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<h2 class="hd hd-2 unit-title">Introduction</h2>
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<p>Read <a href="/assets/courseware/v1/ceb69c541ec977575fd512c3b7d164f1/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_Session35.pdf" target="[object Object]">Chapter 20.2 (PDF)</a> of <em>Mathematics for Computer Science</em> for 4.8 Random Walks & PageRank.</p>
<p>View the <a href="/assets/courseware/v1/f863f6e940b9300e427e6651b7d77168/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_cp35.pdf" target="[object Object]">Section 4.8 In-Class Questions (PDF)</a></p>
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<h2 class="hd hd-2 unit-title">Lecture Video | Random Walks</h2>
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<h2 class="hd hd-2 unit-title">Lecture Video | Stationary Distributions</h2>
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<h2 class="hd hd-2 unit-title">Random Walks I</h2>
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Random Walks I
</h3>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_894ecda580e34f3283c393c3bc41ac96_2_1" class=" capa_inputtype textline">
<div class="unanswered ">
<label class="problem-group-label" for="input_894ecda580e34f3283c393c3bc41ac96_2_1" id="label_894ecda580e34f3283c393c3bc41ac96_2_1">Consider the following random-walk graphs:
<br/>
<br/>
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<br/>
<br/>
<em> Please enter your answer in the form of x/y</em>
<br/>
<br/>
1. Find <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>d</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math> for a stationary distribution for graph 1.</label>
<input type="text" name="input_894ecda580e34f3283c393c3bc41ac96_2_1" id="input_894ecda580e34f3283c393c3bc41ac96_2_1" aria-describedby="status_894ecda580e34f3283c393c3bc41ac96_2_1" value="" size="20"/>
<span class="trailing_text" id="trailing_text_894ecda580e34f3283c393c3bc41ac96_2_1"/>
<span class="status unanswered" id="status_894ecda580e34f3283c393c3bc41ac96_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
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<label class="problem-group-label" for="input_894ecda580e34f3283c393c3bc41ac96_3_1" id="label_894ecda580e34f3283c393c3bc41ac96_3_1">2. Find <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>d</mi>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
</math> for a stationary distribution for graph 1.</label>
<input type="text" name="input_894ecda580e34f3283c393c3bc41ac96_3_1" id="input_894ecda580e34f3283c393c3bc41ac96_3_1" aria-describedby="status_894ecda580e34f3283c393c3bc41ac96_3_1" value="" size="20"/>
<span class="trailing_text" id="trailing_text_894ecda580e34f3283c393c3bc41ac96_3_1"/>
<span class="status unanswered" id="status_894ecda580e34f3283c393c3bc41ac96_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_894ecda580e34f3283c393c3bc41ac96_3_1" class="answer"/>
</div>
</div></div>
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<div class="unanswered ">
<label class="problem-group-label" for="input_894ecda580e34f3283c393c3bc41ac96_4_1" id="label_894ecda580e34f3283c393c3bc41ac96_4_1">3. Find <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>d</mi>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
</math> for a stationary distribution for graph 2. Please answer in a fraction of the form x/y.</label>
<input type="text" name="input_894ecda580e34f3283c393c3bc41ac96_4_1" id="input_894ecda580e34f3283c393c3bc41ac96_4_1" aria-describedby="status_894ecda580e34f3283c393c3bc41ac96_4_1" value="" size="20"/>
<span class="trailing_text" id="trailing_text_894ecda580e34f3283c393c3bc41ac96_4_1"/>
<span class="status unanswered" id="status_894ecda580e34f3283c393c3bc41ac96_4_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_894ecda580e34f3283c393c3bc41ac96_4_1" class="answer"/>
</div>
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<legend id="894ecda580e34f3283c393c3bc41ac96_5_1-legend" class="response-fieldset-legend field-group-hd">4. If you start at node <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>x</mi>
</math> in graph 1 and take a (long) random walk, does the distribution over nodes ever get close to the stationary distribution?</legend>
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<input type="radio" name="input_894ecda580e34f3283c393c3bc41ac96_5_1" id="input_894ecda580e34f3283c393c3bc41ac96_5_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="894ecda580e34f3283c393c3bc41ac96_5_1-choice_0-label" for="input_894ecda580e34f3283c393c3bc41ac96_5_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_894ecda580e34f3283c393c3bc41ac96_5_1"> Yes
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<input type="radio" name="input_894ecda580e34f3283c393c3bc41ac96_5_1" id="input_894ecda580e34f3283c393c3bc41ac96_5_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="894ecda580e34f3283c393c3bc41ac96_5_1-choice_1-label" for="input_894ecda580e34f3283c393c3bc41ac96_5_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_894ecda580e34f3283c393c3bc41ac96_5_1"> No
</label>
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<span id="answer_894ecda580e34f3283c393c3bc41ac96_5_1"/>
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<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
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<h2 class="hd hd-2 unit-title">Random Walks (cont.)</h2>
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<h3 class="hd hd-3 problem-header" id="252d5b0baf0648669240b6c08943609f-problem-title" aria-describedby="block-v1:OCW+6.042J+2T2019+type@problem+block@252d5b0baf0648669240b6c08943609f-problem-progress" tabindex="-1">
Random Walks (cont.)
