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Explore the dynamics of complex systems, focusing on potential critical transitions, hysteresis, and early warning signals. Dive into emergence from local interactions to global phenomena, with network models and proof of emergent properties. Delve into systemic risk in cascading processes on networks, analyzing risk levels and interactions between nodes. Investigate aggregation tools, different load distributions, and node failure variants to understand the fragility and dynamics of interconnected systems. Discuss model extensions, stochastic cascading, and approximation methods for network analysis in financial systems.
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Complexity reading group Presentation by: Thomas de Haan Paper by: J. Lorenz, S. Battison and F. Schweitzer
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…
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?
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
Micro-dynamics • 3 model classes • Constant load • Load distribution • Overload distribution
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.
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’.
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
Results • Let failing thresholds be normally distributed such that initial net fragility distribution is equal over all cases.
Results Constand load. Load redistribution Load redistribution Overload redistribution
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.
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?
