2 . 127 6.4 Eccentricity estimation based on Proposition 9. 129 6.5 Distribution of distortion values listed in Table 6.4 . 130 6.6 Comparison of the three proposed eccentricity approximating BFS-trees. 132 7.1 Graph datasets and their parameters. 151 7.2 Percent of vertices with localities equal to 1 and larger than 1 in each graph of the datasets. 155 7.3 The weak diameters and weak radii of the centers of each graph in the datasets.157 7.4 A spanning tree T constructed by Algorithm EAST. 160 7.5 A tree T constructed by Algorithm EAT. 162 8.1 Comparison of the main characteristics of the three eccentricity k-approximating tree algorithms. ..