(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 10.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 2315429, 41053] NotebookOptionsPosition[ 2307289, 40803] NotebookOutlinePosition[ 2308768, 40847] CellTagsIndexPosition[ 2308686, 40842] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[TextData[{ "The Mathematics of Web Search\n", StyleBox["Random walk on a graph\nSearch Engine\nPage Rank Algorithm\n\ Perron-Frobenius Theorem\nPower Method", FontSize->24, FontSlant->"Italic", FontColor->GrayLevel[0]] }], "Title", CellChangeTimes->{{3.62039095741765*^9, 3.620390960896666*^9}, { 3.620396250090518*^9, 3.620396283409547*^9}, {3.654487429700385*^9, 3.6544875150759706`*^9}}], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Transition Matrices", "Section", CellChangeTimes->{{3.61932925989791*^9, 3.619329263671877*^9}}], Cell[TextData[{ StyleBox["Definition:", FontColor->GrayLevel[0]], " In a directed graph, for every vertex i there is a number of edges that \ enter that vertex (i is a head) and a number of edges that exit that vertex \ (i is a tail). Thus we define the ", StyleBox["indegree of vertex", FontColor->GrayLevel[0]], " i as the number of edges for which vertex i is a head. Similarly, the ", StyleBox["outdegree of vertex", FontColor->GrayLevel[0]], " i as the number of edges for which i is a tail.\n\n", StyleBox["Mathematica", FontSlant->"Italic", FontColor->GrayLevel[0]], StyleBox[" has built-in functions for the indegree and the outdegree of the \ vertices of a graph: ", FontColor->GrayLevel[0]], "\n", StyleBox["VertexInDegree[graph]", FontColor->GrayLevel[0]], " -- returns a list with the indegrees of the vertices of the graph \ specified in the argument\n", StyleBox["VertexOutDegree[graph] ", FontColor->GrayLevel[0]], "-- returns a list with the indegrees of the vertices of the graph\n" }], "Subsubsection", CellChangeTimes->{{3.619329275000368*^9, 3.6193292848631973`*^9}, { 3.619331320051779*^9, 3.619331554089898*^9}, {3.619333575347495*^9, 3.619333585383073*^9}, 3.651317205759639*^9, 3.696849284114362*^9}, TextJustification->1.], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Definition:", "Subsection", FontColor->GrayLevel[0]], StyleBox[" ", "Subsection"], StyleBox["The transition matrix A of a directed graph", FontSize->20, FontColor->GrayLevel[0]], StyleBox[" is defined as follows.", FontSize->20], " ", StyleBox["If there is an edge from i to j and the outdegree of vertex i is ", FontSize->20], Cell[BoxData[ FormBox[ SubscriptBox["d", "i"], TraditionalForm]], FontSize->20], StyleBox[", then on column i and row j we put ", FontSize->20], Cell[BoxData[ FormBox[ StyleBox[ FractionBox["1", SubscriptBox["d", "i"]], FontSize->36], TraditionalForm]], FontSize->20, FontColor->GrayLevel[0]], StyleBox[". 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It provides a \ ranking system for the nodes of the graph. Each node is assigned an \ importance factor, based on the amount and relative importance of the nodes \ that link to it.\ \>", "Subsection", CellChangeTimes->{{3.6204012953724194`*^9, 3.6204013421796103`*^9}, { 3.6204013748738728`*^9, 3.6204014074421377`*^9}, {3.6204421117893467`*^9, 3.620442123195847*^9}}], Cell["\<\ In our example, Node 1(Page 1) has page rank 0.38 so it is the most important \ page. Page 3 has a score of 0.29. Page 2 has importance factor 0.12, so it is \ the least important, and Page 4 has importance factor 0.19.