3FQSFTFOUBUJPO-FBSOJOHGPS 4DBMF'SFF/FUXPSLT εέʔϧϑϦʔωοτϫʔΫʹର͢Δදݱֶश ৘ใཧ޻ֶӃ৘ใ޻ֶίʔε Ҫ্ݚڀࣨ.ࢁޱॱ໵ʢ.ʣ

4DBMFGSFF/FUXPSL εέʔϧϑϦʔʢTMBDFGSFFʣωοτϫʔΫͱ͸ɺϊʔυͷ࣍਺ʢJFۙ๣ϊʔυ਺ʣͷ෼෍͕ɺ΂ ͖৐ଇʢQPXFSMBXʣʹै͏ωοτϫʔΫͷ͜ͱ DGࢦ਺͕ͷ΂͖৐ଇ ྫʣҰ෦ͷਓ͸ඇৗʹͨ͘͞Μ༑ୡ͕͍ͯɺେଟ਺͸༑ୡ͕গͳ͍ ස౓ ࣍਺ ࢦ਺ؔ਺తʹݮগ

࣍਺෼෍ͷҧ͍ http://www.network-science.org/

ैདྷͷάϥϑຒΊࠐΈख๏ ࣗવݴޠॲཧͷք۾Ͱ༗໊ͳXPSEWFD<.JLPMPWFUBM>ΛɺϊʔυຒΊࠐΈʢ/PEF&NCFEEJOHʣ ʹద༻ͨ͠OPEFWFD<(SPWFSBOE-FTLPWFD>͕੒ޭΛऩΊ͍ͯΔ ୯ޠϊʔυ ηϯςϯεάϥϑ্ͷϥϯμϜ΢ΥʔΫྻ ·ͨɺϊʔυຒΊࠐΈ͸ɺଟ༷ମֶशͱݺ͹ΕΔ ඇઢܗ࣍ݩ࡟ݮख๏ͷҰ෦ͷύʔτͱͯ͠ɺੲ͔ Β੝Μʹݚڀ͞Ε͍ͯΔλεΫͷͻͱͭ ϥϓϥγΞϯݻ༗Ϛοϓ๏ʢ-BQMBDJBO&JHFONBQʣ <#FMLJOBOE/JZPHJ>͕༗໊ σʔλؒͷྨࣅ౓ΛΤοδͷॏΈͱͨ͠ྨࣅ౓ά ϥϑʢTJNJMBSJUZHSBQIʣΛߏஙͯ͠ɺ͜ͷྡ઀ߦ ྻͷݻ༗஋෼ղͰ௿࣍ݩຒΊࠐΈΛಘΔ https://www.cs.cmu.edu/~aarti/Class/10701/slides/Lecture21_2.pdf

ैདྷख๏ͷ໰୊఺ ঺հͨ͠ैདྷख๏͸ɺ఺ؒͷؔ܎ͷΑ͏ͳɺωοτϫʔ Ϋ͕࣋ͭہॴతͳߏ଄ͷ৘ใΛอ࣋͢ΔΑ͏ͳຒࠐΛߦ͏ ͔͠͠εέʔϧϑϦʔੑ͸େҬతͳ৘ใͷͻͱͭ ݱ࣮ͷωοτϫʔΫʹසൟʹݟΒΕΔ͜ͷ৘ใΛແࢹ͢Δͷ ͸೗Կͳ΋ͷ͔ʁ ࣮ࡍɺಘΒΕͨຒΊࠐΈͰ΋ͱͷάϥϑΛ࠶ߏஙͨ͠ͱ͖ɺ ैདྷख๏͸IJHIEFHSFFͳϊʔυ਺͕ଟ͘ͳΔ܏޲ʹ͋Δ ͭ·ΓɺຒΊࠐΈۭؒͰσʔλಉ͕࢜ͻ͖ͬͭ͗ͯ͢ɺશମ తʹ͙ͪΌͬͱͳͬͯ͠·͍ͬͯΔ ʮଟ͘ͷϊʔυͷۙ͘ʹډΕΔݖརΛɺগ਺ϊʔυʹ͚ͩ༩͑Δʯ੍໿͕ඞཁ

͜ͷ࿦จͰ΍Γ͍ͨຒΊࠐΈ Node Embedding Reconstruction ޡϦϯΫʢΤοδʣͷ࠷খԽ ϢʔΫϦουڑ཭͕ᮢ஋ΑΓ ΋খ͚͞Ε͹ΤοδΛߏ੒

