takaguchi_15th_neteco.pdf

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ωοτϫʔΫՊֶͷݚڀಈ޲ ʙߏ଄ͷ͋Δࢀরج४ϞσϧΛத৺ʹ ߴޱ ଠ࿕ (Taro Takaguchi) ୈ15ճωοτϫʔΫੜଶֶγϯϙδ΢Ϝ 2018/11/18 Recent topics in network science;  Through a lens of structured reference models 1

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͸͡Ίʹ • ຊߨԋͷ಺༰͸ɺֶज़ݚڀͷαʔϕΠͰ͢ɻ • ߨԋऀͷݱࡏͷۀ຿ͱͷ௚઀తͳؔ࿈͸͋Γ·ͤΜɻ 2

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ࣗݾ঺հ ߴޱ ଠ࿕ʢ͔͙ͨͪ ͨΖ͏ʣ LINE Corp., Data Labs σʔλαΠΤϯςΟετʢ2017/10 -ʣ ܦྺ 2010 - 2013 ౦ژେֶେֶӃ ৘ใཧ޻ֶܥݚڀՊ ത࢜՝ఔ 2013 - 2016 ࠃཱ৘ใֶݚڀॴ ಛ೚ݚڀһ 2016 - 2017 ৘ใ௨৴ݚڀػߏ ςχϡΞτϥοΫݚڀһ ઐ໳෼໺ ωοτϫʔΫՊֶ ର৅ɿ࣮ࣾձͷͭͳ͕Γߏ଄ʢιʔγϟϧάϥϑɺ΢Σϒʣ खஈɿཧ࿦Ϟσϧ + σʔλ෼ੳ Network data from Rocha, Liljeros, & Holme, PLOS Comput. Biol. 7, e1001109 (2011) 3

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2 2 2 1 1 1 3 3 4 4 3 4 • ࣮σʔλ෼ੳ • ఻೻Ϟσϧʢײછ঱ɺ৘ใ֦ࢄʣͷཧ࿦Ϟσϧղੳ ςϯϙϥϧɾωοτϫʔΫͷɿ ࣌ࠁ ࠃࡍձٞϓϩάϥϜҕһ աڈͷओͳݚڀ಺༰ Physical Review E / X, EPJ Data Science,  ICDM, J. Theoretical Biology, J. Banking & Finance, etc. NetSci-X 2016, NetSci 2017, SocInfo 2017, NetSci-X 2018, IC2S2 2018 ࿦จ 4

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ຊߨԋͷείʔϓ ࿩͢͜ͱ ࿩͞ͳ͍͜ͱ • ωοτϫʔΫσʔλͷߏ଄෼ੳʢωοτϫʔΫղੳʣ • ωοτϫʔΫ্ͷμΠφϛΫεʢײછ঱ etc.ʣ • ςϯϙϥϧɾωοτϫʔΫಛ༗ͷղੳʢόʔετੑ etc.ʣ 5

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ωοτϫʔΫղੳʁ 6

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ϥϯμϜͳ ωοτϫʔΫ ݱ࣮ੈքʹݟΒΕΔ ωοτϫʔΫ ೚ҙͷάϥϑ ֤ ௖ ఺ ͷ ஋ ωοτϫʔΫղੳ = “ྑ͍”௿࣍ݩԽ • ࣍਺ • த৺ੑ • ίϛϡχςΟ 7

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ϥϯμϜͳ ωοτϫʔΫ ݱ࣮ੈքʹݟΒΕΔ ωοτϫʔΫ ೚ҙͷάϥϑ ౷ܭ෺ཧ Ұൠੑ ීวੑ ػցֶश ૉੑ ਫ਼౓ άϥϑ ΞϧΰϦζϜ άϥϑ σʔλϕʔε ߴ଎Խ ਫ਼౓ɺεέʔϧ ֤ ௖ ఺ ͷ ஋ ωοτϫʔΫղੳ = “ྑ͍”௿࣍ݩԽ 8

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ม׵ઌ Ұൠੑ ࣍਺ εέʔϧϑϦʔੑ ฏۉڑ཭ εϞʔϧϫʔϧυੑ ॴଐάϧʔϓͷϥϕϧ ີʹͭͳ͕ͬͨάϧʔϓ ʹ෼͔ΕΔ Ұൠੑɾීวੑͷྫ 9

