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An important challenge in several disciplines is to understand how sudden changes can propagate among coupled systems. Examples include the synchronization of business cycles, population collapse in patchy ecosystems, markets shifting to a new technology platform, collapses in prices and in confidence in financial markets, and protests erupting in multiple countries. A number of mathematical models of these phenomena have multiple equilibria separated by saddlenode bifurcations. We study this behaviour in its normal form as fast–slow ordinary differential equations. In our model, a system consists of multiple subsystems, such as countries in the global economy or patches of an ecosystem. Each subsystem is described by a scalar quantity, such as economic output or population, that undergoes sudden changes via saddlenode bifurcations. The subsystems are coupled via their scalar quantity (e.g. trade couples economic output; diffusion couples populations); that coupling moves the locations of their bifurcations. The model demonstrates two ways in which sudden changes can propagate: they can cascade (one causing the next), or they can hop over subsystems. The latter is absent from classic models of cascades. For an application, we study the Arab Spring protests. After connecting the model to sociological theories that have bistability, we use socioeconomic data to estimate relative proximities to tipping points and Facebook data to estimate couplings among countries. We find that although protests tend to spread locally, they also seem to ‘hop' over countries, like in the stylized model; this result highlights a new class of temporal motifs in longitudinal network datasets. Coupled catastrophes: sudden shifts cascade and hop among interdependent systems Charles D. Brummitt, George Barnett, Raissa M. D'Souza J. R. Soc. Interface 2015 12 20150712; DOI: 10.1098/rsif.2015.0712. Published 11 November 2015. Open Access.
A model of a banking network predicts the balance of high and lowpriority debts that ensures financial stability. Highlighted in a synopsis: http://physics.aps.org/synopsisfor/10.1103/PhysRevE.91.062813 Cascades in multiplex financial networks with debts of different seniority The seniority of debt, which determines the order in which a bankrupt institution repays its debts, is an important and sometimes contentious feature of financial crises, yet its impact on systemwide stability is not well understood. We capture seniority of debt in a multiplex network, a graph of nodes connected by multiple types of edges. Here an edge between banks denotes a debt contract of a certain level of seniority. Next we study cascading default. There exist multiple kinds of bankruptcy, indexed by the highest level of seniority at which a bank cannot repay all its debts. Selfinterested banks would prefer that all their loans be made at the most senior level. However, mixing debts of different seniority levels makes the system more stable in that it shrinks the set of network densities for which bankruptcies spread widely. We compute the optimal ratio of senior to junior debts, which we call the optimal seniority ratio, for two uncorrelated ErdősRényi networks. If institutions erode their buffer against insolvency, then this optimal seniority ratio rises; in other words, if default thresholds fall, then more loans should be senior. We generalize the analytical results to arbitrarily many levels of seniority and to heavytailed degree distributions. Charles D. Brummitt and Teruyoshi Kobayashi Phys. Rev. E 91, 062813 (2015) Published June 24, 2015
We explore a model of the interaction between banks and outside investors in which the ability of banks to issue inside money (shortterm liabilities believed to be convertible into currency at par) can generate a collapse in asset prices and widespread bank insolvency. The banks and investors share a common belief about the future value of certain longterm assets, but they have different objective functions; changes to this common belief result in portfolio adjustments and trade. Positive belief shocks induce banks to buy risky assets from investors, and the banks finance those purchases by issuing new shortterm liabilities. Negative belief shocks induce banks to sell assets in order to reduce their chance of insolvency to a tolerably low level, and they supply more assets at lower prices, which can result in multiple marketclearing prices. A sufficiently severe negative shock causes the set of equilibrium prices to contract (in a manner given by a cusp catastrophe), causing prices to plummet discontinuously and banks to become insolvent. Successive positive and negative shocks of equal magnitude do not cancel; rather, a banking catastrophe can occur even if beliefs simply return to their initial state. Capital requirements can prevent crises by curtailing the expansion of balance sheets when beliefs become more optimistic, but they can also force larger price declines. Emergency asset price supports can be understood as attempts by a central bank to coordinate expectations on an equilibrium with solvency. Brummitt CD, Sethi R, Watts DJ (2014) Inside Money, Procyclical Leverage, and Banking Catastrophes. PLoS ONE 9(8): e104219. doi: 10.1371/journal.pone.0104219
