Social networks support efficient decentralized search: people can
collectively construct short paths to a specified target in the
network. \emph{Rank-based friendship}---where the probability that
person~$u$ befriends person $v$ is inversely proportional to the
number of people who are closer to $u$ than $v$ is---is an empirically
validated model of acquaintanceship that provably results in efficient
decentralized search via greedy routing, even in networks with
variable population densities. In this paper, we introduce
\emph{cautious-greedy routing}, a variant of greedy that avoids taking
large jumps unless they make substantial progress towards the target.
Our main result is that cautious-greedy routing finds a path of short
expected length from an arbitrary source to a randomly chosen target,
independent of the population densities. To quantify the expected
length of the path, we define the \emph{depth of field} of a metric
space, a new quantity that intuitively measures the ``width'' of
directions that leave a point in the space. Our main result is that
cautious-greedy routing finds a path of expected length $O(\log^2 n)$
in $n$-person networks that have aspect ratio polynomial in $n$,
bounded doubling dimension, and bounded depth of field. Specifically,
in $k$-dimensional grids under Manhattan distance with arbitrary
population densities, the $O(\log^2 n)$ expected path length that we
achieve with the cautious-greedy algorithm improves the best previous
bound of $O(\log^3 n)$ with greedy routing.