Temporal Modularity in Networks and Learning

• Temporal modularity is a framework that integrates time into the analysis and optimization of modular community structures in networks and learning systems.
• It encompasses multilayer networks, link streams, and neural architectures, adapting null models, interlayer coupling, and volatility penalties to dynamic settings.
• The concept informs both theoretical development and practical applications, including brain network analysis, evolutionary dynamics, and temporal learning architectures.

Temporal modularity denotes the incorporation of temporal structure into the definition, measurement, optimization, or interpretation of modular organization. Across the cited literature, the temporal variable may be developmental age, a sliding-window index, a multilayer slice index, exact event time in a link stream, the time evolution of a population of architectures, or the separation of functional roles across modules in a learning system. The shared motif is that modularity still concerns the contrast between within-community organization and an appropriate baseline, but time changes either the partition itself, the null model, the coupling between successive states, or the performance consequences of modular structure (Chen et al., 2015, Pamfil et al., 2018, Brabant et al., 2024).

1. Core meanings and conceptual scope

The literature does not use a single standardized meaning of temporal modularity. Instead, several technically distinct constructions recur. In multilayer temporal networks, modularity is extended to layer-indexed node copies and coupled across adjacent times. In link streams, modularity is defined directly on node-time memberships without imposing fixed snapshots. In some neuroscience papers, “temporal” refers not to within-scan reconfiguration but to developmental change across age. In evolutionary theory, temporal modularity is the dynamics of a population-level distribution over modular architectures. In architectural work on learning systems, it refers to a separation of temporal functions across modules rather than to community detection in graphs (Pamfil et al., 2018, Chen et al., 2015, Park et al., 2012, Voce et al., 11 May 2026).

Setting	Temporal variable	Distinctive mechanism
Developmental fMRI	Age	Modularity related to childhood-to-adulthood change
Time-resolved fMRI	Sliding windows	High- and low-modularity periods
Multilayer temporal networks	Layer index $t$	Intralayer modularity plus interlayer coupling
Link streams	Exact event times or intervals	Longitudinal null models and switch penalties
Stable-community detection	Time-varying weighted layers	Volatility penalization
Evolutionary theory	Population time	Replicator–mutator dynamics over modularity

A common mathematical backbone remains visible. Static Newman–Girvan modularity compares observed within-community connectivity with a null expectation; temporal variants modify that backbone by adding either interlayer persistence terms, duration-sensitive expectations, volatility penalties, or dynamical laws for how average modularity changes over time. This suggests a family of related formalisms rather than a single universal object.

2. Multilayer temporal networks and statistically grounded parameterization

In temporal multilayer network analysis, the canonical construction represents a temporal network as a sequence of layers with the same physical nodes, layer-specific adjacency matrices $A_{ij}^{(t)}$, and layer-specific community labels $g_i^{(t)}$. The modularity objective used in this setting is

$$Q \;=\; \sum_{t=1}^T \sum_{i,j} \Big(A_{ij}^{(t)} - \gamma_t\,P_{ij}^{(t)}\Big)\,\delta\big(g_i^{(t)},g_j^{(t)}\big) \;+\; \sum_{t=2}^T \sum_i \omega_i^{(t-1,t)}\,\delta\big(g_i^{(t-1)},g_i^{(t)}\big),$$

where $P_{ij}^{(t)}$ is typically the Newman–Girvan null model, $\gamma_t$ is a layer-specific resolution, and $\omega_i^{(t-1,t)}$ is an interlayer coupling that rewards label persistence across consecutive times (Pamfil et al., 2018).

A central result is that, under a degree-corrected Poisson planted-partition SBM in each layer together with an interlayer copying prior, maximizing temporal modularity is equivalent to maximizing the posterior probability of the community assignment. In the layer-uniform case, the parameters acquire explicit statistical meanings:

$$\gamma \;=\; \frac{\theta_{\mathrm{in}}-\theta_{\mathrm{out}}}{\log \theta_{\mathrm{in}}-\log \theta_{\mathrm{out}}}, \qquad \omega \;=\; \frac{1}{\log\theta_{\mathrm{in}}-\log\theta_{\mathrm{out}}}\, \log\Big(1+\frac{p}{1-p}K\Big).$$

The same framework yields layer-weighted modularity when block strength varies across layers, through weights $\beta_t$ proportional to $$\log\theta_{\mathrm{in}}^{(t)}-\log\theta_{\mathrm{out}}^{(t)}$$ (Pamfil et al., 2018).

