࣍਺૬ؔͷଌఆ ࠃࡍਓؒՊֶ෦άϩʔόϧจԽֶՊ ؠӬ༔ر

"TTPSUBUJWJUZ ࣾձֶɿྨ͸༑ΛݺͿʢIPNPQIJMZʣ ԿΒ͔ͷ؍఺Ͱྨࣅ͍ͯ͠Δϊʔυಉ͕࢜ͭͳ͕͍ͬͯΔ͜ͱ Peter Reid. “Birds of a feather fl ock together — collaboration and Svandis”. DataDrivenInvestor. 2018. https://onl.sc/AcC7nyq, ʢࢀর 2023-01-22ʣ w ଐੑʹجͮ͘BTTPSUBUJWJUZ w ࣍਺ʹجͮ͘BTTPSUBUJWJUZ

ଐੑʹجͮ͘BTTPSUBUJWJUZ Zachary's karate club Political blogs network (2004) https://en.wikipedia.org/wiki/Zachary's_karate_club Barabási, A.-L. Network science (Cambridge University Press, 2016). http://barabasi.com/networksciencebook/.

࣍਺ʹجͮ͘BTTPSUBUJWJUZ Menczer, F., Fortunato, S. & Davis, C. A. A First Course in Network Science (Cambridge University Press, 2020). Core-periphery-structure Hub-and-spoke

ʮࣅͨ΋ͷಉ͕࢜ͭͳ͕͍ͬͯΔʯΛ 
 ఆྔతʹଊ͍͑ͨ

࣍਺૬ؔߦྻʢઅʣ ϝϦοτ • ࣍਺૬ؔͷ৘ใΛ͢΂ؚͯΉ σϝϦοτ • ղऍ͕೉͍͠ • ߦྻͷαΠζ͕େ͖͘ͳΔ ʹͭΕͯՄࢹԽ͕ࠔ೉ Barabási, A.-L. Network science (Cambridge University Press, 2016). http://barabasi.com/networksciencebook/.

ຊઅͷఏҊɿ࣍਺૬ؔؔ਺knn (k) ࣍਺਌࿨త χϡʔτϥϧ ࣍਺ഉଞత Barabási, A.-L. Network science (Cambridge University Press, 2016). http://barabasi.com/networksciencebook/.

ྡ઀ϊʔυͷฏۉ࣍਺ ϊʔυ ͷྡ઀ϊʔυͷฏۉ࣍਺ɿ i knn (ki ) = 1 ki N ∑ j=1 Aij kj (7.6) A11 … A1j … A1N ⋮ ⋮ ⋮ Ai1 … Aij … AiN ⋮ ⋮ ⋮ AN1 … ANj … ANN ɿྡ઀ߦྻͷཁૉɽ ·ͨ͸ ɽ Aij 0 1 ɿϊʔυ ͷྡ઀ϊʔυͷ࣍਺͚ͩΛՃࢉ͢Δɽ i N ∑ j=1 Aij kj import networks as nx knn_i = nx.average_neighbor_degree(G)

࣍਺૬ؔؔ਺ͷఆٛ ࣍਺૬ؔؔ਺ʢdegree correlation functionʣΛ࣍ͷΑ͏ʹఆٛ͢Δɿ knn (k) = ∑ k′  k′  P(k′  |k) (7.7) ɿແ࡞ҝʹબΜͩϦϯΫͷยํ͕࣍਺kͷϊʔυͰ͋Δ৔߹ʹɼ΋͏ยํ͕࣍਺k’ͷϊʔυͰ͋Δ֬཰ɽ P(k′  |k) ɿ࣍਺ ͷ͢΂ͯͷϊʔυ ʹؔ͢Δ ͷฏۉɽ knn (k) k i (i = 1,…, n) knn (ki ) ࣍਺૬ؔؔ਺ ࣜͷղऍɿ (7.7) • ࣍਺ ͷϊʔυͷྡ઀ϊʔυ͕ฏۉతʹ࣋ͭฏۉ࣍਺ɽ • ࣍਺ ͷϊʔυ͕༩͑ΒΕͨͱ͖ɼ͍͍ͩͨ ຊͷϦϯΫΛ࣋ͭϊʔυʹғ·Ε͍ͯΔɽ k k knn (k)

࣍਺૬ؔؔ਺ͷخ͍͠ϙΠϯτ ࣍਺૬ؔؔ਺ ͸ ʹ͍ͭͯͷ૿Ճؔ਺ʗఆ਺ؔ਺ʗݮগؔ਺ knn (k) k

χϡʔτϥϧωοτϫʔΫ ࣍਺૬͕ؔͳ͍ ͕ ʹؔͯ͠ҰఆͰ͋ͬͯ΄͍͠ → knn (k) k knn (k) = ⟨k2⟩ ⟨k⟩ (7.9) ࣜͷܗΛ໨ࢦͯ͠ɼ৚݅෇͖֬཰ ʢແ࡞ҝʹબΜͩϦϯΫͷยํ͕ 
 ࣍਺kͷϊʔυͰ͋Δ৔߹ʹɼ΋͏ยํ͕࣍਺k’ͷϊʔυͰ͋Δ֬཰ʣΛมܗ͍ͯ͘͠ɽ (7.9) P(k′  |k)

৚݅෇͖֬཰ ࣜ (7.8) ࣄ৅AɿϦϯΫʹ࣍਺ ͷϊʔυ͕͋Δ k′  (7.8) ࣄ৅BɿϦϯΫʹ࣍਺ ͷϊʔυ͕͋Δ k

ࣜͷಋग़ (7.9) ࣜͱ ࣜɼ࣍਺෼෍ͷ ࣍Ϟʔϝϯτͷܭࢉ͔Β (7.7) (7.8) n (7.9)

