The traditional approach to understanding node importance is via centrality indicators.
Centrality indices are designed to produce a ranking which accurately identifies the most influential nodes.
Since the mid 2000s, however, social scientists and network physicists have begun to question the suitability of centrality indices for understanding node influence.
Borgatti and Everett's 2006 review article[1] showed that the accuracy of centrality indices is highly dependent on network topology.
In 2012, Bauer and colleagues reminded us that centrality indices only rank nodes but do not quantify the difference between them.
[4] In 2013, Sikic and colleagues presented strong evidence that centrality indices considerably underestimate the power of non-hub nodes.
Hence a centrality measure which is appropriate for identifying highly influential nodes will most likely be inappropriate for the remainder of the network.
[3] This has inspired the development of novel methods designed to measure the influence of all network nodes.
The most general of these are the accessibility, which uses the diversity of random walks to measure how accessible the rest of the network is from a given start node,[6] and the expected force, derived from the expected value of the force of infection generated by a node.
The original work used simulated walks of length 60 to characterize the network of urban streets in a Brazilian city.
[6] It was later formalized as a modified form of hierarchical degree which controls for both transmission probabilities and the diversity of walks of a given fixed length.
[7] The accessibility has been shown to reveal community structure in urban networks,[6] corresponds to the number of nodes which can be visited in a defined time period,[7] and is predictive of the outcome of epidemiological SIR model spreading processes on networks with large diameter and low density.
[2] The expected force measures node influence from an epidemiological perspective.
The definition naturally extends to directed networks by limiting the enumeration
[3] The expected force has been shown to strongly correlate with SI, SIS, and SIR epidemic outcomes over a broad range of network topologies, both simulated and empirical.
[3][8] It has also been used to measure the pandemic potential of world airports,[9] and mentioned in the context of digital payments,[10] ecology,[11] fitness,[12] and project management.
[13] Others suggest metrics which explicitly encode the dynamics of a specified process unfolding on the network.
The dynamic influence is the proportion of infinite walks starting from each node, where walk steps are scaled such that the linear dynamics of the system are expected to converge to a non-null steady state.