Distance-sum heterogeneity in graphs and complex networks

Ernesto Estrada, Eusebio Vargas Estrada

Research output: Contribution to journalArticlepeer-review

9 Citations (Scopus)


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-sumheterogeneity in networks. Thus, we motivate an index of distance-sumheterogeneity 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-sumheterogeneity index φ(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 φ(G) is maximized for a graph with a structure resembling the agave plant. We also found the graphs that maximize φ(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-sumheterogeneity index reveals important structural characteristics of networks which can be important for understanding the functional and dynamical processes in complex systems.

Original languageEnglish
Pages (from-to)10393-10405
Number of pages13
JournalApplied Mathematics and Computation
Issue number21
Publication statusPublished - 1 Jul 2012


  • distance distributions
  • distance-sum
  • complex networks
  • Balaban index
  • graph distances

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