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5 gusev

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Workshop on Random graphs 24—26oct
Доклад Глеба Гусева

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5 gusev

  1. 1. en snEdepth vook —t €referenti—l ett—™hment wodel of the ‡e˜X hegree —nd idge histri˜utions Workshop on Random Graphs and their Applications G.G. Gusev et al. Yandex LLC October 26, G.G. Gusev et al. 2013 Preferential Attachment and the Web: Edge Distribution
  2. 2. yutline 1 u—ntit—tive —spe™ts of l—rge networks 2 fu™kley!ysthus r—ndom gr—ph model 3 ‡e˜ host gr—ph implement—tion 4 pitting the model to the we˜ host gr—ph 5 gomp—rison with —nother r—ndom gr—ph models G.G. Gusev et al. Preferential Attachment and the Web: Edge Distribution
  3. 3. u—ntit—tive —spe™ts of l—rge networks histri˜ution of degrees #(d ) num˜er of verti™es with degree d F st follows the power l—w for m—ny re—lEworld networksF histri˜ution of edges X (d1, d2) tot—l num˜er of edges ˜etween p—irs of verti™es with degrees d1 D d2 F st is not wellEstudied for re—lEword networksF G.G. Gusev et al. Preferential Attachment and the Web: Edge Distribution
  4. 4. u—ntit—tive —spe™ts of l—rge networks gumul—tive fun™tions of degree —nd edge distri˜utions #(d ) = j >d #(j ) ™umul—tive degrees X ,d ρ(d1 , d2 ) = #(d(d)1#(2d) ) ™umul—tive edges 1 2 where X (d1 , d2 ) = j1 >j2 , j1 >max(d1 ,d2 ), j2 >min(d1 ,d2 ) F por re—lEworld networksD these ™umul—tive fun™tions h—ve lesser v—ri—tion —nd thus —re more ™onvenient to study their regul—ritiesF G.G. Gusev et al. Preferential Attachment and the Web: Edge Distribution
  5. 5. u—ntit—tive —spe™ts of l—rge networks essort—tivity dnn (d ) expe™ted —ver—ge degree of neigh˜ors of r—ndom vertex with degree d F d1 X (d ,d1 ) we h—ve dnn (d ) = d1 X (d ,d1 ) d1 thereforeD edge distri˜ution X (d , d1 ) is something more inform—tive th—n —ssort—tivity isF G.G. Gusev et al. Preferential Attachment and the Web: Edge Distribution
  6. 6. €referenti—l —tt—™hment Buckley-Osthus implementation Generate a series of graphs and H H mn 1 a ,1 : t −1 a ,1 from n Ha,m n = , , ,..., 1 2 with n vertices edges one vertex with a self-loop. t → Ha,1 t via adding a new vertex to random j ∈{ , , . . . , t}   dt −1 (s ) + a − 1   P (j = s ) =  (a +a1)t − 1  for  (a + 1)t − 1 dt (s ) nm → H n Ha,1 a ,m k= where is degree of s in t Ha,1 by merging vertices one vertex, 0 t and an edge going 1 2 , . . . , n − 1. G.G. Gusev et al. for 1 ≤s ≤t −1 s = t, . km + 1 , . . . , (k + 1)m into Preferential Attachment and the Web: Edge Distribution
  7. 7. €referenti—l —tt—™hment ristoryFFF f—r—˜¡si —nd el˜ertX preferenti—l —tt—™hment — expl—ins the €ower v—w for degree distri˜ution of the we˜gr—phF follo˜¡s —nd ‚iord—n pre™isely de(ned — model — whose degrees follow €ower v—w with the exponent QD whi™h is unre—listi™ for the we˜gr—phF fu™kley —nd ysthus gener—lized the model ˜y introdu™ing —n initi—l —ttr—™tiveness p—r—meter th—t —llows to (t the exponentF G.G. Gusev et al. a Preferential Attachment and the Web: Edge Distribution
  8. 8. €referenti—l —tt—™hment FFF—nd our ™ontri˜ution „here —re di'erent r—ndom gr—ph models th—t ™—pture powerEl—w degree distri˜utionF fut preferenti—l —tt—™hment is still — ˜etter expl—n—tionD sin™e it ™—n —lso ™—pture the edge distri˜ution of the we˜gr—phF G.G. Gusev et al. Preferential Attachment and the Web: Edge Distribution
