Mechanisms of Systemic Risk Contagion, Reinforcement, - PowerPoint PPT Presentation

Mechanisms of Systemic Risk Frank Schweitzer ZIF Workshop · Bielefeld, Germany 01-11 September 2009 1 / 36 Mechanisms of Systemic Risk – Contagion, Reinforcement, Redistribution Frank Schweitzer fschweitzer@ethz.ch in collaboration with: S. Battiston (Zurich), J. Lorenz (Zurich) J. Lorenz, S. Battiston, F. Schweitzer: Systemic Risk in a Unifying Framework for Cascading Processes on Networks, European Physical Journal B (2009, forthcoming), http://arxiv.org/abs/0907.5325 Chair of Systems Design http://www.sg.ethz.ch/

Mechanisms of Systemic Risk Frank Schweitzer ZIF Workshop · Bielefeld, Germany 01-11 September 2009 2 / 36 Motivation Motivation systemic risk ◮ system: comprised of many interacting agents ◮ risk that whole system fails Chair of Systems Design http://www.sg.ethz.ch/

Mechanisms of Systemic Risk Frank Schweitzer ZIF Workshop · Bielefeld, Germany 01-11 September 2009 2 / 36 Motivation Motivation systemic risk ◮ system: comprised of many interacting agents ◮ risk that whole system fails examples ◮ financial sector (banks, companies) ◮ epidemics (humans: SARS, plaque, animals: bird flu) ◮ power grids (blackout due to overload) ◮ material science (bundles of fibers) common features ◮ failure of few agents is amplified ⇒ system failure ◮ individual agent dynamics: fragility, threshold for failure ◮ interaction: network topology Chair of Systems Design http://www.sg.ethz.ch/

Mechanisms of Systemic Risk Frank Schweitzer ZIF Workshop · Bielefeld, Germany 01-11 September 2009 3 / 36 Motivation Aim: develop a common framework for systemic risk cover examples from different areas ◮ what do they have in common?, what makes them unique? highlight critical conditions ◮ role of heterogeneity?, leads diversification to larger systemic risk? allow prediction and prevention ◮ how does the fraction of failed nodes evolve over time? ◮ Can we counterbalance failure propagation? Chair of Systems Design http://www.sg.ethz.ch/

Mechanisms of Systemic Risk Frank Schweitzer ZIF Workshop · Bielefeld, Germany 01-11 September 2009 4 / 36 Complex Systems Theory of Complex Systems system comprised of a large number of strongly interacting (similar) subsystems (entities, processes, or ’ agents ’) - - ◮ examples: brain, insect societies (ants, bees, termites), ... , , - , - , , , , , - - complex network: agents ⇒ nodes , interactions ⇒ links - - - - , , , , - - Micro Level Macro Level ⇔ challenge: The micro-macro link ◮ How are the properties of the elements and their interactions (“microscopic” level) related to the dynamics and the properties of the whole system (“macroscopic” level)? Chair of Systems Design http://www.sg.ethz.ch/

Mechanisms of Systemic Risk Frank Schweitzer ZIF Workshop · Bielefeld, Germany 01-11 September 2009 5 / 36 Micro and Macro Description Micro Dynamics: Individual Agent node i with interaction matrix A ◮ state s i ( t ) ∈ { 0 , 1 } : ’healthy’, ’failed’ ⇒ s ( t ) = s 1 ( t ) , ..., s i ( t ) , ..., s n ( t ) ◮ fragility φ i ( t ) > 0: susceptibility to fail, may depend on other nodes ◮ (individual) threshold θ i for failure key variable: net fragility : z i ( t ) = φ i ( t , s , A ) − θ i deterministic dynamics s i ( t + 1) = Θ[ z i ( t )] ◮ s i = 1 if z i ( t ) ≥ 0; s i = 0 if z i ( t ) < 0 Chair of Systems Design http://www.sg.ethz.ch/

Mechanisms of Systemic Risk Frank Schweitzer ZIF Workshop · Bielefeld, Germany 01-11 September 2009 6 / 36 Micro and Macro Description Macro Dynamics: System Level global fraction of failed nodes ⇒ prediction n X ( t ) = 1 � s i ( t ) n i =1 dynamics ◮ assumption: probability distribution p ( z ), ( z i = φ i − θ i ) � ∞ � 0 X ( t + 1) = p z ( t ) ( z ) dz = 1 − p z ( t ) ( z ) dz 0 −∞ ◮ cascading process: failures modify net fragility of other nodes p z ( t +1) = F ( p z ( t ) ) systemic risk: X ( t → ∞ ) = X ⋆ → 1 ◮ iterate X ( t ) dependent on φ (0), θ (0) Chair of Systems Design http://www.sg.ethz.ch/

Mechanisms of Systemic Risk Frank Schweitzer ZIF Workshop · Bielefeld, Germany 01-11 September 2009 7 / 36 Different Model Classes Models with constant load assumptions: ◮ ’load’ of nodes is constant (equals one) ◮ changes in fragility φ i do not depend on φ j (i) ’inward’ variant : increase of fragility depends on in-degree φ i ( t ) = 1 � s j ( t ) k in i j ∈ nb in ( i , A ) examples: ◮ model of social activation (Granovetter, 1978) ◮ model of bankrupcy cascades (Battiston et. al , 2009): firms characterized by robustness ρ i ⇒ φ i , θ i = ρ 0 i / a i − a � ρ i ( t + 1) = ρ 0 s i ( t ) k in i j ∈ nb in ( i , A ) Chair of Systems Design http://www.sg.ethz.ch/

