Longitudinal Modularity: Temporal Dynamics

• Longitudinal modularity is a framework that defines network modules with explicit temporal dimensions, capturing dynamic behavior through state dependence and perturbation spread.
• Methodologies include perturbation propagation in dynamical systems, quality functions on link streams, and repeated modularity detection in rolling correlation networks.
• Applications span financial market analysis, neural network training, and evolutionary studies, illustrating how modular structures persist, shift, and inform system behavior over time.

Longitudinal modularity denotes modular organization defined with an explicit temporal dimension. In the literature, the expression is used for several non-equivalent constructions: a dynamical-systems notion in which modules are groups of variables that constrain the spread of perturbations over time; a modularity quality function on link streams in which communities are sets of node–time pairs and the score compares observed internal temporal interactions to a longitudinal null model while penalizing community switches; and a longitudinal analysis protocol in which modular partitions are recomputed through time and then compared for persistence, instability, or regime change (Kolchinsky et al., 2015, Brabant et al., 2024, Wirth et al., 2024). Across these usages, the common departure from static modularity is that modular structure is indexed by time horizon, state, community lifetime, training epoch, or evolutionary history rather than treated as a single timeless partition.

1. Conceptual scope and relation to static modularity

Static modularity is typically written as

$$Q = \frac{1}{2m}\sum_{i,j}\left(A_{ij}-\frac{k_i k_j}{2m}\right)\delta(c_i,c_j),$$

with $A_{ij}$ an adjacency matrix, $k_i=\sum_j A_{ij}$, $m=\frac12\sum_{ij}A_{ij}$, and $\delta(c_i,c_j)=1$ when $i$ and $j$ are assigned to the same community (Brabant et al., 2024). Longitudinal variants retain the observed-minus-expected logic but alter the object being partitioned, the null model, or the way temporal continuity is scored.

One line of work replaces static edges by perturbation propagation in a multivariate dynamical system, so that modularity becomes a property of how perturbation mass remains localized across a time horizon $t$ (Kolchinsky et al., 2015). A second line defines communities directly on link streams, with $C\subseteq V\times T$, and evaluates temporal edge counts inside such communities against longitudinal expectations such as Joint-Membership and Mean-Membership, together with a penalty on Community Switch Count (Brabant et al., 2024, Brabant et al., 1 Oct 2025). A third line applies modularity-based community detection repeatedly to a sequence of time-indexed correlation networks and then studies persistence or reorganization by measures such as the Adjusted Rand Index (Wirth et al., 2024). In still other settings, the phrase is used behaviorally or developmentally: modularity is the ability to reuse independently learned components across temporal gaps, or an order parameter that emerges, drifts, saturates, and restructures over evolutionary time (Portegys, 2021, Lorenz et al., 2012).

A common source of confusion is that these constructions are not interchangeable. Some are intrinsic quality functions on temporal objects, some are state-dependent dynamical diagnostics, and some are longitudinal analysis schemes built on repeated static modularity. The shared theme is temporalized mesoscopic organization, not a single universal formula.

2. Perturbation-based longitudinal modularity in dynamical systems

In "Modularity and the spread of perturbations in complex dynamical systems" (Kolchinsky et al., 2015), the basic object is an $N$-dimensional dynamical system with evolution operator $A_{ij}$0. Given an initial state $A_{ij}$1 and perturbation $A_{ij}$2, the relative perturbation size in a subsystem $A_{ij}$3 at time $A_{ij}$4 is

$A_{ij}$5

For a partition $A_{ij}$6, the coarse-grained perturbation profile is

$A_{ij}$7

Perturbation modularity is then defined as the autocovariance

$A_{ij}$8

with expectations over a perturbation distribution $A_{ij}$9. The optimal decomposition at horizon $k_i=\sum_j A_{ij}$0 is

$k_i=\sum_j A_{ij}$1

This construction is longitudinal in three explicit senses. First, $k_i=\sum_j A_{ij}$2 is a resolution parameter: small $k_i=\sum_j A_{ij}$3 emphasizes fast local propagation, intermediate $k_i=\sum_j A_{ij}$4 reveals mesoscopic retention, and large $k_i=\sum_j A_{ij}$5 tends toward coarse or degenerate partitions as perturbations spread throughout the system. Second, the score is state-dependent because both $k_i=\sum_j A_{ij}$6 and $k_i=\sum_j A_{ij}$7 are defined relative to the current initial condition $k_i=\sum_j A_{ij}$8. Third, it is perturbation-dependent because the expectations are taken over a chosen perturbation class. The paper states that when there is not a time scale of a priori interest, optimal decompositions can be identified at multiple resolutions by sweeping across a range of time scales (Kolchinsky et al., 2015).