This statistical equivalence also yields a parameter-selection workflow. One initializes $A_{ij}^{(t)}$0, optimizes modularity with GenLouvain or GenLouvainRand, estimates $A_{ij}^{(t)}$1, $A_{ij}^{(t)}$2, persistence $A_{ij}^{(t)}$3, and the current number of communities $A_{ij}^{(t)}$4, then updates $A_{ij}^{(t)}$5 and repeats. In synthetic temporal benchmarks, the method recovers accurate $A_{ij}^{(t)}$6 profiles and can identify change points where $A_{ij}^{(t)}$7, implying $A_{ij}^{(t)}$8 (Pamfil et al., 2018).

Because temporal modularity depends strongly on how time is sliced, a separate line of work studies slice selection itself. For a candidate number of slices $A_{ij}^{(t)}$9, one computes $g_i^{(t)}$0 on the original sliced network, $g_i^{(t)}$1 on a degree-preserving randomized baseline, and then defines corrected modularity by

$g_i^{(t)}$2

The proposed rule is to choose the slice count maximizing $g_i^{(t)}$3, because raw multilayer modularity typically increases with the number of slices even when the underlying structure is unchanged (Seiron et al., 2023).

Optimization remains difficult in general. For temporal graphs with small underlying treewidth, a recent algorithmic result defines temporal modularity with a loyalty term

$g_i^{(t)}$4

where $g_i^{(t)}$5 counts unchanged community assignments across consecutive timesteps. The paper proves a $g_i^{(t)}$6-approximation in time $g_i^{(t)}$7 when the underlying graph has treewidth at most $g_i^{(t)}$8, and an exact algorithm for non-normalized temporal $g_i^{(t)}$9-modularity on short windows in time $$Q \;=\; \sum_{t=1}^T \sum_{i,j} \Big(A_{ij}^{(t)} - \gamma_t\,P_{ij}^{(t)}\Big)\,\delta\big(g_i^{(t)},g_j^{(t)}\big) \;+\; \sum_{t=2}^T \sum_i \omega_i^{(t-1,t)}\,\delta\big(g_i^{(t-1)},g_i^{(t)}\big),$$0 (Agdur et al., 23 Jul 2025).

3. Volatility-sensitive objectives and exact-time link-stream formulations

Not all temporal modularity formalisms allow communities to change freely across layers. One contrasting approach assumes that node-to-community affiliations are stable across time and evaluates them using a dynamic modularity $$Q \;=\; \sum_{t=1}^T \sum_{i,j} \Big(A_{ij}^{(t)} - \gamma_t\,P_{ij}^{(t)}\Big)\,\delta\big(g_i^{(t)},g_j^{(t)}\big) \;+\; \sum_{t=2}^T \sum_i \omega_i^{(t-1,t)}\,\delta\big(g_i^{(t-1)},g_i^{(t)}\big),$$1 that rewards strong within-community connectivity while penalizing unstable internal edges. For a weighted, undirected time-varying network, the proposed quality function is

$$Q \;=\; \sum_{t=1}^T \sum_{i,j} \Big(A_{ij}^{(t)} - \gamma_t\,P_{ij}^{(t)}\Big)\,\delta\big(g_i^{(t)},g_j^{(t)}\big) \;+\; \sum_{t=2}^T \sum_i \omega_i^{(t-1,t)}\,\delta\big(g_i^{(t-1)},g_i^{(t)}\big),$$2

where $$Q \;=\; \sum_{t=1}^T \sum_{i,j} \Big(A_{ij}^{(t)} - \gamma_t\,P_{ij}^{(t)}\Big)\,\delta\big(g_i^{(t)},g_j^{(t)}\big) \;+\; \sum_{t=2}^T \sum_i \omega_i^{(t-1,t)}\,\delta\big(g_i^{(t-1)},g_i^{(t)}\big),$$3 is edge volatility. This construction differs from Mucha-style multilayer modularity by keeping community assignments constant across time and encoding temporal information through volatility rather than explicit interlayer coupling (Wang et al., 2018).

