Representation Learning for Scale-free Networks: スケールフリーネットワークに対する表現学習

Representation Learning for Scale-free Networks [AAAI 2018]

https://arxiv.org/abs/1711.10755
https://pdfs.semanticscholar.org/fc1e/99aaab5831fc75c7390051b3a7fff51fdeef.pdf

Transcript

1. 3FQSFTFOUBUJPO
   -FBSOJOH
   GPS
   4DBMF'SFF
   /FUXPSLT εέʔϧϑϦʔωοτϫʔΫʹର͢Δදݱֶश ৘ใཧ޻ֶӃ
   ৘ใ޻ֶίʔε
   Ҫ্ݚڀࣨ
   .
   ࢁޱॱ໵ʢ.ʣ
2. 4DBMFGSFF
   /FUXPSL εέʔϧϑϦʔʢTMBDFGSFFʣωοτϫʔΫͱ͸ɺϊʔυͷ࣍਺ʢJF
   ۙ๣ϊʔυ਺ʣͷ෼෍͕ɺ΂ ͖৐ଇʢQPXFS
   MBXʣʹै͏ωοτϫʔΫͷ͜ͱ DG
   ࢦ਺͕ͷ΂͖৐ଇ ྫʣҰ෦ͷਓ͸ඇৗʹͨ͘͞Μ༑ୡ͕͍ͯɺେଟ਺͸༑ୡ͕গͳ͍ ස౓ ࣍਺ ࢦ਺ؔ਺తʹݮগ
3. ࣍਺෼෍ͷҧ͍ http://www.network-science.org/
4. ैདྷͷάϥϑຒΊࠐΈख๏  ࣗવݴޠॲཧͷք۾Ͱ༗໊ͳ
   XPSEWFD
   <.JLPMPW
   FU
   BM
   >
   ΛɺϊʔυຒΊࠐΈʢ/PEF
   &NCFEEJOHʣ ʹద༻ͨ͠
   OPEFWFD
   <(SPWFS
   BOE
   -FTLPWFD
   >
   ͕੒ޭΛऩΊ͍ͯΔ
    ୯ޠϊʔυ
    ηϯςϯεάϥϑ্ͷϥϯμϜ΢ΥʔΫྻ  ·ͨɺϊʔυຒΊࠐΈ͸ɺଟ༷ମֶशͱݺ͹ΕΔ ඇઢܗ࣍ݩ࡟ݮख๏ͷҰ෦ͷύʔτͱͯ͠ɺੲ͔ Β੝Μʹݚڀ͞Ε͍ͯΔλεΫͷͻͱͭ
    ϥϓϥγΞϯݻ༗Ϛοϓ๏ʢ-BQMBDJBO
