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A Degeneracy Framework for Graph Similarity: グラフ類似度のための縮退フレームワーク

C1d3178df919e19ab4ef5dba1aeaf425?s=47 OpenJNY
November 04, 2018

A Degeneracy Framework for Graph Similarity: グラフ類似度のための縮退フレームワーク

C1d3178df919e19ab4ef5dba1aeaf425?s=128

OpenJNY

November 04, 2018
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  1. "%FHFOFSBDZ'SBNFXPSLGPS (SBQI4JNJMBSJUZ ౦ژ޻ۀେֶҪ্ݚ. ࢁޱॱ໵ .  άϥϑྨࣅ౓ͷͨΊͷॖୀϑϨʔϜϫʔΫ

  2. ࿦จʹ͍ͭͯ

  3. "CPVU1BQFS ‣ ஶऀ৘ใ w ΤίʔϧɾϙϦςΫχʔΫʢ¬DPMFQPMZUFDIOJRVFʣͱ Ξςωେֶͷڞಉݚڀ ‣ *+*$"*Ͱ࠾୒ ‣ બΜͩཧ༝

    w άϥϑΧʔωϧʹ
 ڵຯ͕͋ͬͨ
  4. Χʔωϧͱ͸ͳΜͧ΍ ‣ Χʔωϧؔ਺ʢLFSOFMGVODUJPOʣ͸σʔλ఺ಉ࢜ͷྨࣅ౓ΛଌΔؔ਺ w ڭࢣ͋ΓֶशͷҝͷػցֶशΞϧΰϦζϜʹ͸ɺڭࢣσʔλͱͷۙ͞ͷ৘ใ͚ͩΛཔΓ ʹֶशɾ༧ଌΛߦ͏΋ͷ΋ʢFHαϙʔτϕΫτϧϚγϯʣ w ਓؒ΋ಉ༷ɿະ஌ͳ΋ͷʹରͯ͠ɺྨࣅ౓͕ߴ͍ط஌৘ใͰਪ࿦ ‣ ਖ਼֬ʹ͸ɺΧʔωϧؔ਺ɹɹɹɹɹɹɹɹɹ͸ɺ࣍ͷ৚݅Λຬͨؔ͢਺

    w ରশੑɿ w ਖ਼ఆ஋ੑɿ k : × → ℝ+ ∀x, y ∈ : k(x, y) = k(y, x) ∀n ∈ ℕ, x1 , …, xn ∈ : (Gij ) ≜ (k(xi , xj )) ∈ ℝn×n (͸άϥϜߦྻʢ(SBNNBUSJY (SBNJBOʣͱݺ͹ΕΔ ͕൒ਖ਼ఆ஋ߦྻ ͞Βʹݫີʹ͸ɺ͜Ε͸ʮਖ਼ఆ஋Χʔωϧʯ΍ʮϚʔαʔΧʔ ωϧʯͱݺ͹ΕΔಛघͳΧʔωϧؔ਺Ͱ͋Δ͕ɺඇৗʹศརͳ ͷͰҰൠతͳఆٛͱͳ͍ͬͯΔ
  5. ‣ άϥϑΧʔωϧ͸άϥϑͷϖΞΛೖྗͱ͢ΔΧʔωϧؔ਺ w ͭ·ΓɺάϥϑΧʔωϧͰάϥϑಉ࢜ͷྨࣅ౓Λܭࢉ͢Δ͜ͱ͕Ͱ͖Δ ‣ ͳͥάϥϑΧʔωϧ͕ॏཁͳͷ͔ʁ w ੈͷதͷσʔλͷଟ͘͸ɺہॴతʹ΋େҬతʹ΋ԿΒ͔ͷߏ଄Λ͍࣋ͬͯΔ͜ͱ͕ଟ͘ɺ άϥϑ͸ϩεϨεͳσʔλදݱͷྑ͍ۙࣅ w

    w w w w άϥϑΧʔωϧ͸άϥϑΛ௚઀ೖྗͱͯ͠ѻ͑ΔΞϧΰϦζϜͷઃܭʹ໾ཱͭ άϥϑΧʔωϧͱ͸ʁ k( , ) = 100
  6. άϥϑΧʔωϧͷԠ༻ྫ https://art.ist.hokudai.ac.jp/~takigawa/data/fpai94_takigawa.pdf

