Nature Communications (2015) • Mahadevan et al., “Systematic Topology Analysis and Generation Using Degree Correlations”, ACM SIGCOMM ’06 dk-ϥϯμϜɾάϥϑ ϊʔυ d ݸ͔ΒͳΔ࿈݁ͳ෦άϥϑͷ͕ ʮ࣍ͷใࠐΈͰʯҰఆͰɺଞϥϯμϜ d อͨΕΔߏ 0 ฏۉ࣍ 1 ϊʔυͷ࣍ 2 ϖΞͷ࣍ 3 ̏ମͷ࣍ k k k′ k′′ k k′ k′′ k k′ 18
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-ϥϯμϜɾάϥϑͷੜΛ࣮ݱ͢Δํ๏ΒΕ͍ͯͳ͍ Image from Figure 1 19
แׅతͳΈʁ 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
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 ॴͷߏతಛΛอͭΑ͏ʹάϥϑͷੜ֬ΛॏΈ͚͢Δ : ن֨Խఆ : ωοτϫʔΫͷߏతͳಛྔ A Z(θ) c r (A) P(A|θ) = 1 Z(θ) exp (∑ r θ r c r (A) ) ྫ: ↓ͷϞνʔϑͷݸ 25
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
Review E (2004) • [Review] Cimini et al., “The Statistical Physics of Real-World Networks”, Preprint arXiv:1810.05095 ࢦϥϯμϜɾάϥϑ ฏߧ౷ܭྗֶ ϘϧπϚϯ ੜ͞ΕΔάϥϑू߹ ΧϊχΧϧΞϯαϯϒϧ ϋϛϧτχΞϯ ؔ ࣗ༝ΤωϧΪʔ P(A|θ) Z(θ) F ≡ ln Z [c r] = 1 Z ∂Z ∂θr = ∂F ∂θr • ɹ͕ߏͷใΛͭ • ౷ܭྗֶͰ։ൃ͞Εͨܭࢉख๏Λԉ༻Ͱ͖Δ Z H(θ) ≡ ∑ r θ r c r (A) 27
E (2004) ֤ϊʔυͷ࣍ɹ ͕อͨΕΔ k i H(θ) = ∑ i θ i k i (A) = ∑ i θ i ∑ j A ij = ∑ i<j (θ i + θ i) A ij Z(θ) = ∑ A exp H(θ) = Π i<j ( 1 + eθ i +θ j ) p ij = ⟨A ij ⟩ = ∂ ln Z ∂Θij = 1 1 + e−Θij 1. ϋϛϧτχΞϯΛܭࢉ͢Δ 2. ن֨ԽఆΛܭࢉ͢Δ 3. ϊʔυϖΞɹɹ ͷྡ֬Λܭࢉ͢Δ (i, j) ≡ Θ ij P(A ij = 1) = k i k j 2M ͜ΕҰൠԽϥϯμϜɾάϥϑʁ 29
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 ͷྡ֬ (i, j) ฏۉ࣍ Τοδ 30
r c r (A*) − ln Z(θ) ͷܭࢉҰൠʹ͍͠ˠ MCMC Λ༻͍ͯۙࣅ͢Δ Z(θ) Snijders, “Markov Chain Monte Carlo Estimation of Exponential Random Graph Models,” Journal of Social Structure (2002) ظΛ 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 ) ) ͦͷޙܭࢉख๏ͷൃల༷ʑʹ͋Δ 31
for directed graphs (with discussion)” , Journal of the American Statistical Association (1981) ԾఆɿϖΞɹɹ ͷྡ͕֬ҎԼͷΈʹґଘ͢Δ • ݸਓ͝ͱͷʮࣾަੑʯ • ݸਓؒͷཧతڑ (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<j (α + β(η i + η j ) − γd ij )A ij) θ ij ྡ֬ͷϩδεςΟοΫճؼ ֤ϖΞ͕ಠཱͩͱԾఆͯ͠άϥϑશମͰ·ͱΊΔͱ 34
