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4.7 ⋅ algorithm analysis 91 α with ∣α∣ < 1. See Horn and Serra-Capizzano [2007] for the first study of PageRank with complex α. Thus, 11 ρa+ρb=+ 2−1 rr slips into (4.41) for the PageRank case. We have ρa + ρb; let’s now find ω. In α-space where α = αR + iαI, the ellipse is Figure 4.8 – The Gauss quadrature error analysis applied to PageRank. When integrating xi (α), we use the ellipse given in red to bound the error in a Gauss quadrature approximation. Note that xi (α) is clearly analytic in this region as it’s enclosed inside ∣α∣ < 1. ( r / 2 − α R ) 2 (1/2)2 α I2 + 1√ =1. ( 1−r2)2 2 This ellipse is centered at r/2 with semi-major axis length 1/2, as illustrated in figure 4.8. In (4.41), the value of ω is an upper bound on f (z) inside the ellipse. Thus, we must bound the magnitude of PageRank components for a complex α. First, x=αPx+(1−α)v gives ∥x∥≤∣α∣∥x∥+∣1−α∣. For complex α, this bound yields √ ( 1 − α R ) 2 + α I2 =√ 1− αR2+αI2 ≡ F(αR,αI). ∣xi (α)∣ ≤ ∥x(α)∥ ≤ 1−∣α∣ When αI = 0, this bound respects the property that xi (α) ≤ 1 for 0 ≤ αR < 1. WhenαI =/0,theboundisconsiderablymoreinteresting.Infigure4.9atright, we see that as αI increases, F increases. Analytically, we find that ∂F/∂αI > 0 for αI > 0 and ∂F/∂αI < 0 for αI < 0. Consequently, the maximum ω is going to occur on the boundary of the ellipse. In this case, 2 1 2 (r/2 − αR)2 αI= (1−r)(1− 2 ). 4 (1/2) LetFR(αR)=F(αR,αI(αR))bethevalueofFontheellipse.Thecritical points of F are 0.8 0.6 0.4 0.2 0 −0.5 0 0.5 Figure 4.9 – PageRank magnitude foracomplexdampingparameter. In this contour plot, we show an upper bound on ∥x(α)∥ when α ∈ C. Darker red indicates larger magnitude and white indicates a magnitude near 0. The magnitudes increase as α veers off the real line, or when the real component is negative. r2 −1 r2 −2r−3 αR=; ; √ ∣1 − α∣ r2 +2r−1 2r 2r 2r (4.42)PDF Image | Instagram Cheat Sheet
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