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  1. ❉②♥❛♠✐❝ ❤❛r✈❡st✐♥❣ ♦❢ s♦❝✐❛❧ ❣r❛♣❤s ❏✳ ■✳ ❖r❧✐❝❦✐ ✶ ✶ ■❚❇❆ ❲❉❈❙ ✷✵✵✾✱ ❇✉❡♥♦s ❆✐r❡s✱ ❆r❣❡♥t✐♥❛

  2. ▼✐♥✐ ❙❝✐ ❇✐♦ ◮ ▲✐tt❧❡ ❜❛❝❦❣r♦✉♥❞ ✐♥ t❤❡♦r❡t✐❝❛❧ ❝♦♠♣✉t❡r s❝✐❡♥❝❡✳ ❘❡✈❡rs✐❜❧❡ ❚✉r✐♥❣ ♠❛❝❤✐♥❡s ✭❄✮✳ ◮ ▲✐tt❧❡ ❜❛❝❦❣r♦✉♥❞ ✐♥ ❛♣♣❧✐❡❞ s❡❝✉r✐t② r❡s❡❛r❝❤✿ ✐♥❢♦r♠❛t✐♦♥ ❣❛t❤❡r✐♥❣ ❛♥❞ ❛tt❛❝❦ s✐♠✉❧❛t✐♦♥✳ ◮ ❲♦r❦✐♥❣ ✇✐t❤ P❤❞ ❛❞✈✐s♦rs ❏✳ ■✳ ❆❧✈❛r❡③✲❍❛♠❡❧✐♥ ❛♥❞ P✳ ■✳ ❋✐❡r❡♥s ❛t ■❚❇❆ ♦♥ r❡♣✉t❛t✐♦♥ ❛♥❞ ♣r✐✈❛❝② ✐♥ s♦❝✐❛❧ ♥❡t✇♦r❦s✳

  3. ❆❣❡♥❞❛ ◮ ❲❤❛t❄ ◮ ❆♥s✇❡r✿ st✉❞②✐♥❣ s♦❝✐❛❧ ❣r❛♣❤s✴♥❡t✇♦r❦s ❞②♥❛♠✐❝s ◮ ❍♦✇❄ ◮ ❆♥s✇❡r✿ ❝♦❧❧❡❝t✐♥❣ ❞②♥❛♠✐❝ s♦❝✐❛❧ ❞❛t❛ ◮ ❲❤❡♥❄ ◮ ❆♥s✇❡r✿ ✷✵✵✾✲✷✵✶✵✱ ✈❡r② ✐♥t❡r❡st✐♥❣ q✉❡st✐♦♥s ❛r✐s❡✳ ◮ ❉❛♥❣❡rs❄ ◮ ✏ ❚❤❡r❡ ❛r❡ ✜❡❧❞s✱ ❡♥❞❧❡ss ✜❡❧❞s✱ ✇❤❡r❡ ❤✉♠❛♥ ❜❡✐♥❣s ❛r❡ ♥♦ ❧♦♥❣❡r ❜♦r♥✱ ✇❡ ❛r❡ ❣r♦✇♥✳ ✑ ▼♦r♣❤❡✉s✱ ▼❛tr✐① ✭✶✾✾✾✮

  4. ❚❤❡ ❙♦❝✐❛❧ ●r❛♣❤ ✭❲❤❛t❄✮ ◮ ■s ❛ r❡❛❧ ♦❜❥❡❝t✦ ❲❡ ✉s❡ ❣r❛♣❤s ❛s ❛❜str❛❝t✐♦♥s✳ ◮ ✭♥♦t❛t✐♦♥✮ ❙♦❝✐❛❧ ❣r❛♣❤✿ t❤❡ s♦❝✐❛❧ ✐♥t❡r❛❝t✐♦♥s ❜❡t✇❡❡♥ ❤✉♠❛♥ ❜❡✐♥❣s✳ ◮ ✭♥♦t❛t✐♦♥✮ ❙♦❝✐❛❧ ♥❡t✇♦r❦ ✭s❡r✈✐❝❡✮✿ ❛ ❝♦♠♠✉♥✐❝❛t✐♦♥ s②st❡♠ ❞❡s✐❣♥❡❞ ❢♦r ✉s✐♥❣ ❛♥❞ ❡①♣❛♥❞✐♥❣ t❤❡ s♦❝✐❛❧ ❣r❛♣❤✳ ❤tt♣✿✴✴❜✐t✳❧②✴♦r❦✉t✲❦✲❝♦r❡s✲❜✐❣

