File:PageRanks-Example.svg

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Summary

Numeric examples of PageRank values in a small graph with a damping factor of 0.85. The exact solution is:

<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34cd9a9eec43fa2856c2efc531d4225f981b6119" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -17.171ex; width:67.567ex; height:35.509ex;" alt="{\displaystyle \mathbf {R} ={\begin{bmatrix}PR(A)\\PR(B)\\\vdots \\PR(K)\end{bmatrix}}={\dfrac {1}{579662461}}\cdot {\begin{pmatrix}19002201\\222822800\\198772220\\22657320\\46886400\\22657320\\9372840\\9372840\\9372840\\9372840\\9372840\end{pmatrix}}\approx {\begin{pmatrix}0,03278149\\0,38440095\\0,34291029\\0,03908709\\0,08088569\\0,03908709\\0,01616948\\0,01616948\\0,01616948\\0,01616948\\0,01616948\end{pmatrix}}}">

And here’s the solution for any arbitrary damping factor d:

<img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e1b2181c0e75aa12abc8def6dbf9997f28d34b0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -19.906ex; margin-bottom: -0.265ex; width:101.872ex; height:41.509ex;" alt="{\displaystyle \mathbf {R_{d}} ={\begin{bmatrix}PR_{d}(A)\\PR_{d}(B)\\\vdots \\PR_{d}(K)\end{bmatrix}}=\left({\dfrac {1}{(d+1)\cdot (7\,d^{4}+28\,d^{2}+12\,d-132)}}\right)\cdot {\begin{bmatrix}\left({d}+1\right)\left(7\,{d}^{3}+6\,{d}+12\right)\left({d}-1\right)\\-9\,{d}^{3}-18\,{d}^{2}-46\,{d}-12\\-11\,{d}^{4}-12\,{d}-12-18\,{d}^{3}-32\,{d}^{2}\\2\,\left({d}+1\right)\left(7\,{d}^{2}+2\,{d}+6\right)\left({d}-1\right)\\12\,\left({d}+1\right)\left(4\,{d}^{2}-3\,{d}-1\right)\\2\,\left({d}+1\right)\left(7\,{d}^{2}+2\,{d}+6\right)\left({d}-1\right)\\-2\,\left({d}+1\right)\left(-6+{d}^{2}\right)\left({d}-1\right)\\-2\,\left({d}+1\right)\left(-6+{d}^{2}\right)\left({d}-1\right)\\-2\,\left({d}+1\right)\left(-6+{d}^{2}\right)\left({d}-1\right)\\-2\,\left({d}+1\right)\left(-6+{d}^{2}\right)\left({d}-1\right)\\-2\,\left({d}+1\right)\left(-6+{d}^{2}\right)\left({d}-1\right)\end{bmatrix}}}">