</h3>
<div class="problem-progress" id="block-v1:OCW+6.042J+2T2019+type@problem+block@252d5b0baf0648669240b6c08943609f-problem-progress"></div>
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<div>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_252d5b0baf0648669240b6c08943609f_2_1">
<fieldset aria-describedby="status_252d5b0baf0648669240b6c08943609f_2_1">
<legend id="252d5b0baf0648669240b6c08943609f_2_1-legend" class="response-fieldset-legend field-group-hd">Consider the following random-walk graphs:
<br/>
<br/>
<div align="center">
<img height="300" width="378" src="/assets/courseware/v1/3f7c637767a0735428425a7c47b4c36d/asset-v1:OCW+6.042J+2T2019+type@asset+block/6.042J_UnitIV_4.8_Image.jpg" alt="Random walks "/>
</div>
<br/>
<br/>
1. If you start at node <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>w</mi>
</math> in graph 2 and take a (long) random walk, does the distribution over nodes ever get close to the stationary distribution? Try a few steps to see what is happening.</legend>
<div class="field">
<input type="radio" name="input_252d5b0baf0648669240b6c08943609f_2_1" id="input_252d5b0baf0648669240b6c08943609f_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="252d5b0baf0648669240b6c08943609f_2_1-choice_0-label" for="input_252d5b0baf0648669240b6c08943609f_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_252d5b0baf0648669240b6c08943609f_2_1"> Yes
</label>
</div>
<div class="field">
<input type="radio" name="input_252d5b0baf0648669240b6c08943609f_2_1" id="input_252d5b0baf0648669240b6c08943609f_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="252d5b0baf0648669240b6c08943609f_2_1-choice_1-label" for="input_252d5b0baf0648669240b6c08943609f_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_252d5b0baf0648669240b6c08943609f_2_1"> No
</label>
</div>
<span id="answer_252d5b0baf0648669240b6c08943609f_2_1"/>
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<span class="status unanswered" id="status_252d5b0baf0648669240b6c08943609f_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
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<fieldset aria-describedby="status_252d5b0baf0648669240b6c08943609f_3_1">
<legend id="252d5b0baf0648669240b6c08943609f_3_1-legend" class="response-fieldset-legend field-group-hd">2. How many stationary distribitions are there for graph 3?</legend>
<div class="field">
<input type="radio" name="input_252d5b0baf0648669240b6c08943609f_3_1" id="input_252d5b0baf0648669240b6c08943609f_3_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="252d5b0baf0648669240b6c08943609f_3_1-choice_0-label" for="input_252d5b0baf0648669240b6c08943609f_3_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_252d5b0baf0648669240b6c08943609f_3_1"> 0
</label>
</div>
<div class="field">
<input type="radio" name="input_252d5b0baf0648669240b6c08943609f_3_1" id="input_252d5b0baf0648669240b6c08943609f_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="252d5b0baf0648669240b6c08943609f_3_1-choice_1-label" for="input_252d5b0baf0648669240b6c08943609f_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_252d5b0baf0648669240b6c08943609f_3_1"> 1
</label>
</div>
<div class="field">
<input type="radio" name="input_252d5b0baf0648669240b6c08943609f_3_1" id="input_252d5b0baf0648669240b6c08943609f_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="252d5b0baf0648669240b6c08943609f_3_1-choice_2-label" for="input_252d5b0baf0648669240b6c08943609f_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_252d5b0baf0648669240b6c08943609f_3_1"> 2
</label>
</div>
<div class="field">
<input type="radio" name="input_252d5b0baf0648669240b6c08943609f_3_1" id="input_252d5b0baf0648669240b6c08943609f_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="252d5b0baf0648669240b6c08943609f_3_1-choice_3-label" for="input_252d5b0baf0648669240b6c08943609f_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_252d5b0baf0648669240b6c08943609f_3_1"> 4
</label>
</div>
<div class="field">
<input type="radio" name="input_252d5b0baf0648669240b6c08943609f_3_1" id="input_252d5b0baf0648669240b6c08943609f_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="252d5b0baf0648669240b6c08943609f_3_1-choice_4-label" for="input_252d5b0baf0648669240b6c08943609f_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_252d5b0baf0648669240b6c08943609f_3_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<msup>
<mn>2</mn>
<mn>4</mn>
</msup>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_252d5b0baf0648669240b6c08943609f_3_1" id="input_252d5b0baf0648669240b6c08943609f_3_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="252d5b0baf0648669240b6c08943609f_3_1-choice_5-label" for="input_252d5b0baf0648669240b6c08943609f_3_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_252d5b0baf0648669240b6c08943609f_3_1"> infinitely many
</label>
</div>
<span id="answer_252d5b0baf0648669240b6c08943609f_3_1"/>
</fieldset>
