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. Community detection is an important task in network analysis. It is used to uncover structural patterns and to reduce the complexity of large-scale graphs. Community detection has applications in many domains where systems can be modeled as networks, such as social science, economics, and biology. In the static setting, leading approaches such as Louvain [1], Infomap [2], or Leiden [3] typically rely on defining an objective function and optimizing it using greedy algorithms. This approach offers two main advantages: it produces communities that are meaningful according to a well-defined quality measure, and it scales efficiently to large graphs due to the computational simplicity of greedy methods. Real-world data often involves temporal dynamics, where interactions occur at specific timestamps.

Temporal networks are commonly used to model real-life phenomena. When these phenomena represent interactions and are captured at a fine-grained temporal resolution, they are modeled as link streams. Community detection is an essential network analysis task. Although many methods exist for static networks, and some methods have been developed for temporal networks represented as sequences of snapshots, few works can handle link streams. This article introduces the first adaptation of the well-known Modularity quality function to link streams. Unlike existing methods, it is independent of the time scale of analysis. After introducing the quality function, and its relation to existing static and dynamic definitions of Modularity, we show experimentally its relevance for dynamic community evaluation.

A link stream is a set of triplets $(t, u, v)$ indicating that $u$ and $v$ interacted at time $t$. Link streams model numerous datasets and their proper study is crucial in many applications. In practice, raw link streams are often aggregated or transformed into time series or graphs where decisions are made. Yet, it remains unclear how the dynamical and structural information of a raw link stream carries into the transformed object. This work shows that it is possible to shed light into this question by studying link streams via algebraically linear graph and signal operators, for which we introduce a novel linear matrix framework for the analysis of link streams. We show that, due to their linearity, most methods in signal processing can be easily adopted by our framework to analyze the time/frequency information of link streams. However, the availability of linear graph methods to analyze relational/structural information is limited. We address this limitation by developing (i) a new basis for graphs that allow us to decompose them into structures at different resolution levels; and (ii) filters for graphs that allow us to change their structural information in a controlled manner. By plugging-in these developments and their time-domain counterpart into our framework, we are able to (i) obtain a new basis for link streams that allow us to represent them in a frequency-structure domain; and (ii) show that many interesting transformations to link streams, like the aggregation of interactions or their embedding into a euclidean space, can be seen as simple filters in our frequency-structure domain.

Graph theory provides a language for studying the structure of relations, and it is often used to study interactions over time too. However, it poorly captures the both temporal and structural nature of interactions, that calls for a dedicated formalism. In this paper, we generalize graph concepts in order to cope with both aspects in a consistent way. We start with elementary concepts like density, clusters, or paths, and derive from them more advanced concepts like cliques, degrees, clustering coefficients, or connected components. We obtain a language to directly deal with interactions over time, similar to the language provided by graphs to deal with relations. This formalism is self-consistent: usual relations between different concepts are preserved. It is also consistent with graph theory: graph concepts are special cases of the ones we introduce. This makes it easy to generalize higher-level objects such as quotient graphs, line graphs, k-cores, and centralities. This paper also considers discrete versus continuous time assumptions, instantaneous links, and extensions to more complex cases.