2 2.1 i j i j (i, j) (i, j) i (out-going degree) j (in-coming degree) n N = {1, 2,..., n} E(N) i(lender) j(borrower) (loans) i j w ji = w(j, i) W w ij

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1 : in progress version 1 CDS CDO Haldane(2009) Haldane and May(2011) May, Levin and Sugihara(2008) (critical phenomena) (phase transitions) (percolation theory) *1 Callaway, et al.(2000) Newman, et al.(2001) Newman(2002) Watts(2002) *2 *1 Newman(2010) *2 López-Pintado(2006,2008) Newman(2010) (2014) 1

2 2 2.1 i j i j (i, j) (i, j) i (out-going degree) j (in-coming degree) n N = {1, 2,..., n} E(N) i(lender) j(borrower) (loans) i j w ji = w(j, i) W w ij = w(i, j) (i, j) (adjacency matrix) j i a ij = 1 { 1 w ij > 0 a ij = 0 ij {a ij } A (1) Bank_A money_flow money_flow Bank_B money_flow money_flow Bank_C Bank_D Bank_E Bank_F Fig.2.1 2

3 i (predecessors) P i i (successors) S i P i = {j : (i, j) E(N)}, S i = {j : (j, i) E(N)} i N i = P i S i i i k in i k in i = P i = j a ji k out i k out i = S i = j a ij A A n i=1 k in i = n i=1 k out i = m m k = m/n k world wide web world wide web * Google Yahoo GSCC(giant strongly connected component) 2 3 IN (GIN giant in-component) OUT (GOUT giant out-component) GIN GSCC GSCC GIN GIN GSCC GSCC GIN GSCC GOUT GOUT GSCC GOUT GSCC WWW GOUT GIN tendrils (tube) GIN GOUT 2 GIN GOU GSCC DCs(disconnected components) *3 Broder et al.(2000) Dorogovtsez(2001) 3

4 GWCC(giant weakly connected componetns) G j N = j G j h k {(i, j), (j, i) : i G h j G k } = GIN GSCC DCs 1 A C cc 2 3 B E H ee ff 6 5 G I dd 7 4 D seven tube_1 F K GOUT six tube_2 J one three two four five Fig.2.2 Broder et al(2000) (2003) Soramäki et al.(2006) 4

5 k k p k = Ck α, α > 0 C (right-skewed) ln p k = ln C α ln k α world wide web Fig.2.3 world wide web hyperlinks (the complement of the cumulative distribution function) P k = r=k p r P k k k 5

6 k min k k min P k = C r=k r α k P k k r α dr = C α 1 k (α 1) (α 1) *4 world wide web Fig.2.4 world wide web hyperlinks world wide web hyperlinks Broder et al.(2000) Broder *4 Newman(2010) 6

7 3 3.1 fire-sale 2 *5 Arinaminpathy, et al.(2012) Battiston, et al.(2012a,b) Gai and Kapadia(2010) May and Arinaminpathy(2011) Nier et al.(2007) Gai and Kapadia(2010) 3 < z < z = 8 Nier et al.(2007) May and Arinaminpathy(2011) Nier et al(2007) Gai and Kapadia(2010) May and Arinaminpathy(2011) Nier et al(2007) Gai and Kapadia(2010) Battiston, et al.(2012a,b) *5 Allen and Gale(2000) 7

8 3.2 Gai and Kapadia(20010) N N = {1, 2,..., N} G = {g ij }, g ij [0, 1], g ii = 0, i, j N g ij = g ji g ij j i g ij i (in-coming links/borrowing) j (out-going links/lending) p z = p(n 1) i (assets)a i (external assets)e i (IB) l i a i = e i + l i e i (liabilities) d i IB b i c i IB (solvency) c i = (e i + l i ) (d i + b i ) 0 i i e i + l i d i b i < 0 i IB γ i IB θ i c i = γ i a i, θ i = l i a i IB w i = θ ia i z z z = z E = i e i i i IB ϕ i e i q i i q i e i + (1 ϕ i )l i d i b i 0 8

9 IB ϕ i l i q ϕ i ϕ i c i (1 q i )e i l i,, l i 0 i i i j i j i (out-going links) i (in-coming links) i i j i i IB 1/j i c i (1 q i )e i l i < 1 j i i i i (vulnerable) (out-dgree)j i v j = Pr{ c i (1 q i )e i l i < 1 j } c i, d i, e i j k v j p jk p jk j k v j p jk G(x, y) G(x, y) = j,k v j p jk x j y k 1 N j i = jp jk = 1 k i = kp jk = z N i j,k i j,k k i i G 0 (y) = G(1, y) = j,k v j p jk y k G 0 (1) = j,k v j p jk G 1 (y) 9