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ม׵ઌ λεΫ ࣍਺ ϊʔυΛ࢝఺ͱͨ͠ͱ͖ͷ ֦ࢄͷαΠζ ॴଐάϧʔϓͷϥϕϧ Τοδ͕͋Δ͔Ͳ͏͔ͷ  ༧ଌʢlink predictionʣ ૉੑͱͯ͠ͷਫ਼౓ͷྫ ۙ೥ͷαʔϕΠ࿦จ “Representation Learning on Graphs: Methods and Applications” Hamilton, Ying & Leskovec, IEEE Data Engineering Bulletin (2017) 10

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ࢀরج४Ϟσϧͷඞཁੑ ௿࣍ݩԽ → ৘ใ͕ࣦΘΕΔɺϊΠζͷհࡏ ௚ײʹ߹Θͳ͍݁Ռ͕ಘΒΕͨͱ͖ɿ • ਅͷʮൃݟʯͳͷ͔ʁ • ϥϯμϜͳ͹Β͖ͭͷൣғͰઆ໌͞ΕΔͷ͔ʁ 㱺 ൺֱͷج४ͱͳΔਓ޻తͳϞσϧʢࢀরج४Ϟσϧʣ͕ඞཁ ΧϥςΫϥϒɾωοτϫʔΫ ʢϊʔυͷ৭ = ೿ൊʣ 11

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“ߏ଄ͷ͋Δ”ࢀরج४Ϟσϧ ׬શʹϥϯμϜͳࢀরج४ϞσϧͰΑ͍ʁ = Erdős-Renyí ϥϯμϜɾάϥϑ No. ͢Ͱʹʮ΋ͬͱ΋Β͍͠ωοτϫʔΫߏ଄ʯ͕஌ΒΕ͍ͯΔ • ඇҰ༷ͳ࣍਺෼෍ɺ࣍਺૬ؔ • ίϛϡχςΟߏ଄ • ֊૚ߏ଄, etc. ͜ΕΒͷߏ଄తͳಛ௃Λ;·্͑ͨͰɿ • ਪఆͨ͠ύϥϝʔλ͔ΒσʔλΛಡΈղ͖͍ͨ • Ϟσϧ + ͹Β͖ͭͰઆ໌Ͱ͖ͳ͍ࣄ৅ΛऔΓग़͍ͨ͠ 㱺 ߏ଄ͷ͋Δࢀরج४Ϟσϧ͕ඞཁ 12

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ߏ଄ͷ͋Δࢀরج४Ϟσϧ͕ͳ͍ͱ… • ΦʔόʔϑΟοςΟϯάɿϊΠζʹҾ͖ͣΒΕΔ • ૬ޓࢀরੑͷ௿Լɿଞͷݚڀͱͷൺֱ͕೉͍͠ • ༗ҙͳߏ଄ͷෆ໌֬ԽɿϥϯμϜͷൣғͰઆ໌Ͱ͖ͳ͍ࣄ͸ʁ ࣮ωοτϫʔΫͷղੳʹࢀরج४Ϟσϧ͸໾ཱͭ ωοτϫʔΫσʔλͷಛघੑ • ͨͩ̍ͭͷαϯϓϧ͔Β෼෍Λਪఆ͢Δඞཁ͕͋Δ • ΢Σϒ͸̍ͭɺߤۭ໢͸̍ͭ… 13

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શମͷߏ੒ 3. ࢀরج४ϞσϧͷԠ༻  ɹ- static ͷ৔߹  ɹ- temporal ͷ৔߹ ɾ ɾ ɾ ฏۉ࣍਺ P(k) ࣍਺෼෍ P(k, k′) ࣍਺૬ؔ άϧʔϓߏ଄ 1. Τοδͷ  ͭͳ͗ସ͑ 2. ߏ੒తͳϞσϧ ωοτϫʔΫσʔλͷ΋ͭߏ଄ ⟨k⟩ 14

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1 & 2 અͷ·ͱΊʛtakeaways 1. Τοδͷͭͳ͗ସ͑ 2. ߏ੒తͳϞσϧ ɾ ɾ ɾ ฏۉ࣍਺ P(k) ࣍਺෼෍ P(k, k′) ࣍਺૬ؔ άϧʔϓߏ଄ ⟨k⟩ dk-γϦʔζ d = 0 ϥϯμϜͭͳ͗ସ͑ d = 1 Maslov-Sneppen d = 2 ࢦ਺ϥϯμϜɾάϥϑ Erdős-Renyí Ϟσϧ ҰൠԽϥϯμϜɾάϥϑ ֬཰తϒϩοΫϞσϧ • statnet https://cran.r-project.org/web/packages/statnet/index.html • graph-tool https://graph-tool.skewed.de/ 15