The central dynamical interpretation is separation of fast intramodular from slow intermodular spreading. Modules are subsets of variables that trap perturbations over the chosen horizon. In the special case of diffusion dynamics $k_i=\sum_j A_{ij}$9 with single-variable perturbations $m=\frac12\sum_{ij}A_{ij}$0 and the $m=\frac12\sum_{ij}A_{ij}$1 norm, perturbation modularity coincides with Markov stability for random walks on graphs (Kolchinsky et al., 2015). The same paper also shows that, with $m=\frac12\sum_{ij}A_{ij}$2, perturbation modularity can be rewritten exactly as directed weighted Newman's modularity on an induced graph with weights

$m=\frac12\sum_{ij}A_{ij}$3

The empirical examples make the longitudinal point explicit. In 80 coupled logistic maps with planted hierarchy, scanning over $m=\frac12\sum_{ij}A_{ij}$4 recovers 8 low-level modules at short times, 4 mid-level modules at intermediate times, and 2 high-level modules at longer times, before $m=\frac12\sum_{ij}A_{ij}$5 falls toward zero once perturbations spread globally (Kolchinsky et al., 2015). In homogeneous coupled map lattices, the same framework detects self-organized modularity that depends on initial state, dynamical parameters, and perturbation type, including a case with $m=\frac12\sum_{ij}A_{ij}$6, $m=\frac12\sum_{ij}A_{ij}$7, and fixed horizon $m=\frac12\sum_{ij}A_{ij}$8 where optimal perturbation modularity increases stepwise along a 12,000-step trajectory as spatial domains emerge (Kolchinsky et al., 2015). This suggests a notion of modularity tied directly to dynamical propagation rather than to a pre-existing static graph.

3. Longitudinal Modularity on link streams

"Longitudinal Modularity, a Modularity for Link Streams" (Brabant et al., 2024) introduces a modularity-style quality function defined directly on a link stream

$m=\frac12\sum_{ij}A_{ij}$9

with $\delta(c_i,c_j)=1$0 a finite set of instantaneous, undirected interactions. A dynamic community structure $\delta(c_i,c_j)=1$1 is a set of mutually exclusive communities $\delta(c_i,c_j)=1$2. For a community $\delta(c_i,c_j)=1$3, the membership times of node $\delta(c_i,c_j)=1$4 are

$\delta(c_i,c_j)=1$5

and the number of interactions between $\delta(c_i,c_j)=1$6 and $\delta(c_i,c_j)=1$7 while both are in $\delta(c_i,c_j)=1$8 is

$\delta(c_i,c_j)=1$9

The abstract temporal form is

$i$0

where $i$1 counts observed internal temporal interactions, $i$2 is a longitudinal null-model expectation, and $i$3 penalizes discontinuous community membership (Brabant et al., 2024). In the paper’s concrete formulation,

$i$4

with $i$5 and

$i$6

where $i$7 is the community switch count for node $i$8.

Three longitudinal expectations are defined. The Co-membership model is

$i$9

The Joint-Membership model is

$j$0

The Mean-Membership model is

$j$1

These satisfy $j$2 termwise (Brabant et al., 2024).

The paper establishes three properties that motivate the recommended variants. First, L-Modularity is independent of time aggregation: aggregating interactions into weighted or multi-edge snapshots without losing information leaves the score unchanged. Second, adding the switch penalty yields a smoothness incentive, so a temporally continuous community is preferred to an unnecessary split across two identical time intervals. Third, using Joint-Membership or Mean-Membership yields robustness to topochrone disconnections, meaning that a community formed by components sharing neither nodes nor time is better split (Brabant et al., 2024). When $j$3, or when communities are temporally stationary, the temporal terms collapse and the score reduces to static modularity.

The conceptual difference from multislice modularity is direct. L-Modularity works on the event-level link stream, not on a chosen snapshot sequence; it uses a longitudinal configuration-model logic rather than per-snapshot null models; and it penalizes switching through Community Switch Count rather than artificial interslice edges (Brabant et al., 2024). This makes temporal resolution part of the formalism rather than a preprocessing choice.

4. Continuous-time optimization and generalized L-Modularity

"Discovering Communities in Continuous-Time Temporal Networks by Optimizing L-Modularity" (Brabant et al., 1 Oct 2025) specializes L-Modularity to continuous-time link streams and introduces LAGO, a greedy optimizer for dynamic communities with exact join and leave times. In this formulation, a dynamic community structure is again a set of non-overlapping communities $j$4, and the score is

$j$5

with $j$6. Here $j$7 is a time-smoothness or resolution parameter: $j$8 allows rapid switching, while larger $j$9 favors fewer, longer memberships (Brabant et al., 1 Oct 2025).

