We study expansion and information diffusion in dynamic networks, that is in networks in which nodes and edges are continuously created and destroyed. We consider information diffusion by {em flooding}, the process by which, once a node is informed, it broadcasts its information to all its neighbors. We study models in which the network is {em sparse}, meaning that it has $mathcal{O}(n)$ edges, where $n$ is the number of nodes, and in which edges are created randomly, rather than according to a carefully designed distributed algorithm. In our models, when a node is born, it connects to $d=mathcal{O}(1)$ random other nodes. An edge remains alive as long as both its endpoints do. If no further edge creation takes place, we show that, although the network will have $Omega_d(n)$ isolated nodes, it is possible, with large constant probability, to inform a $1-exp(-Omega(d))$ fraction of nodes in $mathcal{O}(log n)$ time. Furthermore, the graph exhibits, at any given time, a large-set expansion property. We also consider models with {em edge regeneration}, in which if an edge $(v,w)$ chosen by $v$ at birth goes down because of the death of $w$, the edge is replaced by a fresh random edge $(v,z)$. In models with edge regeneration, we prove that the network is, with high probability, a vertex expander at any given time, and flooding takes $mathcal{O}(log n)$ time. The above results hold both for a simple but artificial streaming model of node churn, in which at each time step one node is born and the oldest node dies, and in a more realistic continuous-time model in which the time between births is Poisson and the lifetime of each node follows an exponential distribution.
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