Abstract
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 Erdős 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.
Highlights
The study of complex networks has increased enormously in recent years due to their applicability to a wide range of physical [1,2], biological [3], epidemiological [4,5] and sociological [6] systems
The infection time can be compared to a variety of measures of topological closeness, and we focus on the generalized Erdo ́s numbers (GENs) (Eji), the mean first passage times (MFPT) in a random walk and the resistance distance (Rij)
Derived from simple principles based on a conceptual picture of nodes sharing finite resources, the GENs incorporate the global topology of the network into a pairwise measure of closeness for connected and disconnected nodes alike
Summary
The study of complex networks has increased enormously in recent years due to their applicability to a wide range of physical [1,2], biological [3], epidemiological [4,5] and sociological [6] systems. The centrality of individual nodes can be measured incorporating the global topology of the network in a variety of ways, including PageRank [18], betweenness [15] or random walk [13] centralities. Each of these measures reduces the global properties of the network into an individualized local measure of importance, permitting a rank-ordering of their importance in the network [19,20]. This work illustrates that the GENs are a useful measure of the topological closeness between pairs of nodes in a complex network, and illustrates that a meaningful definition of closeness has the potential to bridge the gap between the topology of a network and the dynamics on the network in multiple contexts
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