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1. ࢬͷͭͳ͗ସ͑ 16

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ࢬͷͭͳ͗ସ͑ɿੲ͔Β͋ΔΞΠσΟΞ ฏۉ࣍਺ɹ Λอͭͭͳ͗ସ͑ ࣍਺ྻɹɹΛอͭͭͳ͗ସ͑ {k i} k k′ k′′ k k′ Maslov & Sneppen, “Speciﬁcity and stability in topology of protein networks”, Science (2002) แׅతͳ࿮૊Έ͸ʁ k′′′ k′′ k′′′ ⟨k⟩ 17

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dk-γϦʔζ • Orsini et al., “Quantifying randomness in real networks”, Nature Communications (2015) • Mahadevan et al., “Systematic Topology Analysis and Generation Using Degree Correlations”, ACM SIGCOMM ’06 dk-ϥϯμϜɾάϥϑ ϊʔυ d ݸ͔ΒͳΔ࿈݁ͳ෦෼άϥϑͷ෼෍͕ ʮ࣍਺ͷ৘ใࠐΈͰʯҰఆͰɺଞ͸ϥϯμϜ d อͨΕΔߏ଄ 0 ฏۉ࣍਺ 1 ϊʔυͷ࣍਺ 2 ϖΞͷ࣍਺ 3 ̏ମͷ࣍਺ k k k′ k′′ k k′ k′′ k k′ 18

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k- ⊃ k-ϥϯμϜάϥϑ dk-γϦʔζͷੑ࣭ Orsini et al., “Quantifying randomness in real networks”, Nature Communications (2015)  DOI: 10.1038/ncomms9627 แؚੑ d < d′ 㱺 d d′ ऩଋੑ d = N 㱺 k-ϥϯμϜάϥϑ ɹ= ݩͷάϥϑ N ೉఺ 3k-ϥϯμϜɾάϥϑͷੜ੒Λ࣮ݱ͢Δํ๏͸஌ΒΕ͍ͯͳ͍ Image from Figure 1 19

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2.1 / 2.5k-ϥϯμϜɾάϥϑ Orsini et al., “Quantifying randomness in real networks”, Nature Communications (2015) d อͨΕΔߏ଄ 2.1 ϖΞͷ࣍਺ + ฏۉΫϥελϦϯά܎਺ 2.5 ϖΞͷ࣍਺ + ฏۉΫϥελϦϯά܎਺ʢ࣍਺ґଘʣ αϯϓϦϯάํ๏ k k′ k′′ k k′ 1. 2k-ϥϯμϜɾάϥϑʢϖΞ࣍਺อଘʣͷͭͳ͗ସ͑ީิ k k′′ k 2. ֬཰ min (1, exp(−βΔH))Ͱͭͳ͗ସ͑Λ࠾୒ ͭͳ͗ସ͑ޙͷΫϥελϦϯά܎਺ͱ໨ඪ஋ͱͷζϨ ※஫ҙ※ αϯϓϦϯάͷҰ༷ੑ͸ཧ࿦తʹอূ͞Ε͍ͯͳ͍ 20

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࣮ωοτϫʔΫσʔλ΁ͷద༻ Orsini et al., “Quantifying randomness in real networks”, Nature Communications (2015)  DOI: 10.1038/ncomms9627 • 2.5k-Ϟσϧ͸ 3, 4 αΠζϞνʔϑ΋࠶ݱʢσʔλʹΑΔʣ 2.5k-Ϟσϧ͔Β֎ΕΔྫ • σʔλɿ೴ͷ fMRI ϘΫηϧؒ׆ಈ૬ؔωοτϫʔΫ • ੑ࣭ɿίϛϡχςΟߏ଄ Image from Figure 4 Image from Figure 6 21

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2. ߏ੒తͳϞσϧ 22

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ɾ ɾ ɾ P(k) ࣍਺෼෍ P(k, k′) ࣍਺૬ؔ άϧʔϓߏ଄ ωοτϫʔΫσʔλͷ΋ͭߏ଄ ߏ੒తͳϞσϧʹର͢Δཁ੥ 1. ʮ੍໿Λ଍͢ʯ͜ͱΛ਺ࣜͱͯ͠໌ࣔతʹॻ͖ද͍ͨ͠ 2. ؍ଌ͞ΕΔม਺ʢe.g., ෺ཧతڑ཭ʣͷґଘੑ΋औΓೖΕ͍ͨ 3. ؍ଌ͞Εͳ͍ม਺ʢe.g., άϧʔϓॴଐʣ΋औΓΕ͍ͨ ฏۉ࣍਺ ⟨k⟩ 23

