Exploring sandpile cascades on oscillator networks
Abstract
Problem overview: Cascading failures are a hallmark of complex systems and the BTW sandpile model of cascading dynamics forms a cornerstone for our understanding of such failures in systems ranging from avalanches and forest fires to electric power grids and brain networks. The latter two are systems of tremendous practical importance, more so they are examples of oscillator networks. Yet, the BTW model does not account for this. Hence there is a crucial need to understand the interplay between the oscillatory and sandpile dynamics to identify the range of new failure mechanisms and opportunities to enhance system performance and develop novel control interventions for the broad range of systems that have both oscillatory and cascading dynamics. Goals: In this project we propose to lay the foundation for understanding SOC sandpile cascades on networks of oscillators. The goals are: (1) to discover and analyze the new emergent behaviors that arise from the interactions; (2) to understand how the oscillator nodal dynamics impact the distribution of sandpiles; and (3) to identify parameters that we can tune to influence the cascading dynamics and potential local intervention strategies to control the cascades. Approach: Our focus is on coupling the quintessential model of cascades, the BTW model, to the quintessential model of synchronization, the Kuramoto model on a network system. Thus each node in the network is an oscillator with a particular natural frequency that has capacity to handle a certain amount of load. The load is modeled as discrete grains of sand that arrive on nodes chosen uniformly at random. Once the load on a node exceeds its capacity, the node "topples" and sheds its load to neighbors who may in-turn topple and so on. There are several natural choices for coupling together the dynamics. Inspired by electric power networks, where synchronization between elements is of fundamental importance, it is natural to lower the sand carrying capacity of nodes that are out of synchronization with their neighbors, as they are more susceptible to failure. When a node does topple the simplest scenario is that it sheds its load to neighbors and its capacity is subsequently fully restored with its phase reset at random. Our preliminary work shows that this leads to long-time oscillatory behavior with a long and desirable build-up phase where oscillators are fully synchronized and cascades are largely avoided, but ultimately the system reaches a tipping point where, in contrast to the BTW model, a large cascade triggers an even larger cascade, leading to a "cascade of cascades" dragon king event, after which the system has a short transient dynamic that restores full synchrony and returns the system to the build-up stage. The frequency of the long-term oscillations is much less than the natural frequency of the oscillators, introducing a new emergent timescale and long-range order in the system, opening opportunities for self-organized interventions and control. We will also explore a slightly more complicated combination of the BTW and Kuramoto models accounting for inertia. Here when a node gains more load its natural frequency slows down, and as it sheds load, its natural frequency speeds up. This dampens the long-time oscillations described above and leads to novel phenomena that we are currently mapping out including reducing large cascades. We will use a combination of numerical simulation and analytic treatment using branching processes and generating functions. In preliminary work we have developed the numerical studies to establish the behavior of the first model and the analytic treatment to predict the frequency of the long-term oscillations and cascade size distributions.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Apr 19, 2023
- Source ID
- W911NF2310087
Entities
People
- Raissa D Souza
Organizations
- Army Contracting Command
- United States Army
- University of California, Davis