\ \>", "Subsection", CellChangeTimes->{{3.620401479945607*^9, 3.6204015862716713`*^9}, { 3.620401669080421*^9, 3.620401705823455*^9}, {3.620401816621235*^9, 3.620401820227655*^9}}], Cell[TextData[{ StyleBox["Random Surfer Model: ", FontColor->GrayLevel[0]], "The page rank of Page i represents the probability that a random surfer on \ the Internet that opens a browser to any page and starts following \ hyperlinks, visits Page i. " }], "Subsection", CellChangeTimes->{{3.620401727912456*^9, 3.620401732591213*^9}, { 3.620401828232712*^9, 3.620401854731892*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Perron-Frobenius Theorem", "Section", CellChangeTimes->{{3.620395501505601*^9, 3.620395502985736*^9}, { 3.620400088473547*^9, 3.620400114341177*^9}}], Cell[CellGroupData[{ Cell[TextData[StyleBox["If M is a positive, column stochastic nxn matrix, \ then the following statements are true:", FontColor->GrayLevel[0]]], "Subsection", CellChangeTimes->{{3.6204001278932056`*^9, 3.620400184066436*^9}, 3.620441915075162*^9}, FontSize->22], Cell[CellGroupData[{ Cell["\<\ The number 1 is an eigenvalue of M, of multiplicity one (that is, if u and v \ are two eigenvectors corresponding to the eigenvalue 1, then u is a scalar \ multiple of v).\ \>", "ItemNumbered", CellChangeTimes->{{3.620400208082406*^9, 3.620400306668828*^9}, 3.6204003551611156`*^9, {3.620400641006122*^9, 3.620400650937354*^9}}, FontSize->20], Cell["\<\ The eigenvalue 1 is the largest eigenvalue of M; all the other eigenvalues of \ M are strictly less than 1 in absolute value. \ \>", "ItemNumbered", CellChangeTimes->{{3.620400208082406*^9, 3.620400306668828*^9}, 3.6204003551611156`*^9, 3.620400641006122*^9}, FontSize->20], Cell["\<\ Any eigenvector corresponding to 1 has either positive or negative entries. \ \>", "ItemNumbered", CellChangeTimes->{{3.620400208082406*^9, 3.620400322306418*^9}, { 3.620400375193266*^9, 3.620400420930415*^9}}, FontSize->20], Cell["\<\ There exists a unique probabilistic eigenvector w corresponding to the \ eigenvalue 1. \ \>", "ItemNumbered", CellChangeTimes->{{3.620400208082406*^9, 3.620400322306418*^9}, { 3.620400375193266*^9, 3.6204004747471943`*^9}}, FontSize->20] }, Open ]] }, Open ]], Cell[TextData[{ "Recall the following definitions:\n A square matrix is called ", StyleBox["column stochastic", FontSlant->"Italic"], " if all its entries are greater than or equal to 0, and the sum of the \ entries in each column is 1.\n A matrix is called ", StyleBox["positive", FontSlant->"Italic"], " if all of its entries are strictly greater than 0.\n A vector is \ called ", StyleBox["probabilistic", FontSlant->"Italic"], " (or a p", StyleBox["robability distribution vector", FontSlant->"Italic"], ") if all its entries are greater than or equal to 0 and the sum of all \ entries is 1. " }], "Subsection", CellChangeTimes->{{3.6204009066315603`*^9, 3.62040091984623*^9}, { 3.620400957677389*^9, 3.620401130635096*^9}, {3.620401215778606*^9, 3.6204012372737494`*^9}, {3.620439262109868*^9, 3.620439277523871*^9}}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Power Method Convergence Theorem", "Section", CellChangeTimes->{{3.620400602790394*^9, 3.6204006050725803`*^9}}], Cell[TextData[{ "Let M be a positive, column stochastic nxn matrix. Denote by w its unique \ probabilistic eigenvector corresponding to the eigenvalue 1. \nLet v be the \ column vector with all entries equal to 1/n . Then the sequence v, Mv, ... , \ M", Cell[BoxData[ FormBox[ SuperscriptBox["", "k"], TraditionalForm]]], "v converges to the vector w as k goes to \[Infinity].