# Tight Analysis of Asynchronous Rumor Spreading in Dynamic Networks

The asynchronous rumor algorithm spreading propagates a piece of information, the so-called rumor, in a network. Starting with a single informed node, each node is associated with an exponential time clock with rate 1 and calls a random neighbor in order to possibly exchange the rumor. Spread time is the first time when all nodes of a network are informed with high probability. We consider spread time of the algorithm in any dynamic evolving network, ๐ข={G^(t)}_t=0^โ, which is a sequence of graphs exposed at discrete time step t=0,1.... We observe that besides the expansion profile of a dynamic network, the degree distribution of nodes over time effect the spread time. We establish upper bounds for the spread time in terms of graph conductance and diligence. For a given connected simple graph G=(V,E), the diligence of cut set E(S, S) is defined as ฯ(S)=min_{u,v}โ E(S,S)max{dฬ /d_u, dฬ /d_v} where d_u is the degree of u and dฬ is the average degree of nodes in the one side of the cut with smaller volume (i.e., ๐๐๐(S)=โ_uโ Sd_u). The diligence of G is also defined as ฯ(G)=min_โ โ Sโ Vฯ(S). We show that the spread time of the algorithm in ๐ข is bounded by T, where T is the first time that โ_t=0^Tฮฆ(G^(t))ยทฯ(G^(t)) exceeds Clog n, where ฮฆ(G^(t)) denotes the conductance of G^(t) and C is a specified constant. We also define the absolute diligence as ฯ(G)=min_{u,v}โ Emax{1/d_u,1/d_v} and establish upper bound T for the spread time in terms of absolute diligence, which is the first time when โ_t=0^Tโฮฆ(G^(t))โยทฯ(G^(t))> 2n. We present dynamic networks where the given upper bounds are almost tight.

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