Discovering Communities in Continuous-Time Temporal Networks by Optimizing L-Modularity

TL;DR

This work tackles dynamic community detection in continuous-time networks by introducing LAGO, a greedy optimization framework for Longitudinal Modularity on link streams. LAGO avoids time discretization, leveraging the Trimmed Communities Property to focus on active time nodes and employing a Recursive Time Module Mover with refinement and substitution strategies. Across synthetic and real datasets, LAGO demonstrates the ability to recover temporally coherent communities and provides practical guidance on which variant configurations perform best under different modularity objectives. The approach is applicable beyond L-Modularity, enabling optimization of other quality functions defined over link streams, and is supported by open-source code for reproducibility.

Abstract

Community detection is a fundamental problem in network analysis, with many applications in various fields. Extending community detection to the temporal setting with exact temporal accuracy, as required by real-world dynamic data, necessitates methods specifically adapted to the temporal nature of interactions. We introduce LAGO, a novel method for uncovering dynamic communities by greedy optimization of Longitudinal Modularity, a specific adaptation of Modularity for continuous-time networks. Unlike prior approaches that rely on time discretization or assume rigid community evolution, LAGO captures the precise moments when nodes enter and exit communities. We evaluate LAGO on synthetic benchmarks and real-world datasets, demonstrating its ability to efficiently uncover temporally and topologically coherent communities.

Figures (6)

• Figure 1: Representation of a link stream with two dynamic communities. Nodes are represented in ordinates, and interactions between them occur through time.
• Figure 2: Illustration of trimmed communities. L-Modularity assigns a higher value to the trimmed community structure that starts and ends on active time nodes represented with black dots.
• Figure 3: Illustration of the start of the Time Module Mover at the finest time modules level, where each active time node is first assigned to its own time module. Each active time node (e.g. $et_3$), is affiliated to the community that increases L-Modularity the most. Candidates for affiliation changes are topological neighbors (e.g. $dt_3$) and temporally adjacent time nodes (e.g. $et_1$, $et_4$). This process is repeated until no further affiliation change improves L-Modularity, leading to two time modules represented by the two colors
• Figure 4: Performance of LAGO variants in recovering the ground truth community structure (Fig. \ref{['figure:naive:gtcs']}) across different link stream sizes. Both versions of L-Modularity are optimized. The following metrics are reported: time of execution, final L-Modularity values, and the accuracy of community recovery as measured by the Normalized Variation of Information (NVI) between the detected and ground truth communities. The size of a link stream is the number of its active time nodes.
• Figure 5: Relative comparisons of LAGO variants (see Table \ref{['tab1']}) when optimizing both L-Modularity versions. For a given variant, the cross indicates the mean ranking of time execution and the mean L-Modularity value ranking, areas are for the values between the 1st and 3rd quartile of each axe, and the whiskers indicates values between the 1st and 9th deciles. Best LAGO variants are in the bottom left corner.

Theorems & Definitions (6)

• Definition 1
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• Definition 6