   &JHFONBQʣ <#FMLJO
   BOE
   /JZPHJ
   >
   ͕༗໊
    σʔλؒͷྨࣅ౓ΛΤοδͷॏΈͱͨ͠ྨࣅ౓ά ϥϑʢTJNJMBSJUZ
   HSBQIʣΛߏஙͯ͠ɺ͜ͷྡ઀ߦ ྻͷݻ༗஋෼ղͰ௿࣍ݩຒΊࠐΈΛಘΔ https://www.cs.cmu.edu/~aarti/Class/10701/slides/Lecture21_2.pdf
5. ैདྷख๏ͷ໰୊఺  ঺հͨ͠ैདྷख๏͸ɺ఺ؒͷؔ܎ͷΑ͏ͳɺωοτϫʔ Ϋ͕࣋ͭہॴతͳߏ଄ͷ৘ใΛอ࣋͢ΔΑ͏ͳຒࠐΛߦ͏
    ͔͠͠εέʔϧϑϦʔੑ͸େҬతͳ৘ใͷͻͱͭ
    ݱ࣮ͷωοτϫʔΫʹසൟʹݟΒΕΔ͜ͷ৘ใΛແࢹ͢Δͷ ͸೗Կͳ΋ͷ͔ʁ
    ࣮ࡍɺಘΒΕͨຒΊࠐΈͰ΋ͱͷάϥϑΛ࠶ߏஙͨ͠ͱ͖ɺ ैདྷख๏͸
   IJHI
   EFHSFF
   ͳϊʔυ਺͕ଟ͘ͳΔ܏޲ʹ͋Δ
    ͭ·ΓɺຒΊࠐΈۭؒͰσʔλಉ͕࢜ͻ͖ͬͭ͗ͯ͢ɺશମ తʹ͙ͪΌͬͱͳͬͯ͠·͍ͬͯΔ ʮଟ͘ͷϊʔυͷۙ͘ʹډΕΔݖརΛɺগ਺ϊʔυʹ͚ͩ༩͑Δʯ੍໿͕ඞཁ
6. ͜ͷ࿦จͰ΍Γ͍ͨຒΊࠐΈ Node Embedding Reconstruction ޡϦϯΫʢΤοδʣͷ࠷খԽ ϢʔΫϦουڑ཭͕ᮢ஋ΑΓ ΋খ͚͞Ε͹ΤοδΛߏ੒
7. Node Embedding Reconstruction ؾ࣋ͪɿ͍͍ײ͡ʹ֤ϊʔυΛݻఆ௕ϕΫτϧ ΁ରԠ෇͚͍ͨʢ೚ҙͷάϥϑʹରͯ͠ରԠ෇ ͚Δؔ਺Λֶश͢ΔΘ͚Ͱ͸ͳ͍ʣ
8. ͜ͷ࿦จͰ΍Γ͍ͨຒΊࠐΈ Node Embedding Reconstruction ޡϦϯΫʢΤοδʣͷ࠷খԽ ϢʔΫϦουڑ཭͕ᮢ஋ΑΓ ΋খ͚͞Ε͹ΤοδΛߏ੒
9. Ξϓϩʔν
10. ͦ΋ͦ΋εέʔϧϑϦʔੑ͸
    FNCFE
    Ͱ͖Δͷ͔໰୊  ௚ײతʹ͸ɺຒΊࠐΈۭؒʹ͓͍ͯσʔλ͕ΰνϟͬͱͯ͠͠·͏͔Βɺάϥϑ࠶ߏங࣌ʹߴ࣍਺ϊʔ w w w υ͕૿͑ͯ͠·͏ͱਪଌͰ͖Δ
     ΰνϟͬͱ͍ͯ͠Δ
    
