The identification of relationships in complex networks is critical
in a variety of scientific contexts. This includes the identification of
globally central nodes and analysing the importance of pairwise
relationships between nodes. In this paper, we consider the
concept of topological proximity (or ‘closeness’) between nodes
in a weighted network using the generalized Erdo´´s numbers
(GENs). This measure satisfies a number of desirable properties
for networks with nodes that share a finite resource. These
include: (i) real-valuedness, (ii) non-locality and (iii) asymmetry.
We show that they can be used to define a personalized
measure of the importance of nodes in a network with a natural
interpretation that leads to new methods to measure centrality.
We show that the square of the leading eigenvector of an
importance matrix defined using the GENs is strongly correlated
with well-known measures such as PageRank, and define a
personalized measure of centrality that is also well correlated
with other existing measures. The utility of this measure
of topological proximity is demonstrated by showing the
asymmetries in both the dynamics of random walks and the
mean infection time in epidemic spreading are better
predicted by the topological definition of closeness provided
by the GENs than they are by other measures.