\nLet z be any \ probabilistic vector of size n. Then the sequence z, Mz, ... , M", Cell[BoxData[ FormBox[ SuperscriptBox["", "k"], TraditionalForm]]], "z converges to the vector w as k goes to \[Infinity]." }], "Subsection", CellChangeTimes->{{3.620400695663224*^9, 3.6204007938072653`*^9}, { 3.620401923518289*^9, 3.620401943324905*^9}, {3.620439197229843*^9, 3.620439235763359*^9}}, FontColor->GrayLevel[0]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Sketch of the proof", "Section", CellChangeTimes->{{3.620405119361874*^9, 3.6204051230567837`*^9}, { 3.620439386668187*^9, 3.6204394186044073`*^9}}], Cell[TextData[{ StyleBox["Proposition 1:", FontColor->GrayLevel[0]], " If v is a probabilistic vector and M is a positive and column stochastic \ matrix, then Mv is a probabilistic vector with only positive entries." }], "Subsection", CellChangeTimes->{{3.620405274498014*^9, 3.620405340541543*^9}, { 3.6204063149500237`*^9, 3.620406321206286*^9}, {3.620442148246285*^9, 3.620442150132*^9}}], Cell[TextData[{ StyleBox["idea of proof:", FontColor->GrayLevel[0]], " If v is a probabilistic vector of dimension n, and M is a column \ stochastic matrix, then Mv is also a probabilistic vector. \n \ If v is a non-negative vector and M is a positive matrix, then Mv has \ only positive entries." }], "Subsection", CellChangeTimes->{{3.6204051481070833`*^9, 3.620405260863097*^9}, { 3.620405344581667*^9, 3.620405359421688*^9}, {3.620405394999675*^9, 3.620405401022058*^9}, {3.620407564517074*^9, 3.620407579348198*^9}}], Cell[TextData[{ StyleBox["Definition: ", FontColor->GrayLevel[0]], "Let u and v be two vectors of size n. We define the distance between u and \ v to be the non-negative real number ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"d", RowBox[{"(", RowBox[{"u", ",", "v"}], ")"}]}], "=", FractionBox["1", "2"], " "}], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ UnderoverscriptBox["\[Sum]", RowBox[{"i", "=", "1"}], "n"], FontSize->24], RowBox[{"|", RowBox[{ SubscriptBox["u", "i"], "-"}]}]}], TraditionalForm]]], Cell[BoxData[ FormBox[ SubscriptBox["v", "i"], TraditionalForm]]], "|. Here ", Cell[BoxData[ FormBox[ SubscriptBox["u", "i"], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ SubscriptBox["v", "i"], TraditionalForm]]], " denote the i-th entries in the vectors u, and respectively v. \n", StyleBox["Definition:", FontColor->GrayLevel[0]], " Consider now a sequence of vectors ", Cell[BoxData[ FormBox[ SubscriptBox["v", "k"], TraditionalForm]]], " , k>0, of size n. We say that the sequence of vectors ", Cell[BoxData[ FormBox[ SubscriptBox["v", "k"], TraditionalForm]]], " converges to w as k tends to \[Infinity], if d(", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["v", "k"], ",", " ", "w"}], ")"}], TraditionalForm]]], "\[Rule]0 as k\[Rule]\[Infinity]." }], "Subsection", CellChangeTimes->{{3.62040566442898*^9, 3.62040570862087*^9}, { 3.620405874098691*^9, 3.62040623836305*^9}}], Cell[TextData[{ StyleBox["Proposition 2 (Contraction):", FontColor->GrayLevel[0]], " Let M be a positive and column stochastic nxn matrix. There exists a \ number r, 0{{3.620405410021886*^9, 3.620405577478216*^9}, { 3.620405632449492*^9, 3.6204056340572577`*^9}, {3.620405710724511*^9, 3.620405877992111*^9}, {3.620406282155445*^9, 3.620406310013853*^9}, { 3.6204076443250113`*^9, 3.62040764616366*^9}, {3.6204421566761227`*^9, 3.620442160923958*^9}, {3.620442438671063*^9, 3.620442448919436*^9}, { 3.620442608747281*^9, 3.620442640183628*^9}}], Cell[TextData[{ StyleBox["Proposition 3: ", FontColor->GrayLevel[0]], "Let M be a positive and column stochastic nxn matrix and v any \ probabilistic vector of size n. Denote by ", Cell[BoxData[ FormBox[ SubscriptBox["v", "k"], TraditionalForm]]], "=", Cell[BoxData[ FormBox[ SuperscriptBox["M", "K"], TraditionalForm]]], "v. Then the sequence ", Cell[BoxData[ FormBox[ SubscriptBox["v", "k"], TraditionalForm]]], " converges exponentially fast to a unique limit vector w.