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Distribution of PageRank Mass Among Principle Components of the Web
 in "Proc. of 5th Workshop On Algorithms And Models For The WebGraph (WAW 2007
, 2007
"... We study the PageRank mass of principal components in a bowtie Web Graph, as a function of the damping factor c. Using a singular perturbation approach, we show that the PageRank share of IN and SCC components remains high even for very large values of the damping factor, in spite of the fact that ..."
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We study the PageRank mass of principal components in a bowtie Web Graph, as a function of the damping factor c. Using a singular perturbation approach, we show that the PageRank share of IN and SCC components remains high even for very large values of the damping factor, in spite of the fact that it drops to zero when c→1. However, a detailed study of the OUT component reveals the presence “deadends” (small groups of pages linking only to each other) that receive an unfairly high ranking when c is close to one. We argue that this problem can be mitigated by choosing c as small as 1/2. 1
A singular perturbation approach for choosing PageRank damping factor
 2006, in arXiv:math.PR/0612079. G. Kollias, E. Gallopoulos
, 2006
"... Abstract. We study the PageRank mass of principal components in a bowtie Web Graph, as a function of the damping factor c. It is known that the Web graph can be divided into three principal components: SCC, IN and OUT. The Giant Strongly Connected Component (SCC) contains a large group of pages all ..."
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Abstract. We study the PageRank mass of principal components in a bowtie Web Graph, as a function of the damping factor c. It is known that the Web graph can be divided into three principal components: SCC, IN and OUT. The Giant Strongly Connected Component (SCC) contains a large group of pages all having a hyperlink path to each other. The pages in the IN (OUT) component have a path to (from) the SCC, but not back. Using a singular perturbation approach, we show that the PageRank share of IN and SCC components remains high even for very large values of the damping factor, in spite of the fact that it drops to zero when c tends to one. However, a detailed study of the OUT component reveals the presence of “deadends ” (small groups of pages linking only to each other) that receive an unfairly high ranking when c is close to one. We argue that this problem can be mitigated by choosing c as small as 1/2. 1
Inversion of analytically perturbed linear operators that are singular at the origin
, 2010
"... Let H and K be Hilbert spaces and for each z ∈ C let A(z) ∈ L(H, K) be a bounded but not necessarily compact linear map with A(z) analytic on a region z  < a. If A(0) is singular we find conditions under which A(z) −1 is well defined on some region 0 < z  < b by a convergent Laurent series with ..."
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Let H and K be Hilbert spaces and for each z ∈ C let A(z) ∈ L(H, K) be a bounded but not necessarily compact linear map with A(z) analytic on a region z  < a. If A(0) is singular we find conditions under which A(z) −1 is well defined on some region 0 < z  < b by a convergent Laurent series with a finite order pole at the origin. We show that by changing to a standard Sobolev topology the method extends to closed unbounded linear operators and also that it can be used in Banach spaces where complementation of certain closed subspaces is possible. Our method is illustrated with several key examples 1. 1
JOINT QUEUE–PERTURBED AND WEAKLY COUPLED POWER CONTROL FOR WIRELESS BACKBONE NETWORKS
"... Wireless Backbone Networks (WBNs) equipped with MultiRadio MultiChannel (MRMC) configurations do experience power control problems such as the interchannel and cochannel interference, high energy consumption at multiple queues and unscalable network connectivity. Such network problems can be con ..."
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Wireless Backbone Networks (WBNs) equipped with MultiRadio MultiChannel (MRMC) configurations do experience power control problems such as the interchannel and cochannel interference, high energy consumption at multiple queues and unscalable network connectivity. Such network problems can be conveniently modelled using the theory of queue perturbation in the multiple queue systems and also as a weak coupling in a multiple channel wireless network. Consequently, this paper proposes a queue perturbation and weakly coupled based power control approach for WBNs. The ultimate objectives are to increase energy efficiency and the overall network capacity. In order to achieve this objective, a Markov chain model is first presented to describe the behaviour of the steady state probability distribution of the queue energy and buffer states. The singular perturbation parameter is approximated from the coefficients of the Taylor series expansion of the probability distribution. The impact of such queue perturbations on the transmission probability, given some transmission power values, is also analysed. Secondly, the interchannel interference is modelled as a weakly coupled wireless system. Thirdly, Nash differential games are applied to derive optimal power control signals for each user subject to power constraints at each node. Finally, analytical models and numerical examples show the efficacy of the proposed model in solving power control problems in WBNs.