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ʮߏ଄Λอଘ͢Δʯҙຯͷ؇࿨ ͭͳ͗ସ͑ɿݩͷάϥϑͱݫີʹҰக ÷ ߏ੒తϞσϧɿݩͷάϥϑͱظ଴஋ͷҙຯͰҰக P(A ij = 1) = p แׅతͳ࿮૊Έ͸ʁ อͨΕΔߏ଄ Ϟσϧ໊ ྡ઀֬཰ ฏۉ࣍਺ Erdős-Renyí  ϥϯμϜɾάϥϑ ϊʔυͷ࣍਺ ҰൠԽ  ϥϯμϜɾάϥϑ P(A ij = 1) = k i k j 2M • Erdős & Rényi, “On the evolution of random graphs”, Publications of the Mathematical Institute of the Hungarian Academy of Sciences (1960) • Newman, Strogatz & Watts, “Random graphs with arbitrary degree distributions and their applications”, Physical Review E (2001) 24

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ࢦ਺ϥϯμϜɾάϥϑϞσϧ ͳͥ͜ͷ෼෍͕ग़ͯ͘Δʁ • p1 model: Holland & Leinhardt, “An exponential family of probability distributions for directed graphs (with discussion)” , Journal of the American Statistical Association (1981) • p* model: Frank & Strauss, “Markov graphs”, Journal of the American Statistical Association (1986) : ྡ઀ߦྻ = ωοτϫʔΫશମ ॴ๬ͷߏ଄తಛ௃ΛอͭΑ͏ʹάϥϑͷੜ੒֬཰ΛॏΈ෇͚͢Δ : ن֨Խఆ਺ : ωοτϫʔΫͷߏ଄తͳಛ௃ྔ A Z(θ) c r (A) P(A|θ) = 1 Z(θ) exp (∑ r θ r c r (A) ) ྫ: ↓ͷϞνʔϑͷݸ਺ 25

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ࢦ਺ϥϯμϜɾάϥϑϞσϧͷಋग़ 2ͭͷ৚݅Λຬ্ͨͨ͠Ͱ׬શʹϥϯμϜʹάϥϑΛੜ੒͢Δ • ֬཰෼෍ͱͯ͠ن֨Խ͞Ε͍ͯΔ • ॴ๬ͷߏ଄Λอͭ ∑ A P(A|θ) = 1 ⟨c r ⟩ = ∑ A c r (A)P(A|θ) = c* r ׬શʹϥϯμϜ 㱺 Τϯτϩϐʔͷ࠷େԽ S = − ∑ A P(A)log P(A) ϥάϥϯδϡະఆ৐਺ ∂ ∂P(A) S + α (∑ A P(A) − 1 ) + ∑ r θ r (∑ A c r (A)P(A) − c* r ) = 0 P(A|θ) = 1 Z(θ) exp (∑ r θ r c r (A) ) 26

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ฏߧ౷ܭྗֶͱͷ૬ࣅ • Park & Newman, “Statistical mechanics of networks”, Physical Review E (2004) • [Review] Cimini et al., “The Statistical Physics of Real-World Networks”,   Preprint arXiv:1810.05095 ࢦ਺ϥϯμϜɾάϥϑ ฏߧ౷ܭྗֶ ϘϧπϚϯ෼෍ ੜ੒͞ΕΔάϥϑू߹ ΧϊχΧϧΞϯαϯϒϧ ϋϛϧτχΞϯ ෼഑ؔ਺ ࣗ༝ΤωϧΪʔ P(A|θ) Z(θ) F ≡ ln Z [c r] = 1 Z ∂Z ∂θr = ∂F ∂θr • ɹ͕ߏ଄ͷ৘ใΛ΋ͭ • ౷ܭྗֶͰ։ൃ͞Εͨܭࢉख๏Λԉ༻Ͱ͖Δ Z H(θ) ≡ ∑ r θ r c r (A) 27

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ྫ̍ʛErdős-Renyí ϥϯμϜɾάϥϑ Park & Newman, “Statistical mechanics of networks”, Physical Review E (2004) ฏۉ࣍਺ɹ ͚ͩΛอͭ → ϊʔυ਺Ұఆͱ͢ΔͱΤοδ਺͕Ұఆ ⟨k⟩ M(A) = ∑ i

