Complexity reading group

1. Complexity reading group Presentation by: Thomas de Haan Paper by: J. Lorenz, S. Battison and F. Schweitzer

2. Short recap • Focus on dynamical systems with potential critical transitions, hysteresis, etc. • Examples, discussion on how to find statistical evidence and presence of ‘early warning signals’ for coming transitions. • But what about…

3. Emergence • How to go from local interactions to global phenomenon. • ‘Aggregation’, or what comes before the dynamical system is written down. • Simulations: Network model, game of life • Proof of emergent properties: Cucker and Smale • What other tools do we have to analyze these systems?

4. Article of today • Systemic risk in a unifying framework for cascading processes on networks. • Using network analysis to model systemic risk. • Risk level depends on presence of ‘cascades’ • Different kind of interactions between nodes. (which induce difference in global dynamics) • Aggregation tools in this paper: • From individual node characteristics to distributions • Mean field approach

5. The Network

6. The Network (2)

7. Micro-dynamics • 3 model classes • Constant load • Load distribution • Overload distribution

8. Constant load • Failure of node i causes a predetermined increase of fragility to it’s neighbors • 2 variants • Inward variant: fragility of node i depends on the number of failed neighbors • Outward variant: increase in fragility of i if neighbor j fails depends on number of neighbors of j.

9. Load distribution • When node i fails, all of it’s load (fragility) is redistributed among “neighbors”. • 2 variants • LLSC variant: load is redistributed via failed nodes to the nearest surviving nodes. (which could just be the neighbors) • LLSS variant: load is redistributed only to the surviving neighbors. In case of no surviving neighbors, load is ‘shed’.

10. Overload distribution • When node i fails, only difference between the load and the capacity, the net fragility, is distributed among neighbors. (Think of bankruptcy). • Also here 2 variants • LLSC • LLSS

11. Aggregation, step 1.

12. Aggregation, step 2

13. Aggregation (finish)

14. Aggregation (finish)

15. Results • Let failing thresholds be normally distributed such that initial net fragility distribution is equal over all cases.

16. Results Constand load. Load redistribution Load redistribution Overload redistribution

17. Extensions • Taking into account network topology (for example by assuming each node has k neighbors). • Writing system in terms of net fragility distribution, instead of X(t). • Stochastic cascading: probability of failed state at time t dependent on net fragility (and perhaps also directly dependent on previous state). • Extension to Voter/herding model and contagion/epidemic model.

18. Discussion • How good of an approximation is the mean field approach? (are there any known theorems on the accuracy of a mean field prediction?) • What approximation methods are there for aggregation apart from mean field? • What elements should be added to the network model to make it a serious model candidate for the financial/banking network?

19. Thank you!