  7. άϥϑΧʔωϧ͕΍͍ͬͯΔ͜ͱ k( , ) = ⟨ϕ( ), ϕ( )⟩ℋ =

    100 ࠶ੜ֩ώϧϕϧτۭؒ 3,)4 σʔλۭؒʢू߹ʣ ℋ = (ℝd, ⟨ ⋅ , ⋅ ⟩ℋ ) ϕ : → ℋ ϕ( ) ϕ( ) ໌ࣔతʹಛ௃ྔΛੜ੒ʢJFࣸ૾ПΛఆٛʣͯ͠΋ྑ͍͕ɺΧʔωϧؔ਺Λఆٛ͢Δ͜ͱͰɺରԠ͢Δ 3,)4ٴͼП͕ʢඇ໌ࣔతʹʣҰҙʹܾఆ͞ΕΔ͜ͱ͕஌ΒΕ͍ͯΔʢΧʔωϧτϦοΫʣɻ ಛ௃ϕΫτϧ΁ͷม׵ʢҰൠʹ͸ඇઢܗࣸ૾ʣ Ұൠʹ͸࣍ݩ਺E͸ແݶେ ಺ ੵ ػցֶशք۾Ͱ͸ಛ௃ۭؒʢGFBUVSFTQBDFʣͱݺ͹ΕΔ΍ͭ
  8. Χʔωϧؔ਺ͷ࢖ΘΕ͔ͨ ‣ Χʔωϧؔ਺͕ྗΛൃش͢Δͷ͸ɿ w ಛ௃ϕΫτϧʢ΁ࣹӨ͢Δؔ਺ʣͷઃܭ͕೉͍͠ͱ͖ w ֶशΞϧΰϦζϜͰඞཁͳܭࢉ͕ɺಛ௃ۭؒ಺Ͱͷσʔλಉ࢜ͷ಺ੵʢJFΧʔωϧؔ਺ͷग़ྗ஋ʣ ͷΈʹґଘ͢Δͱ͖ ‣ ·ͨɺΧʔωϧؔ਺Λ࢖͏ͱઢܗͳֶशΞϧΰϦζϜΛඇઢܗԽͰ͖Δʂ

    w ໨తؔ਺Λࣜมܗͨ͠Γ࠷దԽ໰୊ͷ૒ର໰୊Λղ͘͜ͱͰɺಛ௃ϕΫτϧ͕಺ੵͷܗͰ͔͠ݱΕ ͳ͍ࣜͷΈͷΞϧΰϦζϜΛߏ੒͢Δ w ʲྫʳΧʔωϧԽLNFEPJET๏ɺΧʔωϧओ੒෼෼ੳʢ,FSOFM1$"ʣɺαϙʔτϕΫτϧϚγϯ ʢ47.ʣɺΧʔωϧԽϦοδճؼɺಈܘجఈؔ਺ωοτϫʔΫʢ3#'/FUXPSLʣɺFUD ̂ f(x) = ̂ w⊤ϕ(x) = ( N ∑ i=1 ̂ αi ϕ(xi ) ) ⊤ ϕ(x) = N ∑ i=1 ̂ αi k (x, xi) ಛʹάϥϑΧʔωϧ͸͜͜Ͱॏཁ
  9. ຊ୊ʹ໭Δ

  10. ͜ͷ࿦จͰఏҊ͢Δ΋ͷ ‣ άϥϑͷ֊૚ߏ଄Λ໌ࣔతʹར༻͢Δ৽ͨͳάϥϑΧʔωϧΛఏҊ w ֊૚ߏ଄Λௐ΂ΔͷʹL$PSFͱݺ͹ΕΔ֓೦Λ׆༻ w ఏҊ͢Δख๏ʢJFL$PSFϑϨʔϜϫʔΫʣ͸ɺطଘͷάϥϑΧʔωϧʹదԠՄೳͰ͋ Γɺ͞ΒʹҰൠͷάϥϑϚονϯάख๏ʹ΋దԠͰ͖Δ w ͭ·Γɺ