    ߴ࣍਺ϊʔυ͕ଟ͘ͷ௿࣍਺ϊʔυΛۙ͘ʹҾ͖دͤΔ͕ނʹɺҾ͖دͤΒΕͨ௿࣍਺ϊʔ υಉ࢜ʹ΋άϥϑ࠶ߏங࣌ʹΤοδ͕݁͹Εͯ͠·͏Α͏ͳঢ়ଶ
     Ͱ͸ԿΒ͔ͷ੍໿ΛՃ͑ͯɺΰνϟͬͱ͠ͳ͍Α͏ʹͰ͖Δͷ͔ʁ
     ྫ͑͹
    
    ࣍ݩϢʔΫϦουۭؒʹຒΊࠐΈΛ͢ΔͳΒɺc7c
    
    
    Ͱ͢Βແཧͦ͏
     ͔ͱ͍ͬͯ
    c7c
    
    
    ʹରͯ͠ɺ.
    ࣍ݩͷۭؒΛ༻ҙ͢Δͷ͸΍Γա͗ʜ
     ͦ͜Ͱɺ,
    ࣍ݩͷϢʔΫϦουۭؒʹຒΊࠐΜͩ࣌ʹɺ໰୊ͳ͘εέʔϧϑϦʔੑΛ࠶ߏஙͰ͖ΔΑ͏ ͳϊʔυ਺ͷ࠷େ஋ʢʹର͢ΔԼքʣʹ͍ͭͯɺཧ࿦తͳղੳΛߦͬͨ
11. ݁࿦͚ͩड़΂Δͱ
 ໰୊ͳͦ͞͏ k 10 20 50 100 lower bound 57 3325 637M 4E+17 .@L
    