  5. ❙♦❝✐❛❧ ◆❡t✇♦r❦ ❙❡r✈✐❝❡s ✭❲❤❛t❄✮ ✶✳ Pr♦✈✐❞❡ ❛❜str❛❝t✐♦♥ ♦❢ ✉s❡r ♣r♦✜❧❡ ❛♥❞ s♦❝✐❛❧ ❝♦♥t❛❝ts✳ ✷✳ Pr♦✈✐❞❡ ❝♦♠♠✉♥✐❝❛t✐♦♥ ✇✐t❤ s♦❝✐❛❧ ❝♦♥t❛❝ts✳ ✸✳ Pr♦✈✐❞❡ ✇❛②s t♦ st❛② ✐♥ t♦✉❝❤ ✇✐t❤ ❝♦♥t❛❝ts✳ ✹✳ Pr♦✈✐❞❡ ♥❛✈✐❣❛t✐♦♥ ❛♥❞ r❡❝♦♠♠✳ ♦❢ ♥❡✇ ❝♦♥t❛❝ts✳ Cyworld Flickr LiveJournal Orkut parameter #nodes 12,048,186 1,846,198 5,284,457 3,072,441 %nodes 100% 26.9% 95.4% 11.3% #links 190,589,667 22,613,981 77,402,652 223,534,301 %simmetric links 100% 62.0% 73.5% 100.0% ave. deg � ❦ � 31.6 12.24 16.97 106.1 in/out α - 1.74 / 1.78 1.59 / 1.65 1.50 / 1.50 clustering ❈ 0.16 0.31 0.33 0.17 assortativity r -0.13 0.20 0.18 0.07 ave. distance 3.2 5.67 5.88 4.25 diameter - 27 6 9 ❛♥❛❧②s✐s ❢r♦♠ ❬▼✐s❧♦✈❡ ❡t ❛❧✳✱ ✷✵✵✼❪ ❛♥❞ ❬❆❤♥ ❡t ❛❧✳✱ ✷✵✵✼❪

  6. ❙♦❝✐❛❧ ◆❡t✇♦r❦ ❙❡r✈✐❝❡s✱ ❝♦♥t✳ ✭❲❤❛t❄✮ ◮ ▼✐❝r♦✴♥❛♥♦✲❜❧♦❣❣✐♥❣✿ ♥❡t✇♦r❦ s✐♠♣❧✐❝✐t②✱ s❤♦rt ♠❡ss❛❣❡s✱ r❛♣✐❞ ❞✐✛✉s✐♦♥✳ ◮ ❉②♥❛♠✐❝s✿ ✇❡✐❣❤t❡❞ ♠❡♥t✐♦♥ ♥❡t✇♦r❦s ❬❙❤❛♠♠❛ ❡t ❛❧✳✱ ✷✵✵✾❪ ❛♥❞ r❡t✇❡❡t✐♥❣ ❞②♥❛♠✐❝s ✭♠❡ss❛❣❡ ❢♦r✇❛r❞✐♥❣✮✳ ❊①❛♠♣❧❡✿ ❚✇✐tt❡r✱ ❏❛✐❦✉✱ ❡t❝✳

  7. ❖♣❡♥♥❡ss ✈❡rs✉s Pr✐✈❛❝② ✭❲❤❛t❄✮ ◮ ❙♦❝✐❛❧ ♥❡t✇♦r❦s ❛r❡ ♠❡❛♥s ♦❢ ❝♦♠♠✉♥✐❝❛t✐♦♥✱ s♦ ❛r❡ ✉s❡❢✉❧✳ ◮ ❚❡♥s✐♦♥ ❜❡t✇❡❡♥ t❤♦s❡ ✇❤♦ ❧✐❦❡ ♣r✐✈❛❝② ❛♥❞ t❤♦s❡ ✇❤♦ ❧✐❦❡ ♦♣❡♥♥❡ss✳ ◮ ▲✐♠✐t❡❞ ✈✐s✐❜✐❧✐t② ✉s✉❛❧❧② ✈✐❛ ❧✐♠✐t❡❞ ❧♦♦❦❛❤❡❛❞ ✭t❤❡ ✐♥❢❛♠♦✉s ❖♥❧② ❋r✐❡♥❞s ✈s✳ ❋r✐❡♥❞s ♦❢ ❋r✐❡♥❞s ❢❡❛t✉r❡s ✐♥ ❋❛❝❡❜♦♦❦✮✳ ◮ ❊①❛♠♣❧❡ ♦❢ ♣r✐✈❛❝②✿ ▲✐♥❦❡❞■♥ ✇❡❜s✐t❡✱ ♣r✐✈❛t❡ ❞❛t❛ ❛♥❞ ❧✐♠✐t❡❞ ❧♦♦❦❛❤❡❛❞ ✭❞✐st❛♥❝❡ ✸✮✳ ◮ ❊①❛♠♣❧❡ ♦❢ ♦♣❡♥♥❡ss✿ ●♦♦❣❧❡ ❙♦❝✐❛❧●r❛♣❤ ❬❋✐t③♣❛tr✐❝❦✱ ✷✵✵✼❪✱ ♣✉❜❧✐❝ ✐♥t❡r❢❛❝❡✳

