30 5 2015 10 JOURNAL OF SYSTEMS ENGINEERING Vol.30 No.5 Oct. 2015 1,2, 1 (1., 610054; 2., 401120) :.,. D-SIS,. :, ;,,. : ; ; ; ; D-SIS : F830 : A : 1000 5781(2015)05 0575 09 doi: 10.13383/j.cni.jse.2015.05.001 Contagion delayed effects of associated credit ris based on scale-free networ Li Yongui 1,2, Zhou Zongfang 1 (1. School of Econoic and Manageent, University of Electronic Science & Technology of China, Chengdu 610054, China; 2. School of Econoics, Southwest University of Political Science & Law, Chongqing 401120, China) Abstract: The associated credit ris is a hotspot and a difficult proble in odern credit ris anageent field. Based on the average field theory of coplex networ and infectious diseases odel, the contagion delayed effects of associated credit ris between associated subjects in the case of the associated assets are studied. Through the establishent of a D-SIS odel, i.e., a scale-free networ based associated credit ris contagion odel, the equilibriu state of the associated credit ris contagion between the associated credit subjects in this networ is analyzed. Researches show that the assets correlation between associated credit subjects contributes to ris sharing, thus delaying the outbrea of the relative credit ris. The contagion of correlation credit ris has the delayed effect, and the longer the delayed tie, the stronger the contagion intensity of the associated credit ris. Key words: associated credit subjects; associated credit ris; delay effect; scale-free networs; D-SIS odel 1,,.,, : 2013 05 30; : 2013 09 19. : (71271043); (20110185110021).
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5 : 577.,,,, ; ;.,, BA.,. 2 N,,., P()., η., : 1) S, ; 2) I,.,,.,,. t,, s(t), ρ(t), ρ(t) + s(t) = 1. t,,, ρ. t,, t + 1, γ, γ. SIS, t I,, t+t +1, δ S, p = γ/δ.,, δ = 1, γ p..,,,.,. t, ρ (t), ρ.,,,,, ṡ (t) = ρ,t (t) γη θ(ρ(t))s (t) ρ,0 (t) = ρ,0 (t) + γη ρ (t)s (t) ρ,1 (t) = ρ,1 (t) + ρ,0 (t) (1). ρ,t (t) = ρ,t (t) + ρ,t 1 (t), ρ,τ (t) t τ, T ρ (t) = ρ,τ (t), (2) τ=0
578 30 θ(ρ(t)), θ. (1) T, S ; t,. (1), ρ,τ (t)., D-SIS., θ(ρ(t)) <> θ(ρ(t)) = P()ρ (t), (3) s sp(s) <>= P(). (4) t, ρ(t) = P()ρ (t), (5), ρ,t η γ(1 ρ )θ = 0. (6) ρ η γ T, θ η,γ T. ρ,τ,τ = 0,1,...,T ρ,τ (t). D-SIS,, ρ,τ (t) = 0, ρ,0 = ρ,1 = = ρ,t, (2) ρ,t = ρ T + 1, ρ,t (6) ρ T + 1 η γ(1 ρ )θ = 0, ρ = (T + 1)γθη. 1 + (T + 1)γθη, (3) (4) θ = P()ρ s sp(s) = 1 <> (T + 1)γθη P(). (7) 1 + (T + 1)γθη,θ = 0 (7),. ( ), (7), θ 0, d 1 (T + 1)γθη P() dθ <> 1 + (T + 1)γθη 1. θ=0 η P()(T + 1)γ 1. <>,, γ C = <> < 2 > η (T + 1), (8) < 2 >= 2 P(). (8), T = 0,,,.,.,,, ;,, ;
5 : 579,.,,,. 3 BA 3.1 BA 3 BA. BA, t, P(), P() = 2 2 3,., < >= + P()d = 2. BA K C, K C +,< 2 > 2 2 ln(k C /), (8) γ C 1 η (T + 1)ln(K C /). (9) BA, γ C η, T, K C. K C BA N,, K C N 1 2 (9), γc N γ C 2 η (T + 1)ln(N). (10) (10), γ C T η N.,,, ;, ;,,.,, BA., γ > γ C,,. η, P() <> (7), 1 = + (T + 1)γθη (1 + (T + 1)γθη ) d = η θγ(t + 1)ln P() (5) ρ = 2 2 (T + 1)γθη (Aln + B 1 + A = η γθ(t + 1),B = 1,C = η 2 γ 2 θ 2 (T + 1) 2. ( 1 + (T + 1)γθη C + γθη (T + 1) ln (1 + (T + 1)γθη ) ). (11) + ), (12) A,B,C (14) (11) ρ = 2 2 (T + 1)γθη (γθη (T + 1)ln + 1 ) = 2(T + 1)γθ(1 θ)η, (13) 1 + (T + 1)γθη θ (11). (11) (13), BA,. BA,. 3,, ;,.
