Temporal Imbalance in Systems

• Temporal Imbalance is defined as the uneven allocation of events, such as interactions, supervision, or computational loads over time, affecting system dynamics.
• It shows that static or aggregate descriptions can be misleading when temporal ordering and intra-period variability are critical in applications like network spreading and reinforcement learning.
• It informs interventions—like retiming dynamics, reweighting evidence, or altering temporal units—to correct performance gaps in simulations, machine learning, and market designs.

Temporal imbalance, as it appears across contemporary research, denotes a family of phenomena in which interactions, supervision, labels, computational load, or economic incentives are distributed unevenly across time, or in which the timescale of observation, optimization, and settlement is mismatched to the timescale of the underlying process. In different literatures it appears as temporal sparsity in temporal networks, non-simultaneous link usage in structural balance dynamics, time synchronization mismatch in non-stationary reinforcement learning, temporal misalignment between training and deployment corpora, temporal load imbalance in parallel simulation, and a disconnect between sub-quarter-hourly balancing dynamics and quarter-hourly settlement in electricity markets (Perotti et al., 2014, Nishi et al., 2014, Lee et al., 2023, Luu et al., 2021, Solórzano et al., 2024, Madahi et al., 20 May 2025). A common consequence is that aggregate or static descriptions can be misleading: systems that appear balanced in counts, topology, or net energy can remain imbalanced once temporal ordering, recency, and intra-period structure are made explicit.

1. Scope and recurrent structure

The literature does not present a single universal formalism for temporal imbalance. Instead, closely related works define domain-specific temporal asymmetries: uneven availability of links in spreading processes, one-link-at-a-time social interaction, earlier classes receiving stronger negative supervision toward the end of class-incremental learning, train-test time mismatch in NLP, load variation across iterations in scientific simulation, and sub-quarter-hourly price-setting moments inside quarter-hour settlement intervals (Perotti et al., 2014, Nishi et al., 2014, Ma et al., 2 Mar 2026, Luu et al., 2021, Solórzano et al., 2024, Madahi et al., 20 May 2025). This suggests that temporal imbalance is better understood as a structural property of when influence is exerted than as a single metric.

Setting	Temporal object	Reported effect
Temporal networks	effective number of links active in a window of length equal to the average inter-event time	slowdown of SI spreading
Social balance	only one edge is active at a time for duration $\tau$	longer time to global balance
Class-incremental learning	earlier classes receive stronger negative supervision toward the end of training	asymmetric precision and recall
NLP	model trained on one time period and tested on another	degraded end-task performance
Ondes3D	load variation between execution times of different iterations	architecture-dependent temporal load imbalance
Imbalance settlement	sub-quarter-hourly dynamics determine prices, quarter-hourly basis determines settlement	gaming opportunity for BRPs

Two broad patterns recur. First, temporal imbalance often arises even when aggregate counts appear balanced. The class-incremental learning analysis proves that equal positive sample counts do not imply equal effective supervision if one class’s positives were concentrated earlier and another’s later (Ma et al., 2 Mar 2026). Second, the governing temporal unit is often endogenous to the system: the average inter-event time in temporal-network spreading, the activation duration $\tau$ in social balance, the interaction interval $A_T$ in reinforcement learning, and the imbalance settlement period in electricity markets all change the phenomenon being measured (Perotti et al., 2014, Nishi et al., 2014, Lee et al., 2023, Madahi et al., 20 May 2025).

2. Dynamical formulations in networks and communication systems

In structural balance dynamics on a complete signed graph, temporal imbalance is induced by sequential rather than simultaneous edge updates. The model uses continuous signed weights $x_{ij}=x_{ji}$ on an undirected complete graph, and the bounded dynamics

$\dfrac{\rm d}x_{ij}}{\rm d t} = \dfrac{1}{N-2}\left(1-\dfrac{x_{ij}^2}{R^2}\right)\sum_{k\neq i,j} x_{ik}x_{kj}.$

A triad is balanced if $x_{ij}x_{jk}x_{ki}>0$, and the temporal protocol activates only one edge at a time for duration $\tau$ while all other links remain frozen. The reported result is that the time to global social balance is larger on the temporal than aggregate complete graph, the fraction of unbalanced triads decays more slowly for larger $\tau$, and the distribution of convergence times broadens (Nishi et al., 2014). The paper attributes the slowdown to non-simultaneity: overlapping triads cannot relax cooperatively when only one relationship updates while every other edge is fixed.