    ఏҊάϥϑΧʔωϧ  طଘάϥϑΧʔωϧ ʷ L$PSFϑϨʔϜϫʔΫ  ‣ ఏҊख๏Λ༻͍Δͱɺ47.Λ༻͍ͨάϥϑ෼ྨλεΫʹ͓͍ͯɺطଘ ͷάϥϑΧʔωϧΑΓฏۉBDDVSBDZ͕޲্ͨ͠
  11. άϥϑͷॖୀʢEFHFOFSBDZʣ ‣ ॖୀʢEFHFOFSBDZʣ͸άϥϑʹର͢Δੑ࣭ w ແ޲άϥϑ͕Lॖୀ LEFHFOFSBUF Ͱ͋Δͱ͸ɺ೚ҙͷ ෦෼άϥϑ͕ߴʑLͷ࣍਺ͷ௖఺ΛؚΉͱ͖Λ͍͏ ‣ LDPSF<4FJENBO>

    w άϥϑ( 7 & ͷLDPSFͱ͸ɺશͯͷ௖఺ͷ࣍਺͕L Ҏ্Ͱ͋Δ(ͷ࠷େ༠ಋ෦෼άϥϑCk = (S, E(S)) ∀v ∈ S : degree(v) ≥ k ∀(u, v) ∈ E : u, v ∈ S ⟹ (u, v) ∈ E(S) ˢʮ4 㱪7 ʹΑΔ༠ಋ෦෼άϥϑ 4 & 4 ʯͷఆٛ ˢ஫ҙ༠ಋ෦෼άϥϑʹ͓͚Δ࣍਺ ؆୯ʹݴͬͯ͠·͏ͱɺ4ʹؔ܎ͳ͍ʢ4ʹͳ͍ϊʔυΛ࢖ͬͯΔʣΤοδΛ࡟আͯ͠ಘΒΕΔ෦෼άϥϑ
  12. LDPSFͷྫ ΋ͱͷάϥϑͰ࣍਺ͷϊʔυʢC D V ʜʣ͸͋Δ͕ɺ ༠ಋ෦෼άϥϑ͚ͩͰ࣍਺Λୡ੒͢Δͷ͕ෆՄೳ DPSF͸ଘࡏ͠ͳ͍

  13. LDPSF෼ղΞϧΰϦζϜ ‣ ࣍਺ͷখ͍͞ॱʹɺશͯͷϊʔυʹ ͍ͭͯௐ΂͍ͯ͘ w ࠓߏங͍ͯ͠ΔLDPSFΑΓ΋࣍਺͕খ͚͞ Ε͹ͦΕΛLDPSF͔Β࡟আ w ͢΂ͯͷ࣍਺͕LҎ্ʹͳͬͨͷΛ֬ೝ͠ ͨΒLDPSFΛొ࿥

    ‣ ܭࢉͷΦʔμʔ͸0 / .  w /ϊʔυ਺ w .Τοδ਺ ઢܗ࣌ؒͰܭࢉՄೳͳͷͰɺޙड़ͷఏҊ ख๏Ͱ΋͜ͷܭࢉ͕ൺֱతϘτϧωοΫ ʹͳΓʹ͍͘
  14. LDPSFͷಛ௃ ‣ LDPSFͷಛ௃ɿ෦෼ू߹ੑ ‣ L͕େ͖͘ͳΔʹͭΕɺΑΓॏཁͳ৘ ใΛؚΜͰ͍Δͱߟ͑ΒΕΔ w ྫ͑͹ιʔγϟϧωοτϫʔΫͰ͸ɺத৺త ਓ෺Ͱߏ੒͞ΕΔίϛϡχςΟʔ͕֘౰ Cδ*(G)

    ⊆ … ⊆ C1 ⊆ C0 = G LDPSF͕ߏஙͰ͖Δ࠷େͷLάϥϑͷॖୀ਺ άϥϑͷྨࣅ౓΋LDPSF͝ͱͷྨࣅ౓ͰଌΔͷ͕ྑ͍ͷͰ͸ʁ
  15. ఏҊख๏

  16. ʲఏҊख๏ʳ$PSF7BSJBOUPG#BTF,FSOFM ‣ ϕʔεͱͳΔάϥϑΧʔωϧΛ༻ҙ͢Δ w ྫʣLϫΠεϑΝΠϥʔɾϦʔϚϯʢ8FJTGFJMFS-FINBOʣΧʔωϧ ‣ ༩͑ΒΕͨͭͷάϥϑʹରͯ͠ɺͦΕͧΕͷશLDPSFΛܭࢉ͢Δ w ྫʣ(ͷLDPSFT\$ $