  9. 9. €referenti—l —tt—™hmentX —symptoti™s „heoreti™—l results ‘qre™hnikov“ por the fu™kleyEysthus modelD we h—ve #(d ) ∼ c d −2−a D 1 ) X (d1, d2) ∼ c2 (d1+12dd222 F d for l—rge n, d , d1 , d2 —nd d1 d 2 @ d2 1−a d1AF goroll—ries #(d ) ∼ c3 d −1−a D a a ρ(d1 , d2 ) ∼ c4 (d1 + d2 )1−a d1 d2 F for l—rge n, d , d1 , d2 —nd d1 G.G. Gusev et al. d2 @ d2 d1AF Preferential Attachment and the Web: Edge Distribution
  10. 10. ‡e˜ host gr—ph ‡e˜ host gr—ph represents we˜ hosts —s verti™es —nd hyperlinks —s edgesF yur m—in —ssumptionsX the we˜ host gr—ph is o˜t—ined ˜y using the fu™kleyEysthus gr—ph model —nd thus #Host (d ) ∼ b1 d −1−a1 D a a ρHost (d1 , d2 ) ∼ b2 (d1 + d2 )1−a2 d1 2 d2 2 for some ™onst—nts ai D bi in —ppropri—te r—nges of degrees d , d1 , d2 F G.G. Gusev et al. Preferential Attachment and the Web: Edge Distribution
  11. 11. ‡e˜ host gr—ph h—t— used h—t— ˜—se ™r—wled ˜y ‰—ndex in xovem˜er PHIIF gle—ned from sp—m —nd dupli™—tes †erti™es of the we˜ host gr—ph ™orrespond to dom—in owners en edge ˜etween two verti™esEowners is dr—wn if there is — link from — p—ge of one owner to — p—ge of —nother ownerF „he ™onstru™ted we˜ host gr—ph ™onsists of VTFVw verti™es —nd IFQQf edgesa a To obtain the graph, please see http://events.yandex.ru/events/publications/ or contact the authors. G.G. Gusev et al. Preferential Attachment and the Web: Edge Distribution
  12. 12. pitting the model to the we˜ host gr—ph „he ˜est (t of the initi—l —ttr—™tiveness p—r—meter ‡e (nd it for the re—l we˜ host gr—ph ˜y P independent methodsX 1 2 a estim—te a1 , b1 ˜y deriving the ˜est (t of o˜served ™umul—tive degree distri˜ution #Host to the predi™ted b1 d −1−a1 F estim—te a2 , b2 ˜y deriving the ˜est (t of o˜served ™umul—tive edge distri˜ution ρHost (d1 , d2 ) to the a a predi™ted b2 (d1 + d2 )1−a2 d1 2 d2 2 F G.G. Gusev et al. Preferential Attachment and the Web: Edge Distribution
  13. 13. pitting the model to the we˜ host gr—ph „he o˜t—ined v—lues p—r—meter a σ σs degree distri˜ution edge distri˜ution a1 = H.PUTP PFTQI HFHHSTTT a2 = H.PUUR HFHSWW V.SIV · IH−6 σ me—n squ—red error of predi™tive fun™tions σs st—nd—rd devi—tion of ai o˜t—ined vi— ˜ootstr—ppingF Surprisingly, we obtained the same value a ≈ H, PV! G.G. Gusev et al. Preferential Attachment and the Web: Edge Distribution
  14. 14. pitting the model to the we˜ host gr—ph gon™lusion „he s—me spe™i(™ fu™kleyEysthus model with a ≈ H, PV —™™ur—tely —pproxim—tes the two ˜—si™ —spe™ts of the we˜ host gr—phX degree —nd edge distri˜utions @—nd hen™e —ssort—tivity —s wellAF G.G. Gusev et al. Preferential Attachment and the Web: Edge Distribution
  15. 15. cumulative frequency pitting the model to the we˜ host gr—ph 1010 109 108 107 106 105 104 103 102 101 100 0 10 101 102 103 104 degree 105 106 107 Figure : Cumulative degree distribution (black), approximation obtained by using our method (red) and linear regression in log-log scales (green). G.G. Gusev et al. Preferential Attachment and the Web: Edge Distribution
  16. 16. pitting the model to the we˜ host gr—ph 4 e( 3 1 0 0 1 4 5 2 3 ee (log10) degr 6 10) ) 10 log 2 / degree product (log 5 0 1 2 3 degree 4 (log10) 5 6 6 4 2 1 0 (log 10) cumm edge frequency gre de ct (log10) 6 1 0 1 2 3 4 5 6 7 3 deg ree cumm edge frequency / degree produ 1 0 1 2 3 4 5 6 7 5 Figure : Cumulative edge distribution (blue) and approximation obtained by using our method (green) in logarithmic scales . G.G. Gusev et al. Preferential Attachment and the Web: Edge Distribution