for directed graphs (with discussion)” , Journal of the American Statistical Association (1981) ԾఆɿϖΞɹɹ ͷྡ͕֬ҎԼͷΈʹґଘ͢Δ • ݸਓ͝ͱͷʮࣾަੑʯ • ݸਓؒͷཧతڑ (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<j (α + β(η i + η j ) − γd ij )A ij) θ ij ྡ֬ͷϩδεςΟοΫճؼ ֤ϖΞ͕ಠཱͩͱԾఆͯ͠άϥϑશମͰ·ͱΊΔͱ જࡏมʹґଘ͢Δ߹ϞσϧԽͰ͖Δʁ 35
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 ϒϩοΫԽͨ͠ྡߦྻ P(A|p, b) = ∏ i<j pA ij bi ,bj ( 1 − p bi ,bj ) 1−A ij 1 ≤ b i ≤ B ϒϩοΫ ͕ॴ༩ͷͱͰ B ≃ C ⋅ exp ( ∑ i<j ln p bi ,bj ⋅ A ij) άϧʔϓॴଐͷਪఆ = ؍ଌɹ ͔Βɹɹ Λਪఆ͢Δ A p, b 36
Network Clustering and Blockmodeling” (2018). Preprint arXiv:1705.10225 ͦͦϒϩοΫ ະɺͲ͏ܾΊΔʁ B • ɹ͍ͯ͠Δͱߟ͑ΔʢϕΠζԽʣ • Ϟσϧͷෳࡶ͞ͱͯ·Γͷྑ͞ͷόϥϯεΛݟͯ దͳɹ Λબ͢Δʢ࠷খهड़ݪཧɿ࣍ϖʔδʣ 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
networks”, Physical Review E (2011) DOI:10.1103/PhysRevE.83.016107 γϯϓϧͳ֬తϒϩοΫϞσϧͷऑ̍ • ϒϩοΫͷ࣍ = ඞͣೋ߲ ղܾࡦɿ ֤ϊʔυͷ࣍Λσʔλʹ߹ΘͤΔ P(A|θ, p, b) = ∏ i<j ( θ i θ j p bi,bj) Aij Aij ! exp ( −θ i θ j p bi ,bj ) × ∏ i ( 1 2 θ2 i p bi,bi) Aii/2 Aij ! exp ( − 1 2 θ2 i p bi ,bi ) ϊʔυͷ࣍Λௐ͢Δύϥϝʔλ Image from Fig 1 of the arXiv preprint 39
in Large Networks”, Physical Review X (2014) DOI: 10.1103/PhysRevX.4.011047 • ࠷খهड़ݪཧͰݟ͚ͭΒΕΔ࠷খάϧʔϓαΠζ • ίϛϡχςΟݕग़ͷղ૾ݶք O ( N ) ղܾࡦɿ େ͖͍ϒϩοΫ͔Βϊʔυ୯Ґ·Ͱ ೖΕࢠߏͷϒϩοΫϞσϧΛߟ͑Δ ݕग़Ͱ͖Δ࠷খάϧʔϓαΠζ O (ln N) Image from Fig 1 40
likelihood methods for community detection”, Physical Review E (2016) ղ૾ύϥϝʔλ͖ͭϞδϡϥϦςΟ 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. ϒϩοΫͱ γ ͕ॴ༩ → 2. ʮϞδϡϥϦςΟ࠷େԽϒϩοΫࣗಈతʹܾ·Δʯ = ͨ·ͨ·ɺͱ͋ΔϒϩοΫͰ࠷େͱͳΔ 3. ֬తϒϩοΫϞσϧͷ࠷ਪఆ͕͑ΔέʔεͰ ϞδϡϥϦςΟ࠷େԽΛબ͢Δ߹ཧతͳཧ༝ͳ͍ argmax b Q(γ) = argmax b ln P(A|p, b) 44
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
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. ҙͷɹɹɹͯ͢ʹରͯ͠࠷దͳਪఆΛ͢Δ ख๏ɹ ଘࡏ͠ͳ͍ 2. ϝλσʔλͱਅͷίϛϡχςΟΛಉҰࢹ͢Δ 㱺 ʮϝλσʔλ͕ίϛϡχςΟʹແؔͰ͋Δʯͱ ʮਪఆख๏͕͏·͘ػೳ͠ͳ͍ʯΛ۠ผͰ͖ͳ͍ (G, T) f 46
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