  8. ❚♦♦❧s ❢♦r ❞②♥❛♠✐❝ ❤❛r✈❡st✐♥❣ ✭❍♦✇❄✮ ◮ ❊①♦♠✐♥❞ ❬❏■❖✱ ✷✵✵✽❪ ✭ ❤tt♣✿✴✴❜✐t✳❧②✴❡①♦♠✐♥❞ ✮✿ s✉♣♣♦rt❡❞ ❜② ❈♦r❡ ❙❚ ✱ ♠♦st❧② ❢♦❝✉s❡❞ ✐♥ st❛t✐❝ ❖❙■◆❚ ❛♥❞ s♦❝✐❛❧ ✐♥✜❧tr❛t✐♦♥ ✳ ■♥❝❧✉❞❡s ❞②♥❛♠✐❝ ✉♣❞❛t❡s ◆♦r♠❛❧✐③❡❞ ●♦♦❣❧❡ ❉✐st❛♥❝❡ ✭ ◆●❉ ✮ ❛♥❞ ♦t❤❡r s❡❛r❝❤ ❡♥❣✐♥❡ ♠✐♥✐♥❣ ✐♥❢♦r♠❛t✐♦♥✳

  9. ❚♦♦❧s ❢♦r ❞②♥❛♠✐❝ ❤❛r✈❡st✐♥❣✱ ❝♦♥t✳ ✭❍♦✇❄✮ ◮ ❚✉♣❧❡ts ❬❏■❖✱ ✷✵✵✾❪ ✭ ❤tt♣✿✴✴t✉♣❧❡ts✳❛♣♣s♣♦t✳❝♦♠ ✮✿ ❛♥ ❡①♣❡r✐♠❡♥t❛❧ ❝❧♦✉❞ s❤❡❧❧ ❢♦r s♦❝✐❛❧ ♥❡t✇♦r❦ ❝r❛✇❧✐♥❣ ❛♥❞ ✐♥t❡r❛❝t✐♦♥ ✭❲❡❜ ✹✳✵❄✮✳ ◮ ◆❛♥♦✲❜❧♦❣❣✐♥❣ ✇❡✐❣❤t❡❞ ♠❡♥t✐♦♥ ❧✐♥❦s✿ ❃✗❃✗❃✗❃ t✇✐tt❡r ❅❛s❞❢❛s❞❢ ❳ ❄ ②❡s t✇✐tt❡r ❅❛s❞❢❛s❞❢ ❅q✇❡rq✇❡r✶ ✭✸✸✳✾ ✪✮ t✇✐tt❡r ❅❛s❞❢❛s❞❢ ❅q✇❡rq✇❡r✷ ✭✶✵✳✼ ✪✮ t✇✐tt❡r ❅❛s❞❢❛s❞❢ ❅q✇❡rq✇❡r✸ ✭✽✳✾ ✪✮ t✇✐tt❡r ❅❛s❞❢❛s❞❢ ❅q✇❡rq✇❡r✹ ✭✼✳✶ ✪✮ t✇✐tt❡r ❅❛s❞❢❛s❞❢ ❅q✇❡rq✇❡r✺ ✭✼✳✶ ✪✮ t✇✐tt❡r ❅❛s❞❢❛s❞❢ ❅q✇❡rq✇❡r✻ ✭✺✳✹ ✪✮ t✇✐tt❡r ❅❛s❞❢❛s❞❢ ❅q✇❡rq✇❡r✼ ✭✺✳✹ ✪✮ ✳✳✳ ❃✗❃✗❃✗❃ s♦❝✐❛❧❣r❛♣❤ ❛s❞❢❛s❞❢✳❜❧♦❣s♣♦t✳❝♦♠ ❳ ❄ ✳✳✳ ❃✗❃✗❃✗❃ ❛❧♣❤❛ ✏❛r❣❡♥t✐♥❛ ❡①t❡r♥❛❧ ❞❡❜t✑ ❳ ❄ ✳✳✳ ❃✗❃✗❃✗❃ ❛❧♣❤❛ ✏❛r❣❡♥t✐♥❛ ♣❛t❡♥ts✑ ❳ ❄ ✳✳✳ ❃✗❃✗❃✗❃ ❛❧♣❤❛ ✧t❤❡ ♠❡❛♥✐♥❣ ♦❢ ❧✐❢❡ t❤❡ ✉♥✐✈❡rs❡ ❛♥❞ ❡✈❡r②t❤✐♥❣✧