That volatility-based framework defines temporal stable communities by three criteria: one node belongs to one community alone and such relationship does not change over time; nodes in the same community are connected much more tightly than nodes in different communities; and inter-community edges fluctuate more severely than inner-community edges. An extended Louvain method maximizes $$Q \;=\; \sum_{t=1}^T \sum_{i,j} \Big(A_{ij}^{(t)} - \gamma_t\,P_{ij}^{(t)}\Big)\,\delta\big(g_i^{(t)},g_j^{(t)}\big) \;+\; \sum_{t=2}^T \sum_i \omega_i^{(t-1,t)}\,\delta\big(g_i^{(t-1)},g_i^{(t)}\big),$$4 using a dynamic modularity matrix $$Q \;=\; \sum_{t=1}^T \sum_{i,j} \Big(A_{ij}^{(t)} - \gamma_t\,P_{ij}^{(t)}\Big)\,\delta\big(g_i^{(t)},g_j^{(t)}\big) \;+\; \sum_{t=2}^T \sum_i \omega_i^{(t-1,t)}\,\delta\big(g_i^{(t-1)},g_i^{(t)}\big),$$5 and a contribution matrix $$Q \;=\; \sum_{t=1}^T \sum_{i,j} \Big(A_{ij}^{(t)} - \gamma_t\,P_{ij}^{(t)}\Big)\,\delta\big(g_i^{(t)},g_j^{(t)}\big) \;+\; \sum_{t=2}^T \sum_i \omega_i^{(t-1,t)}\,\delta\big(g_i^{(t-1)},g_i^{(t)}\big),$$6. The method was reported to be robust under perturbations, and volatility-derived features were used for ADHD classification in NYU and PKU datasets (Wang et al., 2018).

A different tradition avoids snapshots entirely and works directly with link streams. In the original longitudinal formulation, a link stream is $$Q \;=\; \sum_{t=1}^T \sum_{i,j} \Big(A_{ij}^{(t)} - \gamma_t\,P_{ij}^{(t)}\Big)\,\delta\big(g_i^{(t)},g_j^{(t)}\big) \;+\; \sum_{t=2}^T \sum_i \omega_i^{(t-1,t)}\,\delta\big(g_i^{(t-1)},g_i^{(t)}\big),$$7 with dynamic communities defined as sets of node-time pairs. For a choice $$Q \;=\; \sum_{t=1}^T \sum_{i,j} \Big(A_{ij}^{(t)} - \gamma_t\,P_{ij}^{(t)}\Big)\,\delta\big(g_i^{(t)},g_j^{(t)}\big) \;+\; \sum_{t=2}^T \sum_i \omega_i^{(t-1,t)}\,\delta\big(g_i^{(t-1)},g_i^{(t)}\big),$$8, longitudinal modularity is

$$Q \;=\; \sum_{t=1}^T \sum_{i,j} \Big(A_{ij}^{(t)} - \gamma_t\,P_{ij}^{(t)}\Big)\,\delta\big(g_i^{(t)},g_j^{(t)}\big) \;+\; \sum_{t=2}^T \sum_i \omega_i^{(t-1,t)}\,\delta\big(g_i^{(t-1)},g_i^{(t)}\big),$$9

where $P_{ij}^{(t)}$0 normalizes community-switch counts. The three expectation models—co-membership, joint membership, and mean membership—encode different assumptions about synchrony and stationarity. A key property is time-scale independence under lossless aggregation: if event counts are preserved and durations scale uniformly, $P_{ij}^{(t)}$1 remains unchanged for a fixed partition (Brabant et al., 2024).

Continuous-time community detection in link streams builds on this definition. LAGO optimizes L-modularity directly on active time nodes, using the Trimmed Communities Property: trimming inactive boundary times cannot decrease $P_{ij}^{(t)}$2. The optimizer combines a Recursive Time Module Mover with refinement rules such as STNM, STMM, and STEM, and uses fast exploration heuristics to restrict attention to promising local moves. On synthetic data, different variants perform best for $P_{ij}^{(t)}$3 and $P_{ij}^{(t)}$4; on SocioPatterns primary-school data, the method yields temporally coherent communities with precise entry and exit times (Brabant et al., 1 Oct 2025).

The generalized L-modularity framework extends this approach beyond simple instantaneous, undirected, unweighted streams. For directed, weighted, multipartite, delayed, and interval-based interactions, the objective becomes

$P_{ij}^{(t)}$5

where $P_{ij}^{(t)}$6 is a multipartite mask and $P_{ij}^{(t)}$7 is either a joint-membership or mean-membership temporal factor. Generalized LAGO then works with active time-segment nodes obtained by segmenting interval interactions so that no interaction starts or ends inside a segment (Brabant et al., 23 May 2026).