    L
    ࣍ݩϢʔΫϦουۭؒʹ͓͍ͯɺத৺
    Y
    ൒ܘЏͷ௒ٿ
    # Y Џ
    ͷ தʹɺ͓ޓ͍͕
    Џ
    Ҏ্཭ΕͯΔ఺Λ࠷େͰԿݸ഑ஔͰ͖Δ͔ʁ ʢ࣍ݩͳΒʣ࣍਺͕
    & 
    ͷεʔύʔϋϒ ͷपΓʹɺ࣍਺͕ͷ༿ϊʔυΛ& ݸຒΊ ࠐΈͰ͖Δ େମͷݱ࣮ੈքͷωοτϫʔΫ͸ɺे෼ʹεέʔ ϧϑϦʔੑΛอ࣋ͨ͠··ຒΊࠐΈՄೳ ͪͳΈʹ ఆཧ͸ɺϊʔυ͕࠷େͰԿݸ഑ஔͰ͖Δ͔ ໰୊Λɺٿॆరʢ4QIFSF
    1BDLJOHʣ໰୊ɺ ͋Δ͍͸ٿମͷ࠷ີॆర໰୊ͱͯ͠஌ΒΕΔ ໰୊ʹؼண͢Δ͜ͱͰূ໌͞Ε͍ͯΔɻ
12. طଘख๏ͷ cons εέʔϧϑϦʔੑΛߟྀ͠ͳ͍ͱͲΜͳ໰୊͕ੜ͡Δͷ͔ʁ
13. ϏʔόʔͱΦόϚɺ஥ྑ͠໰୊ ैདྷख๏Ͱ͸ۙ͘ʹ ຒΊࠐΈ͞ΕΔ྆ऀ
14. ϏʔόʔͱΦόϚɺ஥ྑ͠໰୊΁ͷରॲࡦ  ैདྷख๏ͷଟ͘͸
    TU
    OE
    PSEFS
    ͷ
    QSPYJNJUZ
    Λอͭ͜ͱʹઐ೦
     TU
    PSEFS
    QSPYJNJUZ
    ˺
    ʮ༑ୡʯ౓߹͍
     OE
    PSEFS
    QSPYJNJUZ
    ˺
    ʮ༑ୡͷ༑ୡʯ౓߹͍
     ͦͷ݁ՌɺεέʔϧϑϦʔ
    ωοτϫʔΫʹଘࡏ͢Δ
    lCJH
    IVCz
    ಉ࢜͸ྨࣅ౓͕ߴ͘ͳΓ΍͍͢
     ྫʣΦόϚͷ༑ୡʹ͸ɺδϟεςΟϯɾϏʔόʔͱ༑ୡͳਓ͕ਓͰ΋ډΔ֬཰͸ߴ͍
     ྨࣅ౓͕ߴ͘ͳΓ΍͍͢ͳΒɺ࣍਺͕ߴ͍ϊʔυͷ
    QSPYJNJUZ
    ʹϖφϧςΟΛ༩͑Ε͹ྑ͍ جຊతํ਑
    ཧ࿦ղੳͷ݁Ռʹج͖ͮɺεέʔϧϑϦʔੑΛ୲อͰ͖ ΔຒΊࠐΈΞϧΰϦζϜͷͨΊͷ࣍਺േଇʢEFHSFF
    QFOBMUZʣݪଇ ΛఏҊ͢Δɻ࣍਺േଇ͸ɺ
    ࣍ٴͼ
    
    ࣍ͷ
    QSPYJNJUZ
    Λ୲อ্ͨ͠Ͱɺ ߴ͍࣍਺Λ΋ͭ௖఺ಉ࢜ͷ
    QSPYJNJUZ
    ʹରͯ͠േΛՊ͢ݪଇͰ͋Δɻ
15. ఏҊख๏
    %14QFDUSBM 2nd-order proximity 1st & 2nd-order proximity weighted adjacency matrix ྡ઀ߦྻ ࣍਺ߦྻ
    %
    