ࣜͷղऍ (7.9) knn (k) = ⟨k2⟩ ⟨k⟩ (7.9) • ࣍਺ ͷϊʔυͷྡ઀ϊʔυ͕ฏۉతʹ࣋ͭฏۉ࣍਺͸ɼ࣍਺ ʹґଘͤͣҰఆͰ͋Δɽ • ࣜ͸ɼϑϨϯυγοϓɾύϥυοΫεʢฏۉతʹࣗ෼ͷ༑ୡ͸ࣗ෼ΑΓ΋ଟ͘ͷ༑ୡΛ 
 ࣋ͭʣΛ͍ࣔͯ͠Δɽ k k (7.9) ⟨k2⟩ ⟨k⟩

࣍਺૬͕ؔ͋Δ৔߹ χϡʔτϥϧ ʹ͍ͭͯͷఆ਺ؔ਺ k ࣍਺਌࿨త ʹ͍ͭͯͷ૿Ճؔ਺ k ࣍਺ഉଞత ʹ͍ͭͯͷݮগؔ਺ k

/FUXPSL9ʹΑΔ࣮૷ import networks as nx # G: graph object def get_k_knn(G): knn_dict = nx.k_nearest_neighbors(G) return list(knn_dict.keys()), list(knn_dict.values()) Before NetworkX v3.0 NetworkX v3.0 import networks as nx # G: graph object def get_k_knn(G): knn_dict = nx.average_degree_connectivity(G) return list(knn_dict.keys()), list(knn_dict.values())

/FUXPSL9ʹΑΔ࣮૷ ࣍਺਌࿨త ʹ͍ͭͯͷ૿Ճؔ਺ k χϡʔτϥϧ ʹ͍ͭͯͷఆ਺ؔ਺ k ࣍਺ഉଞత ʹ͍ͭͯͷݮগؔ਺ k

࣍਺૬ؔΛԿΒ͔ͷࢦඪͰදݱ͍ͨ͠ ૬ؔࢦ਺ ૬ؔ܎਺ μ r

૬ؔࢦ਺μ knn (k) ∝ kμ

૬ؔࢦ਺μ ࣍਺૬ؔؔ਺ ͸࣍ͷ ࣜʹΑͬͯۙࣅͰ͖ɼ૬ؔͷ༗ແΛ૬ؔࢦ਺ Ͱ֬ೝͰ͖Δɽ knn (k) (7.10) μ knn (k) = akμ (7.10) log knn (k) = log a + μ log k (7.10′  ) ૬ؔͷ༗ແͷ൑அ w ࣍਺਌࿨తωοτϫʔΫͷ৔߹ɿ w χϡʔτϥϧωοτϫʔΫͷ৔߹ɿ w ࣍਺ഉଞతͳωοτϫʔΫͷ৔߹ɿ μ > 0 μ = 0 μ < 0

૬ؔࢦ਺ ͷਪఆʢ0-4ʣ μ log knn (k) = log a + μ log k (7.10′  ) μ = − 0.23 a = exp(6.77) μ = 0.22 a = exp(2.68)

૬ؔ܎਺r ࣍਺૬ؔ܎਺ ʹΑͬͯɼҟͳΔωοτϫʔΫؒͰ૬ؔͷ౓߹͍ΛൺֱͰ͖Δɽ r Barabási, A.-L. Network science (Cambridge University Press, 2016). http://barabasi.com/networksciencebook/.

࣍਺૬ؔ܎਺r ແ࡞ҝʹબΜͩϦϯΫͷ྆୺ʹ͋Δϊʔυͷ࣍਺ʢ֬཰ม਺ ͱ ʣͷ૬ؔ܎਺ j k ૬ؔͷ༗ແͷ൑அ w ࣍਺਌࿨తωοτϫʔΫͷ৔߹ɿ w χϡʔτϥϧωοτϫʔΫͷ৔߹ɿ w ࣍਺ഉଞతͳωοτϫʔΫͷ৔߹ɿ r > 0 r = 0 r < 0 (7.11) (7.12)

ϐΞιϯͷ૬ؔ܎਺ ϐΞιϯͷ฼૬ؔ܎਺ʢpopulation Pearson correlation coefficientʣ͸࣍ͷΑ͏ʹఆٛ͞ΕΔɽ ρX,Y = Cov(X, Y) σX σY Cov(X, Y) = E[(X − E[X])(Y − E[Y])] = E[XY] − E[X]E[Y], ͜͜Ͱɼ ˡڞ෼ࢄ ˡඪ४ภࠩͷੵ σ2 X = E[X2] − (E[X])2, σ2 Y = E[Y2] − (E[Y])2 .

࣍਺૬ؔ܎਺ ͷಋग़ r ͱ ͷڞ෼ࢄ ͱඪ४ภࠩͷੵ ͸࣍ͷΑ͏ʹද͞ΕΔɽ j k sjk sj sk Αͬͯɼ૬ؔ܎਺ ͸ɼ r

/FUXPSL9ʹΑΔ࣮૷ ૬ؔͷ༗ແͷ൑அ w ࣍਺਌࿨తωοτϫʔΫͷ৔߹ɿ w χϡʔτϥϧωοτϫʔΫͷ৔߹ɿ w ࣍਺ഉଞతͳωοτϫʔΫͷ৔߹ɿ r > 0 r = 0 r < 0 import networks as nx import scipy as sp # for pearson correlation r = nx. degree_assortativity_coefficient(G) #r = nx. degree_pearson_correlation_coefficient(G)