Node Embedding Reconstruction ؾ࣋ͪɿ͍͍ײ͡ʹ֤ϊʔυΛݻఆ௕ϕΫτϧ ΁ରԠ෇͚͍ͨʢ೚ҙͷάϥϑʹରͯ͠ରԠ෇ ͚Δؔ਺Λֶश͢ΔΘ͚Ͱ͸ͳ͍ʣ

͜ͷ࿦จͰ΍Γ͍ͨຒΊࠐΈ Node Embedding Reconstruction ޡϦϯΫʢΤοδʣͷ࠷খԽ ϢʔΫϦουڑ཭͕ᮢ஋ΑΓ ΋খ͚͞Ε͹ΤοδΛߏ੒

Ξϓϩʔν

ͦ΋ͦ΋εέʔϧϑϦʔੑ͸FNCFEͰ͖Δͷ͔໰୊ ௚ײతʹ͸ɺຒΊࠐΈۭؒʹ͓͍ͯσʔλ͕ΰνϟͬͱͯ͠͠·͏͔Βɺάϥϑ࠶ߏங࣌ʹߴ࣍਺ϊʔ w w w υ͕૿͑ͯ͠·͏ͱਪଌͰ͖Δ ΰνϟͬͱ͍ͯ͠Δߴ࣍਺ϊʔυ͕ଟ͘ͷ௿࣍਺ϊʔυΛۙ͘ʹҾ͖دͤΔ͕ނʹɺҾ͖دͤΒΕͨ௿࣍਺ϊʔ υಉ࢜ʹ΋άϥϑ࠶ߏங࣌ʹΤοδ͕݁͹Εͯ͠·͏Α͏ͳঢ়ଶ Ͱ͸ԿΒ͔ͷ੍໿ΛՃ͑ͯɺΰνϟͬͱ͠ͳ͍Α͏ʹͰ͖Δͷ͔ʁ ྫ͑͹࣍ݩϢʔΫϦουۭؒʹຒΊࠐΈΛ͢ΔͳΒɺc7cͰ͢Βແཧͦ͏ ͔ͱ͍ͬͯc7cʹରͯ͠ɺ.࣍ݩͷۭؒΛ༻ҙ͢Δͷ͸΍Γա͗ʜ ͦ͜Ͱɺ,࣍ݩͷϢʔΫϦουۭؒʹຒΊࠐΜͩ࣌ʹɺ໰୊ͳ͘εέʔϧϑϦʔੑΛ࠶ߏஙͰ͖ΔΑ͏ ͳϊʔυ਺ͷ࠷େ஋ʢʹର͢ΔԼքʣʹ͍ͭͯɺཧ࿦తͳղੳΛߦͬͨ

݁࿦͚ͩड़΂Δͱ
 ໰୊ͳͦ͞͏ k 10 20 50 100 lower bound 57 3325 637M 4E+17 .@LL࣍ݩϢʔΫϦουۭؒʹ͓͍ͯɺத৺Y൒ܘЏͷ௒ٿ# Y Џ ͷ தʹɺ͓ޓ͍͕ЏҎ্཭ΕͯΔ఺Λ࠷େͰԿݸ഑ஔͰ͖Δ͔ʁ ʢ࣍ݩͳΒʣ࣍਺͕&ͷεʔύʔϋϒ ͷपΓʹɺ࣍਺͕ͷ༿ϊʔυΛ&ݸຒΊ ࠐΈͰ͖Δ େମͷݱ࣮ੈքͷωοτϫʔΫ͸ɺे෼ʹεέʔ ϧϑϦʔੑΛอ࣋ͨ͠··ຒΊࠐΈՄೳ ͪͳΈʹ ఆཧ͸ɺϊʔυ͕࠷େͰԿݸ഑ஔͰ͖Δ͔ ໰୊Λɺٿॆరʢ4QIFSF1BDLJOHʣ໰୊ɺ ͋Δ͍͸ٿମͷ࠷ີॆర໰୊ͱͯ͠஌ΒΕΔ ໰୊ʹؼண͢Δ͜ͱͰূ໌͞Ε͍ͯΔɻ

