##### Concentric Characterization of Complex Networks

## Concentric Measurements Page

Filipi N. Silva, Bruno A. N. Travencolo, Lucas Antiqueira, Matheus Palhares Viana, Paulino R. Villas Boas, Francisco A. Rodrigues and Luciano da F. Costa

## Introduction

The study and characterization of complex networks has often relied on simple measurements such as the average node degree, clustering coefficient and average path lengths. However, such features do not provide direct insights about several relevant properties of the analyzed networks, especially about the connectivity of the network around specific nodes or subgraphs of special interest. Although such limitations have been acknowledged from time to time and complementary measures have been duly proposed in the literature, including betweeness centrality and spectral approaches, relatively lesser attention has been given to measurements or algorithms capable of comprehensively expressing the hierarchical (or concentric) structure of complex networks Costa, L. da F., arXiv:cond-mat/0412761.

This website provides complete resources for more comprehensive analysis of the hierarchical properties of complex networks with respect to specific nodes of subgraphs of interest. They comprehend a series of concentric measurements which can often be understood as extensions of traditional concepts (such as node degree and clustering coefficient) to incorporate progressive information about the surrounding context. This site starts by presenting the basics concepts regarding the node hierarchy, rings and balls and follows by describing and illustrating a set of hierarchical measures. Links to the papers where the presented concepts and measures were introduced can be found at the References section. In addition, a software has been made available, for non-profit academic use, containing all the described measures were implemented. Three versions of this software are available: an online and a standalone version.

## Basics Concepts

*Concentric level* - The first concentric (or hierarchical) level of the node \(i\) is composed by the immediate neighbors of the node \(i\). The second hierarchical level is composed by the nodes that are immediate neighbors of the nodes of the first hierarchical level. In similar fashion, the remaining hierarchical levels are composed by the nodes that are immediate neighbors of the nodes belonging to the previous hierarchical level. Examples:

The following examples show three hierarchical levels of the node \(i\):

The same hierarchies, now shown in a hierarchical layout starting at the node \(i\):

#### Concentric Rings and Balls

\(R_d(i)\) (*Concentric Ring*) - set of nodes inside the concentric level \(d\) of node \(i\). These nodes are present topological distance \(d\) from node \(i\). Also note that the ring at level \(0\) represent the set containing only the reference node \(i\), i.e. \(R_0(i) = \{i\}\).

\(B_d(i)\) (Concentric Ball) - set of nodes between the zeroth concentric level up to level \(d\) for a node \(i\), which can also be defined as the union of all rings up to \(R_d(i)\), i.e. \(B_d(i)=\bigcup _{j=0}^dR_j(i)\).