In temporal spreading, the corresponding object is not sequential update but temporal sparsity. The event distribution across links in a time window is summarized by

$$H_e=-\sum_e p_e \ln p_e,\qquad M_\mathrm{eff}^{H}=\exp(H_e),$$

and normalized against the Uniform Times reference model as

$$\mu^H = \frac{M_\mathrm{eff}^{H}}{M_{\mathrm{eff},\mathrm{UT}}^{H}}.$$

Deterministic SI slowdown is measured by

$\tau$0

Across ten empirical datasets, the paper reports an almost linear dependence $\tau$1 with $\tau$2, interpreting temporal heterogeneity as reducing the effective number of links available at the timescale of average inter-event times and thereby lengthening relay times along fastest temporal paths (Perotti et al., 2014). Here temporal imbalance is not a deficit of total connectivity but an uneven allocation of activity over time.

A related but distinct communication-theoretic formulation appears in mixed RF/FSO relaying with outdated CSI. There the temporal asymmetry is a mismatch between relay-selection time and transmission time, modeled through a temporal correlation coefficient $\tau$3 between outdated and current first-hop CSI. The consequence is not slower convergence but degraded relay-selection diversity. The paper derives diversity order

$\tau$4

and states that with hardware impairments the outage floor implies $\tau$5 (Balti et al., 2021). Temporal imbalance here is temporal CSI mismatch: outdated estimates no longer represent the channel at the instant when the selected relay is used.

3. Machine learning: supervision timing, temporal coherence, and time-misaligned generalization

In class-incremental learning, temporal imbalance is made explicit as an imbalance of positive and negative supervision over training time. Standard cross-entropy is decomposed into a positive term and a negative supervision term, and each class receives a polarity sequence $\tau$6. With decay kernel $\tau$7, the temporal supervision strength is

$\tau$8

Theorem 1 shows that if two classes have equal total positive sample counts but one class’s positives are shifted later in time, then the later class has larger $\tau$9 at the end of training. The proposed Temporal-Adjusted Loss reweights only the negative supervision term through $A_T$0, so that classes with low temporal supervision strength receive less negative pressure (Ma et al., 2 Mar 2026). The empirical interpretation is asymmetric precision and recall: earlier classes become conservative, with higher precision and lower recall, while later classes are more permissive.

A different form of temporal imbalance occurs when time-series samples are generated by overlapping windows. In solar flare forecasting, the observation window spans 24 hours with $A_T$1, and the $A_T$2-th slice starts at $A_T$3. The resulting temporal coherence means that adjacent slices are too similar to be treated as independent samples. The paper reports that random 10-fold CV within one partition yields TSS in $A_T$4 with average approximately $A_T$5, whereas training on one temporally non-overlapping partition and testing on another yields TSS in $A_T$6 with average approximately $A_T$7 (Ahmadzadeh et al., 2019). The claimed lesson is methodological rather than algorithmic: imbalance remedies cannot be evaluated correctly unless temporal dependence is respected.

In NLP, the temporal object is train-test time mismatch. The paper defines performance degradation relative to a temporally aligned model as

$A_T$8

and summarizes degradation rate through Temporal Degradation. Reported TD scores vary sharply by task: 7.72 for political affiliation classification, 5.46 for publisher classification, and 0.26 for Yelp review classification, with the introduction noting that over five years classifiers’ $A_T$9 can deteriorate by as much as 40 points for Twitter political affiliation or as little as 1 point for Yelp review ratings (Luu et al., 2021). Continued pretraining on temporally matched unlabeled data can help, but the paper states that these gains are small compared to task-specific finetuning on data from the target time period.

Non-stationary reinforcement learning makes the timescale mismatch explicit. The environment changes over wall-clock time $x_{ij}=x_{ji}$0, while the agent acts in episodes indexed by $x_{ij}=x_{ji}$1. ProST chooses interaction times $x_{ij}=x_{ji}$2, or equivalently a training interval $x_{ij}=x_{ji}$3, to balance environment drift against policy optimization time. The dynamic regret is decomposed into a forecasting term $x_{ij}=x_{ji}$4 and a policy optimization term $x_{ij}=x_{ji}$5, and the optimal cadence is the one that trades off fresher observations against stronger optimization between interactions (Lee et al., 2023). Temporal imbalance in this setting is a synchronization problem: the wrong tempo creates a systematic performance gap even if the underlying RL optimizer is otherwise adequate.

