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- 1. JClub 2013.12.12 by Kazutoshi Sasahara Collective dynamics of small-world networks D. J. Watts and S. Strogatz Nature, 1998
- 2. 研究背景 自己組織化システムのモデルにはネットワーク結合力学系が使わ れ、通常は正則かランダムな結合が仮定される しかし、現実のネットワークはこれらの中間ぐらい 正則とランダムの間を補完して、中間領域のネットワークの構造を 調べる必要がある
- 3. mapped neural network. Table 1 shows that all three graphs are small-world networks. These examples were not hand-picked; they were chosen because of their inherent interest and because complete wiring diagrams were available. Thus the small-world phenomenon is not merely a curiosity of social networks13,14 nor an artefact of an idealized acted in a ﬁlm together. We restrict attention t graph, which includes ,90% of all actors listed http://us.imdb.com), as of April 1997. For the transformers and substations, and edges r between them. For C. elegans, an edge joins t a synapse or a gap junction. We treat all edg vertices as identical, recognizing that these ar show the small-world phenomenon: L ) Lrandom WSモデル： 正則グラフをランダムに再配線 Regular Small-world Random 1 C(p) / 0.8 0.6 0.4 0.2 p=0 p=1 Increasing randomness 0 0.0001 L(p) / L(0) 0.001 0.0 p Figure 1 Random rewiring procedure for interpolating between a regular ring 初期状態：環状の正則グラフ（図はn=20, k=4） Figure 2 Characteristic path length L(p) lattice and a random network, without altering the number of vertices or edges in family of randomly rewired graphs descri the graph. We start with a ring of n vertices, each connected to its k nearest number of edges in the shortest path be examples shown here, but much larger n and k are used in the rest of this Letter.) that a vertex v has kv neighbours; then 再配線：ランダムに選んだリンクとつなぎ換える（確率p） of vertices. The clustering coefﬁcien neighbours by undirected edges. (For clarity, n ¼ 20 and k ¼ 4 in the schematic pairs We中間領域のネットワークの条件：n >>nearest ln(n) >>in a choose a vertex and the edge that connects it to its k >> neighbour 1 between them (this occurs when every nei clockwise sense. With probability p, we reconnect this edge to a vertex chosen neighbour of v). Let Cv denote the fraction uniformly at random over the entire ring, with duplicate edges forbidden; other- exist. Deﬁne C as the average of Cv ove wise we leave the edge in place. We repeat this process by moving clockwise statistics have intuitive meanings: L is the around the ring, considering each vertex in turn until one lap is completed. Next, shortest chain connecting two people; C
- 4. ネットワークの特徴量 平均距離L 2つのノードがどれだけはなれているか（大局的な特徴） 求め方：すべてのノード間距離の平均 2Ei クラスター係数C Ci = , ki (ki 1) クリーク度合い（局所的な特徴） n X 1 求め方：右式 Ci . C= n i=1 中間領域にあるネットワークの構造をL(p)とC(p)で特徴付ける
- 5. mapped neural network. Table 1 shows that all three graphs are small-world networks. These examples were not hand-picked; they were chosen because of their inherent interest and because complete wiring diagrams were available. Thus the small-world phenomenon is not merely a curiosity of social networks13,14 nor an artefact of an idealized WSモデルの理論的考察 acted in a ﬁlm together. We restrict attention t graph, which includes ,90% of all actors listed http://us.imdb.com), as of April 1997. For the transformers and substations, and edges r between them. For C. elegans, an edge joins t a synapse or a gap junction. We treat all edg vertices as identical, recognizing that these ar show the small-world phenomenon: L ) Lrandom 1 Regular Small-world Random C(p) / 0.8 0.6 0.4 0.2 p=0 p=1 0 0.0001 L(p) / L(0) 0.001 0.0 Increasing randomness Figure 1 Random rewiring procedure for interpolating between a regular ring p Figure 2 Characteristic path length L(p) lattice and a random network, without altering the number of vertices or edges in3/4） family of randomly rewired graphs descri p→0：ラージワールドでクラスター性が高い（L n/2k >> 1, C the graph. We start with a ring of n vertices, each connected to its k nearest number of edges in the shortest path be neighbours by undirected edges. (For clarity, n ¼ 20 and k ¼ 4 in the schematic pairs of p→1：スモールワールドだがクラスター性が低い（L Lrand ln(n)/ln(k), C vertices. The clustering coefﬁcien Crand k/n） examples shown here, but much larger n and k are used in the rest of this Letter.) that a vertex v has kv neighbours; then We choose a vertex and the edge that connects it to its nearest neighbour in a between them (this occurs when every nei nを大きくしたときLの増え方は、正則グラフは線形、ランダムグラフはlog neighbour of v). Let Cv denote the fraction clockwise sense. With probability p, we reconnect this edge to a vertex chosen →ランダムグラフのLが小さい理由 exist. Deﬁne C as the average of C ove uniformly at random over the entire ring, with duplicate edges forbidden; otherv wise we leave the edge in place. We repeat this process by moving clockwise statistics have intuitive meanings: L is the around the ring, considering each vertex in turn until one lap is completed. Next, shortest chain connecting two people; C
