Hend Alrasheed

Core-Periphery Models for Graphs Based on their δ-Hyperbolicity: An Example Using Biological Networks

Hyperbolicity is a global property of graphs that measures how close their structures are to trees in terms of their distances. It embeds multiple properties that facilitate solving several problems that found to be hard in the general graph form. In this paper, we investigate the hyperbolicity of graphs not only by considering Gromov’s notion of δ-hyperbolicity but also by analyzing its relationship to other graph’s parameters. This new perspective allows us to classify graphs with respect to their hyperbolicity and to show that many biological networks are hyperbolic. Then we introduce the eccentricity-based bending property which we exploit to identify the core vertices of a graph by proposing two models: the maximum-peak model and the minimum cover set model. In this extended version of the paper, we include some new theorems, as well as proofs of the theorems proposed in the conference paper. Also, we present the algorithms we used for each of the proposed core identification models, and we provide more analysis, explanations, and examples.

On the Δ-hyperbolicity in complex networks

δ-Hyperbolicity is a graph parameter that shows how close to a tree a graph is metrically. In this work, we propose a method that reduces the size of the graph to only a subset that is responsible for maximizing its δ-hyperbolicity using the local dominance relationship between vertices. Furthermore, we empirically show that the hyperbolicity of a graph can be found in a set of vertices that are in close proximity. That is, the hyperbolicity in graphs is, to some extent, a local property. Moreover, we show that this set is close to the graphs center. Our observations have crucial implications on computing the value of the δ-hyperbolicity of graphs.

On the Eccentricity Function in Graphs

Given a graph \(G=(V,E)\), the eccentricity of a vertex u is the distance from u to a vertex farthest from u. The set of vertices that minimizes the maximum distance to every other vertex (has minimum eccentricity) constitutes the center of the graph. The minimum eccentricity value represents the graph’s radius. The eccentricity function of a graph can be unimodal or non-unimodal. A graph with unimodal eccentricity function has the property that the eccentricity of every vertex equals its distance to the center plus the radius. A graph with non-unimodal eccentricity function lacks this property. In this work, we characterize each type of eccentricity function and study the impact of each type on the intersection of shortest paths among distant vertex pairs with the center. A shortest path intersects the center if it includes at least one vertex that belongs to the center. In particular, we show that if the eccentricity function is unimodal, all shortest paths among distant vertex pairs intersect the graph’s center. We also discuss when those paths do not intersect the center in graphs with non-unimodal eccentricity functions.

Eccentricity approximating trees

Using the characteristic property of chordal graphs that they are the intersection graphs of subtrees of a tree, Erich Prisner showed that every chordal graph admits an eccentricity 2-approximating spanning tree. That is, every chordal graph G has a spanning tree T such that eccT(v)eccG(v)2 for every vertex v, where eccG(v) (eccT(v)) is the eccentricity of a vertex v in G (in T, respectively). Using only metric properties of graphs, we extend that result to a much larger family of graphs containing among others chordal graphs, the underlying graphs of 7-systolic complexes and plane triangulations with inner vertices of degree at least 7. Furthermore, based on our approach, we propose two heuristics for constructing eccentricity k-approximating trees with small values of k for general unweighted graphs. We validate those heuristics on a set of real-world networks and demonstrate that all those networks have very good eccentricity approximating trees.

Eccentricity Approximating Trees

Using the characteristic property of chordal graphs that they are the intersection graphs of subtrees of a tree, Erich Prisner showed that every chordal graph admits an eccentricity 2-approximating spanning tree. That is, every chordal graph G has a spanning tree T such that

Scope of Selection Nodes in Object Oriented Programs

ecc_Tv-ecc_Gv\le 2

Understanding and Measuring Nesting

for every vertex v, where