4. Temporal imbalance in systems, simulation, and market design

In MPI-parallel seismic simulation, temporal imbalance refers to variation of computational load across iterations rather than across ranks alone. Ondes3D measures computational load as time spent in four macro-kernels and analyzes rank-resolved, iteration-resolved, and kernel-resolved traces. The paper states that temporal imbalance comes from the load variation between execution times of different iterations, examines two earthquake scenarios, and finds that the effect depends on the multicore architecture (Solórzano et al., 2024). For Chuetsu-Oki, most architectures show a clear temporal imbalance increase between iterations about 100 and 150, while for Sichuan the profile is more stable overall but older Intel systems exhibit a sustained increase. AMD Zen 2 is reported as the best platform overall, whereas ARM ThunderX2 is the worst in total execution time but shows the least temporal imbalance in several views.

In electricity balancing, temporal imbalance becomes a market-design vulnerability. The physical system is balanced using sub-quarter-hourly dynamics, the paper models those dynamics at the minute level, but BRP settlement is performed over a 15-minute imbalance settlement period. The BRP controls a battery with fixed quarter-hour energy position

$x_{ij}=x_{ji}$6

while the final imbalance price $x_{ij}=x_{ji}$7 is determined from minute-level balancing outcomes (Madahi et al., 20 May 2025). The case studies show that a BRP can worsen the instantaneous system imbalance in the minute(s) that influence the quarter-hour imbalance price and then compensate over the remaining minutes. Reported profit changes include a Belgian case with a profit increase of 692.9 €, a Belgian mFRR case shifting from a loss of 304.6 € to a profit of 195.3 €, and a Dutch regulation-state case with a gain of 550.39 €. The proposed remedies are shorter imbalance settlement periods, such as 5 minutes, and a modified BRP imbalance position calculation with a counteracting imbalance factor.

A closely related applied perspective appears in spatio-temporal crime prediction. There the target is defined over daily cell-day observations on a $x_{ij}=x_{ji}$8 grid, producing 11,123,304 observations with only 6,266 positives, or approximately 0.06% prevalence. Weekly aggregation raises prevalence to 0.56%. The proposed imbalance-aware hyper-ensemble repeatedly balances the training data by random undersampling and averages predictions across ensemble members, increasing hit ratio from 18.1% to 24.6% for the top 5% hotspots and from 53.1% to 60.4% for the top 20% hotspots (Kadar et al., 2019). This is not a formal temporal theory, but it shows that temporal resolution itself controls the severity of imbalance.

5. Measurement and intervention patterns

Temporal imbalance is rarely measured by a single universal scalar. Instead, each literature couples its temporal hypothesis to process-specific observables. Structural balance uses the number and fraction of unbalanced triads and the time to first global balance $x_{ij}=x_{ji}$9 (Nishi et al., 2014). Temporal-network spreading uses the entropy-based effective number of active links $\dfrac{\rm d}x_{ij}}{\rm d t} = \dfrac{1}{N-2}\left(1-\dfrac{x_{ij}^2}{R^2}\right)\sum_{k\neq i,j} x_{ik}x_{kj}.$0, the temporal sparsity coefficient $\dfrac{\rm d}x_{ij}}{\rm d t} = \dfrac{1}{N-2}\left(1-\dfrac{x_{ij}^2}{R^2}\right)\sum_{k\neq i,j} x_{ik}x_{kj}.$1, and the slowdown coefficient $\dfrac{\rm d}x_{ij}}{\rm d t} = \dfrac{1}{N-2}\left(1-\dfrac{x_{ij}^2}{R^2}\right)\sum_{k\neq i,j} x_{ik}x_{kj}.$2 (Perotti et al., 2014). Non-stationary RL uses dynamic regret and the suboptimal interaction interval $\dfrac{\rm d}x_{ij}}{\rm d t} = \dfrac{1}{N-2}\left(1-\dfrac{x_{ij}^2}{R^2}\right)\sum_{k\neq i,j} x_{ik}x_{kj}.$3 (Lee et al., 2023). NLP uses Temporal Degradation, while solar flare forecasting emphasizes TSS under temporally separated evaluation (Luu et al., 2021, Ahmadzadeh et al., 2019). Crime prediction evaluates hit rate, Prediction Accuracy Index, and surveillance-curve AUC (Kadar et al., 2019). Class-incremental learning reports average incremental accuracy $\dfrac{\rm d}x_{ij}}{\rm d t} = \dfrac{1}{N-2}\left(1-\dfrac{x_{ij}^2}{R^2}\right)\sum_{k\neq i,j} x_{ik}x_{kj}.$4 and final-task accuracy $\dfrac{\rm d}x_{ij}}{\rm d t} = \dfrac{1}{N-2}\left(1-\dfrac{x_{ij}^2}{R^2}\right)\sum_{k\neq i,j} x_{ik}x_{kj}.$5, but its distinctive temporal observable is the supervision strength vector $\dfrac{\rm d}x_{ij}}{\rm d t} = \dfrac{1}{N-2}\left(1-\dfrac{x_{ij}^2}{R^2}\right)\sum_{k\neq i,j} x_{ik}x_{kj}.$6 (Ma et al., 2 Mar 2026).

