  We consider the following dynamic load-balancing process: given an underlying
graph $G$ with $n$ nodes, in each step $t\geq 0$, one unit of load is created,
and placed at a randomly chosen graph node. In the same step, the chosen node
picks a random neighbor, and the two nodes balance their loads by averaging
them. We are interested in the expected gap between the minimum and maximum
loads at nodes as the process progresses, and its dependence on $n$ and on the
graph structure.
  Similar variants of the above graphical balanced allocation process have been
studied by Peres, Talwar, and Wieder, and by Sauerwald and Sun for regular
graphs. These authors left as open the question of characterizing the gap in
the case of \emph{cycle graphs} in the \emph{dynamic} case, where weights are
created during the algorithm's execution. For this case, the only known upper
bound is of $\mathcal{O}( n \log n )$, following from a majorization argument
due to Peres, Talwar, and Wieder, which analyzes a related graphical allocation
process.
  In this paper, we provide an upper bound of $\mathcal{O} ( \sqrt n \log n )$
on the expected gap of the above process for cycles of length $n$. We introduce
a new potential analysis technique, which enables us to bound the difference in
load between $k$-hop neighbors on the cycle, for any $k \leq n / 2$. We
complement this with a "gap covering" argument, which bounds the maximum value
of the gap by bounding its value across all possible subsets of a certain
structure, and recursively bounding the gaps within each subset. We provide
analytical and experimental evidence that our upper bound on the gap is tight
up to a logarithmic factor.
