  Given a graph $\mathcal{G}$, the spanning centrality (SC) of an edge $e$
measures the importance of $e$ for $\mathcal{G}$ to be connected. In practice,
SC has seen extensive applications in computational biology, electrical
networks, and combinatorial optimization. However, it is highly challenging to
compute the SC of all edges (AESC) on large graphs. Existing techniques fail to
deal with such graphs, as they either suffer from expensive matrix operations
or require sampling numerous long random walks. To circumvent these issues,
this paper proposes TGT and its enhanced version TGT+, two algorithms for AESC
computation that offers rigorous theoretical approximation guarantees. In
particular, TGT remedies the deficiencies of previous solutions by conducting
deterministic graph traversals with carefully-crafted truncated lengths. TGT+
further advances TGT in terms of both empirical efficiency and asymptotic
performance while retaining result quality, based on the combination of TGT
with random walks and several additional heuristic optimizations. We
experimentally evaluate TGT+ against recent competitors for AESC using a
variety of real datasets. The experimental outcomes authenticate that TGT+
outperforms the state of the arts often by over one order of magnitude speedup
without degrading the accuracy.