Interventions also fall into recurring classes. Some methods re-time the dynamics themselves: smaller $\dfrac{\rm d}x_{ij}}{\rm d t} = \dfrac{1}{N-2}\left(1-\dfrac{x_{ij}^2}{R^2}\right)\sum_{k\neq i,j} x_{ik}x_{kj}.$7 approaches aggregate dynamics in social balance, and ProST explicitly optimizes interaction cadence (Nishi et al., 2014, Lee et al., 2023). Some methods reweight temporally biased evidence: TAL reweights negative supervision according to $\dfrac{\rm d}x_{ij}}{\rm d t} = \dfrac{1}{N-2}\left(1-\dfrac{x_{ij}^2}{R^2}\right)\sum_{k\neq i,j} x_{ik}x_{kj}.$8, while class-weighted SVM in solar flare forecasting uses

$\dfrac{\rm d}x_{ij}}{\rm d t} = \dfrac{1}{N-2}\left(1-\dfrac{x_{ij}^2}{R^2}\right)\sum_{k\neq i,j} x_{ik}x_{kj}.$9

and is reported to outperform under- and oversampling in that study (Ma et al., 2 Mar 2026, Ahmadzadeh et al., 2019). Others change the temporal unit: weekly rather than daily aggregation in crime prediction, or target-period finetuning rather than target-period continued pretraining in NLP (Kadar et al., 2019, Luu et al., 2021). The electricity-market literature instead proposes redesign of the settlement timescale itself (Madahi et al., 20 May 2025).

A boundary case is marked temporal point processes. There the paper explicitly addresses mark imbalance rather than imbalance in event timing. It proposes prior-normalized thresholding

$x_{ij}x_{jk}x_{ki}>0$0

and a mark-first, time-second prediction pipeline (Liu et al., 23 Oct 2025). This clarifies an important distinction: not every imbalance problem in temporal data is itself a temporal imbalance problem. Some are label-frequency asymmetries attached to temporal sequences.

6. Limitations, assumptions, and unresolved questions

Across domains, temporal imbalance is typically isolated under strong modeling assumptions. Structural balance is studied on an undirected complete graph with independent edge activation and no burstiness or temporal correlations in link activation (Nishi et al., 2014). Temporal sparsity in SI spreading is evaluated at the average inter-event time and is argued to work best when the effective temporal degree is low (Perotti et al., 2014). The solar flare study uses only SVM with RBF kernel and explicitly states that temporal coherence is empirically demonstrated rather than developed into a full statistical theory of grouped dependence (Ahmadzadeh et al., 2019). The NLP benchmark trains on single temporal partitions rather than cumulative histories and does not disentangle task shift from language shift (Luu et al., 2021).

The newer learning formulations also remain selective. TAL assumes an exponential decay kernel $x_{ij}x_{jk}x_{ki}>0$1 and presents its strongest evidence in rehearsal-based CIL regimes (Ma et al., 2 Mar 2026). ProST derives its main theory in the tabular setting and relies on estimates of environment tempo rather than fully general online schedule adaptation (Lee et al., 2023). Ondes3D analyzes compute-side temporal imbalance and explicitly excludes communication time from the reported load metric (Solórzano et al., 2024). The imbalance-settlement case study assumes perfect foresight and models balancing data at 1-minute resolution even though TSO dispatch in reality can occur as fast as 4 seconds (Madahi et al., 20 May 2025). The crime study uses chronological train-test splitting, but its resampling is not temporally structured inside the training set (Kadar et al., 2019).

These limitations do not weaken the central cross-domain pattern. They indicate that temporal imbalance is not exhausted by any one mechanism such as burstiness, recency bias, or concept drift. The literature instead points to a broader principle: systems become temporally imbalanced when the timing of influence, evidence, or control is unevenly allocated relative to the timescale on which outcomes are optimized, evaluated, or settled. This suggests that future theory will need to connect temporal ordering, effective supervision or interaction strength, and decision horizons more directly than static imbalance analyses permit.