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2 : (CS):

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5 マクロ経済学の概念図

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7 l etc etc 6 6 l Boss el al. 24 Quant. Finan.

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9 1. Fabio Caccioli, Paolo Barucca, Teruyoshi Kobayashi Network models of financial systemic risk: a review Journal of Computational Social Science, Handbook on Systemic Risk (213)

10 l l l

11 l 28 l l

12 1. l (i.e., ) l l Economic networks: The new challenges Frank Schweitzer et al. Science, /3/7

13 2. ( ) l l l l 216/3/7

14 Poledna et al. (215, J.Finan.Stability) 216/3/7

15 3. l l GDP 216/3/7

16 4. l l Huang et al. 213, Sci. Rep. Cascading Failures in Bi-partite Graphs: Model for Systemic Risk Propagation. Time

17 l ltoo big to fail l l

18

19 i. e., m k > ϕ k : m : active

20 φ = 1/3 1/4 < φ 2/4 > φ 8/17/18 33

21 Recursion equation for the cascade size ρ: ρ = ρ + (1 ρ ) p k k=1 k m= ( k m) qm (1 q) k m F ( m k ), q = ρ + (1 ρ ) k=1 k z p k k 1 ( k m 1 ) qm (1 q) k 1 m m F ( k ), m= Gleeson and Cahalane, PRE (27) Caccioli et al., JCSS (218)

22 Recursion equation for the cascade size ρ: ρ = ρ + (1 ρ ) p k k=1 q = ρ + (1 ρ ) k=1 k z p k k m= k 1 m= ( k m) qm (1 q) k m F ( m k ), ( k 1 m ) qm (1 q) k 1 m F ( m k ), First-order cascade condition: (1 ρ ) k=1 k(k 1) z p k [ F ( ρ = 1 k ) F() ] > 1. Gleeson and Cahalane, PRE (27) Caccioli et al., JCSS (218)

23 Watts (22, PNAS)

24 ( )! ) 1!! Threshold model A node is activated if % & > R (e.g., Watts, 22 PNAS)!:

25 ( ) Gai and Kapadia (21, Proc. Roy.Soc. A) Watts (22, PNAS)

26 e. g., m 1 + m 2 k 1 + k 2 > ϕ 1 Figure 1 from Discrete-time distributed consensus on multiplex networks I Trpevski et al 214 New J. Phys doi:1.188/ /16/11/11363 m 1 + m 2 k 1 + k 2 > ϕ 2 ϕ 1 < ϕ 2

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28 a multiplex financial network l Watts (22,PNAS) c.f. Gai and Kapadia (21, Proc.Roy.Soc.A) l l

29 Balance sheet (2-layer model: senior and junior)

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31 Tree-like approximation for the cascade size (Gleeson et al. 27, 28, PRE) t = 2 t = 1 Cascade condition: t =

32 Tree-like approximation for the cascade size (Gleeson et al. 27, 28, PRE) t = 2 t = 1 Cascade condition: t =

33 Simulation Simulation

34 Cascade region and the optimal seniority ratio

35

36 Identification of relationship lending in the interbank market

37 Question Substitutability of trading partners!

38 Question Measuring the substitutability of trading partners Number of trades? Frequency of trades? Volume of trades?

39 Question Measuring the substitutability of trading partners Number of trades? Frequency of trades? Volume of trades? Null model:

40 Question Measuring the substitutability of trading partners Number of trades? Frequency of trades? Volume of trades? Null model: Data:

41 u(a i,a j )=a i a j / # contacts a i a j

42 If random (= null hypothesis): If there is a strong partnership: t =1 t =2 t =3

43 Under random matching, # bilateral transactions should follow a binomial distribution: p({m ij } ~a) = Y i,j:i6=j m ij u(a i,a j ) m ij (1 u(a i,a j )) m ij, Maximum-likelihood estimator: F i (~a ) X j:j6=i m ij (a i a j ) 1 (a i a j ) =, 8 i =1,...,N,

44 # trades # unique partners # trades: real model # unique partners: real < model preferential relationship?

45 Edge-based test Under the null, m ij should follow a binomial distribution: g(m ij a* i, a* j ) = ( τ mij ) u(a* i, a* j )m ij (1 u(a*i, a* j )) τ m ij, i, j = 1,, N. m ij >m C ij indicates the presence of a significant tie.

46 Node-based test Under the null, aggregate degree Ki should follow a Poisson binomial distribution: f(k i a *) λ*k i i e λ* i K i! K i <K C i indicates node i depends on significant ties (i.e., relationship-dependent).

47 <latexit sha1_base64="jsi1/emryrkh92yyrje7d7+kua=">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</latexit> <latexit sha1_base64="jsi1/emryrkh92yyrje7d7+kua=">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</latexit> <latexit sha1_base64="jsi1/emryrkh92yyrje7d7+kua=">aaacphichvfntxrbeh2milgqlhix8tjha1mibgoiicbesnqdbw/aukdcwmrm6iww+upp7yywmt/gh/dgsrnijd/diwlxopatiediuhiwznysastyxs9fl2v+nw3g/syuqna8anwztdt4zvv+7cvtcywh27v5jexewjlhf5kvpznut4mhqtlbuv1milnmd1xaq7+ypfx+jlcgofkv3yreronuh7ejpuzz1rdmbkfyxbaztlvghpesflbx+zzs91rjpe6dkpzlicre8tjbpa6qk/rgws6e1yxqzvqubhmdwcvoiyyfqpqf7sxhqgeugggeeiz9ueg4w8dfggxcxtimvomzleukkhc2i5xca5wmn3lcztn6yub8jzvmrrqj3fx+vesndfjx/svzughfant+vxxxmnri/eyx9ntavsj75/lz4ryrgrlhzr/vpzxodpc28svyefx+cq+v7+1/ogs+w55mp+gz/wt/n+ievvmjwt65d7aklj8ifwdr6nvfbyuzdysa1tjcbf5l+rtdeigj1pm+n2aec1hei/c9wcgocgxmgw+mpthqlxodpwycl8ly/ameagd</latexit> <latexit sha1_base64="jsi1/emryrkh92yyrje7d7+kua=">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</latexit> Experiments on synthetic networks 1. Create random temporal networks 2. Assign a fraction of pairs as relationship pairs 3. Decreasing hazard prob for terminating a relationship: p norel ij (t) = b b 1 + b 2 D ij (t 1),

