The Curious Case of Complex Systems

Der Fliege den Ausweg aus dem Fliegenglas zeigen.
(Show the flies a way out of the fly bottle)

Ludwig Wittgenstein, Philosophical Investigations (1952)

Since a young age I have been fascinated with understanding the underlying mechanisms of our world. From the intricate workings of toys and machines to grand-scale phenomena like tectonic plate movement, from the complexity of language to the mysteries of living organisms; I would immerse myself in these topics, eager to uncover how their behaviors and properties emerged. This intellectual curiosity crystallized at the end of high school when I stood at a crossroads: pursue biology or mathematics. I was particularly captivated by neuroscience and the profound question of how the brain generates consciousness, which led me to explore the philosophy of mind and the hard problem of consciousness. I sought to understand how networks of neurons could give rise to our sense of self and create meaning. Most importantly, I wanted to understand how these processes emerged from both mechanistic and computational perspectives.

Over time, my interest expanded beyond neuroscience to encompass other natural systems exhibiting emergent behaviors. The mesmerizing murmurations of starlings and the synchronized flashing of fireflies became compelling examples of how simple interactions could produce complex, coordinated behaviors. This journey reflects my evolving passion for tackling genuine computational problems in complex systems.

This body of work represents my intellectual maturation from curious observer to complexity scientist. It embodies my commitment to unraveling systems where simple components interact to produce phenomena greater than—and often qualitatively different from—the sum of their parts. Through this lens, I’ve found my place at the intersection of multiple disciplines, using complexity science to illuminate nature’s most captivating puzzles.

Yet complex systems present a unique challenge: they offer both intuitive understanding and persistent elusiveness. Unlike the purer domains of physics and mathematics, grasping the essence of complex systems often feels like navigating quicksand — resisting clear and precise understanding, in part caused by imprecise definitions. Traditionally, causal interventions are difficult to apply because they can alter the system in nonlinear ways, complicating the identification of cause and effect.

At the heart of this thesis was to understand the relationship between form and function within complex systems. It touched upon a diverse range of topics to weave a combined story from the perspective of interventions. To understand how the form relates to the function of a complex system, we must first understand what a complex system is, how function relates to forms and vice versa, and how potentially these two crucial phenomena relate to each other over time.

The opening chapter provides an overview of the history of complexity science and offers a modern definition of complex systems. At its core, a complex system is defined by nonlinear interactions among its parts. These interactions create constraints where the whole is different from, and often less predictable than, the sum of its parts. Complex systems are characterized by simpler system-level correlations compared to the complex joint degrees of freedom of the individual components.

This discussion ties into the broader holism versus reductionism debate in complexity science. Complexity scientists aim to understand how the correlations among parts produce macroscopic behavior. By breaking phenomena into their necessary and sufficient parts, we achieve a form of compression that reflects the macroscopic description. From this perspective, reductionism and holism are not opposed but are complementary, reflecting the degree to which a problem can be simplified. Furthermore, complexity arises not necessarily from a large number of interactions but from the nonlinear nature of these interactions. Thus, a complex system requires at least two interacting parts to manifest complexity.

In subsequent chapters, we explored how complexity arises within various systems. These chapters serve as a comprehensive introduction to the field, focusing on the outcomes and impacts of interventions in complex environments. These findings reveal that system dynamics are governed not merely by discrete components but by the complex interplay between them—a relationship in which form and function achieve parity.

The connection between system behavior and its components is explored in detail through novel applications of information theory. In particular, we discussed in Chapter 2 how the connectivity of complex systems does not tell the full story of the dynamic behavior of the system. The simple Ising model was used as a model system as its one of the simplest dynamics that generate complex behavior. Instead, the information flow between components provides a more nuanced understanding of the system’s dynamics. The dynamic importance of a nodes is a function of the environment a system is situated in as well as the rules that exist between these agents. That is, only through the joint of both can we understand the intricacies between the relation of the parts to the whole. Furthermore, it implies that a similar structure can yield different dynamical properties when the dynamical rules change.

The relationship between form and function is examined in greater depth in Chapter 3. This chapter investigates metastable transitions in spin glass systems by analyzing the information flow of individual nodes as they approach a tipping point. This analysis provides a more sophisticated perspective on the findings presented in Chapter 2, revealing that a node’s dynamic significance can vary depending on the system’s current regime.

Notably, weakly connected nodes can drive a system toward a tipping point, triggering a cascade effect that forces more strongly connected nodes to change state. During the actual transition, strongly connected nodes undergo state changes that ultimately stabilize the system in a new low-energy configuration. These strongly connected nodes exhibit high information sharing with the system’s future state. Paradoxically, weakly connected nodes also demonstrate significant information sharing, but through a different mechanism: their relative freedom from constraints serves as an indicator that the system has returned to a stable regime characterized by minor fluctuations.