4. Functional brain networks: developmental time and within-scan fluctuation

In neuroimaging, temporal modularity has been used in two distinct senses. One refers to developmental time. In a study of children and young adults watching 20 minutes of Sesame Street, modularity was computed once per subject from the full fMRI-derived functional network, so “temporal” referred to age rather than to time-varying modularity within the scan. Using Newman–Girvan modularity on thresholded or weighted correlation networks, adults showed higher modularity than children across thresholds, including a significant difference at $P_{ij}^{(t)}$8 with $P_{ij}^{(t)}$9. Within children, the average Pearson correlation between modularity and age was $\gamma_t$0 with $\gamma_t$1 for $\gamma_t$2, and motion analyses indicated that age predicted modularity independently of head motion (Chen et al., 2015).

The same work connected developmental change to cognition and evolution. A Hopfield-type model with modular connectivity and clustered memories showed that higher modularity improves overlap with a target memory at short times, whereas less modular architectures can outperform at longer times. Cognitive performance was then treated as a fitness function, and quasispecies theory was used to predict a rise of modularity from childhood to young adulthood followed by decline in older age, qualitatively matching the empirical developmental pattern and prior reports of a roughly $\gamma_t$3–$\gamma_t$4 decline from young to older adults (Chen et al., 2015).

A second neuroimaging meaning concerns within-scan fluctuations. Using resting-state fMRI from the Human Connectome Project, weighted signed modularity was computed in tapered sliding windows of width $\gamma_t$5 TRs ($\gamma_t$6 s), step size $\gamma_t$7 TRs, and Louvain optimization repeated $\gamma_t$8 times per window. High- and low-modularity periods were defined relative to a null distribution from stationary VAR models. High-modularity periods were characterized by increased dissociation of the default mode network from task-positive modules, whereas low-modularity periods showed flatter, less differentiated connectivity patterns (Fukushima et al., 2015).

These high- and low-modularity periods were not mere opposites in average strength. High-modularity periods were temporally homogeneous, with higher within-period similarity in both edge weights and partitions; in HCP run 2LR, the reported effect sizes were Cohen’s $\gamma_t$9 for edge-weight similarity and $\omega_i^{(t-1,t)}$0 for partition similarity. Their occurrence also showed fair to moderate test–retest reliability, and long-timescale modularity was strongly related to how often individuals entered these regimes: in HCP run 2LR, long-timescale $\omega_i^{(t-1,t)}$1 correlated with high-period frequency at $\omega_i^{(t-1,t)}$2 and with low-period frequency at $\omega_i^{(t-1,t)}$3 (Fukushima et al., 2015).

Together, these studies show that brain-network temporal modularity can index either a developmental trajectory or a within-scan alternation between segregated and integrated configurations. The distinction is methodological, not terminological: in one case, modularity is static within each scan but changes across age; in the other, modularity itself becomes a time series.

5. Evolutionary dynamics and multiple dynamical time-scales

In evolutionary theory, temporal modularity is the time evolution of a population-level distribution over modular architectures. A connection matrix $\omega_i^{(t-1,t)}$4 with fixed block size $\omega_i^{(t-1,t)}$5 defines a modularity

$\omega_i^{(t-1,t)}$6

and the probability $\omega_i^{(t-1,t)}$7 of modularity class $\omega_i^{(t-1,t)}$8 follows a quasispecies master equation balancing selection via a fitness function $\omega_i^{(t-1,t)}$9 and mutation via rewiring rate $$\gamma \;=\; \frac{\theta_{\mathrm{in}}-\theta_{\mathrm{out}}}{\log \theta_{\mathrm{in}}-\log \theta_{\mathrm{out}}}, \qquad \omega \;=\; \frac{1}{\log\theta_{\mathrm{in}}-\log\theta_{\mathrm{out}}}\, \log\Big(1+\frac{p}{1-p}K\Big).$$0. Averaging the master equation yields

$$\gamma \;=\; \frac{\theta_{\mathrm{in}}-\theta_{\mathrm{out}}}{\log \theta_{\mathrm{in}}-\log \theta_{\mathrm{out}}}, \qquad \omega \;=\; \frac{1}{\log\theta_{\mathrm{in}}-\log\theta_{\mathrm{out}}}\, \log\Big(1+\frac{p}{1-p}K\Big).$$1

and, under a narrow-distribution approximation,

$$\gamma \;=\; \frac{\theta_{\mathrm{in}}-\theta_{\mathrm{out}}}{\log \theta_{\mathrm{in}}-\log \theta_{\mathrm{out}}}, \qquad \omega \;=\; \frac{1}{\log\theta_{\mathrm{in}}-\log\theta_{\mathrm{out}}}\, \log\Big(1+\frac{p}{1-p}K\Big).$$2

This fluctuation–dissipation relation links the rate of modularity change to the selection gradient, population variance, and mutation (Park et al., 2012).

