Yksiulotteisesta moniulotteiseen: kriittiset hyperpinnat itsejärjestäytyvissä systeemeissä

Self-organized criticality describes the spontaneous evolution of a system to the verge of a phase transition, where the system balances between two macroscopically distinct phases. At this critical threshold, microscopic dynamics give rise to universal behavior characterized by scale-invariant spatial and temporal correlations spanning the whole system. Traditionally, self-organized criticality has been depicted as a one-dimensional phenomenon in which a single parameter is driven to its critical value. However, as the number of parameters in most real-world systems is vast, the states of the system corresponding to criticality can be expected to form a highdimensional manifold. This thesis explores the consequences of this multidimensionality, illustrating the topological and dynamical flexibility inherent in the self-organized critical state. Positioned at the intersection of network science and dynamical systems theory, this doctoral thesis makes both theoretical and methodological contributions to the study of self-organized criticality and its signatures. Using simulations and theoretical analysis, the first study demonstrates the phenomenon of a self-organized critical drift, where the self-organization process drives an adaptive network to drift on the critical manifold. The second study shows that this topological flexibility corresponds to dynamical richness, providing the first demonstration of an adaptive system self-organizing to multicriticality, where the system resides simultaneously on the verge of two different phase transitions. Finally, the third study addresses the methodological challenge of identifying the power-law distribution, a core hallmark of criticality. Power laws are notoriously difficult to identify, and this task is further complicated when data is subsampled. Our results show that two state-of-the-art methods fail to reliably distinguish between subsamples from power-law and some other heavy-tailed distributions, calling for caution in their use. Together, the results of this thesis provide a novel framework for understanding self-organized criticality that both opens new perspectives and unifies research directions previously regarded as competing. In previous research, it has been suggested that the brain would tune itself to criticality, but a lack of consensus on the phase transition in question has slowed down progress in the field. By demonstrating the inherent flexibility of the criticality hypothesis, this thesis shows how a system can seamlessly move to a new transition without leaving another, providing new insight into systems that may rely on multiple phase transitions to support complex adaptive behavior. As this work concerns the theoretical foundations of self-organized criticality, the findings have broad relevance across disciplines, generally applying to all self-organizing systems from biological brains and artificial neural networks to ecosystems and human contact networks.