- 6. Characteristic path length L and clustering coefﬁcient C for three real networks, compared tion here is that at the local level (as reﬂected by C(p)), the transition connectivity for dynamical systems. Our test case is a deliberately to random graphs withworld is almost undetectable. To check the robustness of to a small the same number of vertices (n) and average number of edges per simpliﬁed model for the spread of an infectious disease. The vertex (k). (Actors: n ¼ 225;226, k ¼ 61. Power grid: n ¼ 4;941, k ¼ 2:67. C. elegans: n ¼ 282, these results, we have tested many different types of initial regular population structure is modelled by the family of graphs described k ¼ 14.) The graphs are deﬁned as follows. Two actors are joined by an edge if they have graphs, as well as different algorithms for random rewiring, and all in Fig. acted in a ﬁlm together. We restrict attention to the giant connected component16 of this 1. At time t ¼ 0, a single infective individual is introduced give qualitatively similar results. in the Internet Movie Database (available atan otherwise healthy population. Infective individuals are into graph, which includes ,90% of all actors listed The only requirement is that the rewired as of April 1997. For the power grid, vertices represent generators, removed permanently (by immunity or death) after a period of http://us.imdb.com),edges must typically connect vertices that would otherwise transformers and substations, and edges represent high-voltage transmission sickness that lasts one unit of dimensionless time. During this time, lines be much farther apart than Lrandom. between them. For idealized construction above neurons the key role of short by eitherinfective individual can infect each of its healthy neighbours The C. elegans, an edge joins two reveals if they are connected each a synapse or a gap junction. We treat all edges as undirected and unweighted, and allprobability r. On subsequent time steps, the disease spreads cuts. It suggests that the small-world phenomenon might be with vertices as identical, recognizing that these are crude approximations. All three networks common in sparse networks with many vertices, as even a tiny along the edges of the graph until it either infects the entire show the small-world phenomenon: L ) Lrandom but C q Crandom . 8 WSモデルの数値実験の結果 fraction of short cuts would sufﬁce. To test this idea, we have computed L and C for the collaboration graph of actors in feature 1 ﬁlms (generated from data available at http://us.imdb.com), the electrical power grid of the western United States, and the neural network of the nematode worm C. elegans17. All three graphs are of C of ﬁlm actors 0.8 scientiﬁc interest. The graph (p) / C(0) is a surrogate for a social network18, with the advantage of being much more easily speciﬁed. It is also akin to the graph of mathematical collaborations centred, 0.6 ¨ traditionally, on P. Erdos (partial data available at http:// www.acs.oakland.edu/,grossman/erdoshp.html). The graph of the power grid is relevant to the efﬁciency and robustness of 0.4 power networks19. And C. elegans is the sole example of a completely mapped neural network. L(p) / L(0) 0.2 Table 1 shows that all three graphs are small-world networks. These examples were not hand-picked; they were chosen because of their inherent interest and because complete wiring diagrams were 0 available. Thus the small-world phenomenon is not merely a 0.0001 0.001 0.01 0.1 1 curiosity of social networks13,14 nor an artefact of an idealized p population, or it dies out, having infected some fraction of the population in the process. スモールワールド(SW)・ネットワーク Table 1 Empirical examples of small-world networks LがLrandと同程度に小さく、CがCrand L L C C ............................................................................................................................................................................. Film