## Abstract

The heterogeneity of the sum of all distances from one node to the rest of nodes in a graph (distance-sum or status of the node) is analyzed. We start here by analyzing the cumulative statistical distributions of the distance-sum of nodes in random and real-world networks. From this analysis we conclude that statistical distributions do not reveal the distance-sum heterogeneity in networks. Thus, we motivate an index of distance-sum heterogeneity based on a hypothetical consensus model in which the nodes of the network try to reach an agreement on their distance-sum values. This index is expressed as a quadratic form of the combinatorial Laplacian matrix of the network. The distance-sum heterogeneity index (G) gives a natural interpretation of the Balaban index for any kind of graph/network. We conjecture here that among graphs with a given number of nodes (G) is maximized for a graph with a structure resembling the agave plant. We also found the graphs that maximize (G) for a given number of nodes and links. Using this index and a normalized version of it we studied random graphs as well as 57 real-world networks. Our findings indicate that the distance-sum heterogeneity index reveals important structural characteristics of networks which can be important for understanding the functional and dynamical processes in complex systems.

Original language | English |
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Pages (from-to) | 10393-10405 |

Number of pages | 13 |

Journal | Applied Mathematics and Computation |

Volume | 218 |

Issue number | 21 |

DOIs | |

State | Published - 1 Jul 2012 |

## Keywords

- Balaban index
- Complex networks
- Distance distributions
- Distance-sum
- Graph distances