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Date/TimeThumbnailDimensionsUserComment
current21:12, 3 January 2017Thumbnail for version as of 21:12, 3 January 2017758 × 611 (33 KB)127.0.0.1 (talk)<p>Numeric examples of PageRank values in a small graph with a damping factor of 0.85. The exact solution is: </p> <dl><dd><span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">R</mi></mrow><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi>P</mi><mi>R</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi>P</mi><mi>R</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>⋮<!-- ⋮ --></mo></mtd></mtr><mtr><mtd><mi>P</mi><mi>R</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mtd></mtr></mtable><mo>]</mo></mrow></mrow><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mn>579662461</mn></mfrac></mstyle></mrow><mo>⋅<!-- ⋅ --></mo><mrow class="MJX-TeXAtom-ORD"><mrow><mo>(</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>19002201</mn></mtd></mtr><mtr><mtd><mn>222822800</mn></mtd></mtr><mtr><mtd><mn>198772220</mn></mtd></mtr><mtr><mtd><mn>22657320</mn></mtd></mtr><mtr><mtd><mn>46886400</mn></mtd></mtr><mtr><mtd><mn>22657320</mn></mtd></mtr><mtr><mtd><mn>9372840</mn></mtd></mtr><mtr><mtd><mn>9372840</mn></mtd></mtr><mtr><mtd><mn>9372840</mn></mtd></mtr><mtr><mtd><mn>9372840</mn></mtd></mtr><mtr><mtd><mn>9372840</mn></mtd></mtr></mtable><mo>)</mo></mrow></mrow><mo>≈<!-- ≈ --></mo><mrow class="MJX-TeXAtom-ORD"><mrow><mo>(</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>0</mn><mo>,</mo><mn>03278149</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>,</mo><mn>38440095</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>,</mo><mn>34291029</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>,</mo><mn>03908709</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>,</mo><mn>08088569</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>,</mo><mn>03908709</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>,</mo><mn>01616948</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>,</mo><mn>01616948</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>,</mo><mn>01616948</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>,</mo><mn>01616948</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>,</mo><mn>01616948</mn></mtd></mtr></mtable><mo>)</mo></mrow></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ={\begin{bmatrix}PR(A)\\PR(B)\\\vdots \\PR(K)\end{bmatrix}}={\dfrac {1}{579662461}}\cdot {\begin{pmatrix}19002201\\222822800\\198772220\\22657320\\46886400\\22657320\\9372840\\9372840\\9372840\\9372840\\9372840\end{pmatrix}}\approx {\begin{pmatrix}0,03278149\\0,38440095\\0,34291029\\0,03908709\\0,08088569\\0,03908709\\0,01616948\\0,01616948\\0,01616948\\0,01616948\\0,01616948\end{pmatrix}}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34cd9a9eec43fa2856c2efc531d4225f981b6119" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -17.171ex; width:67.567ex; height:35.509ex;" alt="{\displaystyle \mathbf {R} ={\begin{bmatrix}PR(A)\\PR(B)\\\vdots \\PR(K)\end{bmatrix}}={\dfrac {1}{579662461}}\cdot {\begin{pmatrix}19002201\\222822800\\198772220\\22657320\\46886400\\22657320\\9372840\\9372840\\9372840\\9372840\\9372840\end{pmatrix}}\approx {\begin{pmatrix}0,03278149\\0,38440095\\0,34291029\\0,03908709\\0,08088569\\0,03908709\\0,01616948\\0,01616948\\0,01616948\\0,01616948\\0,01616948\end{pmatrix}}}"></span></dd></dl> <p>And here’s the solution for any arbitrary damping factor d: </p> <dl><dd><span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><msub><mi mathvariant="bold">R</mi><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">d</mi></mrow></msub></mrow><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi>P</mi><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi>P</mi><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow></msub><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>⋮<!-- ⋮ --></mo></mtd></mtr><mtr><mtd><mi>P</mi><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow></msub><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mtd></mtr></mtable><mo>]</mo></mrow></mrow><mo>=</mo><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>⋅<!-- ⋅ --></mo><mo stretchy="false">(</mo><mn>7</mn><mspace width="thinmathspace"></mspace><msup><mi>d</mi><mrow class="MJX-TeXAtom-ORD"><mn>4</mn></mrow></msup><mo>+</mo><mn>28</mn><mspace width="thinmathspace"></mspace><msup><mi>d</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mo>+</mo><mn>12</mn><mspace width="thinmathspace"></mspace><mi>d</mi><mo>−<!-- − --></mo><mn>132</mn><mo stretchy="false">)</mo></mrow></mfrac></mstyle></mrow><mo>)</mo></mrow><mo>⋅<!-- ⋅ --></mo><mrow class="MJX-TeXAtom-ORD"><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mn>7</mn><mspace width="thinmathspace"></mspace><msup><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>3</mn></mrow></msup><mo>+</mo><mn>6</mn><mspace width="thinmathspace"></mspace><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mo>+</mo><mn>12</mn><mo>)</mo></mrow><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mo>−<!-- − --></mo><mn>1</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mo>−<!-- − --></mo><mn>9</mn><mspace width="thinmathspace"></mspace><msup><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>3</mn></mrow></msup><mo>−<!-- − --></mo><mn>18</mn><mspace width="thinmathspace"></mspace><msup><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mo>−<!-- − --></mo><mn>46</mn><mspace width="thinmathspace"></mspace><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mo>−<!-- − --></mo><mn>12</mn></mtd></mtr><mtr><mtd><mo>−<!-- − --></mo><mn>11</mn><mspace width="thinmathspace"></mspace><msup><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>4</mn></mrow></msup><mo>−<!