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ǰһάάȳΪ硣ָƽƶԳԾκһĽĿͬȻֻһϰ÷ ¶壬Carley TreeȻ磬Ҳûй涨˵ڹ硣һˣN㹹ɵͼУ ԴC2NߣǴMɵͽ硣һķǣһp C2NκһӣǶһԸpӡѡ M = pC2Nģ;Ϳϵ ˼򵥵ģͣ伸ʵоȴͬļ򵥡缸ʵоPaul ErdösAlfréd Rényi Béla Bollobásʮʮ֮ɵġΪǿоͳѧϵǴϵ۵ĽǶ± ͳʵоǣΪһͻģ͵ͳʶǴǰһרŽܷ뼼 һС

ĵͼʰȷֲƽ۳̶ƽ̾롣ж㶼ֵͬͬȷֲ Ϊ $ \delta$$ \left(\vphantom{k-k_{0}}\right.$k - k0$ \left.\vphantom{k-k_{0}}\right)$ƽ۳̶ҲֻҪһ C = $ \left<\vphantom{C_{v}}\right.$Cv$ \left.\vphantom{C_{v}}\right>$ = Cv ̾Ҳֻijһ㿪ʼ֮ľ֮ L $ \sim$ N2Ȼƽֵ d = $ {\frac{{L\times N}}{{2\times C^{2}_{N}}}}$ $ \sim$ N G$ \left(\vphantom{N,p}\right.$N, p$ \left.\vphantom{N,p}\right)$˴ӿͼȫͼп ͼļҪÿһֿͼƽǼÿһֿͼ̾룬ȻոԳֵļƽ һ㹫ʽϸļ7ڡо綥ĶֵƽֵΪpNIJɷֲ 伯۳̶Լp̾ d $ \sim$ ln(N)

Աȹ磬Ƿ֣ƽ۳̶ƽ̾룬̬ܹܺõطӳʼ졣 ƽ۳̶ȸ߶ƽ̾볤ƽ۳̶ȵͶƽ̾С ƽ̾d $ \sim$ N伯۳̶ڽĿk0ͼ(1)ʾĹ Уk0 = 4۳̶Ϊ $ {\frac{{1}}{{2}}}$Уƽ۳̶ȷdzСͼ(1) ʾУͬ۳̶Ϊ0.02


Fig 1: Small WorldģͣͼʾSmall WorldͼĹͨߵõġpΪÿһߵʡp = 0ʱΪ磬p = 1.0ʱΪ硣ͼȡ[7]

Ȼ伯۳̶ȷdzСƽ̾СһuĽڣΪaô aڵĽ֮໥ظĸdz٣Դuνڹϵǿҵa2¶㣬ྭlogaN ڹϵǾͿ硣ԣ̾ d $ \sim$ ln Nɼڹ磬Ҳ伯۳̶ȸߣظʺ ƽ̾˿һ໥ìܵļ

ôǷһͬʱи߼۳̶ȣС·أڴȾģͣƽ۳̶ȶӦڴĹȣƽ̾ ǴȡˣʵͬʱڿĹȺʹȵĻϵĴȾȻڹ 硣WattStrogatzΪҵģ--Small World[3,7]


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