  In modern applications of graphs algorithms, where the graphs of interest are
large and dynamic, it is unrealistic to assume that an input representation
contains the full information of a graph being studied. Hence, it is desirable
to use algorithms that, even when only a (large) subgraph is available, output
solutions that are close to the solutions output when the whole graph is
available. We formalize this idea by introducing the notion of average
sensitivity of graph algorithms, which is the average earth mover's distance
between the output distributions of an algorithm on a graph and its subgraph
obtained by removing an edge, where the average is over the edges removed and
the distance between two outputs is the Hamming distance.
  In this work, we initiate a systematic study of average sensitivity. After
deriving basic properties of average sensitivity such as composition, we
provide efficient approximation algorithms with low average sensitivities for
concrete graph problems, including the minimum spanning forest problem, the
global minimum cut problem, the minimum $s$-$t$ cut problem, and the maximum
matching problem. In addition, we prove that the average sensitivity of our
global minimum cut algorithm is almost optimal, by showing a nearly matching
lower bound. We also show that every algorithm for the 2-coloring problem has
average sensitivity linear in the number of vertices. One of the main ideas
involved in designing our algorithms with low average sensitivity is the
following fact; if the presence of a vertex or an edge in the solution output
by an algorithm can be decided locally, then the algorithm has a low average
sensitivity, allowing us to reuse the analyses of known sublinear-time
algorithms and local computation algorithms (LCAs). Using this connection, we
show that every LCA for 2-coloring has linear query complexity, thereby
answering an open question.
