  Spectral algorithms based on matrix representations of networks are often
used to detect communities but classic spectral methods based on the adjacency
matrix and its variants fail to detect communities in sparse networks. New
spectral methods based on non-backtracking random walks have recently been
introduced that successfully detect communities in many sparse networks.
However, the spectrum of non-backtracking random walks ignores hanging trees in
networks that can contain information about the community structure of
networks. We introduce the reluctant backtracking operators that explicitly
account for hanging trees as they admit a small probability of returning to the
immediately previous node unlike the non-backtracking operators that forbid an
immediate return. We show that the reluctant backtracking operators can detect
communities in certain sparse networks where the non-backtracking operators
cannot while performing comparably on benchmark stochastic block model networks
and real world networks. We also show that the spectrum of the reluctant
backtracking operator approximately optimises the standard modularity function
similar to the flow matrix. Interestingly, for this family of non- and
reluctant-backtracking operators the main determinant of performance on
real-world networks is whether or not they are normalised to conserve
probability at each node.
