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

Abstract

Abstract. 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.

On Computing the Hyperbolicity of Real-World Graphs

“…More in general, the hyperbolicity is connected to other important graph quantities, like treelength [7] and chordality [29]. In the field of the analysis of complex networks, the hyperbolicity and its connection with the size and the diameter of a network has been used in [2] in order to classify networks into three different classes, that is, strongly hyperbolic, hyperbolic, and nonhyperbolic, and to apply this classification to a small dataset of small biological networks (a more extensive analysis of the hyperbolicity of real-world networks has been also recently done in [5]). In general, it is still not clear whether the hyperbolicity value is small in all real-world networks (as it seems from [19,2]), or it is a characteristic of specific networks (as it seems from [1]).…”

Core–periphery models for graphs based on their δ-hyperbolicity: An example using biological networks

“…The cores for the networks in group 2 are larger than for those in group 1, yet they are small (9-18%). 57 …”

Structural Properties in δ-Hyperbolic Networks

“…Specifically, analyzing vertex eccentricities in δ-hyperbolic graphs. In [3], we algorithmically analyze the δ-hyperbolicity property and we exploit it to partition a graph into core and periphery parts. To achieve this, we formalize the notion of the eccentricity layering of a graph and employ it to introduce a new property that we find intrinsic to hyperbolic graphs: the eccentricitybased bending property.…”

“…It was also observed that the negative curvature causes most of the shortest paths to bend making the peak of the arc formed by a shortest path to pass through the core. In [3], we formalized this notion of bending in shortest paths by introducing an important property that is intrinsic to δ-hyperbolic graphs (the eccentricitybased bending property). Then we used this property to partition a graph into core and periphery parts.…”

“…We applied both models to a wide set of biological networks. For more details, the reader is referred to [3].…”