  Motivated by the increasing need to understand the distributed algorithmic
foundations of large-scale graph computations, we study some fundamental graph
problems in a message-passing model for distributed computing where $k \geq 2$
machines jointly perform computations on graphs with $n$ nodes (typically, $n
\gg k$). The input graph is assumed to be initially randomly partitioned among
the $k$ machines, a common implementation in many real-world systems.
Communication is point-to-point, and the goal is to minimize the number of
communication {\em rounds} of the computation.
  Our main contribution is the {\em General Lower Bound Theorem}, a theorem
that can be used to show non-trivial lower bounds on the round complexity of
distributed large-scale data computations. The General Lower Bound Theorem is
established via an information-theoretic approach that relates the round
complexity to the minimal amount of information required by machines to solve
the problem. Our approach is generic and this theorem can be used in a
"cookbook" fashion to show distributed lower bounds in the context of several
problems, including non-graph problems. We present two applications by showing
(almost) tight lower bounds for the round complexity of two fundamental graph
problems, namely {\em PageRank computation} and {\em triangle enumeration}. Our
approach, as demonstrated in the case of PageRank, can yield tight lower bounds
for problems (including, and especially, under a stochastic partition of the
input) where communication complexity techniques are not obvious.
  Our approach, as demonstrated in the case of triangle enumeration, can yield
stronger round lower bounds as well as message-round tradeoffs compared to
approaches that use communication complexity techniques.