actors 3.65 2.99 0.79 0.00027 よりもはるかに大きいネットワーク Power grid 18.7 12.4 0.080 0.005 actual C. elegans 2.65 random 2.25 actual random 0.28 0.05 ............................................................................................................................................................................. Characteristic path length L and clustering coefﬁcient C for three real networks, compared to random graphs with the same number of vertices (n) and average number of edges per vertex (k). (Actors: n ¼ 225;226, k ¼ 61. Power grid: n ¼ 4;941, k ¼ 2:67. C. elegans: n ¼ 282, k ¼ 14.) The graphs are deﬁned as follows. Two actors are joined by an edge if they have acted in a ﬁlm together. We restrict attention to the giant connected component16 of this graph, which includes ,90% of all actors listed in the Internet Movie Database (available at http://us.imdb.com), as of April 1997. For the power grid, vertices represent generators, transformers and substations, and edges represent high-voltage transmission lines between them. For C. elegans, an edge joins two neurons if they are connected by either a synapse or a gap junction. We treat all edges as undirected and unweighted, and all vertices as identical, recognizing that these are crude approximations. All three networks show the small-world phenomenon: L ) Lrandom but C q Crandom . SWネットワークになる理由 pが小さい領域で、Lは急激かつ非線形 に減少、しかしCは線形に減少 →ショートカットの存在 1 Figure 2 Characteristic path length Small-world clustering coefﬁcient C(p) for the L(p) and Regular Random family of randomly rewired graphs described in Fig. 1. Here L is deﬁned as the C(p) / C(0) 0.8 初期状態のグラフ構造や再配線のアル 0.6 pairs of vertices. The clustering coefﬁcient C(p) is deﬁned as follows. Suppose ゴリズムを換えても結果は同じ 0.4 number of edges in the shortest path between two vertices, averaged over all that a vertex v has kv neighbours; then at most kv ðkv 2 1Þ=2 edges can exist between them (this occurs when every neighbour of v is connected to every other neighbour of v). Let Cv denote the fraction of these allowable edges that actually exist. Deﬁne C as the average of Cv over all v. For friendship networks, these p=0 p=1 statistics have intuitive meanings: L is the average number of friendships in the Increasing randomness shortest chain connecting two people; Cv reﬂects the extent to which friends of v 0.2 0 0.0001 L(p) / L(0) 0.001 0.01 0.1 1 p Figure 1 Figure are also friends of Random rewiring procedure for interpolating between a regular ringa typical 2 Characteristic path length L(p) and clustering coefﬁcient C(p) for the each other; and thus C measures the cliquishness of lattice and a random network, without altering the number of vertices or edges in family of randomly rewired graphs described in Fig. 1. Here L is deﬁned as the
- 7. 実世界のネットワークの構造 L Lrand C Crand S Film actors 3.65 2.99 0.79 0.00027 2396.854 Power grid 18.7 12.4 0.08 0.005 10.610 C. elegans 2.65 2.25 0.28 0.05 4.755 Small-world-ness: S = (C/Crand) / (L/Lrand) 映画俳優の競演関係（n=225,226, k=61）、アメリカ西海岸の送電網 （n=4,941,k=2.67）、Cエレガンスの神経回路網（n=282, k=14）は すべてSWネットワーク SW結合は大きくてスパースなネットワークに共通する構造
- 8. Kretschmar and Morris24 have shown that increases in the number of concurrent partnerships can signiﬁcantly accelerate the propagation of a sexually-transmitted disease that spreads along the edges of a graph. All their graphs are disconnected because they ﬁx the average number of partners per person at k ¼ 1. An increase in the number of concurrent partnerships causes faster spreading by increasing the number of vertices in the graph’s largest connected component. In contrast, all our graphs are connected; hence the predicted changes in the spreading dynamics are due to more subtle structural features than changes in connectedness. Moreover, with increasing p. (3) Small-world networks of coupled phase oscillators synchronize almost as readily as in the mean-ﬁeld model2, despite having orders of magnitude fewer edges. This result may be relevant to the observed synchronization of widely separated neurons in the visual cortex27 if, as seems plausible, the brain has a small-world architecture. We hope that our work will stimulate further studies of smallworld networks. Their distinctive combination of high clustering with short characteristic path length cannot be captured by traditional approximations such as those based on regular lattices or random graphs. Although small-world architecture has not received much attention, we suggest that it will probably turn out to be widespread in biological, social and man-made systems, often M with important dynamical consequences. スモールワールド結合の効用： 感染症のモデルの場合 設定 a Received 27 November 1997; accepted 6 April 1998. 