Constant Block Jacobi Model

• The paper introduces a framework that integrates node-level memory into temporal network community detection via modularity maximization.
• It employs a Markovian model with constraints on marginal link probability and one-lag persistence to characterize edge evolution over time.
• The model lowers detectability thresholds in dynamic stochastic block models, validated through both synthetic simulations and real-world network experiments.

Constant Block Jacobi Model

The Constant Block Jacobi Model, as formulated in "Linking Through Time: Memory-Enhanced Community Discovery in Temporal Networks," provides a rigorous framework for community detection in temporal networks, specifically addressing limitations of memory-agnostic approaches by explicitly incorporating node-level memory via modularity maximization (Clemente et al., 2024). This methodology is motivated by the Markovian dynamics exhibited in many real-world networks, where structural correlations persist over time and the temporal autocorrelation of links can be exploited to both sharpen community structure and push fundamental performance limits such as the detectability threshold.

1. Formal Model of Markovian Temporal Networks

Consider a temporal network described by a sequence of $T$ undirected graph snapshots

$G_t=(N,E_t), \quad t=1,\ldots, T;\quad |N|=N$

with adjacency matrices $a_{ij}(t)$: $a_{ij}(t)=1$ if $(i,j)\in E_t$, zero otherwise. The Constant Block Jacobi Model assumes each pair $(i,j)$ evolves as a two-state, time-homogeneous Markov chain with transition matrix

$$P_{ij} = \begin{pmatrix} P[a_{ij}(t+1)=0\,|\,0] & P[1\,|\,0] \ P[0\,|\,1] & P[1\,|\,1] \end{pmatrix}$$

The parameterization enforces two key constraints: - Marginal link probability: $p_{ij} = \langle a_{ij}(t) \rangle$ - One-lag persistence: $q_{ij} = \langle a_{ij}(t)a_{ij}(t+1) \rangle$

Community assignment is assumed persistent over $t$: $G_t=(N,E_t), \quad t=1,\ldots, T;\quad |N|=N$0, i.e. the label $G_t=(N,E_t), \quad t=1,\ldots, T;\quad |N|=N$1 is time-independent.

2. Memory-Enhanced Modularity Formulation

The null model is derived from a maximum-entropy construction with the following node-level constraints: 1. Expected average degree: $G_t=(N,E_t), \quad t=1,\ldots, T;\quad |N|=N$2 2. Expected average persisting degree: $G_t=(N,E_t), \quad t=1,\ldots, T;\quad |N|=N$3

The corresponding maximum likelihood Hamiltonian is

$G_t=(N,E_t), \quad t=1,\ldots, T;\quad |N|=N$4

Solving for the null model yields analytical expressions for $G_t=(N,E_t), \quad t=1,\ldots, T;\quad |N|=N$5 and $G_t=(N,E_t), \quad t=1,\ldots, T;\quad |N|=N$6 (see supplementary methods in (Clemente et al., 2024)), providing expected values for both time-averaged edge occurrence and time-averaged edge persistence. The modularity function explicitly exploits these as follows:

• Define $G_t=(N,E_t), \quad t=1,\ldots, T;\quad |N|=N$7
• The modularity reads

$G_t=(N,E_t), \quad t=1,\ldots, T;\quad |N|=N$8

where $G_t=(N,E_t), \quad t=1,\ldots, T;\quad |N|=N$9. The classical “no-memory” modularity is recovered by replacing $a_{ij}(t)$0 with $a_{ij}(t)$1 and $a_{ij}(t)$2 with $a_{ij}(t)$3.

3. Fundamental Detectability Thresholds

The Constant Block Jacobi Model enables a precise analysis of the detectability threshold in the degree-corrected stochastic block model (SBM) regime:

• Static SBM: For two equal-sized blocks, the classical threshold is

$a_{ij}(t)$4

• Dynamic SBM (DSBM) over $a_{ij}(t)$5 i.i.d. realizations: Threshold reduces to $a_{ij}(t)$6
• Markovian temporal SBM with edge relaxation time $a_{ij}(t)$7:

$a_{ij}(t)$8

• Incorporating memory (persisting degree) yields an improved threshold:

$a_{ij}(t)$9

where $a_{ij}(t)=1$0 is the mean persisting-degree signal. When $a_{ij}(t)=1$1, memory always strictly lowers the detectability limit, making previously undetectable communities recoverable.

4. Greedy Optimization and Complexity

Maximization of the Constant Block Jacobi modularity is accomplished via the Clauset–Newman–Moore greedy agglomerative heuristic:

1. Initialize each node in its own singleton community.
2. Compute all inter-community modularity gains $a_{ij}(t)=1$2.
3. Iteratively merge the pair yielding the largest positive $a_{ij}(t)=1$3, updating the modularity increments.
4. Terminate when no further improvement is possible.

For sparse graphs, the per-iteration cost is $a_{ij}(t)=1$4, with the null-model parameters $a_{ij}(t)=1$5 precomputed once in $a_{ij}(t)=1$6 (can be accelerated for sparsity).

5. Numerical Experiments and Empirical Results

Extensive synthetic simulations are conducted:

• $a_{ij}(t)=1$7 nodes, two blocks, average degree $a_{ij}(t)=1$8, trajectory lengths $a_{ij}(t)=1$9, with $(i,j)\in E_t$0 varied over relevant parameter space.
• Metrics: Adjusted Rand Index (ARI) between detected and planted assignments.
  • No-memory static: detection possible only above classical $(i,j)\in E_t$1.
  • No-memory temporal: improved threshold $(i,j)\in E_t$2, but sensitivity to edge autocorrelation ($(i,j)\in E_t$3).
  • Memory-enhanced modularity demonstrates detectability in parameter regions where all memory-agnostic baselines fail, matching the theoretical prediction $(i,j)\in E_t$4.

In real-world networks (e.g., SocioPatterns primary school), the model is used to optimize the aggregation window for constructing snapshot networks, using modularity to reveal temporal segments of maximal community structure. The method identifies the timescale ($(i,j)\in E_t$5 minutes) at which the combination of temporal correlation and modularity is maximized, directly informing analysis and preprocessing choices.

6. Broader Implications and Future Directions

By enforcing constraints on both degrees and link-persistence, the Constant Block Jacobi Model fundamentally extends the classical modularity framework for temporal networks, showing:

• Memory mechanisms (persistence) are critical for reducing the organizational noise floor inherent in dynamic graph data.
• The modularity-based approach naturally accommodates the detection of temporal scales and nontrivial times of maximal community structure.
• The framework could be extended to allow for time-varying community assignments, overlapping clusters, directed or weighted interactions, and incorporation of higher-order or long-lag memory.
• Efficient (possibly approximate) solvers for $(i,j)\in E_t$6 will be required for large-scale, sparse temporal networks.
• Incorporation of memory effects into other classes of community detection methods (beyond modularity) could yield further improvements to temporal network inference.

The results situate the Constant Block Jacobi Model as a principled, physically and information-theoretically motivated method for addressing the interplay of memory and mesoscale organization in complex temporal networks (Clemente et al., 2024).