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ྫ̎ʛҰൠԽϥϯμϜɾάϥϑ Park & Newman, “Statistical mechanics of networks”, Physical Review E (2004) ֤ϊʔυͷ࣍਺ɹ ͕อͨΕΔ k i H(θ) = ∑ i θ i k i (A) = ∑ i θ i ∑ j A ij = ∑ i

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ྫ̎ʛҰൠԽϥϯμϜɾάϥϑʢ̎ʣ Park & Newman, “Statistical mechanics of networks”, Physical Review E (2004) p ij ≪ 1 p ij = 1 1 + e−Θij εύʔεੑΛԾఆ͢Δɿ e−Θ ij ≫ 1 p ij ≃ 1 e−Θij = eΘ ij = eθ i +θ j 2⟨M⟩ = ∑ i,j p ij ≃ ∑ i eθ i ⋅ ∑ j eθ j ⟨k i ⟩ = ∑ j p ij ≃ eθ i ⋅ ∑ j eθ j eθ i = ⟨k i ⟩ 2⟨M⟩ p ij ≃ ⟨k i ⟩⟨k j ⟩ 2⟨M⟩ 㱺 ϊʔυϖΞɹɹ ͷྡ઀֬཰ (i, j) ฏۉ࣍਺ Τοδ਺ 30

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ύϥϝʔλВͷਪఆ جຊతʹ͸໬౓Λ࠷େԽ͢ΔύϥϝʔλΛಘ͍ͨ ℒ(θ) = ln P(A*|θ) = ∑ r θ r c r (A*) − ln Z(θ) ͷܭࢉ͸Ұൠʹ͸೉͍͠ˠ MCMC Λ༻͍ͯۙࣅ͢Δ Z(θ) Snijders, “Markov Chain Monte Carlo Estimation of Exponential Random Graph Models,” Journal of Social Structure (2002) ෼෍ظ଴஋Λ MCMC αϯϓϧฏۉʹΑΓۙࣅ͢Δ ℒ(θ) − ℒ(θ 0 ) = θ0 [ exp ( ∑ r (θ r − θ0 r )c r (A) )] ≃ 1 m ∑m i=1 exp ( ∑ r (θ r − θ0 r )c r (A i ) ) ͦͷޙ΋ܭࢉख๏ͷൃల͸༷ʑʹ͋Δ 31

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Breakʛࢦ਺ϥϯμϜɾάϥϑΛѻ͏ιϑτ΢ΣΞ • R ݴޠͷ statnet ύοέʔδ • https://cran.r-project.org/web/packages/statnet/index.html • MCMC Λ༻͍ͨαϯϓϦϯά΍ύϥϝʔλਪఆ΋͋Γ https://statnet.org/ 32

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Breakʛ2ͭͷαςϥΠτ@ NetSci 2018 ౷ܭਪ࿦ʢStatistical Inferenceʣ ੜ੒෼෍ͷύϥϝʔλΛਪఆͯ͠ɺͦͷҙຯΛղऍ͢Δ ػցֶशʢMachine Learningʣ λεΫͷਫ਼౓͕ߴ͍ύϥϝʔλΛσʔλʹج͍ͮͯਪఆ͢Δ ॏͳΔ෦෼΋ଟ͍ ʢਓɺςʔϚʣ 33

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؍ଌ͞ΕΔม਺ͷґଘੑΛऔΓೖΕΔ Holland & Leinhardt, “An exponential family of probability distributions for directed graphs (with discussion)” , Journal of the American Statistical Association (1981) ԾఆɿϖΞɹɹ ͷྡ઀֬཰͕ҎԼͷΈʹґଘ͢Δ • ݸਓ͝ͱͷʮࣾަੑʯ • ݸਓؒͷ෺ཧతڑ཭ (i, j) η i , η j d ij η i η j d ij ln p ij 1 − pij = α + β(η i + η j ) − γd ij P(A|θ) = 1 Z exp ( ∑ i