طଘख๏ͷ cons εέʔϧϑϦʔੑΛߟྀ͠ͳ͍ͱͲΜͳ໰୊͕ੜ͡Δͷ͔ʁ

ϏʔόʔͱΦόϚɺ஥ྑ͠໰୊ ैདྷख๏Ͱ͸ۙ͘ʹ ຒΊࠐΈ͞ΕΔ྆ऀ

ϏʔόʔͱΦόϚɺ஥ྑ͠໰୊΁ͷରॲࡦ ैདྷख๏ͷଟ͘͸TU OEPSEFSͷQSPYJNJUZΛอͭ͜ͱʹઐ೦ TUPSEFSQSPYJNJUZ˺ʮ༑ୡʯ౓߹͍ OEPSEFSQSPYJNJUZ˺ʮ༑ୡͷ༑ୡʯ౓߹͍ ͦͷ݁ՌɺεέʔϧϑϦʔωοτϫʔΫʹଘࡏ͢ΔlCJHIVCzಉ࢜͸ྨࣅ౓͕ߴ͘ͳΓ΍͍͢ ྫʣΦόϚͷ༑ୡʹ͸ɺδϟεςΟϯɾϏʔόʔͱ༑ୡͳਓ͕ਓͰ΋ډΔ֬཰͸ߴ͍ ྨࣅ౓͕ߴ͘ͳΓ΍͍͢ͳΒɺ࣍਺͕ߴ͍ϊʔυͷQSPYJNJUZʹϖφϧςΟΛ༩͑Ε͹ྑ͍ جຊతํ਑ཧ࿦ղੳͷ݁Ռʹج͖ͮɺεέʔϧϑϦʔੑΛ୲อͰ͖ ΔຒΊࠐΈΞϧΰϦζϜͷͨΊͷ࣍਺േଇʢEFHSFFQFOBMUZʣݪଇ ΛఏҊ͢Δɻ࣍਺േଇ͸ɺ࣍ٴͼ࣍ͷQSPYJNJUZΛ୲อ্ͨ͠Ͱɺ ߴ͍࣍਺Λ΋ͭ௖఺ಉ࢜ͷQSPYJNJUZʹରͯ͠േΛՊ͢ݪଇͰ͋Δɻ

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෇͖ྡ઀ߦྻʢXFJHIUFEBEKBDFODZNBUSJYʣͷྫ

ఏҊख๏%14QFDUSBM ͩTG 2nd-order proximity 1st & 2nd-order proximity weighted adjacency matrix ྡ઀ߦྻ ࣍਺ߦྻ%EJBH E@ ʜ E@O

ఏҊख๏%14QFDUSBM ͩTG 2nd-order proximity 1st & 2nd-order proximity weighted adjacency matrix ྡ઀ߦྻ ࣍਺ߦྻ%EJBH E@ ʜ E@O J K ؒͷQSPYJNJZ͕ߴ͍ʢ8@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՝͢

ఏҊख๏%14QFDUSBM ͩTG 2nd-order proximity 1st & 2nd-order proximity weighted adjacency matrix ྡ઀ߦྻ ࣍਺ߦྻ%EJBH E@ ʜ E@O J K ؒͷQSPYJNJZ͕ߴ͍ʢ8@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

ఏҊख๏%18BMLFS ϥϯμϜ΢ΥʔΫʹجͮ͘طଘख๏ʹ࣍਺ϖφϧςΟΛՃ͑Δ https://towardsdatascience.com/node2vec-embeddings-for-graph-data-32a866340fef ͜͜Λͪΐͬͱ͍͡Δ͚ͩ

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

࣮ݧ

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

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࠾༻

λεΫ ωοτϫʔΫͷ࠶ߏங ʲ໨తʳεέʔϧϑϦʔੑΛอ࣋ͨ͠ຒΊࠐΈ ʲํ๏ʳલड़௨Γʢলུʣ ʲධՁʳೖྗͱ࠶ߏஙωοτϫʔΫͷͦΕͧΕͷ࣍਺෼෍ΛٻΊɺϐΞιϯͷੵ཰૬ؔ܎਺ < >Λܭࢉ͢Δʢߴ͍ͱྑ͍ʣ ʲඋߟʳ֤ख๏Ͱɺ͔Β·Ͱͷൣғ ࠁΈ ͰЏΛಈ͔͠ɺ࠷ྑ࣌ͷ૬ؔ܎਺Λใࠂ ϦϯΫ༧ଌ ʲ໨తʳϊʔυWJ WKؒʹΤοδ͕͋Δ͔ͷ༧ଌ ʲํ๏ʳຒΊࠐΈۭؒͰͷࠩ෼V@JV@KΛಛ௃ྔͱͯ͠ઢܗճؼϞσϧʹೖྗ͢Δ ʲධՁʳਫ਼౓ʢQSFDJTJPOʣɺ࠶ݱ཰ʢSFDBMMʣɺ'஋ʢ'TDPSFʣ ʲඋߟʳແ࡞ҝʹબ͹ΕͨϊʔυϖΞͷू߹Λ܇࿅ධՁσʔλʢͦΕͧΕશϊʔυϖΞͷ ʣͱ͢Δ ϊʔυ෼ྨ ʲ໨తʳϊʔυʹ෇༩͞Ε͍ͯΔϥϕϧͷ༧ଌ ʲํ๏ʳ༩͑ΒΕͨW@Jʹରͯ͠ɺຒΊࠐΈV@JΛಛ௃ϕΫτϧͱͯ͠ઢܗ෼ྨػʹೖྗ ʲධՁʳਖ਼ղ཰ʢBDDVSBDZʣ ʲඋߟʳσʔληοτ$JUBUJPOͰͷΈλεΫΛ࣮ߦɻϥϕϧ͸ͭ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 )