48 Results: synthetic network relationship parameters a b density c d fraction of significant ties Ground truth: 2%

49 Results: empirical network a # significant ties c frac. of significant ties Edge-based test b d # rel.-dependent banks frac. of rel.-dependent banks years 1 Node-based test

50 April 21 June 27 June 214 All ties Significant ties Red: Italian bank Black: foreign bank

51 Difference in interest rates Difference in trade amount (relationship - non-relationship)

52 Duration of a significant tie

53 Intraday pattern a Frequency non-relationship relationship b interest rate (bp) c amount (million Euros) 9: 1: 11: 12: 13: 14: 15: 16: 17: 18: non-relationship relationship am 1-11am11-12am 12-1pm 1-2pm 2-3pm 3-4pm 4-5pm 5-6pm non-relationship relationship 9-1am 1-11am11-12am 12-1pm 1-2pm 2-3pm 3-4pm 4-5pm 5-6pm 9: 1: 11: 12: 13: 14: 15: 16: 17: 18: am 1-11am11-12am 12-1pm 1-2pm 2-3pm 3-4pm 4-5pm 5-6pm am 1-11am11-12am 12-1pm 1-2pm 2-3pm 3-4pm 4-5pm 5-6pm time 9: 1: 11: 12: 13: 14: 15: 16: 17: 18: am 1-11am11-12am 12-1pm 1-2pm 2-3pm 3-4pm 4-5pm 5-6pm am 1-11am11-12am 12-1pm 1-2pm 2-3pm 3-4pm 4-5pm 5-6pm

54

55 Identifying relationship lending in the interbank market: A network approach Journal of Banking & Finance, in press. T. Kobayashi and Taro Takaguchi (LINE Corp),

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57 Non-negative tensor factorization Extracting patterns from a high-dimensional data Representing a tensor by the sum of outer products X xijk <latexit sha1_base64="wpbsaocpcj3huop1bw24fw+j/k=">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</latexit> R X ar cr, R 個の外積の和 br r=1 R X air bjr ckr, r=1 c1 <latexit 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58 Previous applications: Interaction patterns in online games (Sapienza et al, Informaiton, 218 ), Twitter (Panisson, et al, WWW 214 ) Detection of temporal community structure (Gauvin et al, 214, PLOS ONE)

59 Bank A Bank B 21/6/1 Lending: A to B 21/6/2 Repayment: B to A Extract intra- and inter-day trading patterns!

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61 Synthetic network

62 Fitness model: p ij,t = a i,t a j,t (a) fitness group 1 group 2 group 3 8: 1: 12: 14: 16: 18: time (b) participation probability days

63 (a) fitness group 1 group 2 group 3 (b) participation probability : 1: 12: 14: 16: 18: time days (a) (b).4.2 intraday activity interday activity.6 Component 1 8: 1: 12: 14: 16: 18: c 1 c Component 2 b 1 b 2 8: 1: 12: 14: 16: 18: time 5 1 days Component 3 b 3 8: 1: 12: 14: 16: 18: c 3 5 1

64 Determination of #components, R If #component R is large, Overidentification problem If #component R is small, Tensor decomposition will be less accurate

65 Determination of #components PARAFAC or CP model: R R R x ijk = n=1 m=1 p=1 λ nmp a in b jm c kp, λ nmp = δ nm δ mp δ np Tucker3 model: R n R m R p x ijk = n m p g nmp a in b jm c kp

66 Determination of #components PARAFAC or CP model: R R R x ijk = n=1 m=1 p=1 λ nmp a in b jm c kp, λ nmp = δ nm δ mp δ np Tucker3 model: R n R m R p x ijk = g nmp a in b jm c kp n m p

67 Determination of #components Core-consistency: CC = 1 1 R n=1 R m=1 R p=1 (g nmp λ nmp )2 R,

68 Synthetic network 1 5 Core-Consistency

69 Interbank network

70 Interbank network: core-consistency (a) 1 = 15 (b) 6 Core-Consistency

71 Interbank network: activity pattern (a) 1 Component 1 1 Component 2 1 Component activity (b) activity : 1: 12: 14: 16: 18:.1 Activity Peak of US house price Lehman failure : 1: 12: 14: 16: 18: time : 1: 12: 14: 16: 18:.1.5 (c) years activity (share) years

72 Interbank network: banks belongingness to components (a) (b) (c).1 Banks belonging to Component 1.1 Banks belonging to Component 2.1 Banks belonging to Component 3 membership level Component 1 Component 2 Component years Row average of a r c r

73 Interbank network: characterization of components (a) (b)