1. Shintaro TAKEMURA d.hatena.ne.jp/taos twi9er.com/stakemura facebook.com/shintaro.takemura

2. 4.1 [1] (Complete Graph) A L=1, C=1B =N-1E L M=N*(N-1)/2 k 3. 4.1 [2] (Subgraph) A G C G B E (Clique) F D = Complete subgraph ABE 4. 4.2 [1] A B Z2 L C D L N 5. 4.2 [2] A B E Z1 k=2, L N, C=0 A B CN E L N/4 NL N A B E k L N/2k 6. 4.2 [3] D ZD L N1/D DL NL N 7. 4.3 (Tree) N = 1 + k + k (k 1) + + k (k 1) l 1 C=0 l1 (k 1)l l = 1+ k ~ (k 1)1 (k 1) N (k-1)l l logN / log(k-1) log N N + l~ k : fixed lLlog(k 1) L logN 1 L ln( k 1)1k 8. 4.4 [1] (Random Graph) p = 0.0 ; k = 0 N = 12 Erds and Renyi (1959) 1. N 2. 2 p p = 0.09 ; k = 1 k pNp = 1.0 ; k N M pN(N-1)/2 9. 4.4 [2] (Largest Component) p = 0.0 ; = 0 p = 0.045 ; = 0.5 p = 0.09 ; = 1 p = 1.0 ; N2(Size of largest component)151112(Diameter of largest component) 0 4 7 1(Average path length between nodes) 0.02.0 4.2 1.0 10. 4.4 [3] (Phase Transi 1/N p > logN/N 0 1/N p 11. 4.4 [4] kp=kN k 1 N 1 N 1 kN k 1 N 1 k k N 1p(k ) = k p (1 p ) k N 1 = 1 N 1 k kp(k ) k =0N 1 lim G (x ) = exp( k (x 1))G (x ) = P(k )x k N +k =0k N k 1+ k k k k N 1 k N 1k = e x = k N 1 1 N 1 xk k =0 k! k = 0 N 1 k N = 200 = 1 +(x 1) 2 Poisson N 1k =4 k k k p(k ) = k lim G (x ) =e x N + k! 12. 4.5 L C C