    $ $^ (`ͷLDPSFT\$` $` $`^ ‣ ಉ͡ϨϕϧͷLDPSFΛೖྗͱͨ͠άϥϑΧʔωϧͷग़ྗ஋Λ଍͠߹ΘͤΔ w ྫʣL@D ( (` L $ $`  L $ $`  L $ $` L ɾ ɾ ͕άϥϑΧʔωϧ͡Όͳͯ͘΋ɺάϥϑͷϖΞ Λೖྗͱ͢Δ೚ҙͷؔ਺ʹར༻Ͱ͖Δ LDPSFϑϨʔϜϫʔΫ
  17. $PNQVUBUJPOBM$PNQMFYJUZ ‣ LDPSFϑϨʔϜϫʔΫͷܭࢉෳࡶ͞͸ w ɹɹɿάϥϑͷϖΞΛೖྗͱ͢Δؔ਺ʢFHάϥϑΧʔωϧʣͷܭࢉෳࡶ͞ w ɹɹɿೖྗάϥϑͷॖୀ਺ʢJFLDPSF͕ଘࡏ͢Δ࠷େͷLʣͷখ͍͞ํ ‣ Ұൠʹɺάϥϑͷॖୀ਺ͷ্ք͸࣍ͷͲͪΒ͔Ͱ༩͑ΒΕΔ w

    άϥϑͷ࠷େ࣍਺ w ྡ઀ߦྻͷ࠷େݻ༗஋ ‣ ɹɹ͸ϊʔυ਺ΑΓे෼ʹখ͍͞ʢɹɹɹɹʣ͜ͱ͕ଟ͍ͷͰɺLDPSF ϑϨʔϜϫʔΫʹཁ͢Δ௥Ճܭࢉ͸ൺֱత୹ͯ͘ࡁΉ c = A × δ* min A δ* min λmax λmax λmax ≪ n
  18. ࣮ݧ

  19. ࣮ݧͷηοςΟϯά ‣ σʔληοτɿ w όΠΦΠϯϑΥϚςΟΫεͱιʔγϟϧ ωοτϫʔΫ༝དྷͷσʔληοτΛར༻ ‣ ෼ྨ໰୊ɿ w 47.Λར༻ͯ͠෼ྨλεΫΛղ͘

    w ௒ύϥϝʔλ͸GPME$7Ͱܾఆ ‣ ൺֱ͢ΔάϥϑΧʔωϧɿ w ϕʔεάϥϑΧʔωϧछʷఏҊϑϨʔ ϜϫʔΫͷ༗ແछྨ όΠΦΠϯϑΥ ιʔγϟϧωοτ https://ls11-www.cs.tu-dortmund.de/staff/morris/graphkerneldatasets
  20. ݁ՌʢฏۉBDDVSBDZͱͦͷ෼ࢄʣ ଠࣈͷ਺ࣈ͸ɺUݕఆʢ༗ҙਫ ४ʣʹΑͬͯɺ$03&ϑϨʔ ϜϫʔΫͷੑೳ޲্͕༗ҙʹ ೝΊΒΕͨέʔεΛද͢ɻ

  21. ݁ՌʢฏۉBDDVSBDZͱͦͷ෼ࢄʣ ଠࣈͷ਺ࣈ͸ɺUݕఆʢ༗ҙਫ ४ʣʹΑͬͯɺ$03&ϑϨʔ ϜϫʔΫͷੑೳ޲্͕༗ҙʹ ೝΊΒΕͨέʔεΛද͢ɻ όΠΦܥͷσʔλΑΓ΋ιʔγϟϧ ωοτܥͷσʔλͷํ͕ੑೳ޲্͕ ΈΒΕͨ ԾઆʮίΞ਺͕େ͖͍LDPSFͷ ΄͏͕ॏཁʯΛࢧ࣋͢Δ݁Ռ