Two structural ingredients are central. The set of active time nodes is

$t$0

The Trimmed Communities Property shows that trimming inactive boundary times never decreases the score: $t$1 As a result, optimization can be restricted to active time nodes (Brabant et al., 1 Oct 2025).

LAGO initializes each active time node as its own time module and then applies Recursive Time Module Mover, with candidate moves restricted to topological neighbors and temporal neighbors. Refinements include Single Time Node Movements, Sub Time Module Movements, and Single Time Edge Movements; Fast Exploration maintains a dynamic candidate set so that recomputation is concentrated where changes are ongoing (Brabant et al., 1 Oct 2025). On Mosaic benchmarks, LAGO recovers planted dynamic communities, and among 14 variants the paper reports that $t$2 consistently performs well in both quality and speed. On the SocioPatterns primary-school data, LAGO is applied directly to a link stream of 72 students, 1,555 timesteps, and 28,904 active time nodes, detecting 11 dynamic communities for a focal student that align with lecture periods and lunchtime mixing; the comparison multilayer representation contains 111,960 time nodes $t$3 (Brabant et al., 1 Oct 2025).

"Generalized L-Modularity for Community Detection Beyond Simple Temporal Networks" (Brabant et al., 23 May 2026) extends the framework to directed, weighted, multipartite, timestamp-based, delayed, and interval-based interactions. In the generalized setting, the expected intra-community weight is

$t$4

and the full score becomes

$t$5

with $t$6 a structural resolution parameter and $t$7 the temporal smoothness penalty (Brabant et al., 23 May 2026).

The generalized data model supports instantaneous events, delayed interactions with departure and arrival times, and continuous interval-based contacts. It also preserves direction, weights, and multipartite masks $t$8 rather than forcing aggregation, projection, or undirected simplification (Brabant et al., 23 May 2026). LAGO is correspondingly generalized from active time nodes to active time-segment nodes when interval segmentation is required. The empirical cases—directed weighted Eurovision voting, delayed versus continuous Citi Bike trips, and bipartite Bluesky user–hashtag streams—show that modeling choice changes the detected notion of temporal cohesion itself (Brabant et al., 23 May 2026). A plausible implication is that longitudinal modularity is not only time-aware but semantics-aware: different temporal encodings expose different mesoscopic structure even on the same raw system.

5. Longitudinal modularity in financial correlation networks

"Longitudinal market structure detection using a dynamic modularity-spectral algorithm" (Wirth et al., 2024) uses the phrase in a snapshot-plus-comparison sense. For each month $t$9 and look-back horizon $C\subseteq V\times T$0 months, the method computes a Pearson correlation matrix of daily returns,

$C\subseteq V\times T$1

producing 82, 79, 73, and 61 matrices for the 3-, 6-, 12-, and 24-month windows, respectively (Wirth et al., 2024). The optimization target at each snapshot is

$C\subseteq V\times T$2

where $C\subseteq V\times T$3 and $C\subseteq V\times T$4 are the average intra- and inter-cluster correlations.

DynMSA applies a fixed pipeline at each time point: compute the correlation matrix, clean it by Random Matrix Theory using the Marchenko–Pastur bounds, threshold the cleaned matrix by

$C\subseteq V\times T$5

run Leiden modularity optimization on the thresholded graph with sector-based initialization from GICS labels, and then perform local spectral refinement inside selected communities (Wirth et al., 2024). The dynamic aspect is not an interlayer modularity term; it is the repeated estimation of modular structure and the explicit comparison of successive partitions. Stability is quantified with the Adjusted Rand Index,

$C\subseteq V\times T$6

computed between consecutive monthly partitions (Wirth et al., 2024).

The empirical interpretation is explicitly longitudinal. Short look-backs produce more adaptive but more volatile partitions, while longer look-backs produce smoother partitions. Around the onset of COVID-19 in March 2020, the number of clusters drops sharply and ARI plunges for short windows, indicating regime change and a temporary collapse toward a more homogeneous market (Wirth et al., 2024). Twelve-month windows are reported as an attractive balance between responsiveness and stability, and 24-month windows yield a relatively stable structure of about 11–12 clusters with ARI values near 1 for long stretches outside crisis periods (Wirth et al., 2024). Portfolios built from these longitudinal clusters showed higher Sortino and Sharpe ratios, lower downside volatility, reduced maximum drawdown, and higher annualised returns than an equally weighted market benchmark (Wirth et al., 2024). This use of the term therefore denotes repeated modularity inference on evolving correlation networks rather than an intrinsic temporal quality function.