Environmental variability enters through change magnitude $$\gamma \;=\; \frac{\theta_{\mathrm{in}}-\theta_{\mathrm{out}}}{\log \theta_{\mathrm{in}}-\log \theta_{\mathrm{out}}}, \qquad \omega \;=\; \frac{1}{\log\theta_{\mathrm{in}}-\log\theta_{\mathrm{out}}}\, \log\Big(1+\frac{p}{1-p}K\Big).$$3 and period $$\gamma \;=\; \frac{\theta_{\mathrm{in}}-\theta_{\mathrm{out}}}{\log \theta_{\mathrm{in}}-\log \theta_{\mathrm{out}}}, \qquad \omega \;=\; \frac{1}{\log\theta_{\mathrm{in}}-\log\theta_{\mathrm{out}}}\, \log\Big(1+\frac{p}{1-p}K\Big).$$4. In the linear-response regime, the paper derives $$\gamma \;=\; \frac{\theta_{\mathrm{in}}-\theta_{\mathrm{out}}}{\log \theta_{\mathrm{in}}-\log \theta_{\mathrm{out}}}, \qquad \omega \;=\; \frac{1}{\log\theta_{\mathrm{in}}-\log\theta_{\mathrm{out}}}\, \log\Big(1+\frac{p}{1-p}K\Big).$$5 and hence a relation between environmental pressure $$\gamma \;=\; \frac{\theta_{\mathrm{in}}-\theta_{\mathrm{out}}}{\log \theta_{\mathrm{in}}-\log \theta_{\mathrm{out}}}, \qquad \omega \;=\; \frac{1}{\log\theta_{\mathrm{in}}-\log\theta_{\mathrm{out}}}\, \log\Big(1+\frac{p}{1-p}K\Big).$$6 and modularity growth,

$$\gamma \;=\; \frac{\theta_{\mathrm{in}}-\theta_{\mathrm{out}}}{\log \theta_{\mathrm{in}}-\log \theta_{\mathrm{out}}}, \qquad \omega \;=\; \frac{1}{\log\theta_{\mathrm{in}}-\log\theta_{\mathrm{out}}}\, \log\Big(1+\frac{p}{1-p}K\Big).$$7

with $$\gamma \;=\; \frac{\theta_{\mathrm{in}}-\theta_{\mathrm{out}}}{\log \theta_{\mathrm{in}}-\log \theta_{\mathrm{out}}}, \qquad \omega \;=\; \frac{1}{\log\theta_{\mathrm{in}}-\log\theta_{\mathrm{out}}}\, \log\Big(1+\frac{p}{1-p}K\Big).$$8. The same framework produces a least-action characterization of steady-state modularity and reproduces protein-evolution simulations with a fitted fitness function $$\gamma \;=\; \frac{\theta_{\mathrm{in}}-\theta_{\mathrm{out}}}{\log \theta_{\mathrm{in}}-\log \theta_{\mathrm{out}}}, \qquad \omega \;=\; \frac{1}{\log\theta_{\mathrm{in}}-\log\theta_{\mathrm{out}}}\, \log\Big(1+\frac{p}{1-p}K\Big).$$9 near $\beta_t$0 and $\beta_t$1 overall (Park et al., 2012).

A different dynamical tradition links modular topology to multiple relaxation times. In hierarchically nested modular networks, edge probabilities obey $\beta_t$2, where smaller $\beta_t$3 implies stronger separation between levels. Linearizing Kuramoto synchronization around synchrony yields modal decay governed by the normalized Laplacian. Because the hierarchical block structure creates distinct eigenvalue bands and spectral gaps, synchronization proceeds through as many distinct time-scales as there are hierarchical levels: first within the smallest modules, then within successively larger aggregates, and finally globally. As $\beta_t$4, the gaps collapse and the timescales merge (Sinha et al., 2011).

