  The total communicability of a network (or graph) is defined as the sum of
the entries in the exponential of the adjacency matrix of the network, possibly
normalized by the number of nodes. This quantity offers a good measure of how
easily information spreads across the network, and can be useful in the design
of networks having certain desirable properties. The total communicability can
be computed quickly even for large networks using techniques based on the
Lanczos algorithm.
  In this work we introduce some heuristics that can be used to add, delete, or
rewire a limited number of edges in a given sparse network so that the modified
network has a large total communicability. To this end, we introduce new edge
centrality measures which can be used to guide in the selection of edges to be
added or removed.
  Moreover, we show experimentally that the total communicability provides an
effective and easily computable measure of how "well-connected" a sparse
network is.