0.35 WSモデルでノードは人、感染症はリンクを伝わる 0.3 r half t=0でネットワークの1人が感染 0.25 0.2 0.15 0.0001 1. Winfree, A. T. The Geometry of Biological Time (Springer, New York, 1980). 2. Kuramoto, Y. Chemical Oscillations, Waves, and Turbulence (Springer, Berlin, 1984). 3. Strogatz, S. H. & Stewart, I. Coupled oscillators and biological synchronization. Sci. Am. 269(6), 102– 109 (1993). 4. Bressloff, P. C., Coombes, S. & De Souza, B. Dynamics of a ring of pulse-coupled oscillators: a group theoretic approach. Phys. Rev. Lett. 79, 2791–2794 (1997). 5. Braiman, Y., Lindner, J. F. & Ditto, W. L. Taming spatiotemporal chaos with disorder. Nature 378, 465–467 (1995). 6. Wiesenfeld, K. New results on frequency-locking dynamics of disordered Josephson arrays. Physica B 222, 315–319 (1996). 7. Gerhardt, M., Schuster, H. & Tyson, J. J. A cellular automaton model of excitable media including curvature and dispersion. Science 247, 1563–1566 (1990). 8. Collins, J. J., Chow, C. C. & Imhoff, T. T. Stochastic resonance without tuning. Nature 376, 236–238 (1995). 9. Hopﬁeld, J. J. & Herz, A. V. M. Rapid local synchronization of action potentials: Toward computation with coupled integrate-and-ﬁre neurons. Proc. Natl Acad. Sci. USA 92, 6655–6662 (1995). 10. Abbott, L. F. & van Vreeswijk, C. Asynchronous states in neural networks of pulse-coupled oscillators. Phys. Rev. E 48(2), 1483–1490 (1993). 11. Nowak, M. A. & May, R. M. Evolutionary games and spatial chaos. Nature 359, 826–829 (1992). 12. Kauffman, S. A. Metabolic stability and epigenesis in randomly constructed genetic nets. J. Theor. Biol. 22, 437–467 (1969). 13. Milgram, S. The small world problem. Psychol. Today 2, 60–67 (1967). 14. Kochen, M. (ed.) The Small World (Ablex, Norwood, NJ, 1989). 15. Guare, J. Six Degrees of Separation: A Play (Vintage Books, New York, 1990). half ´ 16. Bollabas, B. Random Graphs (Academic, London, 1985). 17. Achacoso, T. B. & Yamamoto, W. S. AY’s Neuroanatomy of C. elegans for Computation (CRC Press, Boca Raton, FL, 1992). 18. Wasserman, S. & Faust, K. Social Network Analysis: Methods and Applications (Cambridge Univ. Press, 1994). 19. Phadke, A. G. & Thorp, J. S. Computer Relaying for Power Systems (Wiley, New York, 1988). 20. Sattenspiel, L. & Simon, C. P. The spread and persistence of infectious diseases in structured populations. Math. Biosci. 90, 341–366 (1988). 21. Longini, I. M. Jr A mathematical model for predicting the geographic spread of new infectious agents. Math. Biosci. 90, 367–383 (1988). 22. Hess, G. Disease in metapopulation models: implications for conservation. Ecology 77, 1617–1632 (1996). 23. Blythe, S. P., Castillo-Chavez, C. & Palmer, J. S. Toward a uniﬁed theory of sexual mixing and pair formation. Math. Biosci. 107, 379–405 (1991). 24. Kretschmar, M. & Morris, M. Measures of concurrency in networks and the spread of infectious disease. Math. Biosci. 133, 165–195 (1996). 25. Das, R., Mitchell, M. & Crutchﬁeld, J. P. in Parallel Problem Solving from Nature (eds Davido, Y., ¨ Schwefel, H.-P. & Manner, R.) 344–353 (Lecture Notes in Computer Science 866, Springer, Berlin, 1994). 26. Axelrod, R. The Evolution of Cooperation (Basic Books, New York, 1984). 感染者はつながりのある健常者を確率rで感染させる 0.001 0.01 0.1 1 p 一定時間後、感染者は取り除かれる（免疫 or 死亡） 結果 b 1 T(p) /T(0) L(p) /L(0) 0.8 a：臨界感染率r （集団半分が感染するのに必要なr）は pを大きくするにしたがい減少 0.6 b：集団全体が感染するのにかかる時間TはLと同じ挙動 0.4 SW結合のもとでは感染症はすぐ広まる 0.2 0 0.0001 0.001 0.01 0.1 1 p Figure 3 Simulation results for a simple model of disease spreading. The community structure is given by one realization of the family of randomly rewired
- 9. スモールワールド結合の効用： 他のモデルの場合 計算タスクを解くセルオートマトン SWネットワーク上の多数決原理は高性能 複数エージェントによる繰り返し囚人のジレンマゲーム TFT戦略をする集団でショートカットを増やすと協調が起こりにくくなる SWネットワーク上の結合位相振動子 平均場モデルと同じぐらい、しかし少ないリンクで同期する
- 10. まとめ 実世界のネットワークは規則的なものとランダムの中間？ でもあまり研究されていなかった WSモデルを提案してその構造を調べた 大きな結合係数（∼正則）と小さな平均距離（∼ランダム）を併せもつ スモールワールド（SW）・ネットワークが存在 実世界のネットワークもSWだった 映画俳優の競演関係、米国西海岸の送電網、Cエレガンスの神経回路網 SW結合の効用 感染症が広まりやすい、信号の伝達速度/計算能力/同期が向上

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