  22. ݁ՌʢฏۉBDDVSBDZͱͦͷ෼ࢄʣ ଠࣈͷ਺ࣈ͸ɺUݕఆʢ༗ҙਫ ४ʣʹΑͬͯɺ$03&ϑϨʔ ϜϫʔΫͷੑೳ޲্͕༗ҙʹ ೝΊΒΕͨέʔεΛද͢ɻ (3Ͱ͸ݦஶʹੑೳ޲্͕ΈΒΕ ΔҰํ 8-Ͱ͸͍·͍ͪޮՌͳ͠ 8-͸֤ϊʔυͷۙ઀Λཁ ໿͢ΔΑ͏ͳΧʔωϧͳͷ

    Ͱɺ௚ײతʹ͸LDPSFͷ ֓೦ͱ͋·Γ૬ҧແ͠
  23. None
  24. ࣮ߦ࣌ؒͷ૿େʹؔ͢Δߟ࡯ ‣ ϕʔεΧʔωϧͷ࣮ߦ࣌ؒʹର͢ΔɺLDPSF֦ு ͷ૬ର࣮ߦ࣌ؒΛࣔͨ͠ද ‣ *.%##*/"3:ͱ*.%#.6-5*Ͱ͸ඇৗʹ࣮ߦ ͕࣌ؒ஗͘ͳ͍ͬͯΔ͕ɺੑೳ޲্Λߟྀ͢Δͱ ܾͯ͠๏֎ͳ΋ͷͰ͸ͳ͍ʢͱओுʣ

  25. ιʔγϟϧωοτϫʔΫͰੑೳ޲্͕ݦஶͳཧ༝ ‣ ωοτϫʔΫͷ࣍਺෼෍͕ҟͳΔ ‣ ΂͖৐ଇʹै͏ωοτϫʔΫ͸ɺΑΓத৺ͷLDPSFʹ༗ӹͳ৘ใ͕٧ ·͍ͬͯΔͱߟ͑ΕΔ

  26. ୯ҰLDPSFΛ࢖ͬͨBDDVSBDZ ‣ ೖྗΛɺΦϦδφϧͷάϥϑͰ͸ͳ͘ɺ LDPSFͱஔ͖׵͑ͨ৔߹ͷBDDVSBDZ w LͷLDPSF͸άϥϑͦͷ΋ͷͳͷͰஔ͖׵ ͑͸͠ͳ͍ ‣ ؍ଌɿ w

    ʲ$PSF(3ʳ୯ௐతʹখ͍͞LͰੑೳ޲্ w ʲ(3ʳL ͰɺΦϦδφϧͷάϥϑΛೖྗ ͤͨ͞৔߹ΑΓ΋ੑೳ͕ྑ͘ͳ͍ͬͯΔ  ʢײ૝ʣඞཁ࠷௿ݶͷ৘ใ͕٧·͍ͬͯΔ࠷খͷ෦ ෼άϥϑ͕͜ͷ͋ͨΓͳͷͰ͸ʁ IMDB-BINARYͰͷGRͱCore GR
  27. ·ͱΊ

  28. ·ͱΊ LDPSF෼ղʹجͮ͘ϑϨʔϜϫʔΫ ‣ LDPSF࠷খ࣍਺͕LͰ͋Δ࠷େ༠ಋ෦෼άϥϑ ‣ LDPSF㱬 L  DPSFͰ͋Δ͜ͱΛར༻ͯ͠ɺάϥϑͷ֊૚ߏ଄͝ͱʹൺֱΛߦ͏ϑϨʔϜϫʔΫΛఏҊ ‣

    ຊ࿦จͰ͸άϥϑΧʔωϧʹద༻͕ͨ͠ɺ೚ҙͷάϥϑϚονϯάΞϧΰϦζϜʹద༻Ͱ͖Δ ͜ͷϑϨʔϜϫʔΫʹΑͬͯɺάϥϑ෼ྨλεΫʹ͓͍ͯطଘͷά ϥϑΧʔωϧͷੑೳΛ޲্ͤͨ͞ ‣ ιʔγϟϧωοτϫʔΫͳͲͷɺεέʔϧϑϦʔωοτϫʔΫͰ༗ޮ ‣ LDPSFʹࣅͨ֓೦ͷطଘάϥϑΧʔωϧʹରͯ͠͸ޮՌ͸͍·͍ͪ