-- − --></mo><mn>12</mn><mspace width="thinmathspace"></mspace><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mo>−<!-- − --></mo><mn>12</mn><mo>−<!-- − --></mo><mn>18</mn><mspace width="thinmathspace"></mspace><msup><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>3</mn></mrow></msup><mo>−<!-- − --></mo><mn>32</mn><mspace width="thinmathspace"></mspace><msup><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup></mtd></mtr><mtr><mtd><mn>2</mn><mspace width="thinmathspace"></mspace><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mn>7</mn><mspace width="thinmathspace"></mspace><msup><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn><mspace width="thinmathspace"></mspace><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mo>+</mo><mn>6</mn><mo>)</mo></mrow><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mo>−<!-- − --></mo><mn>1</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mn>12</mn><mspace width="thinmathspace"></mspace><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mn>4</mn><mspace width="thinmathspace"></mspace><msup><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mo>−<!-- − --></mo><mn>3</mn><mspace width="thinmathspace"></mspace><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mo>−<!-- − --></mo><mn>1</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mn>2</mn><mspace width="thinmathspace"></mspace><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mn>7</mn><mspace width="thinmathspace"></mspace><msup><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn><mspace width="thinmathspace"></mspace><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mo>+</mo><mn>6</mn><mo>)</mo></mrow><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mo>−<!-- − --></mo><mn>1</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mo>−<!-- − --></mo><mn>2</mn><mspace width="thinmathspace"></mspace><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mo>−<!-- − --></mo><mn>6</mn><mo>+</mo><msup><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mo>)</mo></mrow><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mo>−<!-- − --></mo><mn>1</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mo>−<!-- − --></mo><mn>2</mn><mspace width="thinmathspace"></mspace><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mo>−<!-- − --></mo><mn>6</mn><mo>+</mo><msup><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mo>)</mo></mrow><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mo>−<!-- − --></mo><mn>1</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mo>−<!-- − --></mo><mn>2</mn><mspace width="thinmathspace"></mspace><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mo>−<!-- − --></mo><mn>6</mn><mo>+</mo><msup><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mo>)</mo></mrow><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mo>−<!-- − --></mo><mn>1</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mo>−<!-- − --></mo><mn>2</mn><mspace width="thinmathspace"></mspace><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mo>−<!-- − --></mo><mn>6</mn><mo>+</mo><msup><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mo>)</mo></mrow><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mo>−<!-- − --></mo><mn>1</mn><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mo>−<!-- − --></mo><mn>2</mn><mspace width="thinmathspace"></mspace><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mo>−<!-- − --></mo><mn>6</mn><mo>+</mo><msup><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mo>)</mo></mrow><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mi>d</mi></mrow><mo>−<!-- − --></mo><mn>1</mn><mo>)</mo></mrow></mtd></mtr></mtable><mo>]</mo></mrow></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \mathbf {R_{d}} ={\begin{bmatrix}PR_{d}(A)\\PR_{d}(B)\\\vdots \\PR_{d}(K)\end{bmatrix}}=\left({\dfrac {1}{(d+1)\cdot (7\,d^{4}+28\,d^{2}+12\,d-132)}}\right)\cdot {\begin{bmatrix}\left({d}+1\right)\left(7\,{d}^{3}+6\,{d}+12\right)\left({d}-1\right)\\-9\,{d}^{3}-18\,{d}^{2}-46\,{d}-12\\-11\,{d}^{4}-12\,{d}-12-18\,{d}^{3}-32\,{d}^{2}\\2\,\left({d}+1\right)\left(7\,{d}^{2}+2\,{d}+6\right)\left({d}-1\right)\\12\,\left({d}+1\right)\left(4\,{d}^{2}-3\,{d}-1\right)\\2\,\left({d}+1\right)\left(7\,{d}^{2}+2\,{d}+6\right)\left({d}-1\right)\\-2\,\left({d}+1\right)\left(-6+{d}^{2}\right)\left({d}-1\right)\\-2\,\left({d}+1\right)\left(-6+{d}^{2}\right)\left({d}-1\right)\\-2\,\left({d}+1\right)\left(-6+{d}^{2}\right)\left({d}-1\right)\\-2\,\left({d}+1\right)\left(-6+{d}^{2}\right)\left({d}-1\right)\\-2\,\left({d}+1\right)\left(-6+{d}^{2}\right)\left({d}-1\right)\end{bmatrix}}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e1b2181c0e75aa12abc8def6dbf9997f28d34b0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -19.906ex; margin-bottom: -0.265ex; width:101.872ex; height:41.509ex;" alt="{\displaystyle \mathbf {R_{d}} ={\begin{bmatrix}PR_{d}(A)\\PR_{d}(B)\\\vdots \\PR_{d}(K)\end{bmatrix}}=\left({\dfrac {1}{(d+1)\cdot (7\,d^{4}+28\,d^{2}+12\,d-132)}}\right)\cdot {\begin{bmatrix}\left({d}+1\right)\left(7\,{d}^{3}+6\,{d}+12\right)\left({d}-1\right)\\-9\,{d}^{3}-18\,{d}^{2}-46\,{d}-12\\-11\,{d}^{4}-12\,{d}-12-18\,{d}^{3}-32\,{d}^{2}\\2\,\left({d}+1\right)\left(7\,{d}^{2}+2\,{d}+6\right)\left({d}-1\right)\\12\,\left({d}+1\right)\left(4\,{d}^{2}-3\,{d}-1\right)\\2\,\left({d}+1\right)\left(7\,{d}^{2}+2\,{d}+6\right)\left({d}-1\right)\\-2\,\left({d}+1\right)\left(-6+{d}^{2}\right)\left({d}-1\right)\\-2\,\left({d}+1\right)\left(-6+{d}^{2}\right)\left({d}-1\right)\\-2\,\left({d}+1\right)\left(-6+{d}^{2}\right)\left({d}-1\right)\\-2\,\left({d}+1\right)\left(-6+{d}^{2}\right)\left({d}-1\right)\\-2\,\left({d}+1\right)\left(-6+{d}^{2}\right)\left({d}-1\right)\end{bmatrix}}}"></span></dd></dl>
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