  Most previous work of centralities focuses on metrics of vertex importance
and methods for identifying powerful vertices, while related work for edges is
much lesser, especially for weighted networks, due to the computational
challenge. In this paper, we propose to use the well-known Kirchhoff index as
the measure of edge centrality in weighted networks, called $\theta$-Kirchhoff
edge centrality. The Kirchhoff index of a network is defined as the sum of
effective resistances over all vertex pairs. The centrality of an edge $e$ is
reflected in the increase of Kirchhoff index of the network when the edge $e$
is partially deactivated, characterized by a parameter $\theta$. We define two
equivalent measures for $\theta$-Kirchhoff edge centrality. Both are global
metrics and have a better discriminating power than commonly used measures,
based on local or partial structural information of networks, e.g. edge
betweenness and spanning edge centrality.
  Despite the strong advantages of Kirchhoff index as a centrality measure and
its wide applications, computing the exact value of Kirchhoff edge centrality
for each edge in a graph is computationally demanding. To solve this problem,
for each of the $\theta$-Kirchhoff edge centrality metrics, we present an
efficient algorithm to compute its $\epsilon$-approximation for all the $m$
edges in nearly linear time in $m$. The proposed $\theta$-Kirchhoff edge
centrality is the first global metric of edge importance that can be provably
approximated in nearly-linear time. Moreover, according to the
$\theta$-Kirchhoff edge centrality, we present a $\theta$-Kirchhoff vertex
centrality measure, as well as a fast algorithm that can compute
$\epsilon$-approximate Kirchhoff vertex centrality for all the $n$ vertices in
nearly linear time in $m$.