    EJBH E@
    ʜ
    
    E@O https://www.slideshare.net/pecorarista/ss-51761860 ॏΈ෇͖ྡ઀ߦྻʢXFJHIUFE
    BEKBDFODZ
    NBUSJYʣͷྫ
16. ఏҊख๏
    %14QFDUSBM ͩTG 2nd-order proximity 1st & 2nd-order proximity weighted adjacency matrix ྡ઀ߦྻ ࣍਺ߦྻ
    %
    
    EJBH E@
    ʜ
    
    E@O
17. ఏҊख๏
    %14QFDUSBM ͩTG 2nd-order proximity 1st & 2nd-order proximity weighted adjacency matrix ྡ઀ߦྻ ࣍਺ߦྻ
    %
    
    EJBH E@
    ʜ
    
    E@O J K
    ؒͷ
    QSPYJNJZ
    ͕ߴ͍ʢ8@JK ͷ஋͕େ͖͍ʣͳΒɺຒΊࠐΈ ͷڑ཭Λ୹͍ͨ͘͠ ॏཁ౓ʢ%@JʣͰεέʔϧௐઅͨ͠ޙ ͷۭؒͰɺ௨ৗͷݻ༗ϕΫτϧͷੑ࣭ Λຬͨ͠ ͯཉ͍͠ʢҰൠԽݻ༗஋໰୊ʣ ∀i : u⊤ i Dui = 1 ∀i, j(i ≠ j) : u⊤ i Duj = 0 Wij = 1 (di dj )β C′ ij ࣍਺ͷߴ͞ʹରͯ͠ࢦ਺ తͳϖφϧςΟΛ՝͢
18. ఏҊख๏
    %14QFDUSBM ͩTG 2nd-order proximity 1st & 2nd-order proximity weighted adjacency matrix ྡ઀ߦྻ ࣍਺ߦྻ
    %
    
    EJBH E@
    ʜ
    
    E@O J K
    ؒͷ
    QSPYJNJZ
    ͕ߴ͍ʢ8@JK ͷ஋͕େ͖͍ʣͳΒɺຒΊࠐΈ ͷڑ཭Λ୹͍ͨ͘͠ ॏཁ౓ʢ%@JʣͰεέʔϧௐઅͨ͠ޙ ͷۭؒͰɺ௨ৗͷݻ༗ϕΫτϧͷੑ࣭ Λຬͨ͠ ͯཉ͍͠ʢҰൠԽݻ༗஋໰୊ʣ ∀i : u⊤ i Dui = 1 ∀i, j(i ≠ j) : u⊤ i Duj = 0 LͳΒ Wij = 1 (di dj )β C′ ij ࣍਺ͷߴ͞ʹରͯ͠ࢦ਺ తͳϖφϧςΟΛ՝͢ ৽نੑ͸8ͷ࡞Γํ
19. ఏҊख๏
    %18BMLFS  ϥϯμϜ΢ΥʔΫʹجͮ͘طଘख๏ʹ࣍਺ϖφϧςΟΛՃ͑Δ https://towardsdatascience.com/node2vec-embeddings-for-graph-data-32a866340fef ͜͜Λͪΐͬͱ͍͡Δ͚ͩ
20. ఏҊख๏
    %18BMLFS i high degree low degree طଘख๏Ͱ͸ɺϥϯμϜ΢ΥʔΫͷࡍͷ ભҠ֬཰͸ɺۙ๣ϊʔυͰಉ֬͡཰ ࣍਺͕ߴ͍ϊʔυ΁ͷભҠ֬཰Λখ͘͞ ͢ΔΑ͏ͳ੍໿ΛՃ͑Δ i ߴ͍࣍਺ͷϊʔυͳΒɺଞͷϊʔυ ͔ΒͷϥϯμϜ΢ΥʔΫͰ͍͔ͭདྷ ΕΔͰ͠ΐ
21. ࣮ݧ
22. σʔληοτ Vertex Edge |V| |E| Synthetic 10K 400K Facebook Ϣʔβ ༑ୡ 4K 88K Twitter Ϣʔβ ϑΥϩʔ 81K 1.76M Author ஶऀ ڞஶ 5K 29K Citation ஶऀ Ҿ༻ 48K 357K Mobile ొ࿥ऀ ௨࿩ 198K 1.15M
23. ൺֱख๏ ུশ Ҿ༻ ৄࡉ ϥϓϥγΞϯݻ༗ Ϛοϓ๏ LE Belkin and Niyogi 2003 εϖΫτϧϕʔεͷຒΊࠐΈख๏ɻ DP-Spectral ͱͷҧ͍͸ɺྨࣅ౓ߦྻͷߏங๏ํ๏ Deep Walk DW Perozzi, Al-Rfou, and Skiena 2014 skip-gram Ϟσϧʹجͮ͘ຒΊࠐΈɻ֤௖఺͔Β10ݸͷϥϯ μϜ΢ΥʔΫྻʢྻ௕:40ʣΛੜ੒͢Δɻ DP-Walker ͱͷҧ͍͸ɺભҠ֬཰͕Ұ༷Ͱ͋Δ͜ͱ ఏҊख๏1 DP-Spectral - LE + ྨࣅ౓ߦྻʹ࣍਺േଇΛద༻ ఏҊख๏2 DP-Walker - DW + ϥϯμϜ΢ΥʔΫͷભҠ֬཰ʹ࣍਺േଇΛద༻ શख๏ڞ௨ͯ͠ɺຒΊࠐΈ࣍ݩ͸
    
