Lesson6: Modeling the Web as a graph Unit5: Linear Algebra for graphs

Transcription

1 Lesson6: Modeling the We s grph Unit5: Liner Alger for grphs Rene Pikhrdt Introdution to We Siene Prt 2 Emerging We Properties Rene Pikhrdt Institute CC-BY-SA-3. for We Siene nd Tehnologies Modeling the We s University Grph of Kolenz-Lndu, Germny 52

2 Completing this unit you should Be le to red nd uild n djeny mtrix of grph Know some si mtrix vetor multiplitions to generte some sttistis out of the djeny mtrix Understnd wht is enoded in the omponents of the k-th power of the Adjeny mtrix of grph Rene Pikhrdt CC-BY-SA-3. Modeling the We s Grph 53

3 A met model for grph sed models Wht does simple Wikipedi look like? modelling interpreting Grph s model modelling lulting Desriptive sttistis interpreting Vetor spe model lulting Mtrix lultions Rene Pikhrdt CC-BY-SA-3. Modeling the We s Grph 54

4 Modelling grphs with liner lger Given this grph G Whih of the following is n djeny mtrix for G? A = C A B A D A Rene Pikhrdt CC-BY-SA-3. Modeling the We s Grph 55

5 All four mtries represent the sme grph! Mtries re not the sme How n we mke sure to tlk out the sme thing? A = C A B A D A Rene Pikhrdt CC-BY-SA-3. Modeling the We s Grph 56

6 Fix se of our vetor spe We hve out grph With set of verties We hve our Vetor spe With set of se vetors V = {,, } Without loosing generlity we n hoose unit vetors ~e A ~e 2 A ~e 3 A V = { ~e, ~e 2, ~e 3 } Rene Pikhrdt CC-BY-SA-3. Modeling the We s Grph 57

7 Defining the djeny mtrix Fix se i.e. define: hoose f se : V! V With V = {,, } nd V = { ~e, ~e 2, ~e 3 } And f se must e ijetive f se ({x, y}) =f se (x)+f se (y) For exmple one hoie might e f se () = ~e f se () = ~e 2 f se () = ~e 3 ~e ~e 2 ~e 3 Rene Pikhrdt CC-BY-SA-3. Modeling the We s Grph 58

8 Defining the djeny mtrix Fix se i.e. define: hoose f se : V! V With V = {,, } nd V = { ~e, ~e 2, ~e 3 } And f se must e ijetive f se ({x, y}) =f se (x)+f se (y) For exmple one hoie might e Py ttention: Nottion is somehow sloppy. Here f is not defined on sets. f se () = ~e f se () = ~e 2 f se () = ~e 3 ~e ~e 2 ~e 3 Rene Pikhrdt CC-BY-SA-3. Modeling the We s Grph 59

9 Defining the djeny mtrix (II) We re looking for mtrix A : V! V suh tht A(f se ()) = f se (In()) ~e ~e 2 ~e 3 Don t pni! A just enodes the outlink of every vertex It is mde in wy tht it respets the hoie of our se Rene Pikhrdt CC-BY-SA-3. Modeling the We s Grph 6

10 Exmple A(f se ()) = f se A A! A ~e = ~e 2 ~e ~e 2 ~e 3 Multiplying mtrix with the i-th se vetor from right gives the i-th olumn! Rene Pikhrdt CC-BY-SA-3. Modeling the We s Grph 6

11 Exmple A(f se ()) = f se A A! A ~e = ~e A A! A ~e 2 = A A! A ~e 3 = ~e 2 ~e ~e 2 ~e 3 For eh multiplition the result is the vetors of nodes represnting in links Rene Pikhrdt CC-BY-SA-3. Modeling the We s Grph 62

12 Similrly for out nodes we get Left multiply with se vetor Result is vetor representing ~e ~e 2 ~e A =! ~e t 2 A = ~e t t + ~e 3 fse() = ~e2 fse() = ~e Out() ={, } f se () = ~e 3 Multiplying mtrix with the i-th se vetor from left gives the i-th row! Rene Pikhrdt CC-BY-SA-3. Modeling the We s Grph 63

13 Wht hppens if A is pplied severl times? We sw pply it one to se vetor yields ll neighours linking in or out Neighours n e seen s pths of lenght Eh omponent ounts how mny pths of length exist from to the node of the omponent ~e i t So wht does ~e i t A k represent? Wht out? A k ~e i Rene Pikhrdt CC-BY-SA-3. Modeling the We s Grph 64

14 Wht hppens if A is pplied severl times? represents ~e i t A k A vetor where in eh ompnent is the numer of pths of length k from the node represented y to the node represented y tht omponent. ~e i t represents A k ~e i A vetor where in eh ompnent is the numer of pths of length k to the node represented y ~e i from the node represented y tht omponent Rene Pikhrdt CC-BY-SA-3. Modeling the We s Grph 65

15 Quiz question! Who is still with me (: If A represents the djeny mtrix of the strongly onneted omponent of Simiple English Wikipedi For wht power of k will A k hve no zero entries? Rene Pikhrdt CC-BY-SA-3. Modeling the We s Grph 66

16 Degree distriution multiply with ounting vetor we get ~e ~e 2 ~e A = =~e t +~e t t 2 +~e 3 In degree distriution Counting A 2A =~e +2~e 2 +~e 3 Out degree distriution Rene Pikhrdt CC-BY-SA-3. Modeling the We s Grph 67

17 Thnk you for your ttention! Contt: Rene Pikhrdt Institute for We Siene nd Tehnologies Universität Kolenz-Lndu Rene Pikhrdt Institute CC-BY-SA-3. for We Siene nd Tehnologies Modeling the We s University Grph of Kolenz-Lndu, Germny 68

18 Pitures: By User:AzToth (Imge:6n-grf.png simlr input dt) [Puli domin], vi Wikimedi Commons Puli domin By MistWiz (Own work) [Puli domin], vi Wikimedi Commons By Pluke (Own work) [CC], vi Wikimedi Commons By Lkeworks (Own work) [GFDL ( or CC BY-SA ( retiveommons.org/lienses/y-s/ )], vi Wikimedi Commons By Houl78 (Own work) [CC], vi Wikimedi Commons By MrtinThom (Own work) [GFDL ( or CC BY 3. ( retiveommons.org/lienses/y/3.)], vi Wikimedi Commons direted_network_svg.svg By Limner (Own work) [CC BY-SA 4. ( retiveommons.org/lienses/y-s/4.)], vi Wikimedi Commons By Reidpth [Puli domin], vi Wikimedi Commons vi flikr CC-BY 2. y Dnny Sullivn Grph sttistis vi puli domin puli domin (definition of world wide we) By Sestin Shelter [CC BY-SA 3. ( lienses/y-s/3.)], vi Wikimedi Commons Chris 73 / Wikimedi Commons [GFDL.3 ( or CC BY-SA 3. ( vi Wikimedi Commons By Mnipnde (Own work) [CC BY-SA 3. ( vi Wikimedi Commons vi Flikr nd Vlerie Everett CC-BY-SA y CSTAR & Oleg Alexndrov By Alemonroy (Own work) [GFDL ( or CC BY 3. ( retiveommons.org/lienses/y/3.)], vi Wikimedi Commons Rene Pikhrdt CC-BY-SA-3. Modeling the We s Grph 69