  This paper develops upper and lower bounds on the influence measure in a
network, more precisely, the expected number of nodes that a seed set can
influence in the independent cascade model. In particular, our bounds exploit
nonbacktracking walks, Fortuin-Kasteleyn-Ginibre (FKG) type inequalities, and
are computed by message passing implementation. Nonbacktracking walks have
recently allowed for headways in community detection, and this paper shows that
their use can also impact the influence computation. Further, we provide a knob
to control the trade-off between the efficiency and the accuracy of the bounds.
Finally, the tightness of the bounds is illustrated with simulations on various
network models.
