Cascades on a stochastic pulse-coupled network

Sandpile cascades on oscillator networks: The BTW model meets Kuramoto

Cascading failures abound in complex systems and the Bak-Tang-Weisenfeld (BTW) sandpile model provides a theoretical underpinning for their analysis. Yet, it does not account for the possibility of nodes having oscillatory dynamics, such as in power grids and brain networks. Here, we consider a network of Kuramoto oscillators upon which the BTW model is unfolding, enabling us to study how the feedback between the oscillatory and cascading dynamics can lead to new emergent behaviors. We assume that the more out-of-sync a node is with its neighbors, the more vulnerable it is and lower its load-carrying capacity accordingly. Also, when a node topples and sheds load, its oscillatory phase is reset at random. This leads to novel cyclic behavior at an emergent, long timescale. The system spends the bulk of its time in a synchronized state where load builds up with minimal cascades. Yet, eventually, the system reaches a tipping point where a large cascade triggers a "cascade of larger cascades," which can be classified as a dragon king event. The system then undergoes a short transient back to the synchronous, buildup phase. The coupling between capacity and synchronization gives rise to endogenous cascade seeds in addition to the standard exogenous ones, and we show their respective roles. We establish the phenomena from numerical studies and develop the accompanying mean-field theory to locate the tipping point, calculate the load in the system, determine the frequency of the long-time oscillations, and find the distribution of cascade sizes during the buildup phase.

Knowledge transfer across scientific disciplines

Thomas Kuhn suggested that symbolic generalizations are applied to concrete systems by a process involving exemplars and analogical reasoning. Using the related concepts of theoretical and formal templates, I argue that the process of applying templates can in some cases be made explicit and that we do not need to rely on similarity relations and tacit knowledge. In so doing I show how some formal models can be transferred from one scientific field to another. Examples include scale-free networks, the Lotka-Volterra model from biology, and the Goodwin model in economics. I also argue that this explicit approach has advantages over the more psychologically oriented approach of Kuhn and explain the sense in which templates do and do not produce unification.

Pattern phase diagram of spiking neurons on spatial networks

We study an abstracted model of neuronal activity via numerical simulation and report spatiotemporal pattern formation and criticallike dynamics. A population of pulse coupled, discretized, relaxation oscillators is simulated over networks with varying edge density and spatial embeddedness. For intermediate edge density and sufficiently strong spatial embeddedness, we observe a spatiotemporal pattern in the field of oscillator phases, visually resembling the surface of a frothing liquid. Increasing the edge density results in a distribution of neuronal avalanche sizes which follows a power law with exponent one (Zipf's law). Further increasing edge density yields metastability between pattern formation and synchronization, before transitioning entirely into synchrony.

A Financial Market Model Incorporating Herd Behaviour

Herd behaviour in financial markets is a recurring phenomenon that exacerbates asset price volatility, and is considered a possible contributor to market fragility. While numerous studies investigate herd behaviour in financial markets, it is often considered without reference to the pricing of financial instruments or other market dynamics. Here, a trader interaction model based upon informational cascades in the presence of information thresholds is used to construct a new model of asset price returns that allows for both quiescent and herd-like regimes. Agent interaction is modelled using a stochastic pulse-coupled network, parametrised by information thresholds and a network coupling probability. Agents may possess either one or two information thresholds that, in each case, determine the number of distinct states an agent may occupy before trading takes place. In the case where agents possess two thresholds (labelled as the finite state-space model, corresponding to agents’ accumulating information over a bounded state-space), and where coupling strength is maximal, an asymptotic expression for the cascade-size probability is derived and shown to follow a power law when a critical value of network coupling probability is attained. For a range of model parameters, a mixture of negative binomial distributions is used to approximate the cascade-size distribution. This approximation is subsequently used to express the volatility of model price returns in terms of the model parameter which controls the network coupling probability. In the case where agents possess a single pulse-coupling threshold (labelled as the semi-infinite state-space model corresponding to agents’ accumulating information over an unbounded state-space), numerical evidence is presented that demonstrates volatility clustering and long-memory patterns in the volatility of asset returns. Finally, output from the model is compared to both the distribution of historical stock returns and the market price of an equity index option.