The empirical examples in that work make the time-scale interpretation concrete. For a hierarchical network with $\beta_t$5, $\beta_t$6, $\beta_t$7, and $\beta_t$8, the reported ordering times were approximately $\beta_t$9 for lowest-level modules, $$\log\theta_{\mathrm{in}}^{(t)}-\log\theta_{\mathrm{out}}^{(t)}$$0 for meta-modules, $$\log\theta_{\mathrm{in}}^{(t)}-\log\theta_{\mathrm{out}}^{(t)}$$1 for the next level, and $$\log\theta_{\mathrm{in}}^{(t)}-\log\theta_{\mathrm{out}}^{(t)}$$2 for global synchronization. This temporal separation was mirrored by three spectral gaps in the rank-ordered reciprocal eigenvalues of the normalized Laplacian (Sinha et al., 2011).

These results establish a broader meaning of temporal modularity: modular organization is not only something that changes over time, but also something that creates characteristic timescales for dynamics.

6. Architectural modularity for temporal learning and reasoning

Outside community detection, temporal modularity also denotes a decomposition of temporal functions across architectural components. In spatiotemporal prediction, one proposal separates a spatial encoder–decoder from a temporal predictor. A VQ-VAE first maps each frame $$\log\theta_{\mathrm{in}}^{(t)}-\log\theta_{\mathrm{out}}^{(t)}$$3 to a quantized latent $$\log\theta_{\mathrm{in}}^{(t)}-\log\theta_{\mathrm{out}}^{(t)}$$4, then a predictor forecasts future latents, and a frozen decoder reconstructs future frames. Applied to PredRNN and TCTN, this modular design improved LPIPS on MovingMNIST from $$\log\theta_{\mathrm{in}}^{(t)}-\log\theta_{\mathrm{out}}^{(t)}$$5 to $$\log\theta_{\mathrm{in}}^{(t)}-\log\theta_{\mathrm{out}}^{(t)}$$6 for PredRNN and from $$\log\theta_{\mathrm{in}}^{(t)}-\log\theta_{\mathrm{out}}^{(t)}$$7 to $$\log\theta_{\mathrm{in}}^{(t)}-\log\theta_{\mathrm{out}}^{(t)}$$8 for TCTN; on KTH-Action it improved TCTN from LPIPS $$\log\theta_{\mathrm{in}}^{(t)}-\log\theta_{\mathrm{out}}^{(t)}$$9 and SSIM $A_{ij}^{(t)}$00 to LPIPS $A_{ij}^{(t)}$01 and SSIM $A_{ij}^{(t)}$02, while also reducing parameter count, memory, and training time relative to scaled baselines (Pan et al., 2022).

A second use appears in sequence-learning tasks designed to test compositional temporal behavior. In modular mazes, an initial door cue must be remembered across an intervening maze and then reapplied at the end. Morphognosis implements this with a “long-context” and a “short-context” MLP combined by a subsumption rule, whereas the comparison LSTM uses a single recurrent state. Both architectures perform well in training, but Morphognosis generalizes better on composite mazes assembled for the first time at test, so temporal modularity here means successful reuse of independently learned temporal substructures (Portegys, 2021).

A biologically motivated version is the cortico-cerebellar RNN. The recurrent “cortical” core maintains temporal context, while a cerebellar-inspired feedforward module receives both $A_{ij}^{(t)}$03 and $A_{ij}^{(t)}$04 and returns an additive bias $A_{ij}^{(t)}$05 to the recurrent preactivation. On delayed match-to-sample, parity, multi-task, and task-switching curricula, the CB-RNN learns faster and reaches higher maximum difficulty than parameter-matched recurrent-only baselines. In DMS, for example, the CB-RNN reached half of the global maximum difficulty in $A_{ij}^{(t)}$06 epochs versus $A_{ij}^{(t)}$07 for the RNN-only baseline; even a “full reservoir” variant, in which the RNN is frozen after solving the easiest level, still outpaced the recurrent-only model on speed and AUC (Voce et al., 11 May 2026).

Temporal modularity also appears in formal reasoning systems. A temporal module for logical frameworks adds time modularly through a meta-level function $A_{ij}^{(t)}$08 that maps formulas to intervals, reserves the first two arguments of each atomic predicate for timestamps, and adds interval operators such as $A_{ij}^{(t)}$09. The point is not community detection but conservative extension: time is added without redesigning the host logic, so the temporal component is reusable, composable, and framework-agnostic (Pitoni et al., 2019).

Across these architectural and logical examples, the common idea is functional separation across temporal roles. One module preserves or supplies context, another performs local or fast transformations, and the composition is intended to reduce interference, simplify optimization, or preserve host structure. This suggests a broad but precise extension of the term: temporal modularity can refer either to modular structure in time-varying networks or to modular designs that explicitly partition temporal responsibilities across components.