݁ՌɿωοτϫʔΫ ࠶ߏஙλεΫ ϐΞιϯ૬ؔ܎਺ͷ%18BMLFS͸ɺ εϐΞϚϯ૬ؔ܎਺ͩͱ%14QFDUSBM ͸Ͱੑೳ޲্͕ΈΒΕͨ ࠷దͳЏͷ஋΋ɺ%14QFDUSBM͸ ෇ۙͰ҆ఆ͍ͯ͠Δ %1WBSJBOUT͸ɺεέʔϧϑϦʔੑΛ ΑΓอ࣋ͨ͠··ຒΊࠐΈͰ͖Δख๏ Ͱ͋Δͱݴ͑Δ

݁ՌɿωοτϫʔΫ࠶ߏஙλεΫ ຒΊࠐΈ࣍ݩ േଇͷڧ͞ Wij = 1 (didj)β C′ ij %14QFDUSBMͷ΄͏͕ෆ҆ఆͳͷ͸ɺЌ͕௚઀໨తؔ਺ʹ૊Έࠐ·Ε͍ͯ Δ͔Βɻ%18BMLFS͸ϥϯμϜ΢ΥʔΫྻੜ੒ʹ͔͔͔͠ΘΒͳ͍ 4ZOUIFUJDͱ'BDFCPPLͰ࠷దͳЌ͕ҟͳΔ͕ɺ͜Ε͸േଇ͕࣍਺ͷߴ͞ ͦͷ΋ͷʹ՝ͤΒΕ͍ͯΔ͔ΒͰɺτϙϩδʔ͕มΘΕ͹Ќͷޮ͖۩߹͍ ͸มԽ͢Δɻ εέʔϧϑϦʔੑΛ୲อ͢Δͷʹे෼ͳ࣍ݩ਺͕ଘࡏ͢Δ͜ͱ͕෼͔Δɻ %14QFDUSBM͸ຒΊࠐΈ࣍ݩ͕গͳ͍ͱੑೳ͕ඇৗʹѱ͍͕ɺ%18BMLFS ͸࣍ݩ਺͕૿͑ͯ΋͋·ΓมԽͤͣɺ҆ఆͯ͠ྑ͍݁ՌɻϥϯμϜ΢Υʔ Ϋͷઓུ͕ޮ͍͍ͯΔͷ͕ཁҼͰ͋ΔͱਪଌͰ͖Δɻ

݁ՌϦϯΫ༧ଌ ͍͍ͩͨͷέʔεͰ%14QFDUSBM͕࠷ ΋ྑ͍ੑೳ ࣍ݩേଇͷݪଇ͸ɺ͜ͷλεΫͷͨΊ ʹ௚઀ઃܭ͞Ε͍ͯΔΘ͚Ͱ͸ͳ͍͕ɺ ΑΓ༗ӹͳ৘ใΛଟ͘ຒΊࠐΈͰ͖ͯ ͍Δ݁Ռɺ޷੒੷ΛऩΊͨͱਪଌ͞Ε Δ

݁Ռϊʔυ෼ྨ %14QFDUSBM͸ϥϓϥγΞϯݻ༗Ϛοϓ๏Λɺ%1 8BMLFS͸%FFQ8BMLΛ౗ͨ͠ɻ %14QFDUSBM͸҆ఆͯ͠޷੒੷Λ࢒͍ͯ͠Δɻ ඪ४ภࠩ΋%14QFDUSBMͷ΄͏͕গͳ͔ͬͨ %14QFDUSBM -& %FFQ8BML ϊʔυ෼ྨλεΫͷਖ਼ղ཰ʢBDDVSBDZʣΛใࠂͨ͠ද ֤Ϛε͸σʔληοτͰͷBDDVSBDZͷฏۉ஋Λ͍ࣔͯ͠Δ

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