10 j p jk ξ jk = j p jk / j,k j p jk v j ξ jk G 1 (y) = v j ξ jk y k 1 = j,k j,k j p v j j p jk y k jk j,k G(1) = j,k v j ξ jk (vulnerable cluster) *6 H 1 (y) H 1 (y) = Pr{ } + y j,k v j j p jk [H 1 (y)] k / j,k j p jk 1 G(1) H 1 (y) = [1 G 1 (1)] + yg 1 (H 1 (y)) (2) H 0 (y) G 0 (1) j k v j p jk H 1 (y) H 0 (y) = Pr{ } + y j,k v j p jk [H 0 (y)] k = [1 G 0 (1)] + yg 0 (H 1 (y)) (3) (2) H 1 (y) (3) H 0 (y) S S = H 0(1) H 1 (y) H 1 (1) = 1 S = G ( 1) + G 0(1)G 1 (1) 1 G 1 (1) (4) G 1(1) = 1 *6 Callaway, et al.(2000) Newman, et al.(2001) Newman(2002) Watts(2002) 10

11 j k v j p jk = z j,k (5) G 1(1) z (5) jkp jk v j (5) Gai and Kapadia(2010) N = 1000, θ I = 0.2, γ i = < z < 10, p = z/(n 1) 3 < z < z = 8 γ = 0.03, 0.04,

12 [1] Allen, Franklin and Douglas Gale(2000), Financial Contagion, Journal of Political Economy, 108(1), [2] Anand, Kartik, Prasanna Gai, Sujit Kapadia, Simon Brennan, and Matthew Willison(2012), A Network Model of Financial System Resilience, forthcoming to Journal of Economic Behavior and Organization. [3] Arinaminpathy, Nimalan, Sujit Kapadia, and Robert M. May(2012), Size and Complexity in Model Financial Systems, Proc. of National Academy of Science, 109(45), [4] Battiston, Stefano, Domenico Delli Gatti, Mauro Gallegati, Bruce C. Greenwald, and Joseph E. Stiglitz(2012a), Dedault Cascades: When Does Risk Diversification Increase Stability, Journal of Financial Stability, 8, [5] Battiston, Stefano, Domenico Delli Gatti, Mauro Gallegati, Bruce C. Greenwald, and Joseph E. Stiglitz(2012b), Liasons Dangereuses : Increasing Connectivity, Risk Sharing, and Systemic Risk, Journal of Economic Dynamics and Control, 36(8), [6] Martin L. Beck and Enghin Atalay(2008), The topology of the federal funds market, European Central Bank, Working Paper Series No.986. [7] Callaway, Duncan S., M. E. J. Newman, Steve H. Strogatz, and Duncan J. Watts(2000), Network Robustness and Fragility: Percolation on Random Graphs, Physical Review Letters, 85, [8] Dodds, Peter Sheridan and Duncan Watts(2004), Universal Behavior in a Generalizad Model of Contagion, Physical Review Letters, 82(21), [9] Dodds, Peter Sheridan and Duncan Watts(2005), A Generalized Model of Social and Biological Contagion, Journal of Theoretical Biology, 232, [10] Gai, Prasanna and Sujit Kapadia(2010), Contagion in Financial Networks, Proceedings of the Royal Society A, 466(2110), [11] Gai, Prasanna, Andrew G. Haldane, and Sujit Kapadia(2011), Complexity, Concentration and Contagion, Journal of Monetary Economics, 58, [12] Holdane, Andrew G.(2009), Rethinking the Financial Network, Bank of England, Working Paper. [13] Haldane, Andrew G. and Robert M. May(2011), Systemic Risk in Banking Ecosystems, Nature, 469, [14] López-Pintado, Dunia(2006), Contagion and Coordination in Random Networks, Int. J. Game Theory, 34, [15] López-Pintado, Dunia(2008), Diffusion in Compex Social Networks, Games and Economic Behavior, 62, [16] May, Robert M. and Nimalan Arinaminpathy(2010), Systemic Risk: the Dynamics of Model Banking Systems, Journal of the Royal Society Interface, 7, [17] Newman, M. E. J.(2002), The Spread of Epidemic Disease on Networks, Physical Review E, 66,

13 [18] Newman, M. E. J., Steven H. Strogatz, and Duncan J. Watts(2001), Random Graph with Arbitrary Distributions and their Applications, Physical Review E,64, [19] Nier, Elrend, Jing Yang, Tanju Yorulmazer and Amdeo Alentorn(2007), Network Models and Financial Stability, Journal of Economic Dynamics and Control, 31, [20] Pastor-Satorras, Romuald and Alessandro Vespignani(2001), Epidemic Spreading in Scale-Free Networks, Physical Review Letters, 86, [21] Roukny, Tarik, Hugues Bersini, Hugues Pirotte, Guido Caldarelli, and Stefano Battiston(2013), Default Cascades in Complex Networks: Topology and Systemic Risk, Scientific Reports, 3, 1-8. [22] K. Soramäki, M. L. Bech, M. J. Arnold, R. J. Glass, and W. E. Beyeler(2006), The topology of interbank payment flows, Federal Reserve Bank of New York, Staff Report no [23] Vega-Redondo, Fernando(2007), Complex Social Networks, Cambridge University Press. [24] Watts, Duncan J. (2002), A Simple Model of Global Cascades on Random Networks, Proc. of National Academy of Science, 99(9), [25] (2014) mimeo. [26] (2003). [27] (2008). [28] Geroge Akerlof and Robert Shiller(2008) 7 [29] (2015) [30] (2015) 13

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