Expansion and Flooding in Dynamic Random Networks with Node Churn

Abstract

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 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 sparse, meaning that it has 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 = 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 Ω d (n) isolated nodes, it is possible, with large constant probability, to inform a 1 − exp(−Ω(d)) fraction of nodes in O(log n) time. Furthermore, the graph exhibits, at any given time, a "large-set expansion" property.We also consider models with 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 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 continuoustime model in which the time between births is Poisson and the lifetime of each node follows an exponential distribution.Previous work on expansion and flooding studied models in which either the vertex set is fixed and only edges change with time or models in which edge generation occurs according to an algorithm. Our motivation for studying models with random edge generation is to go in the direction of models that may eventually capture the formation of social networks or peer-to-peer networks.

Dynamic Hidden-Variable Network Models

“…Adaptive network models (for instance, SIS-dynamics [161] alongside contactswitching [23,24]), have dynamic node-properties that evolve with time and guide network evolution, a commonality with THVMs. Networks with node-growth and node-removal [162][163][164][165] have dynamic node-properties (degree-values as opposed to hidden variables) that influence link-formation. In the fitness model of growing networks [12], static HVs and dynamic degrees both govern connection probabilities.…”