Distance-sum heterogeneity in graphs and complex networks

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 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-sum heterogeneity index reveals important structural characteristics of networks which can be important for understanding the functional and dynamical processes in complex systems.