  We study the problem of gossip in dynamic networks controlled by an adversary
that can modify the network arbitrarily from one round to another, provided
that the network is always connected. In the gossip problem, $n$ tokens are
arbitrarily distributed among the $n$ network nodes, and the goal is to
disseminate all the $n$ tokens to every node. Our focus is on token-forwarding
algorithms, which do not manipulate tokens in any way other than storing,
copying, and forwarding them. Gossip can be completed in linear time in any
static network, but a basic open question for dynamic networks is the existence
of a distributed protocol that can do significantly better than an easily
achievable bound of $O(n^2)$ rounds.
  In previous work, it has been shown that under adaptive adversaries, every
token forwarding algorithm requires $\Omega(n^2/\log n)$ rounds. In this paper,
we study oblivious adversaries, which differ from adaptive adversaries in one
crucial aspect--- they are oblivious to random choices made by the protocol. We
present an $\tilde{\Omega}(n^{3/2})$ lower bound under an oblivious adversary
for RANDDIFF, a natural algorithm in which neighbors exchange a token chosen
uniformly at random from the difference of their token sets. We also present an
$\tilde{\Omega}(n^{4/3})$ bound under a stronger notion of oblivious adversary
for symmetric knowledge-based algorithms. On the positive side, we present a
centralized algorithm that completes gossip in $\tilde{O}(n^{3/2})$ rounds. We
also show an $\tilde{O}(n^{5/3})$ upper bound for RANDDIFF in a restricted
class of oblivious adversaries, which we call paths-respecting.
