Event-Resolved Criticality

• Event-resolved criticality is the study of discrete events marking system transitions to critical states, characterized by power-law scaling and detailed statistical signatures.
• Methodologies involve precise event detection, statistical quantification, and mechanistic attribution to reveal microscale dynamics underlying macroscopic phenomena.
• Applications span neuroscience, quantum systems, material failure, and risk management, enabling real-time monitoring and targeted intervention of critical events.

Event-resolved criticality refers to the identification, quantification, and mechanistic understanding of system behavior at or near critical points, resolved at the level of individual events—such as neuronal avalanches, acoustic emissions, measurement outcomes in quantum circuits, or rare system failures. This concept underpins modern analyses in neuroscience, quantum many-body physics, material failure, reinforcement learning, and intelligent system safety, combining rigorous statistical measures, mathematical modeling, and event-by-event tracking or control to reveal features not accessible through static, time-averaged, or purely ensemble-based approaches.

1. Definition and Scope of Event-Resolved Criticality

Event-resolved criticality characterizes regimes where large-scale, often system-spanning responses ("critical events") emerge from microscale dynamics, with statistical signatures—such as power-law scaling—indicative of proximity to a critical point. The event-resolved viewpoint means that not only is the aggregate behavior (e.g., average avalanche size, entropy, or energy release) analyzed, but each individual event is detected, classified, and statistically measured. This approach is essential in domains where the temporal structure, sequencing, or conditional statistics of critical events carry key mechanistic or risk-signaling information.

The event-resolved paradigm is operationalized by:

• Detection and delineation of events (e.g., avalanches, quantum jumps, AE events, buffer threshold crossings)
• Statistical quantification (e.g., power-law exponents, mutual information peaks, divergence rates)
• Mechanistic attribution (mapping individual event genesis to underlying dynamic mechanisms)
• Real-time or offline tracking (e.g., via filtering, optimization, or data assimilation)
• Risk/impact assessment (e.g., safety margins, proxy measures, event-based alerting)

2. Mechanisms Underlying Event-Resolved Criticality

Event-resolved analyses expose mechanistic layers that produce and regulate critical transitions. In biological neural networks, event-resolved criticality arises from the interplay of mean-driven subthreshold membrane dynamics, fast-initiating spike events mediated by strong excitatory coupling, and delayed inhibitory feedback, driving the system to the brink of a Hopf bifurcation, with neuronal avalanches emerging through stochastic crossings (Zeng et al., 2023). In sheared amorphous solids, event-resolved tracking reveals precursors and mainshocks as distinct avalanche types, whose statistics and system-size scaling encode the evolution from off-criticality to universal mean-field behavior (Oyama et al., 2020).

In quantum systems under continuous monitoring, event-resolved ("single-trajectory") quantum dynamics provide access to phase transitions in entanglement scaling driven by measurement—a phenomenon obscured in ensemble-averaged Lindblad descriptions. Criticality is tied to volume-to-area law transitions in entanglement, with conditional statistics (e.g., mutual information, entropy, symmetry-resolved quantities) revealing emergent conformal structures (Fuji et al., 2020, Murciano et al., 2023).

Intelligent systems leverage event-resolved criticality for risk prediction: at each time step, the probability of rare, safety-critical events is formalized and estimated ("criticality"), and refined via multi-stage frameworks to focus on events with actual risk profiles of interest (Bai et al., 20 Mar 2024, Grushin et al., 26 Sep 2024). In cyber-physical and crisis-management systems, complex event processing engines track composite patterns in real-time event streams, flagging situations when event-resolved signatures of criticality (crowd anomalies, sensor correlations) are detected (Itria et al., 2014).

3. Statistical Quantification and Power-Law Behavior

Event-resolved criticality is diagnosed by extracting statistical laws from the distribution of event sizes, durations, inter-event times, or related observables. In neural avalanches, size and duration distributions of supra-threshold firing events are quantified by exponents $\tau_s$ and $\tau_t$ in power-law forms $P(S) \sim S^{-\tau_s}$ and $P(T) \sim T^{-\tau_t}$, with mean event size versus duration following $\langle S \rangle(T) \sim T^\pi$, and a $\kappa$ index tracking deviations from threshold behavior (Zeng et al., 2023).

In acoustic emission during fracture, the event rate diverges as $\dot{n}(t_c-t) \sim (t_c-t)^{-\Delta}$ with $\Delta \approx 1.4$, while energy distributions follow $P(E) \sim E^{-\beta}$ (Rosti et al., 2010). In sheared glassy materials, avalanche size $S$ distributions follow $P(S) \sim S^{-\tau} f(S/S_c)$, with universal $\tau \simeq 1.5$ in critical regimes and additional finite-size scaling to extract exponents (Oyama et al., 2020).

