Timeliness Criticality in Complex Systems

The timely delivery of goods, services, or people is paramount. Delays significantly depreciate value, as seen in transportation or food delivery. Socio-technical systems (STSs), where technology and human elements interact, depend heavily on timeliness. Examples include transport, food systems, healthcare, emergency response, and computer services. These systems, termed schedule-based systems, may be dynamic and decentralized. Operators of STSs face incentives to enhance cost and time efficiency, often leading to tighter schedules and reduced temporal buffers (e.g., shorter train stops, fewer standby crews). Extreme reduction of buffers can cause even minor delays to reverberate system-wide. This paper uses a stylized model to demonstrate that efficiency incentives can lead to a critical phenomenon: timeliness criticality.

The authors refer to previous work on temporal criticality in socio-technical systems, cascading disruptions in transport networks, data-driven methods for assessing delay propagation in air transportation, systemic risk assessment in complex supply networks, and the effects of supply fluctuations in production networks. They mention the impact of the Suez Canal blockage as an example of just-in-time supply chain disruptions. Existing literature on the bounded Kardar-Parisi-Zhang equation, which describes the motion of a driven interface with a hard wall, is also relevant to the model's formulation. The mode-coupling theory of glass transition is mentioned as a potential analogy to the observed critical behavior.

The model is formulated using temporal networks, where the delay τᵢ(t) of component i at time t depends on delays of connected components at previous times (t'<t), mitigated by a buffer Bᵢ(t). The general equation (1) is simplified by assuming Markovian dynamics (delay at t depends only on t-1) and a constant buffer B across components, leading to equation (2). Two variants of the adjacency matrix A(t) are considered: a mean-field (MF) case, where each component is affected by K randomly chosen independent components, and a synthetic temporal network (STN) with additional constraints. The noise εᵢ(t) is assumed to be independently and identically distributed (i.i.d.). The MF case with exponentially distributed noise allows for an analytical solution. Equation (3) is derived, representing the distribution of delays, which simplifies to a differential equation (4) in the steady state. Analysis of this equation reveals a critical buffer size B_c, given by equation (5), above which delays remain bounded and below which they accumulate without bound. The order parameter v (mean delay accumulation) exhibits a second-order phase transition at B_c (equation (6)), with a square-root singularity in the exponential tail parameter α near B_c (equation (7)). Simulations verify these analytical results for the MF case, also showing similar results for STN and different noise distributions. Temporal correlations in mean delay are analyzed by investigating the autocorrelation function and identifying critical exponents (Fig. 3). The model is then applied to real-world temporal contact networks from a high school and a workplace. The data is event mapped. Delay development is simulated on these networks with a uniform buffer B, and critical transitions are identified by analyzing the ν versus B curves (Fig. 4).

The analytical solution for the mean-field case with exponentially distributed noise reveals a critical buffer size B_c, which is approximately log K for large K. Above B_c, delays remain bounded and the system exhibits stability, while below B_c, delays accumulate indefinitely, representing a phase transition. Simulations verify this behavior for the mean-field and synthetic temporal network cases. The autocorrelation function of mean delay shows that the correlation time diverges as a power law (B-B_c)^γ as the buffer B approaches B_c from above, indicating the presence of long-range temporal correlations. Analysis of delay avalanches reveals power-law distributions of avalanche sizes and persistence times, characteristic of critical phenomena. The critical exponents are determined by data collapse. The model, when applied to real-world temporal networks from a high school and a workplace, also demonstrates timeliness criticality. Critical transitions occur at lower buffer values than expected for STNs with similar agent numbers, possibly due to network sparsity.

The findings demonstrate the existence of timeliness criticality, a novel critical phenomenon in schedule-based systems. The model successfully captures the trade-off between efficiency (minimizing buffers) and resilience (maintaining timeliness). The presence of power-law distributions and diverging correlation times supports the identification of a critical transition. The application to real-world temporal networks validates the model's generality and relevance to real-world systems. The observed critical behavior provides insights into the vulnerability of complex systems to delays and suggests that operating close to criticality could lead to significant system-wide disruptions.

This paper introduces the concept of timeliness criticality, a novel phase transition observed in schedule-based systems as the delay-mitigating buffer is reduced. The model demonstrates this critical behavior analytically in a mean-field approximation and numerically using both synthetic and real-world temporal networks. Power-law scaling in avalanches and correlation times further establishes the critical nature of the transition. This work highlights the crucial interplay between efficiency and resilience in complex systems and provides a framework for understanding and potentially mitigating the risks associated with operating close to the critical point. Future research should focus on incorporating more realistic buffer dynamics, heterogeneity, and network topology into the model.

The model simplifies certain aspects of real-world systems, including assuming constant and uniform buffers across components and focusing primarily on exponentially distributed noise. The translation of temporal buffers to real-world measures (e.g., inventories, production speed) is not straightforward, requiring further investigation. The study's real-world applications are limited by the availability of appropriate temporal network data. More extensive data collection is required to fully explore the robustness and generalizability of the findings.