Predictability of Complex Systems

TL;DR

The paper surveys the predictability of complex systems across time series, networks, and dynamical systems, introducing three core analytical strands: information-theoretic bounds (e.g., the Fano inequality and its refinements), metric-based measures (such as permutation entropy and the $\kappa$ index), and exact equivalences to Bayes error rate. It highlights advances in AI-augmented forecasting, information bottleneck approaches, and extreme-event analysis, while also detailing spectral and structural methods for networks and multiple perspectives on dynamical predictability. Key contributions include formalizing upper bounds for predictability, linking these bounds to practical metrics, and showing how multi-source information, context, and representation learning can approach intrinsic limits. The findings underscore the practical significance of predictability as a diagnostic tool and design guide for interventions, policy, and forecasting in domains ranging from mobility and finance to climate and culture, while outlining open challenges and opportunities for a unified, robust science of prediction.

Abstract

The study of complex systems has attracted widespread attention from researchers in the fields of natural sciences, social sciences, and engineering. Prediction is one of the central issues in this field. Although most related studies have focused on prediction methods, research on the predictability of complex systems has received increasing attention across disciplines--aiming to provide theories and tools to address a key question: What are the limits of prediction accuracy? Predictability itself can serve as an important feature for characterizing complex systems, and accurate estimation of predictability can provide a benchmark for the study of prediction algorithms. This allows researchers to clearly identify the gap between current prediction accuracy and theoretical limits, thereby helping them determine whether there is still significant room to improve existing algorithms. More importantly, investigating predictability often requires the development of new theories and methods, which can further inspire the design of more effective algorithms. Over the past few decades, this field has undergone significant evolution. In particular, the rapid development of data science has introduced a wealth of data-driven approaches for understanding and quantifying predictability. This review summarizes representative achievements, integrating both data-driven and mechanistic perspectives. After a brief introduction to the significance of the topic in focus, we will explore three core aspects: the predictability of time series, the predictability of network structures, and the predictability of dynamical processes. Finally, we will provide extensive application examples across various fields and outline open challenges for future research.

Figures (68)

• Figure 1: Illustration of the relationship between entropy $S$ and predictability $\Pi$ as mapped by the Fano function $S_F(p)$ under different candidate set sizes $C$. The curves demonstrate the monotonic decreasing and concave properties of $S_F(p)$. obviously, the relationship between entropy and predictability is not linear.
• Figure 2: Illustrative examples of the network structure and temporal patterns of human mobility. (a) Example of a user mobility network, where nodes represent cell tower locations, node size is proportional to visit frequency, and edge width indicates the observed frequency of direct transitions between towers. (b) Distribution of inter-call times $P(\tau)$ for all users, where $\tau$ denotes the time interval between two consecutive calls, showing that call activities exhibit bursty temporal patterns. (c) Distribution $P(q)$ of the hourly proportion of unknown locations, where $q$ represents the fraction of time within an hour during which a user makes no calls and thus location is unobserved. (d) Example of a weekly call sequence, where each vertical bar represents a call and colors correspond to the base station at which the call occurred, reflecting the temporal dependencies of location visits. Source: The figure is reproduced from Ref. song2010limits.
• Figure 3: Entropy distributions, predictability limits, and influencing factors of human mobility. (a) Distributions of the real entropy $S$ (denoted as $S^{\mathrm{real}}$ in the main text), the uncorrelated entropy $S^{\mathrm{unc}}$, and the random entropy $S^{\mathrm{rand}}$ across 45,000 users. (b) Corresponding distributions of the maximum predictability $\Pi^{\mathrm{max}}$, the uncorrelated predictability $\Pi^{\mathrm{unc}}$, and the random predictability $\Pi^{\mathrm{rand}}$. (c) Relationship between the maximum predictability $\Pi^{\mathrm{max}}$ and the radius of gyration $r_g$, which characterizes the typical spatial scale of user activity. When $r_g > 10\ \mathrm{km}$, $\Pi^{\mathrm{max}}$ becomes nearly independent of $r_g$ and saturates at approximately 0.93. (d) Fraction of time $\tilde{\Pi}$ spent by users in their Top-$c$ most frequently visited locations, used as an approximation to the upper bound $\Pi^{\mathrm{max}}$. When $c=2$, $\tilde{\Pi}$ is about 0.6; as $c$ increases, $\tilde{\Pi}$ grows approximately logarithmically. Source: The figure is reproduced from Ref. song2010limits.
• Figure 4: Entropy measures and predictability under different spatio-temporal resolutions for an individual with a radius of gyration of about 15 kilometers. (a) Random entropy $S^{\mathrm{rand}}$, (b) estimated entropy $S^{\mathrm{est}}$, and (c) maximum predictability $\Pi^{\mathrm{max}}$ across varying spatio-temporal scales. Temporal resolution ranges from 5 minutes to 90 minutes, and spatial resolution $s$ ranges from 6 to 18. Source: The figure is reproduced from Ref. lin2012predictability.
• Figure 5: Relationship between visiting behavior and prediction accuracy. (a) Day-to-day prediction accuracy based on historical trajectory data (excluding users with no activity on a given day), comparing Markov chain models of different orders (MC(0)–MC(7)). (b) Trends of prediction accuracy as a function of historical trajectory length $L_{\mathrm{hist}}$, showing that all Markov chain models improve in performance as the history length increases. (c) Distribution of the proportion of time spent at the top-$n$ most frequently visited locations. Users are grouped into 10 categories according to their total number of distinct visited locations $N$. Source: The figure is reproduced from Ref. lu2013approachingTest.