The BakTangWiesenfeld (BTW) sandpile process is an archetypal, stylized model of complex systems with a critical point as an attractor of their dynamics. This phenomenon, called selforganized criticality, appears to occur ubiquitously in both nature and technology. Initially introduced on the twodimensional lattice, the BTW process has been studied on network structures with great analytical successes in the estimation of macroscopic quantities, such as the exponents of asymptotically powerlaw distributions. In this article, we take a microscopic perspective and study the inner workings of the process through both numerical and rigorous analysis. Our simulations reveal fundamental flaws in the assumptions of past phenomenological models, the same models that allowed accurate macroscopic predictions; we mathematically justify why universality may explain these past successes. Next, starting from scratch, we obtain microscopic understanding that enables mechanistic models; such models can, for example, distinguish a cascade's area from its size. In the special case of a 3regular network, we use selfconsistency arguments to obtain a zeroparameter mechanistic (bottomup) approximation that reproduces nontrivial correlations observed in simulations and that allows the study of the BTW process on networks in regimes otherwise prohibitively costly to investigate. We then generalize some of these results to configuration model networks and explain how one could continue the generalization. The numerous tools and methods presented herein are known to enable studying the effects of controlling the BTW process and other selforganizing systems. More broadly, our use of multitype branching processes to capture information bouncing back and forth in a network could inspire analogous models of systems in which consequences spread in a bidirectional fashion. Noël, P. A., Brummitt, C. D., & D'Souza, R. M. (2014). Bottomup model of selforganized criticality on networks. Physical Review E, 89, 012807. doi:10.1103/PhysRevE.89.012807

We introduce a new kind of percolation on finite graphs called jigsaw percolation. This model attempts to capture networks of people who innovate by merging ideas and who solve problems by piecing together solutions. Each person in a social network has a unique piece of a jigsaw puzzle. Acquainted people with compatible puzzle pieces merge their puzzle pieces. More generally, groups of people with merged puzzle pieces merge if the groups know one another and have a pair of compatible puzzle pieces. The social network solves the puzzle if it eventually merges all the puzzle pieces. For an Erdős–Rényi social network with n vertices and edge probability p_n, we define the critical value p_c(n) for a connected puzzle graph to be the p_n for which the chance of solving the puzzle equals 1/2. We prove that for the ncycle (ring) puzzle, p_c(n)=Θ(1/log n), and for an arbitrary connected puzzle graph with bounded maximum degree, p_c(n)=O(1/log n) and ω(1/n^b)for any b>0. Surprisingly, with probability tending to 1 as the network size increases to infinity, social networks with a powerlaw degree distribution cannot solve any boundeddegree puzzle. This model suggests a mechanism for recent empirical claims that innovation increases with social density, and it might begin to show what social networks stifle creativity and what networks collectively innovate.
Brummitt, Charles D.; Chatterjee, Shirshendu; Dey, Partha S.; Sivakoff, David. Jigsaw percolation: What social networks can collaboratively solve a puzzle?. Ann. Appl. Probab. 25 (2015), no. 4, 20132038. doi:10.1214/14AAP1041. http://projecteuclid.org/euclid.aoap/1432212435.
Threshold cascade models have been used to describe the spread of behavior in social networks and cascades of default in financial networks. In some cases, these networks may have multiple kinds of interactions, such as distinct types of social ties or distinct types of financial liabilities; furthermore, nodes may respond in different ways to influence from their neighbors of multiple types. To start to capture such settings in a stylized way, we generalize a threshold cascade model to a multiplex network in which nodes follow one of two response rules: some nodes activate when, in at least one layer, a large enough fraction of neighbors is active, while the other nodes activate when, in all layers, a large enough fraction of neighbors is active. Varying the fractions of nodes following either rule facilitates or inhibits cascades. Near the inhibition regime, global cascades appear discontinuously as the network density increases; however, the cascade grows more slowly over time. This behavior suggests a way in which various collective phenomena in the real world could appear abruptly yet slowly. Lee, K.M., Brummitt, C. D., & Goh, K.I. (2014). Threshold cascades with response heterogeneity in multiplex networks. Physical Review E, 90(6), 062816. doi:10.1103/PhysRevE.90.062816
Controlling selforganizing systems is challenging because the system responds to the controller. Here, we develop a model that captures the essential selforganizing mechanisms of BakTangWiesenfeld (BTW) sandpiles on networks, a selforganized critical (SOC) system. This model enables studying a simple control scheme that determines the frequency of cascades and that shapes systemic risk. We show that optimal strategies exist for generic cost functions and that controlling a subcritical system may drive it to criticality. This approach could enable controlling other selforganizing systems. Noël, P.A., Brummitt, C. D., & D'Souza, R. M. (2013). Controlling SelfOrganizing Dynamics on Networks Using Models that SelfOrganize. Physical Review Letters, 111(7), 078701. doi:10.1103/PhysRevLett.111.078701
