  We have recently proposed a "surplus-based" algorithm which solves the
multi-agent average consensus problem on general strongly connected and static
digraphs. The essence of that algorithm is to employ an additional variable to
keep track of the state changes of each agent, thereby achieving averaging even
though the state sum is not preserved. In this note, we extend this approach to
the more interesting and challenging case of time-varying topologies: An
extended surplus-based averaging algorithm is designed, under which a necessary
and sufficient graphical condition is derived that guarantees state averaging.
The derived condition requires only that the digraphs be arbitrary strongly
connected in a \emph{joint} sense, and does not impose "balanced" or
"symmetric" properties on the network topology, which is therefore more general
than those previously reported in the literature.
