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, $\mathcal{G}=\{G^{(t)}\}_{t=0}^{\infty}$, which is a sequence of graphs exposed at discrete time step $t=0,1\ldots$. 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 $E(S, \overline{S})$ is defined as $\rho(S)=\min_{\{u,v\}\in E(S,\overline{S})}\max\{\bar{d}/d_u, \bar{d}/d_v\}$ where $d_u$ is the degree of $u$ and $\bar{d}$ is the average degree of nodes in the one side of the cut with smaller volume (i.e., ${\mathtt{vol}}{(S)}=\sum_{u\in S}d_u$). The diligence of $G$ is also defined as $\rho(G)=\min_{ \emptyset\neq S\subset V}\rho(S)$. We show that the spread time of the algorithm in $\mathcal{G}$ is bounded by $T$, where $T$ is the first time that $\sum_{t=0}^T\Phi(G^{(t)})\cdot\rho(G^{(t)})$ exceeds $C\log n$, where $\Phi(G^{(t)})$ denotes the conductance of $G^{(t)}$ and $C$ is a specified constant. We also define the absolute diligence as $\overline{\rho}(G)=\min_{\{u,v\}\in E}\max\{1/d_u,1/d_v\}$ and establish upper bound $T$ for the spread time in terms of absolute diligence, which is the first time when $\sum_{t=0}^T\lceil\Phi(G^{(t)})\rceil\cdot \overline{\rho}(G^{(t)})\ge 2n$. We present dynamic networks where the given upper bounds are almost tight.