\n\n", StyleBox["idea of proof: Banach Fixed point Theorem!", FontColor->GrayLevel[0]], " \nLet j and k be any two positive integer. We can estimate the distance \ between any two terms of the sequence ", Cell[BoxData[ FormBox[ SubscriptBox["v", "j"], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ SubscriptBox["v", RowBox[{"j", "+", "k"}]], TraditionalForm]]], " in the following way:\n d(", Cell[BoxData[ FormBox[ SubscriptBox["v", RowBox[{"j", "+", "k"}]], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ SubscriptBox["v", "j"], TraditionalForm]]], ") = d(", Cell[BoxData[ FormBox[ SuperscriptBox["M", RowBox[{"j", "+", "k"}]], TraditionalForm]]], "v, ", Cell[BoxData[ FormBox[ SuperscriptBox["M", "j"], TraditionalForm]]], "v) \[LessSlantEqual] (1-r)", Cell[BoxData[ FormBox[ SuperscriptBox["", "j"], TraditionalForm]]], "d(", Cell[BoxData[ FormBox[ SuperscriptBox["M", "k"], TraditionalForm]]], Cell[BoxData[ FormBox["v", TraditionalForm]]], ", v) \[LessSlantEqual] (1-r)", Cell[BoxData[ FormBox[ SuperscriptBox["", "j"], TraditionalForm]]], "d(", Cell[BoxData[ FormBox[ SubscriptBox["v", "k"], TraditionalForm]]], ",v) \[LessSlantEqual] (1-r)", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["", "j"], "\[Rule]", "0"}], TraditionalForm]]], " as j\[Rule]\[Infinity], because 1-r is strictly less than 1.\nTherefore \ the sequence {", Cell[BoxData[ FormBox[ SubscriptBox[ RowBox[{ SubscriptBox["v", "k"], "}"}], RowBox[{"k", "\[GreaterSlantEqual]", "0"}]], TraditionalForm]]], " is a Cauchy sequence in \[DoubleStruckCapitalR]", Cell[BoxData[ FormBox[ SuperscriptBox["", "n"], TraditionalForm]]], ", so there exists a vector w in \[DoubleStruckCapitalR]", Cell[BoxData[ FormBox[ SuperscriptBox["", "n"], TraditionalForm]]], " such that ", Cell[BoxData[ FormBox[ SubscriptBox["v", "k"], TraditionalForm]]], "\[Rule]w as k\[Rule]\[Infinity]. The limit w is unique! Suppose by \ contradiction, that there would be (at least) two limit vectors, w and z. \ Then both w and z must satisfy Mw=w and Mz=z. Then we must also have the \ inequalities 0 \[LessSlantEqual] d(w, z) = d(Mw, Mz) \[LessSlantEqual] \ (1-r)d(w,z) < d(w,z), because 1-r is strictly less than 1. We reached a \ constradiction, d(w, z){{3.620406273151067*^9, 3.620406302854217*^9}, { 3.620406336494083*^9, 3.620406596863365*^9}, {3.620406634423963*^9, 3.6204066362259283`*^9}, 3.6204066668105927`*^9, {3.620406704659689*^9, 3.6204072407488813`*^9}, {3.62040727852598*^9, 3.620407280516514*^9}, { 3.6204076687081423`*^9, 3.620407742602162*^9}, {3.620432713201961*^9, 3.620432816441785*^9}, {3.620442164052182*^9, 3.620442424653944*^9}, { 3.620442479135594*^9, 3.620442536160034*^9}, {3.654486694690078*^9, 3.6544866953529463`*^9}, {3.654486733078774*^9, 3.654486734292577*^9}, { 3.654487088200108*^9, 3.654487092107321*^9}}, TextJustification->1.] }, Open ]] }, Open ]] }, WindowSize->{1270, 592}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, PrintingCopies->1, PrintingPageRange->{1, Automatic}, FrontEndVersion->"11.0 for Mac OS X x86 (32-bit, 64-bit Kernel) (September \ 21, 2016)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) 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