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֬཰తϒϩοΫϞσϧ ϊʔυͷάϧʔϓॴଐ • ௚઀ʹ͸؍ଌͰ͖ͳ͍જࡏม਺ • ྡ઀֬཰ΛܾΊΔʢe.g., ಉ͡άϧʔϓ͸ͭͳ͕Γ΍͍͢ʣ • Holland et al., “Stochastic blockmodels: First steps”, Social Networks (1981) • [Review] Peixoto, “Bayesian stochastic blockmodeling”, a chapter in“Advances in Network Clustering and Blockmodeling” (2018). Preprint arXiv:1705.10225 p 1,1 p 1,2 p 1,3 p 2,1 p 1,2 p 2,3 p 3,1 p 3,2 p 3,3 ϒϩοΫԽͨ͠ྡ઀ߦྻ P(A|p, b) = ∏ i

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֬཰తϒϩοΫϞσϧʛϕΠζԽ [Review] Peixoto, “Bayesian stochastic blockmodeling”, a chapter in“Advances in Network Clustering and Blockmodeling” (2018). Preprint arXiv:1705.10225 ͦ΋ͦ΋ϒϩοΫ਺ ΋ະ஌ɺͲ͏ܾΊΔʁ B • ɹ΋෼෍͍ͯ͠Δͱߟ͑ΔʢϕΠζԽʣ • Ϟσϧͷෳࡶ͞ͱ౰ͯ͸·Γͷྑ͞ͷόϥϯεΛݟͯ  ద੾ͳɹ Λબ୒͢Δʢ࠷খهड़௕ݪཧɿ࣍ϖʔδʣ B B P(b) = P(b|n) P(n|B) 1 N ( N − 1 B − 1) −1 ∏ r n r ! N! × × B n b ϒϩοΫ਺ άϧʔϓ  αΠζ άϧʔϓ  ॴଐ P(b|A*) ∝ P(A*|b)P(b) ϕΠζߋ৽ 37

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֬཰తϒϩοΫϞσϧʛ࠷খهड़௕ Ϟσϧͷهड़௕ ≡ ಉ࣌֬཰ͷϏοτ௕ Σ ≡ − log 2 P(A*, p, b) = − log 2 P(A*|p, b) + (−log 2 P(p, b)) ύϥϝʔλΛॴ༩ͱͯ͠ ؍ଌΛهड़͢Δͷʹඞཁͳ৘ใ ϞσϧύϥϝʔλΛ  هड़͢Δͷʹඞཁͳ৘ใ [Review] Peixoto, “Bayesian stochastic blockmodeling”, a chapter in“Advances in Network Clustering and Blockmodeling” (2018). Preprint arXiv:1705.10225 ϞσϧΛ  ෳࡶʹ͢Δͱɿ B ϒϩοΫ਺ هड़௕ Σ ̂ B 38

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࣍਺ิਖ਼͖ͭ֬཰తϒϩοΫϞσϧ Karrer & Newman, “Stochastic blockmodels and community structure in networks”, Physical Review E (2011) DOI:10.1103/PhysRevE.83.016107 γϯϓϧͳ֬཰తϒϩοΫϞσϧͷऑ఺̍ • ϒϩοΫ಺ͷ࣍਺෼෍ = ඞͣೋ߲෼෍ ղܾࡦɿ  ֤ϊʔυͷ࣍਺Λσʔλʹ߹ΘͤΔ P(A|θ, p, b) = ∏ i

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ೖΕࢠܕ֬཰తϒϩοΫϞσϧ γϯϓϧͳ֬཰తϒϩοΫϞσϧͷऑ఺̎ Peixoto, “Hierarchical Block Structures and High-Resolution Model Selection in Large Networks”, Physical Review X (2014) DOI: 10.1103/PhysRevX.4.011047 • ࠷খهड़௕ݪཧͰݟ͚ͭΒΕΔ࠷খάϧʔϓαΠζ • ίϛϡχςΟݕग़ͷղ૾౓ݶք O ( N ) ղܾࡦɿ  େ͖͍ϒϩοΫ͔Βϊʔυ୯Ґ·Ͱ  ೖΕࢠߏ଄ͷϒϩοΫϞσϧΛߟ͑Δ ݕग़Ͱ͖Δ࠷খάϧʔϓαΠζ O (ln N) Image from Fig 1 40

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graph-tool • https://graph-tool.skewed.de/ • Tiago P. Peixoto ࢯʢUniversity of Bath / ISI Foundationʣ͕։ൃ • ΠϯλʔϑΣΠε͸ Python, ؔ਺࣮૷͸ C++ https://graph-tool.skewed.de/static/ doc/ inference.html#graph_tool.inference .minimize.minimize_blockmodel_dl 41