  10. ❚♦♣✐❝s✿ ◆❡✇s ❈②❝❧❡ ✭❲❤❡♥❄✮ ◮ ◆❡✇s ❞✐✛✉s✐♦♥ ❜② ♥❡✇s s♦✉r❝❡s ❬▲❡s❦♦✈❡❝ ❡t ❛❧✳✱ ✷✵✵✾❪✳ ◮ ❙t✉❞② ♦❢ ♠❡♠❡s✴♣❤r❛s❡s ✐♥ ❜❧♦❣ ❛♥❞ tr❛❞✐t✐♦♥❛❧ ♥❡✇s ♠❡❞✐❛✳ ◮ ❆♥❛❧②s✐s ♦❢ ✾✵ ♠✐❧❧✐♦♥ ❛rt✐❝❧❡s t♦ ❡①tr❛❝t ✶✶✷ ♠✐❧❧✐♦♥ q✉♦t❡s ❛♥❞ ❣❡♥❡r❛t❡ ✸✺✱✽✵✵ ♣❤r❛s❡ ❝❧✉st❡rs✳ ❤tt♣✿✴✴♠❡♠❡tr❛❝❦❡r✳♦r❣✴

  11. ❚♦♣✐❝s✿ ◆❡✇s ❈②❝❧❡✱ ❝♦♥t✳ ✭❲❤❡♥❄✮ ❙♦♠❡ ❝✉r✐♦✉s ♣r♦♣❡rt✐❡s t❤❡② ❢♦✉♥❞✿

  12. ❚♦♣✐❝s✿ ◆❡✇s ❈②❝❧❡✱ ❝♦♥t✳ ✭❲❤❡♥❄✮ ◮ P❡❛❦ ✈♦❧✉♠❡ ♦❢ ♠❡♠❡s✴♣❤r❛s❡ ❛♥❞ ❞❡❝❛②✿ ② ( t ) ∼ ❛ ❧♦❣ ( t ) ◮ ▼♦❞❡❧✿ ♣❡r✐♦❞s t = ✶ , ✷ ,... ✱❚✱ ✇✐t❤ ◆ ♠❡❞✐❛ s♦✉r❝❡s r❡♣♦rt✐♥❣ ♦♥❡ t❤r❡❛❞ ♣❡r ♣❡r✐♦❞✱ ♦♥❡ ♥❡✇ t❤r❡❛❞ ✐s ❣❡♥❡r❛t❡❞ ❡❛❝❤ ♣❡r✐♦❞ ❛♥❞ ❛❧❧ s♦✉r❝❡ r❡♣♦rt ❞✐✛❡r❡♥t t❤r❡❛❞s ❛t t = ✵✳ ◮ ❙✐♠♣❧❡ ♣r❡❢✳ ❛tt❛❝❤♠❡♥t✿ Pr ( ❥ , t ) = ❢ ( ♥ ❥ ) δ ( t − t ❥ ) ✇❤❡r❡ ♥ ❥ ✐s t❤❡ t♦t❛❧ ♣r❡✈✐♦✉s st♦r✐❡s ❛❜♦✉t t❤r❡❛❞ ❥ ❛♥❞ t ❥ ✐s ✇❤❡♥ t❤r❡❛❞✴❤✐st♦r② ❥ ✇❛s ✜rst ♣r♦❞✉❝❡ ✳ ◮ ❢ ( · ) ♠♦♥♦✳ ✐♥❝r❡❛s✐♥❣ ♦♥ ♥ ❥ ✭✐♠✐t❛t✐♦♥ ❡✛❡❝t✮ ◮ δ ( · ) ♠♦♥♦✳ ❞❡❝r❡❛s✐♥❣ ✐♥ t − t ❥ ✭r❡❝❡♥❝② ❡✛❡❝t✮✳

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