UC Davis

This thesis takes a mathematical journey through concepts of synchronization, cascades, self-organized criticality, and flows on networks, establishing several novel emergent phenomena. It provides new analytical tools insights into the collective behaviors that can arise from the co-evolution of different dynamical processes on complex networks.

Emergence entails the birth of surplus complexity in a macroscopic system, when the whole becomes more than the sum of its parts. For example, ant colonies, composed of many seemingly dull and unimpressive individual ants display surprising structure and organization on the system level. And a cup of coffee that we label as hot and wet, is composed of molecules possessing neither of these properties. Emergent properties of some systems, e.g., in thermodynamics, can be summarized with concise, elegant equations. Yet, in many systems, even the full knowledge of the fundamental rules governing each element isn't enough. The field of Chemistry, for example, relies on quantum mechanics and can in principle be fully simulated given high enough computational power, yet it remains a complex, open ended field of research with centuries of tradition and no end in sight. This is mainly due to two reasons: One is the superiority of direct analytical models for the emergent properties over a blind search with simulations. The second is the lack of remotely enough computational power required for such simulations.

Complexity science studies the question of emergence in general, how the system can be more than the sum of its parts? Or why is it that ``more is different," as the esteemed Philip Anderson put it. The field of complex systems mainly focuses on the types of collective behaviors with wide applicability, including but not limited to flows, consensus formation, synchronization, cascades, self-organized criticality, and extreme events. Emergent phenomena usually rely on the interactions between subsystems, and hence network science with the primary focus on system composition through the relationships (the edges) between the subsystems (the nodes) is particularly well suited for answering such questions.

In this thesis, we will embark on a mathematical journey through these concepts, heavily relying on both, complex systems' and network scientific approaches, finding new analytical insights and supporting our findings with numerical simulations.

We start by coupling two quintessential models in complexity science: the Bak-Tang-Wiesenfeld (BTW) sandpile model of self-organized critical cascades and the Kuramoto model of synchronization. This fusion reveals emergent long-time oscillations not present in either model alone, punctuated by extreme dragon king events in the form of system-spanning cascades of cascades. Our analysis allows the characterization of the tipping point leading to dragon kings, frequency of oscillations, and cascade size distributions.

Abstracting from this specific phenomenon, we next explore different ways through which models of self-organized criticality could give rise to dragon kings. This leads to a natural taxonomy of different types of dragon king events, and provides a so far unknown necessary condition on self-organized criticality in general.

Next, we examine the interplay and co-evolution of diverse mobile agents on a network. Analogous to the social dynamics of intrinsically diverse individuals who navigate between and interact within various physical or virtual locations, agents in our model traverse a complex network of heterogeneous environments and engage with everyone they encounter. The precise nature of agents' internal dynamics and the various interactions that nodes induce are left unspecified and can be tailored to suit the requirements of individual applications. We derive effective dynamical equations for agent states which are instrumental in investigating thresholds of consensus, devising effective attack strategies to hinder coherence, and designing optimal network structures with inherent node variations in mind.

Finally, we consider flows on networks under various symmetries and conservations, such as human transportation, internet packets, power grids, and mailed packages. We construct a hierarchy of field-theoretic models, capable of accurately modeling the throughput, predicting sustainability collapse under growing demand, and allowing us to explore a spectrum of routing strategies, locate flow bottlenecks, and determine the potential benefits of dissipation.