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3. ࢀরج४ϞσϧͷԠ༻ static ͷ৔߹ 43

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Ԡ༻̍ʛϞδϡϥϦςΟ࠷େԽ = ͋ΔϞσϧͷ࠷໬ਪఆ Newman, “Equivalence between modularity optimization and maximum likelihood methods for community detection”, Physical Review E (2016) ղ૾౓ύϥϝʔλ͖ͭϞδϡϥϦςΟ Q(γ) = 1 2M ∑ i,j ( A ij − γ k i k j 2M ) δ bi ,bj Planted partition Ϟσϧͷର਺໬౓ p in p out ln P(A|p, b) = C 1 2M ∑ i,j ( A ij − p in − p out ln pin − ln pout k i k j 2M ) δ bi ,bj + C 2 1. ϒϩοΫ਺ͱ γ ͕ॴ༩ → 2. ʮϞδϡϥϦςΟ࠷େԽ͸ϒϩοΫ਺΋ࣗಈతʹܾ·Δʯ = ͨ·ͨ·ɺͱ͋ΔϒϩοΫ਺Ͱ໬౓࠷େͱͳΔ 3. ֬཰తϒϩοΫϞσϧͷ࠷໬ਪఆ͕࢖͑ΔέʔεͰ  ϞδϡϥϦςΟ࠷େԽΛબ୒͢Δ߹ཧతͳཧ༝͸ͳ͍ argmax b Q(γ) = argmax b ln P(A|p, b) 44

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Ԡ༻̎ʛϝλσʔλ = ਅͷίϛϡχςΟʁ Peel, Larremore & Clauset, “The ground truth about metadata and community detection in networks”, Science Advances (2017) DOI: 10.1126/sciadv.1602548 A B ίϛϡχςΟਪఆ݁Ռɿ A ΑΓ΋ B ͷ΄͏͕ʮྑ͍ʯʁ Images adapted from Fig 1 ϝλσʔλɿਓͷଐੑɺ؍࡯ͷ݁ՌɺͳͲ 45

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Ԡ༻̎ʛϝλσʔλ = ਅͷίϛϡχςΟʁʢ̎ʣ Peel, Larremore & Clauset, “The ground truth about metadata and community detection in networks”, Science Advances (2017) DOI: 10.1126/sciadv.1602548 T G M C ਅͷίϛϡχςΟ f g ؍ଌ͞Εͨάϥϑ ਪఆ݁Ռ ϝλσʔλ ಉҰʁ 1. ೚ҙͷɹɹɹ͢΂ͯʹରͯ͠࠷దͳਪఆΛ͢Δ  ख๏ɹ ͸ଘࡏ͠ͳ͍ 2. ϝλσʔλͱਅͷίϛϡχςΟΛಉҰࢹ͢Δ 㱺  ʮϝλσʔλ͕ίϛϡχςΟʹແؔ܎Ͱ͋Δʯͱ  ʮਪఆख๏͕͏·͘ػೳ͠ͳ͍ʯΛ۠ผͰ͖ͳ͍ (G, T) f 46

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Ԡ༻̎ʛϝλσʔλ = ਅͷίϛϡχςΟʁʢ̏ʣ Peel, Larremore & Clauset, “The ground truth about metadata and community detection in networks”, Science Advances (2017) DOI: 10.1126/sciadv.1602548 ίϛϡχςΟਪఆ݁Ռͱϝλσʔλ͸ผʑͷଆ໘ΛͱΒ͍͑ͯΔʁ neoSBM Ϟσϧ ௨ৗͷ֬཰తϒϩοΫϞσϧ  + Ұ෦ͷϊʔυ͸ϝλσʔλ = ॴଐϒϩοΫ ྫɿ๏཯Ոͷަ༑ؔ܎ • Ofﬁce ͱ Law school ͷϝλ σʔλʹدͤͨਪఆ (1, 3) • ωοτϫʔΫߏ଄͚͔ͩΒ ͷਪఆ (2) Images adapted from Fig 4 47

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3. ࢀরج४ϞσϧͷԠ༻ temporal ͷ৔߹ 48