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<span class="status unanswered" id="status_252d5b0baf0648669240b6c08943609f_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
</div>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div class="choicegroup capa_inputtype" id="inputtype_252d5b0baf0648669240b6c08943609f_4_1">
<fieldset aria-describedby="status_252d5b0baf0648669240b6c08943609f_4_1">
<legend id="252d5b0baf0648669240b6c08943609f_4_1-legend" class="response-fieldset-legend field-group-hd">3. If you start at node <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>b</mi>
</math> in graph 3 and take a (long) random walk, the probabililty you are at node <math xmlns="http://www.w3.org/1998/Math/MathML">
<mi>d</mi>
</math> will be close to:</legend>
<div class="field">
<input type="radio" name="input_252d5b0baf0648669240b6c08943609f_4_1" id="input_252d5b0baf0648669240b6c08943609f_4_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="252d5b0baf0648669240b6c08943609f_4_1-choice_0-label" for="input_252d5b0baf0648669240b6c08943609f_4_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_252d5b0baf0648669240b6c08943609f_4_1"> 0
</label>
</div>
<div class="field">
<input type="radio" name="input_252d5b0baf0648669240b6c08943609f_4_1" id="input_252d5b0baf0648669240b6c08943609f_4_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="252d5b0baf0648669240b6c08943609f_4_1-choice_1-label" for="input_252d5b0baf0648669240b6c08943609f_4_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_252d5b0baf0648669240b6c08943609f_4_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<msup>
<mrow>
<mo>(</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>)</mo>
</mrow>
<mi>n</mi>
</msup>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_252d5b0baf0648669240b6c08943609f_4_1" id="input_252d5b0baf0648669240b6c08943609f_4_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="252d5b0baf0648669240b6c08943609f_4_1-choice_2-label" for="input_252d5b0baf0648669240b6c08943609f_4_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_252d5b0baf0648669240b6c08943609f_4_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mstyle displaystyle="true" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</mstyle>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_252d5b0baf0648669240b6c08943609f_4_1" id="input_252d5b0baf0648669240b6c08943609f_4_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="252d5b0baf0648669240b6c08943609f_4_1-choice_3-label" for="input_252d5b0baf0648669240b6c08943609f_4_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_252d5b0baf0648669240b6c08943609f_4_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mstyle displaystyle="true" scriptlevel="0">
<mfrac>
<mn>5</mn>
<mn>16</mn>
</mfrac>
</mstyle>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_252d5b0baf0648669240b6c08943609f_4_1" id="input_252d5b0baf0648669240b6c08943609f_4_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="252d5b0baf0648669240b6c08943609f_4_1-choice_4-label" for="input_252d5b0baf0648669240b6c08943609f_4_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_252d5b0baf0648669240b6c08943609f_4_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mstyle displaystyle="true" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mstyle>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_252d5b0baf0648669240b6c08943609f_4_1" id="input_252d5b0baf0648669240b6c08943609f_4_1_choice_5" class="field-input input-radio" value="choice_5"/><label id="252d5b0baf0648669240b6c08943609f_4_1-choice_5-label" for="input_252d5b0baf0648669240b6c08943609f_4_1_choice_5" class="response-label field-label label-inline" aria-describedby="status_252d5b0baf0648669240b6c08943609f_4_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mstyle displaystyle="true" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mstyle>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_252d5b0baf0648669240b6c08943609f_4_1" id="input_252d5b0baf0648669240b6c08943609f_4_1_choice_6" class="field-input input-radio" value="choice_6"/><label id="252d5b0baf0648669240b6c08943609f_4_1-choice_6-label" for="input_252d5b0baf0648669240b6c08943609f_4_1_choice_6" class="response-label field-label label-inline" aria-describedby="status_252d5b0baf0648669240b6c08943609f_4_1"> <math xmlns="http://www.w3.org/1998/Math/MathML">
<mstyle displaystyle="true" scriptlevel="0">
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</mstyle>
</math>
</label>
</div>
<div class="field">
<input type="radio" name="input_252d5b0baf0648669240b6c08943609f_4_1" id="input_252d5b0baf0648669240b6c08943609f_4_1_choice_7" class="field-input input-radio" value="choice_7"/><label id="252d5b0baf0648669240b6c08943609f_4_1-choice_7-label" for="input_252d5b0baf0648669240b6c08943609f_4_1_choice_7" class="response-label field-label label-inline" aria-describedby="status_252d5b0baf0648669240b6c08943609f_4_1"> 1
</label>
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