Wiener Index

The Wiener index is a node centrality index. The Wiener index is the summa of the all distances (shortest paths) between a node v and all other nodes in the graph. Basically, it is similar to the closeness but here the score has the opposite meaning, as the reciprocal is not calculated. Thus a low Wiener index means that a node is closer to all other nodes. High and low values are more meaningful when compared to the average Wiener index of the graph G calcu- lated by averaging the Wiener index values of all nodes in the graph. Notably, the graph Wiener index is the summa of the Wiener indexes of all nodes in the graph. Also here, a low graph Wiener index indicates a graph whose nodes are likely to be highly connected. As for the closeness, the Wiener index provides a general evaluation of the average tendency of the node to stay in proximity or isolation, not really informative on the specific nature of the individual node couples. The Wiener index should be always compared to the eccentricity, closeness and radiality.

In biological terms
The Wiener index of a node in a biological network, for instance a protein-signaling network, can be interpreted as the probability of a protein to be functionally relevant for several other proteins, but with the possibility to be irrelevant for few other proteins. Thus, a protein with low Wiener index, compared to the average Wiener index of the network, will be easily central to the regulation of other proteins but with some proteins not influenced by its activity. Notably, in biological networks could be also of interest to analyze nodes with high Wiener index, compared to the average Wiener index of the network, as these nodes, although less relevant for that specific network, are possibly behaving as intersecting boundaries with other networks. Accordingly, a signaling network with a very low average Wiener index or a low graph Wiener index is more likely organizing functional units or modules, whereas a signaling network with very high average Wiener index or graph Wiener index will behave more likely as an open cluster of proteins connecting different regulatory modules.

• Scardoni, G. and Carlo Laudanna, C.B.M.C., 2011. Network centralities for Cytoscape. University of Verona.