    Λ࠾༻
24. λεΫ  ωοτϫʔΫͷ࠶ߏங
     ʲ໨తʳεέʔϧϑϦʔੑΛอ࣋ͨ͠ຒΊࠐΈ
     ʲํ๏ʳલड़௨Γʢলུʣ
     ʲධՁʳೖྗͱ࠶ߏஙωοτϫʔΫͷͦΕͧΕͷ࣍਺෼෍ΛٻΊɺϐΞιϯͷੵ཰૬ؔ܎਺
    < >
    Λܭࢉ͢Δʢߴ͍ͱྑ͍ʣ
     ʲඋߟʳ֤ख๏Ͱɺ͔Β·Ͱͷൣғ ࠁΈ ͰЏΛಈ͔͠ɺ࠷ྑ࣌ͷ૬ؔ܎਺Λใࠂ
     ϦϯΫ༧ଌ
     ʲ໨తʳϊʔυ
    WJ
    WK
    ؒʹΤοδ͕͋Δ͔ͷ༧ଌ
     ʲํ๏ʳຒΊࠐΈۭؒͰͷࠩ෼
    V@J
    
    V@K
    Λಛ௃ྔͱͯ͠ઢܗճؼϞσϧʹೖྗ͢Δ
     ʲධՁʳਫ਼౓ʢQSFDJTJPOʣɺ࠶ݱ཰ʢSFDBMMʣɺ'஋ʢ'TDPSFʣ
     ʲඋߟʳແ࡞ҝʹબ͹ΕͨϊʔυϖΞͷू߹Λ܇࿅ධՁσʔλʢͦΕͧΕશϊʔυϖΞͷ ʣͱ͢Δ
     ϊʔυ෼ྨ
     ʲ໨తʳϊʔυʹ෇༩͞Ε͍ͯΔϥϕϧͷ༧ଌ
     ʲํ๏ʳ༩͑ΒΕͨ
    W@J
    ʹରͯ͠ɺຒΊࠐΈ
    V@J
    Λಛ௃ϕΫτϧͱͯ͠ઢܗ෼ྨػʹೖྗ
     ʲධՁʳਖ਼ղ཰ʢBDDVSBDZʣ
     ʲඋߟʳσʔληοτ
    $JUBUJPO
    ͰͷΈλεΫΛ࣮ߦɻϥϕϧ͸ͭͷݚڀ෼໺ i P((i, j ) ∈ E ) ≜ sigmoid(w⊤(ui − ui) + b) j {Architecture Computer Network Computer Science Data Mining Theory Graphics Unknown i P(i ∈ l ) ≜ softmax(w⊤ l ui + bl )
25. ݁ՌɿωοτϫʔΫ ࠶ߏஙλεΫ ϐΞιϯ૬ؔ܎਺ͷ%18BMLFS͸ɺ εϐΞϚϯ૬ؔ܎਺ͩͱ%14QFDUSBM ͸Ͱੑೳ޲্͕ΈΒΕͨ ࠷దͳЏͷ஋΋ɺ%14QFDUSBM
    ͸
    ෇ۙͰ҆ఆ͍ͯ͠Δ %1WBSJBOUT
    ͸ɺεέʔϧϑϦʔੑΛ ΑΓอ࣋ͨ͠··ຒΊࠐΈͰ͖Δख๏ Ͱ͋Δͱݴ͑Δ
26. ݁ՌɿωοτϫʔΫ࠶ߏஙλεΫ ຒΊࠐΈ࣍ݩ േଇͷڧ͞ Wij = 1 (didj)β C′ ij %14QFDUSBM
    ͷ΄͏͕ෆ҆ఆͳͷ͸ɺЌ͕௚઀໨తؔ਺ʹ૊Έࠐ·Ε͍ͯ Δ͔Βɻ%18BMLFS͸ϥϯμϜ΢ΥʔΫྻੜ੒ʹ͔͔͔͠ΘΒͳ͍ 4ZOUIFUJD
    ͱ
    'BDFCPPL
    Ͱ࠷దͳЌ͕ҟͳΔ͕ɺ͜Ε͸േଇ͕࣍਺ͷߴ͞ ͦͷ΋ͷʹ՝ͤΒΕ͍ͯΔ͔ΒͰɺτϙϩδʔ͕มΘΕ͹Ќͷޮ͖۩߹͍ ͸มԽ͢Δɻ εέʔϧϑϦʔੑΛ୲อ͢Δͷʹे෼ͳ࣍ݩ਺͕ଘࡏ͢Δ͜ͱ͕෼͔Δɻ %14QFDUSBM
    ͸ຒΊࠐΈ࣍ݩ͕গͳ͍ͱੑೳ͕ඇৗʹѱ͍͕ɺ%18BMLFS
    ͸࣍ݩ਺͕૿͑ͯ΋͋·ΓมԽͤͣɺ҆ఆͯ͠ྑ͍݁ՌɻϥϯμϜ΢Υʔ Ϋͷઓུ͕ޮ͍͍ͯΔͷ͕ཁҼͰ͋ΔͱਪଌͰ͖Δɻ
27. ݁Ռ
    ϦϯΫ༧ଌ ͍͍ͩͨͷέʔεͰ
    %14QFDUSBM
    ͕࠷ ΋ྑ͍ੑೳ ࣍ݩേଇͷݪଇ͸ɺ͜ͷλεΫͷͨΊ ʹ௚઀ઃܭ͞Ε͍ͯΔΘ͚Ͱ͸ͳ͍͕ɺ ΑΓ༗ӹͳ৘ใΛଟ͘ຒΊࠐΈͰ͖ͯ ͍Δ݁Ռɺ޷੒੷ΛऩΊͨͱਪଌ͞Ε Δ
28. ݁Ռ
    ϊʔυ෼ྨ %14QFDUSBM
    ͸ϥϓϥγΞϯݻ༗Ϛοϓ๏Λɺ%1 8BMLFS
    ͸
    %FFQ
    8BML
    Λ౗ͨ͠ɻ
    %14QFDUSBM
    ͸҆ఆͯ͠޷੒੷Λ࢒͍ͯ͠Δɻ
    ඪ४ภࠩ΋
    %14QFDUSBM
    ͷ΄͏͕গͳ͔ͬͨ
     %14QFDUSBM
    
     -&
    
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    GPS
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    """*