In measurement-induced quantum criticality, entanglement entropy scales logarithmically with subsystem size at the event-resolved transition, $S(l_A) = \alpha_S \ln x_A + \beta_S$, mutual information decays algebraically $I_{AB}(r) \sim r^{-2\Delta}$, and symmetry-resolved entanglement reflects sector-specific operator content (Fuji et al., 2020).

4. Modeling, Detection, and Event-Resolved Control

Mathematical modeling of event-resolved criticality blends mean-field theory, stochastic differential equations, data assimilation, and control-theoretic approaches. For conductance-based neural networks, Gaussian closure of membrane voltages and stochastic synaptic gating variables yield analytical predictions for population firing rates and bifurcation structure. Event-resolved network state estimations are performed using ensemble Kalman filters (EnKF), which infer effective synaptic parameters and reconstruct temporal patterns of criticality from noisy (BOLD/fMRI) outputs (Zeng et al., 2023).

In safety engineering, event-resolved risk is detected via criticality functions mapping parameter spaces to scalar risk metrics, which are then optimized using hierarchical or simultaneous optimistic optimization algorithms, drastically increasing the resolution of rare (critical) events versus baseline Monte Carlo (Grujic et al., 2021). Mixed-criticality wireless networks implement event-triggered escalation policies with queuing models and optimization of transmit parameters, activating increasingly complex mitigation only when buffer backlogs indicate significant risk for mission-critical traffic (Karacora et al., 2 Dec 2024).

Methodologies in RL define true criticality as the expected loss from randomizing agent actions at particular time steps, with proxy criticality providing low-overhead, statistically monotonic risk indicators. Safety margins ($s^*$) specify the number of mistakes tolerable at an event before exceeding a loss threshold, enabling both post-hoc and real-time supervision and debugging (Grushin et al., 26 Sep 2024).

5. Event-Resolved Criticality Across Domains

Event-resolved criticality unifies diverse domains through a common emphasis on temporally precise, mechanism-based, and statistically robust identification of critical behavior:

• Neural and cognitive systems: Event-resolved avalanche dynamics elucidate the computational benefits of brain dynamics at criticality—maximizing dynamic range, information transfer, and adaptive flexibility (Zeng et al., 2023).
• Quantum systems: Single-trajectory analysis captures phase transitions and operator content inaccessible to ensemble averages, providing direct observable predictions for experimental realization (e.g., in Rydberg arrays or NISQ quantum hardware) (Fuji et al., 2020, Murciano et al., 2023).
• Materials and fracture: Event-resolved AE experiments decouple critical dynamics in event rate from energy release, highlighting limits of existing statistical fracture models and connections to phenomena such as earthquake aftershock statistics (Omori law) (Rosti et al., 2010).
• Complex engineered systems: Rare event simulation, structured around event-resolved criticality metrics, enables efficient, targeted risk analysis in high-dimensional parameter spaces (BMS, automotive safety, intelligent control) (Grujic et al., 2021, Bai et al., 20 Mar 2024).
• Crisis management and cyber-physical infrastructure: CEP engines synthesize large-scale event streams, correlating micro-events into composite signatures of impending threats, supporting rapid and reliable intervention (Itria et al., 2014).
• Transport in quantum materials: Local imaging reveals signatures of quantum criticality in current profiles invisible in bulk averaging, with transitions between Ohmic, hydrodynamic, and quantum critical regimes marked by event-resolved spatial crossovers in current density (Huang et al., 2021).

6. Implications, Applications, and Conceptual Advances

Event-resolved criticality informs both theoretical understanding and practical intervention. In neuroscience and quantum many-body systems, it enables empirical identification of critical points and mechanisms, refining universality claims (e.g., emergence of mean-field exponents versus non-universal signatures tied to protocol details). In engineered systems, criticality metrics support interpretable, statistically guaranteed oversight, risk mitigation, and resource-efficient monitoring and control. The event-resolved approach is crucial for domains where rare but impactful events cannot be inferred reliably from bulk or average behavior.

A direct implication is the necessity of event-level detection and filtering to achieve both high recall and precision in rare event prediction against extreme class imbalance ("curse of rarity"). Multi-stage learning frameworks, hierarchical optimization, and criticality-focused data assimilation tools become central methodological components (Bai et al., 20 Mar 2024, Grujic et al., 2021).

In summary, event-resolved criticality provides the conceptual and methodological bridge linking microdynamic mechanisms to macroscale critical behavior, undergirds statistical diagnosis and real-time control or monitoring of complex systems, and grounds cross-domain universality and diversity in the patterns and consequences of discrete, temporally resolved events.