6. Learning, compositional reuse, and training-time modular structure

"A modularity comparison of Long Short-Term Memory and Morphognosis neural networks" (Portegys, 2021) uses modularity in a functional and behavioral sense. Modularity is defined as the ability to combine reusable components of learned behavior to solve an overarching task rather than learning the task from beginning to end every time a portion of it changes. The experimental system consists of mazes built from rooms and doors, with context begin rooms, maze entry rooms, maze interior rooms, and context end rooms. Door associations and intervening mazes are trained separately; at test time, previously unseen composite mazes are formed by inserting an independently trained intervening maze between a learned context begin/end pair (Portegys, 2021). A run counts as correct only if all door choices are correct.

This is longitudinal modularity in the sense of learned components that must persist over a temporal gap and be recomposed without catastrophic interference. Both the LSTM and Morphognosis perform well during training, but Morphognosis is significantly better than LSTM on the modular testing task (Portegys, 2021). The interpretation given is architectural: Morphognosis records context information modularly in the input rather than non-modularly entangled in recurrent state, and its subsumption-like dual-network scheme separates short-term and long-term temporal contexts (Portegys, 2021). The result is a temporal form of modular reuse rather than a graph-theoretic modularity score.

"Modular Boundaries in Recurrent Neural Networks" (Tanner et al., 2023) examines modularity across training in continuous-time RNNs. Functional connectivity modules are obtained by computing correlation matrices of recurrent activity and then applying modularity maximization with the Louvain algorithm. In a training-time reading, the work treats modular structure as something that appears before training, is transformed by learning, and becomes increasingly stable as performance improves (Tanner et al., 2023). For perceptual decision RNNs, mean correlation between module stability and log-loss across training is $C\subseteq V\times T$7, with analogous values $C\subseteq V\times T$8 for a feedforward MNIST classifier trained with backpropagation and $C\subseteq V\times T$9 for the same architecture trained with predictive coding (Tanner et al., 2023). Networks sharing the same initialized input weights, or matched input and output weights, converge to more similar final modular organizations than networks with different initializations (Tanner et al., 2023).

The same paper also shows that modules matter causally. Output lesions targeting specific modules selectively impair the corresponding behavioral alternative, and lesion-similarity analyses show that neurons within the same functional-connectivity module have similar effects on the phase portrait when individually lesioned (Tanner et al., 2023). This suggests a longitudinal concept of modularity tied to learning trajectories: modules are not merely endpoint clusters but increasingly stable dynamical substructures shaped by initial projection asymmetries and refined by task training.

7. Evolutionary and biological trajectories

"The Emergence of Modularity in Biological Systems" (Lorenz et al., 2012) treats modularity and hierarchy as quantities that emerge and reorganize over evolutionary time. A module is defined operationally as a component that can operate relatively independently of the rest of the system, and modularity emerges when there are more intramodule connections than intermodule connections. The review uses several measures, including Newman's modularity $N$0, a normalized within-module density $N$1, bandedness, and the cophenetic correlation coefficient for hierarchy (Lorenz et al., 2012).

Its central theoretical model is a spin-glass evolutionary system on a rugged fitness landscape with horizontal gene transfer and environmental change. Modularity is represented by excess block structure $N$2, where $N$3 is the baseline value for random wiring, and functions as an order parameter in a symmetry-breaking phase transition (Lorenz et al., 2012). The model has three timescales: fast sequence evolution, intermediate environmental change, and slow evolution of interaction structures. Starting from random contact matrices, modularity increases approximately linearly at early stages, depends on the severity and frequency of environmental change, and approaches a steady state set by the balance between mutational entropy and selection for evolvability (Lorenz et al., 2012).

The review’s longitudinal synthesis is hierarchical. In protein interaction networks, first-order modularity increases over evolutionary time and then saturates, while second-order modularity continues to increase after the first-order plateau (Lorenz et al., 2012). The same temporal logic is invoked across metabolic networks, gene networks, food webs, developmental systems, physiology, and social networks. During treadmill stress, modularity of the interbeat covariance matrix increases before heart rate rises; functional brain networks in young adults are more modular than those of older adults; and hierarchy in the world trade network increases during recessions while globalization lowers hierarchy and slows recovery (Lorenz et al., 2012). This suggests a broad use of longitudinal modularity as a trajectory-level descriptor: modular organization is treated as a variable that grows, saturates, decays, or shifts across evolutionary, developmental, physiological, and socio-technical timescales.

Taken together, these literatures show that longitudinal modularity is best understood as a family resemblance concept. In dynamical systems it is the retention of perturbation mass across time; in link streams it is observed-minus-expected temporal interaction density with explicit switching costs; in rolling correlation studies it is modular structure tracked across successive snapshots; and in learning and evolutionary settings it is modular organization followed across training or historical time (Kolchinsky et al., 2015, Brabant et al., 2024, Wirth et al., 2024, Lorenz et al., 2012). The unifying claim is not that all of these formalisms are identical, but that modularity becomes longitudinal once temporal persistence, state dependence, switching, or historical emergence is part of the object being measured.