Mechanisms of Systemic Risk Frank Schweitzer ZIF Workshop · Bielefeld, Germany 01-11 September 2009 8 / 36 Different Model Classes Example: Inward variant - node C fails 0.55 non-failed node 0 G 0.55 0.55 failing node 0 0.5 0 0 H F E 0.3 failed node 0 D θ φ 0 label 0.55 0 0.7 0 0 C I B z 0.7 0 A -1 0 1 failing! Chair of Systems Design http://www.sg.ethz.ch/

Mechanisms of Systemic Risk Frank Schweitzer ZIF Workshop · Bielefeld, Germany 01-11 September 2009 8 / 36 Different Model Classes Example: Inward variant - node C fails non-failed node 0.55 0 failing node 0.55 0.55 0 0.5 0 0 0.5 0.3 failed node θ φ 0 label 0.55 0 0.7 0 1 z 0.7 1 -1 0 1 failing! Chair of Systems Design http://www.sg.ethz.ch/

Mechanisms of Systemic Risk Frank Schweitzer ZIF Workshop · Bielefeld, Germany 01-11 September 2009 8 / 36 Different Model Classes Example: Inward variant - node C fails non-failed node 0.55 0 failing node 0.55 0.55 0.2 0.5 0 0 0.5 0.3 failed node θ φ 0 label 0.55 0.7 1 0 1 z 0.7 1 -1 0 1 failing! low degree node ⇒ high vulnerability to fail ◮ failure causes little damage, cascade stops after 2 steps ⇒ no ’systemic risk’ Chair of Systems Design http://www.sg.ethz.ch/

Mechanisms of Systemic Risk Frank Schweitzer ZIF Workshop · Bielefeld, Germany 01-11 September 2009 9 / 36 Different Model Classes Example: Inward variant - node E fails 0.55 non-failed node 0 G 0.55 0.55 failing node 0 0 0 0 H F E 0.3 failed node 0 D θ φ 0.3 label 0.55 0 0.7 0 0 C I B z 0.7 0 A -1 0 1 failing! Chair of Systems Design http://www.sg.ethz.ch/

Mechanisms of Systemic Risk Frank Schweitzer ZIF Workshop · Bielefeld, Germany 01-11 September 2009 9 / 36 Different Model Classes Example: Inward variant - node E fails non-failed node 0.55 1 failing node 0.55 0.55 1 0 1 0 failed node 0.5 0.3 θ φ 0.3 label 0.55 0.7 0 1 0 z 0.7 0 -1 0 1 failing! Chair of Systems Design http://www.sg.ethz.ch/

Mechanisms of Systemic Risk Frank Schweitzer ZIF Workshop · Bielefeld, Germany 01-11 September 2009 9 / 36 Different Model Classes Example: Inward variant - node E fails non-failed node 0.55 1 failing node 0.55 0.55 1 0 1 1 failed node 0.5 0.3 θ φ 0.33 0.3 label 0.55 0.7 1 0 z 0.7 0 -1 0 1 failing! Chair of Systems Design http://www.sg.ethz.ch/

Mechanisms of Systemic Risk Frank Schweitzer ZIF Workshop · Bielefeld, Germany 01-11 September 2009 9 / 36 Different Model Classes Example: Inward variant - node E fails non-failed node 0.55 1 failing node 0.55 0.55 0 1 1 1 failed node 0.3 1 θ φ 0.33 0.3 label 0.55 0.7 1 1 z 0.7 1 -1 0 1 failing! Chair of Systems Design http://www.sg.ethz.ch/

Mechanisms of Systemic Risk Frank Schweitzer ZIF Workshop · Bielefeld, Germany 01-11 September 2009 9 / 36 Different Model Classes Example: Inward variant - node E fails non-failed node 0.55 1 failing node 0.55 0.55 0 1 1 1 failed node 0.3 1 θ φ 0.3 label 0.55 0.7 1 1 1 z 0.7 1 -1 0 1 failing! high degree node ⇒ low vulnerability to fail ◮ failure causes big damage (to low degree nodes), cascade involves all nodes ⇒ ’systemic risk’ Chair of Systems Design http://www.sg.ethz.ch/

Mechanisms of Systemic Risk Frank Schweitzer ZIF Workshop · Bielefeld, Germany 01-11 September 2009 10 / 36 Different Model Classes Models with constant load (ii) ’outward variant’: increase of fragility depends on out-degree ◮ load of failing node (i.e. 1) is shared equally among neighbors s j ( t ) � φ i ( t ) = k out j j ∈ nb in ( i , A ) undirected, regular networks: ◮ inward and outward variant equivalent heterogeneous degree: ◮ failing high-degree nodes cause less damage then low-degree nodes high-degree node: ◮ high vulnerability if connected to low-degree nodes (dissortative networks) Chair of Systems Design http://www.sg.ethz.ch/

Mechanisms of Systemic Risk Frank Schweitzer ZIF Workshop · Bielefeld, Germany 01-11 September 2009 11 / 36 Different Model Classes Example: Outward variant - node C fails 0.55 non-failed node 0 G 0.55 0.55 failing node 0 0.5 0 0 H F E 0.3 failed node 0 D θ φ 0 label 0.55 0 0.7 0 0 C I B z 0.7 0 A -1 0 1 failing! Chair of Systems Design http://www.sg.ethz.ch/

Mechanisms of Systemic Risk Frank Schweitzer ZIF Workshop · Bielefeld, Germany 01-11 September 2009 11 / 36 Different Model Classes Example: Outward variant - node C fails non-failed node 0.55 0 failing node 0.55 0.55 0 0.5 0 0 0.33 0.3 failed node θ φ 0 label 0.55 0 0.33 0.7 0 z 0.33 0.7 -1 0 1 failing! Chair of Systems Design http://www.sg.ethz.ch/