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ςϯϙϥϧɾωοτϫʔΫσʔλʹର͢Δ  ͭͳ͗ସ͑ํ๏ͷ໢ཏతͳΧλϩά Gauvin, Génois, Karsai, Kivelä, Takaguchi, Valdano & Vestergaard,   “Randomized reference models for temporal networks”, Preprint arXiv:1806.04032 • Τοδߏ଄ͷͭͳ͗ସ͑   × ࣌ؒํ޲ͷϥϯμϜԽ • طଘݚڀͰ༻͍ΒΕͨํ๏Λ  ମܥతʹ੔ཧ • ͭͳ͗ସ͑ʹର͢Δಛ௃Ͳ͏͠ ͷґଘؔ܎ΛՄࢹԽ • ͭͳ͗ସ͑ૢ࡞ͷ͍͔ͭ͘ͷ  ਺ֶతੑ࣭Λఆཧͱͯ͠ূ໌ Python ؔ਺ͷΈʹΑΔ࣮૷ https://github.com/mgenois/RandTempNet Python & C++ ॲཧͷߴ଎൛ https://github.com/bolozna/Events 49

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ࢀরج४ϞσϧΛؼແԾઆͱͯ͠  ۜߦؒͷ͝ͻ͍͖औҾؔ܎Λநग़͢Δ Kobayashi & Takaguchi, “Identifying relationship lending in the interbank market: A network approach”, Journal of Banking and Finance (2018) DOI: 10.1016/j.jbankﬁn.2018.09.018 • ೔࣍ͷۜߦؒऔҾωοτϫʔΫ • ͝ͻ͍͖औҾؔ܎ = औҾ͕ࢀর ج४ϞσϧͰظ଴͞ΕΔΑΓ΋ ༗ҙʹଟ͍ϖΞ • 2008೥ۚ༥ةػͷ௚ޙ͸  ͝ͻ͍͖औҾؔ܎͕૿Ճ͢Δ • நग़͞Εͨ͝ͻ͍͖औҾ͸ɺܧ ଓ೔਺ɾརଉɾ1೔ͷதͰͷऔҾ ࣌ؒଳ͕௨ৗͷऔҾͱ͸ҟͳΔ 50

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ࢀরج४Ϟσϧͷݶք 1. ωοτϫʔΫͷߏ੒ϝΧχζϜ͸Θ͔Βͳ͍ 2. ܭࢉίετ • ϥϯμϜͭͳ͗ସ͑ɺ੒௕ + ༏ઌతબ୒ɺΤοδίϐʔ triadic closure ͳͲ • ࢀরج४ϞσϧͱͷରൺΛ༻͍ͯ஫໨͢΂͖ߏ଄Λ͋ͿΓ ग़͢ + ͦͷߏ଄Λ࠶ݱ͢ΔϝΧχζϜϞσϧΛߟ͑Δ • ҰൠతʹαϯϓϦϯάϕʔεʢϒʔτετϥοϓɺMCMCʣ • ߏ଄ࢦඪͷ৴པ۠ؒΛݟੵ΋Δํ๏͕͋Ε͹๬·͍͠ 51

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શମͷ·ͱΊʛtakeaways 1. Τοδͷͭͳ͗ସ͑ 2. ߏ੒తͳϞσϧ ɾ ɾ ɾ ฏۉ࣍਺ P(k) ࣍਺෼෍ P(k, k′) ࣍਺૬ؔ άϧʔϓߏ଄ ⟨k⟩ dk-γϦʔζ d = 0 ϥϯμϜͭͳ͗ସ͑ d = 1 Maslov-Sneppen d = 2 ࢦ਺ϥϯμϜɾάϥϑ Erdős-Renyí Ϟσϧ ҰൠԽϥϯμϜɾάϥϑ ֬཰తϒϩοΫϞσϧ • statnet https://cran.r-project.org/web/packages/statnet/index.html • graph-tool https://graph-tool.skewed.de/ 3. ࢀরج४ϞσϧͷԠ༻ σʔλɾख๏ʹର͢Δ҉໧ͷԾఆΛ໌֬Խ͢Δ 52

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࿦จڞஶऀͷօ͞Μ T. Kobayashi (Kobe) L. Gauvin (ISI) M. Karsai (ENS Lyon) E. Valdano (UCLA) M. Kivelä (Aalto) M. Génois (Aix-Marseille) C. L. Vestergaard (Institut Pasteur) 53

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ଞ෼໺ͱͷ઀఺ʁ ߏ੒తͳ ࢀরج४Ϟσϧ ࣾձωοτϫʔΫ෼ੳ ౷ܭతػցֶश ਺ཧ౷ܭ ੴࠇɼྛɼʮؔ܎σʔλֶशʯ ߨஊࣾʢ2016ʣ Robins et al., “An introduction to exponential random graph (p*) models for social networks”, Social Networks (2007) Wang & Bickel, “Likelihood-based model selection stochastic block models”, The Annals of Statistics (2017) 54