Tight Analysis of Asynchronous Rumor Spreading in Dynamic Networks

TL;DR

This paper provides a detailed analysis of the spread time of asynchronous rumor algorithms in dynamic networks, establishing bounds based on network conductance, degree distribution, and diligence, with implications for understanding information dissemination efficiency.

Contribution

It introduces new upper bounds for rumor spread time in dynamic networks considering conductance and diligence, extending previous static network analyses.

Findings

Spread time bounded by sum of conductance and diligence over time

Upper bounds are nearly tight in certain dynamic network models

Degree distribution influences rumor spreading efficiency

Abstract

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.