  17. 17. pitting the model to the we˜ host gr—ph ‚em—rk3 „he —pproxim—tion w—s —˜le to ™—pture — ™on™—ve —re— —round the di—gon—l d1 = d2 F st is due to the term (d + d )1−a predi™ted 1 2 theoreti™—lly —nd would not ˜e possi˜le with — a a simpler —pproxim—tion of the form d1 d2 F G.G. Gusev et al. Preferential Attachment and the Web: Edge Distribution
  18. 18. gomp—rison with —nother r—ndom gr—ph models fut is it re—lly surprisingc hoes —ny re—son—˜le r—ndom gr—ph model whose degrees follow the €ower v—w ™—pture —lso the edge distri˜ution of the we˜ host gr—phc ghoi™e of r—ndom gr—ph models ‡e ™onsidered two models whose degree distri˜utions follow the power l—wX the ™on(gur—tion model @qhƒAF the rolmeEuim model @ruAF G.G. Gusev et al. Preferential Attachment and the Web: Edge Distribution
  19. 19. gomp—rison with —nother r—ndom gr—ph models fut is it re—lly surprisingc hoes —ny re—son—˜le r—ndom gr—ph model whose degrees follow the €ower v—w ™—pture —lso the edge distri˜ution of the we˜ host gr—phc ghoi™e of r—ndom gr—ph models ‡e ™onsidered two models whose degree distri˜utions follow the power l—wX the ™on(gur—tion model @qhƒAF the rolmeEuim model @ruAF G.G. Gusev et al. Preferential Attachment and the Web: Edge Distribution
  20. 20. gomp—rison with —nother r—ndom gr—ph models ixperiments with simul—ted gr—phs por e—™h of the thee modelsD we simul—ted sever—l gr—ph inst—n™es with VTFVw of verti™es @—s in our we˜ host gr—phAD derived degree —nd edge distri˜utions —ver—ged over the inst—n™esD —nd ™omp—red with the we˜ host gr—ph distri˜utionsF G.G. Gusev et al. Preferential Attachment and the Web: Edge Distribution
  21. 21. gomp—rison with —nother r—ndom gr—ph models 2 1 0 0 5 3 4 2 gree (log10) de 1 6 ) 10) (log 3 0 0 1 2 degre 3 4 e (log 10) 5 1 2 3 g10) 4 (lo 5 egree 6 d 6 2 3 degree 4 (log10) 5 6 6 5 4 2 0 10) 3 (log 1 ee 0 1 4 6 0 1 degr2 3 4 5 ee (log1 0) 6 0 1 2 5 3 4 10) degree (log 6 0 1 0 1 2 3 4 5 6 2 deg r uct (log10) ency / degree prod cumm edge frequ 2 1 0 1 2 3 4 5 6 2 4 6 cumm edge frequency / degree product (log10) ree 4 cumm edge frequency / degree product (log10) 5 deg 1 6 4 5 2 3ree (log10) deg 6 2 0 / degree product (log10 2 0 2 4 6 degree product (log10) 6 deg 5 ree 4 3 (log 2 10) 1 0 0 cumm edge frequency cumm edge frequency / cumm edge frequency / degree product (log10) 1 0 1 2 3 4 5 6 0 1 2 3 degree (log4 5 6 10) 0 1 2 3 4 5 6 degree (log10) Figure : Cumulative edge distributions for the web host graph (blue), the BO simulated graph (cyan), the GDS graph (red), and the HK graph (orange) in log10 scales.. G.G. Gusev et al. Preferential Attachment and the Web: Edge Distribution
  22. 22. ‚eferen™es [Zhukovskij] & Vinogradov, Pritykin, Ostroumova, Grechnikov, Gusev, Serdyukov, Raigorodskii Empirical Validation of the BuckleyOsthus Model for the Web Host Graph: Degree and Edge Distributions. http://arxiv.org/abs/1208.2355, 2012. [Grechnikov] The degree distribution and the number of edges between vertices of given degrees in the Buckley-Osthus model of a random web graph. arXiv:1108.4054v1, 2011. G.G. Gusev et al. Preferential Attachment and the Web: Edge Distribution
  23. 23. Thank you!!! G.G. Gusev et al. Preferential Attachment and the Web: Edge Distribution

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