Stability of graph communities across time scales (0812.1811v4)

Abstract: The complexity of biological, social and engineering networks makes it desirable to find natural partitions into communities that can act as simplified descriptions and provide insight into the structure and function of the overall system. Although community detection methods abound, there is a lack of consensus on how to quantify and rank the quality of partitions. We show here that the quality of a partition can be measured in terms of its stability, defined in terms of the clustered autocovariance of a Markov process taking place on the graph. Because the stability has an intrinsic dependence on time scales of the graph, it allows us to compare and rank partitions at each time and also to establish the time spans over which partitions are optimal. Hence the Markov time acts effectively as an intrinsic resolution parameter that establishes a hierarchy of increasingly coarser clusterings. Within our framework we can then provide a unifying view of several standard partitioning measures: modularity and normalized cut size can be interpreted as one-step time measures, whereas Fiedler's spectral clustering emerges at long times. We apply our method to characterize the relevance and persistence of partitions over time for constructive and real networks, including hierarchical graphs and social networks. We also obtain reduced descriptions for atomic level protein structures over different time scales.

• The paper introduces a stability measure based on the clustered autocovariance of a Markov process that quantifies how long partitions persist, ranking graph communities over time.
• The framework unifies standard metrics like modularity and normalized cut, linking short-term measures to long-term spectral clustering techniques.
• Numerical results on networks, including social and protein structure graphs, demonstrate the method's ability to reveal robust, multi-scale community structures.

Stability of Graph Communities Across Time Scales

The paper "Stability of Graph Communities Across Time Scales" by J.-C. Delvenne, S. Yaliraki, and M. Barahona introduces a framework for evaluating the quality of graph partitions by measuring their stability over time using a Markov process. This approach provides a unified view of various community detection methods and offers insights into how partitions can reveal underlying structures in complex networks.

Key Contributions

The authors define the stability of a partition in terms of the clustered autocovariance of a Markov process on a graph. This measure intrinsically depends on time scales, allowing for a comparison and ranking of partitions. As time acts as an intrinsic resolution parameter, it facilitates a hierarchy of clusterings ranging from fine to coarse. The framework unifies standard measures such as modularity and normalized cut size, which correspond to one-step time measures, and relates them to Fiedler’s spectral clustering method at long times.

The method's applicability extends to hierarchical graphs, social networks, and atomic-level models of protein structures, illustrating its generality. The stability measure is particularly useful in overcoming the resolution limit of modularity, enabling finer partitioning.

Analytical Framework

The authors explore the connection between Markov processes and graph partitions, emphasizing the relationship between graph structures and random walks. A stationary distribution is established, and the persistence of clusters is measured through autocovariance. The proposed measure evaluates the quality of clusterings at all times, crucially indicating that the most stable partitions persist longest.

The stability measure bridges the gap among existing heuristics. It considers paths of varying lengths to evaluate clustering quality, offering an interpretation of Markov time as an intrinsic resolution parameter. This approach not only highlights the stability of conventional measures but also extends beyond their limitations, particularly the resolution limit of modularity.

Numerical Results and Applications

The paper provides numerical examples demonstrating the framework's effectiveness. For a network of scientific collaborations, the stability measure identifies relevant meta-community structures indicating strong internal connections over time. The stability curves produced by different algorithms illustrate how well they capture the time-dependent community structures.

In protein structural graphs, the method identifies partitions that align with known biophysical features, providing a compromise between simplicity and rigidity prediction. This highlights the potential for developing reduced models through these community structures.

Implications and Future Directions

The stability framework advances theoretical understanding by linking together disparate clustering methodologies through a time-scale perspective. Practically, it offers robust tools for network analysis, enabling the exploration of dynamic processes occurring over multiple time scales. The methodology’s extension to weighted and directed graphs appears promising, and further research could enhance its applicability to real-world dynamic systems.

In conclusion, the paper presents a compelling framework that enriches the theoretical landscape of community detection in networks and opens pathways for innovative applications in diverse fields such as bioinformatics, social sciences, and engineering networks.