Robert Elsässer — Page 2

arXiv2011

Awareness and Movement vs. the Spread of Epidemics - Analyzing a Dynamic Model for Urban Social/Technological Networks

We consider the spread of epidemics in technological and social networks. How do people react? Does awareness and cautious behavior help? We analyze these questions and present a dynamic model to describe the movement of individuals and/or their mobile devices in a certain (idealistic) urban environment. Furthermore, our model incorporates the fact that different locations can accommodate a different number of people (possibly with their mobile devices), who may pass the infection to each other. We obtain two main results. First, we prove that w.r.t. our model at least a small part of the system will remain uninfected even if no countermeasures are taken. The second result shows that with certain counteractions in use, which only influence the individuals' behavior, a prevalent epidemic can be avoided. The results explain possible courses of a disease, and point out why cost-efficient countermeasures may reduce the number of total infections from a high percentage of the population to a negligible fraction.

cs.SI↗

arXiv2009

Cover Time and Broadcast Time

We introduce a new technique for bounding the cover time of random walks by relating it to the runtime of randomized broadcast. In particular, we strongly confirm for dense graphs the intuition of Chandra et al. \cite{CRRST97} that "the cover time of the graph is an appropriate metric for the performance of certain kinds of randomized broadcast algorithms". In more detail, our results are as follows: For any graph $G=(V,E)$ of size $n$ and minimum degree $δ$, we have $\mathcal{R}(G)= \Oh(\frac{|E|}δ \cdot \log n)$, where $\mathcal{R}(G)$ denotes the quotient of the cover time and broadcast time. This bound is tight for binary trees and tight up to logarithmic factors for many graphs including hypercubes, expanders and lollipop graphs. For any $δ$-regular (or almost $δ$-regular) graph $G$ it holds that $\mathcal{R}(G) = Ω(\frac{δ^2}{n} \cdot \frac{1}{\log n})$. Together with our upper bound on $\mathcal{R}(G)$, this lower bound strongly confirms the intuition of Chandra et al. for graphs with minimum degree $Θ(n)$, since then the cover time equals the broadcast time multiplied by $n$ (neglecting logarithmic factors). Conversely, for any $δ$ we construct almost $δ$-regular graphs that satisfy $\mathcal{R}(G) = \Oh(\max \{\sqrt{n},δ\} \cdot \log^2 n)$. Since any regular expander satisfies $\mathcal{R}(G) = Θ(n)$, the strong relationship given above does not hold if $δ$ is polynomially smaller than $n$. Our bounds also demonstrate that the relationship between cover time and broadcast time is much stronger than the known relationships between any of them and the mixing time (or the closely related spectral gap).

cs.DS↗