Week : DTMC Alictions Rnking Web ges & Slotted ALOHA etwok efonce -
Outline Aly the theoy of discete tie Mkov chins: Google s nking of web-ges Wht ge is the use ost likely seching fo? Foulte web-gh s Mkov chin Does stedy-stte exist? Does use ndoly wlk the web-gh? Cn sech esults be ioved futhe? Slotted ALOHA ediu ccess contol otocol Is the otocol stble fo lge nube of nodes? How should the etnsission obbility be chosen? etwok efonce -2
Rnking of Web-ges oble: how should sech engine nk web-ges? Ide: nk ges bsed on nube of in-links cittions Wekness: not ll in-links e equl Google s ide: ge hs high nk if the su of the nks of its in-link ges is high Foulte oves between web-ges s Mkov chin Solve to obtin stedy-stte obbility of ech stte Stte obbility is ootionl to iotnce of ge Exle with thee web-ges: A M / 2 π / 2 / 2 / 2 2 / 5 2 / 5 / 5 etwok efonce -3
Mkov Model of the Web 5 2 3 Issue : how to choose tnsition obbilities? 4 Assution: ech link is eqully likely to be clicked Cn ccoodte non-unifo obbility if such infotion vilble Issue 2: soe ows e zeo ded ends Assution: on eching ded-end estt t ny stte + is n x colun vecto whose i-th ow is non-zeo fo ded-end nodes v is n x colun vecto whose enties dd to could ll be / unifo v / 2 / 2 could be diffeent fo unifo i.e. esonlized T / 3 / 3 / 3 / 3 / 3 / 3 etwok efonce -4
Mkov Model of the Web contd. 5 2 3 Issue 3: Tnsition obbility tix y still be nonsttiony T α + α uv Solution: inte-connect ll nodes: 4 / 5 / 2 whee u is n x colun vecto with ll enties α is nube between nd tx on iotnce T v /6 4 /6 6 /6 4 /6 /6 α 4 / 5 [ ] Fo nd : /8 /8 /6 /8 33/8 9 / 6 / 2 / 4 / 2 9 / 2 3/ 4 4/2 3/8 7 /8 3/ 4 9 / 6 9 / 6 / 2 / 2 / 3 / 3 / 5 The vey sse initil tix now becoes the dense tix / 4 / 3 / 5 / 2 67 / 24 67 / 24 /6 /8 /8 / 3 / 5 / 3 / 3 / 5 etwok efonce -5
Couting the ge nk Issue 4: Couting Insted tke owes of Itetive ocedue: π[ k π π + ] π[ k] involves solving billion+ equtions! π[ k] α + α + α u v o tix ultiliction, wok with only one vecto Multiliction with sse tix, dense tix not foed Convegence deends on ete α Wht should α be set t? Sll α llows fste convegence why? Lge α eseves bette the tue ntue of the web-gh why? Bin nd ge [Google] cli tht α.85 woks well only 5 to itetions e equied fo convegence T etwok efonce -6
Discussion 2 3 Bsic ide: Rndo wlk on the web-gh The oe often you visit node, the oe oul the ge 5 4 Does you odel of the wlk th tch el use behvio? Insted of connecting evey node to evey othe node tx, cete duy node to which ll othe nodes e connected nd tht connects to ll nodes; this ltes the tue web-gh less. At ded-end, use often hits the bck button; so bis the tnsition obbility towds edecesso ges. How to incese the nking of you web-ge? Cete elics of you ge? Cete ny duy web-ges tht oint to you ge? Mke you web-ges link to ech othe? Futhe eding: The gernk Cittion Rnking: Binging Ode to the Web, 999 Rndo Wlks with Bck Buttons, 2 Deee inside gernk, 24 etwok efonce -7
Slotted Aloh nodes, tie-slotted syste, equl-size ckets obbility of new cket ivl in slot to ny given node is nd the new cket is tnsitted ieditely Collision hens if oe thn one node tnsits in the se slot; detected by ll nodes t end of slot If collision, ech bcklogged node eties in evey slot with obbility until successful tnsission o queueing: new ivls to bcklogged node e doed etwok efonce -8
Slotted Aloh: Mkov chin Stte: nube of bcklogged nodes,, obbility tht i bcklogged nodes tnsit in slot is obbility tht j non-bcklogged nodes tnsit in slot is Mkov chin: u, i b, i i i j j j, k,,, + k j, k, u, b,, 2 u, b, + u,[ b,] u, k, k : 2 k etwok efonce -9
etwok efonce - obbility of successful tnsission in stte : Fo sll nd, nd using fo sll x: Let be the tnsission ttet te in stte, the thoughut successful tnsissions e slot is Slotted Aloh: Efficiency succ b u b u,,,, + + + x x e e x α α α ln ] [ ] [ succ e + + G e G G + Thoughut xiized t G Mx. thoughut /e 36%
This ige cnnot cuently be dislyed. Slotted Aloh: Instbility Does slotted Aloh wok when is lge? Given you e in stte, wht is the obbility of oving bckwds i.e. stte <? bck u b,, Stted nothe wy, when the nube of bcklogged nodes is lge enough, the vege ttet te G becoes > i.e. thee e excessive collisions nd stte kees gowing otentil solution: ensue the ttet te G < li bck li [/ ] G + How? ke the etnsission obbility deendent on stte E.g.: exonentil bckoff: e ice fo king etnsission obbility too sll: lge dely etwok efonce -