The analysis revealed in addition that the size of interventions can have a profound impact on system dynamics (Chapter 2, Chapter 4). This has significant implications for understanding complex systems and unraveling the mechanisms behind emergent phenomena. However, it is essential to carefully consider a system’s inherent characteristics when interpreting intervention results, as oversized or poorly targeted interventions can introduce additional costs or unintended side effects. For example, in Chapter 2, we observed that the intervention size can generate dynamics that are different from the original unperturbed dynamics. Furthermore, the analysis performed in Chapter 4 revealed that the hysteresis effect may profoundly affect policy made to perturb a real-world system. That is, when the size of the intervention increases, it may question our proposed policy on the effectiveness of reaching a particular goal, such as preventing the emergence of criminal organizations. Prevention, in these cases, may be preferred over brute-force interventions with limited effectiveness.

When to intervene may be just as important as where or how strong. Chapter 3 examined the timing of interventions, demonstrating the importance of precise and timely actions to avoid detrimental tipping points. This finding suggests that intervention strategies should be crafted with a nuanced understanding of system dynamics, favoring targeted measures over sweeping changes.

The co-evolutionary dynamics of complex systems proved particularly challenging in Chapter 5. Most research utilizes a separation of time scales to study the dynamics on a network rather than the co-evolution of the two. The results indicate that under co-evolution multi-role coordination may thrive or falter depending on the initial state of the system and external parameters. This chapter discusses the unpredictability of system evolution and how interventions can sometimes lead to negative outcomes over time – that is, it does not achieve the intended result of reducing the level of collaboration but in fact enhance it – emphasizing the need for a deeper understanding of long-term system behavior and potential unintended consequences.

This thesis emphasizes that interactions and dynamics are fundamental to understanding complex systems. The early success of complexity science was rooted in its ability to bridge gaps between fields by applying tools from one domain to another. For instance, using mean-field descriptions to study opinion dynamics revealed similarities between seemingly disparate systems, hinting at a deeper connection. Under certain conditions, we argue that these systems will exhibit identical behavior, allowing us to adapt tools from one field to another. However, this abstraction comes at a significant cost: by generalizing, we risk losing sight of the original problem we sought to solve.

To truly “understand”, in its broadest sense, a complex system, we aim to achieve something practical with it, such as predicting its future behavior or actively controlling its dynamics. This often requires examining the system out of equilibrium rather than just its limiting behavior. Consequently, we seek generality not only in limiting cases but also in out-of-equilibrium situations – there are many paths to Rome, but not all are equal.

Information theory offers a promising approach to creating a general framework for studying multi-variate dependencies. However, it faces three major challenges. Firstly, the computational effort required to calculate multi-variate information measures increases exponentially, making it difficult to apply to real-world systems without reducing the dynamics to a lower-dimensional latent space. Secondly, information-theoretic measures are inherently correlational, and we must be cautious not to fall into the “correlation does not imply causation” trap. Lastly, the interpretation of many information theoretic quantities is often not straightforward. It remains too weak a claim to say that because complex systems consist of many elements we must use tools to quantify these higher-order correlations. We know in practice that complex systems often produce behavior that is less than the sum of its parts, and therefore higher-order correlations may not be tied to a physical mechanism per se, but they may arise through the mutual constrictions generating the whole.

This research not only advances our theoretical understanding of complex systems but also offers practical value for law enforcement agencies dealing with criminal organizations. While the theoretical implications have been discussed throughout this thesis, the practical recommendations for law enforcement agencies are detailed in Implications for Law Enforcement. These recommendations translate our scientific findings into actionable insights, addressing how law enforcement can better understand criminal organizations as complex adaptive systems, anticipate non-linear consequences of interventions, and develop more effective strategies through community partnerships.

Exploring interventions within complex systems has revealed a multifaceted landscape that challenges conventional notions of causality, control, and predictability. The chapters presented here illuminate the delicate relationships between system components and emphasize the importance of timing, targeted interventions, and the difficulty of co-evolutionary dynamics. While this analysis provides a solid foundation for future research, much remains to be discovered about navigating the complexities of interventions in complex systems. The integration of information theory with computational modeling offers promising avenues, but addressing challenges such as feedback loops and the subtleties of causation requires further scholarly inquiry. By developing more nuanced frameworks for intervention strategies and recognizing the potential unintended consequences of our actions, we can work toward designing interventions that promote resilience and adaptability within the complex systems governing our world.

Ik snap er nog steeds helemaal niks van

Casper van Elteren (2024)