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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

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

OpenJNY

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


    άϥϑྨࣅ౓ͷͨΊͷॖୀϑϨʔϜϫʔΫ

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  2. ࿦จʹ͍ͭͯ

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

    ڵຯ͕͋ͬͨ

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  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

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  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

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  6. άϥϑΧʔωϧͷԠ༻ྫ
    https://art.ist.hokudai.ac.jp/~takigawa/data/fpai94_takigawa.pdf

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  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΍ͭ

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  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

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  9. ຊ୊ʹ໭Δ

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  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޲্ͨ͠

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  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

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  12. LDPSFͷྫ
    ΋ͱͷάϥϑͰ࣍਺ͷϊʔυʢC D V ʜʣ͸͋Δ͕ɺ
    ༠ಋ෦෼άϥϑ͚ͩͰ࣍਺Λୡ੒͢Δͷ͕ෆՄೳ
    DPSF͸ଘࡏ͠ͳ͍

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  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

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  14. LDPSFͷಛ௃
    ‣ LDPSFͷಛ௃ɿ෦෼ू߹ੑ
    ‣ L͕େ͖͘ͳΔʹͭΕɺΑΓॏཁͳ৘
    ใΛؚΜͰ͍Δͱߟ͑ΒΕΔ
    w ྫ͑͹ιʔγϟϧωοτϫʔΫͰ͸ɺத৺త
    ਓ෺Ͱߏ੒͞ΕΔίϛϡχςΟʔ͕֘౰
    Cδ*(G)
    ⊆ … ⊆ C1
    ⊆ C0
    = G
    LDPSF͕ߏஙͰ͖Δ࠷େͷLάϥϑͷॖୀ਺
    άϥϑͷྨࣅ౓΋LDPSF͝ͱͷྨࣅ౓ͰଌΔͷ͕ྑ͍ͷͰ͸ʁ

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  15. 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

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  16. $PNQVUBUJPOBM$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

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  17. ࣮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

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  18. ݁ՌʢฏۉBDDVSBDZͱͦͷ෼ࢄʣ
    ଠࣈͷ਺ࣈ͸ɺUݕఆʢ༗ҙਫ
    ४ʣʹΑͬͯɺ$03&ϑϨʔ
    ϜϫʔΫͷੑೳ޲্͕༗ҙʹ
    ೝΊΒΕͨέʔεΛද͢ɻ

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

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  20. ݁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

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  21. ࣮ߦ࣌ؒͷ૿େʹؔ͢Δߟ࡯
    ‣ ϕʔεΧʔωϧͷ࣮ߦ࣌ؒʹର͢ΔɺLDPSF֦ு
    ͷ૬ର࣮ߦ࣌ؒΛࣔͨ͠ද
    ‣ *.%##*/"3:ͱ*.%#.6-5*Ͱ͸ඇৗʹ࣮ߦ
    ͕࣌ؒ஗͘ͳ͍ͬͯΔ͕ɺੑೳ޲্Λߟྀ͢Δͱ
    ܾͯ͠๏֎ͳ΋ͷͰ͸ͳ͍ʢͱओுʣ

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

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  23. 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

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  24. ·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͸͍·͍ͪ

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