0 0 580 30 3.2 BA BA, MATLAB 2012b,,. BA, = 2, θ = 1 2, η = 0.3. 1 N 1 = 10,N 2 = 30,N 3 = 50,. 2 T 1 = 5,T 2 = 10,T 3 = 20,., ;, ;,.,.,,,,.,,,, 4. 0. 8 N 1 = 1 0 N 2 = 3 0 0. 4 T 1 = 5 T 2 = 1 0 N 3 = 5 0 T 3 = 2 0 0. 6 0. 3 γ 0. 4 γ 0. 2 0. 2 0. 1 0 5 1 0 1 5 2 0 0 1 0 2 0 3 0 4 0 5 0 T N 1 N, γ T 2 T, γ N Fig. 1 Given N, the relation between γ and T Fig. 2 Given T, the relation between γ and N 3 4,., 1 3,,,.,,,., 3 4,,., 3 4,.,,.,,,. [7], γ 1 = 0.01,γ 2 = 0.03,γ 3 = 0.06, 5 6,.,,, ;.,,. 5 6, 4.
0 T 0 1 T 5 : 581.,,. 3 γ N, T Fig. 3 The relation between γ, N and T in considering directly related assets ratio 4 γ N, T Fig. 4 The relation between γ, N and T without consideration directly related assets ratio 0. 3 5 γ1 = 0. 0 1 γ1 = 0. 0 1 0. 2 5 γ2 = 0. 0 3 γ3 = 0. 0 6 0. 8 γ2 = 0. 0 3 γ3 = 0. 0 6 0. 6 ρ ρ 0. 1 5 0. 4 0. 2 0. 0 5 0 4 8 1 2 1 5 0 4 8 1 2 1 5 5 ρ T 6 ρ T Fig. 5 The relation between ρ and T in Fig. 6 The relation between ρ and T without considering directly related assets ratio consideration directly related assets ratio 7 8 T 1 = 3,T 2 = 6,T 3 = 15,.,,,.,,.,,. 7 8,,., 9 10. 5 8, 9 10,,,.,,,.,.
0 0 1 582 30 0. 3 5 T 1 = 3 T 1 = 3 T 2 = 6 T 2 = 6 T 3 = 1 5 0. 8 T 3 = 1 5 0. 2 5 0. 6 ρ ρ 0. 1 5 0. 4 0. 2 0. 0 5 0. 0 1 5 0. 0 3 0. 0 4 5 0. 0 6 0. 0 1 5 0. 0 3 0. 0 4 5 0. 0 6 γ γ 7 ρ γ 8 ρ γ Fig. 7 The relation between ρ and γ in Fig. 8 The relation between ρ and γ without considering directly related assets ratio consideration directly related assets ratio 9 ρ γ,t Fig. 9 The relation between ρ,γ and T in considering directly related assets ratio 10 ρ γ, T Fig. 10 The relation between ρ, γ and T without consideration directly related assets ratio,,.,,, ;,, ;,,,.,,. 4,,,.,,.,, ;,, ;.,,.
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