On the conductance of order Markov chains

Karzanov, Alexander V.; Khachiyan, Leonid

doi:10.1007/bf00385809

Cited by 85 publications

(68 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For example, computing the number of linear extensions of a given partial order is known to be #P-complete (Brightwell and Winkler (1991)). While there are algorithms for approximately sampling linear extensions given a partial order (Karzanov and Khachiyan (1991)), Shah and Zaman (2010) proposed the following simpler alternative for general graphs.…”

Section: Rumor Centrality: An Estimatormentioning

confidence: 99%

Finding Rumor Sources on Random Trees

Shah

Zaman

2016

Operations Research

View full text Add to dashboard Cite

We consider the problem of detecting the source of a rumor which has spread in a network using only observations about which set of nodes are infected with the rumor and with no information as to when these nodes became infected.In a recent work (Shah and Zaman 2010) this rumor source detection problem was introduced and studied. The authors proposed the graph score function rumor centrality as an estimator for detecting the source. They establish it to be the maximum likelihood estimator with respect to the popular Susceptible Infected (SI) model with exponential spreading times for regular trees. They showed that as the size of the infected graph increases, for a path graph (2-regular tree), the probability of source detection goes to 0 while for d-regular trees with d ≥ 3 the probability of detection, say α d , remains bounded away from 0 and is less than 1/2. However, their results stop short of providing insights for the performance of the rumor centrality estimator in more general settings such as irregular trees or the SI model with non-exponential spreading times. This paper overcomes this limitation and establishes the effectiveness of rumor centrality for source detection for generic random trees and the SI model with a generic spreading time distribution. The key result is an interesting connection between a continuous time branching process and the effectiveness of rumor centrality. Through this, it is possible to quantify the detection probability precisely. As a consequence, we recover all previous results as a special case and obtain a variety of novel results including the universality of rumor centrality in the context of tree-like graphs and the SI model with a generic spreading time distribution.

show abstract

Section: Rumor Centrality: An Estimatormentioning

confidence: 99%

Finding Rumor Sources on Random Trees

Shah

Zaman

2016

Operations Research

View full text Add to dashboard Cite

show abstract

“…The problem of quantifying this simple notion was formulated in 1989 by Dyer, Frieze & Kannan [15], who also conjectured a lower bound. A first result states that for convex Ω and any subset S ⊆ Ω H n−1 (∂S ∩ Ω) ≥ 1 diam(Ω) min (|S|, |Ω \ S|) and was, using different methods, independently shown by Lovász & Simonovits [20] and Karzanov & Khachiyan [17]. Note the similarity to Polya's longest shortest fence problem (recently solved by Esposito, Ferone, Kawohl, Nitsch & Trombetti [9]).…”

Section: Hypersurface Cutsmentioning

confidence: 95%

Sharp L 1-Poincaré inequalities correspond to optimal hypersurface cuts

Steinerberger

2015

Arch. Math.

View full text Add to dashboard Cite

“…The results present a nice example of the usefulness of recent powerful randomized methods for volume computation (Dyer, Frieze, and Kannan [1989], Karzanov and Khachiyan [1991]) based on rapidly mixing iarkov chains, and the trick is to allow randomization.…”

Section: Probability and Algorithmsmentioning

confidence: 97%