Higher-order organization of complex networks (1612.08447v1)

Abstract: Networks are a fundamental tool for understanding and modeling complex systems in physics, biology, neuroscience, engineering, and social science. Many networks are known to exhibit rich, lower-order connectivity patterns that can be captured at the level of individual nodes and edges. However, higher-order organization of complex networks---at the level of small network subgraphs---remains largely unknown. Here we develop a generalized framework for clustering networks based on higher-order connectivity patterns. This framework provides mathematical guarantees on the optimality of obtained clusters and scales to networks with billions of edges. The framework reveals higher-order organization in a number of networks including information propagation units in neuronal networks and hub structure in transportation networks. Results show that networks exhibit rich higher-order organizational structures that are exposed by clustering based on higher-order connectivity patterns.

• The paper introduces a novel framework that clusters networks based on higher-order connectivity patterns by minimizing motif conductance.
• It employs spectral ordering on a motif-normalized Laplacian combined with a sweep process to efficiently detect clusters in large-scale networks.
• Empirical analyses in neuronal and transportation networks demonstrate that the approach uncovers hidden modular structures and identifies key functional hubs.

Higher-order Organization of Complex Networks

The paper of complex networks has long focused on lower-order structures, such as individual nodes and edges, to understand the intricate connectivity patterns inherent in various domains, such as physics, biology, and social sciences. However, this paper significantly shifts the perspective to the higher-order organization of complex networks. The authors, Benson, Gleich, and Leskovec, present a novel and generalized framework for clustering networks based on higher-order connectivity patterns, particularly focusing on small network subgraphs or network motifs.

Framework and Methodology

The crux of the framework lies in identifying clusters of nodes that participate in many instances of a given network motif while simultaneously minimizing the disruption of these motifs. This is mathematically framed as the minimization of the motif conductance, an extension of the conductance metric used in spectral graph theory. Calculating the exact set of nodes that minimizes the motif conductance is computationally infeasible due to its NP-hard nature. To circumvent this, the authors developed an optimization framework that offers near-optimal solutions and operates efficiently even for networks with billions of edges.

Specifically, the clustering method involves:

1. Forming the motif adjacency matrix by counting motif co-occurrences.
2. Computing the spectral ordering based on the eigenvector of the motif-normalized Laplacian matrix.
3. Finding the set of nodes that minimizes motif conductance through a sweep process.

Empirical Findings

The authors applied their framework to a variety of networks:

• Neuronal Networks: For instance, they analyzed the C. elegans neuronal network using the bi-fan motif and uncovered clusters indicative of information propagation mechanisms. This provided insights into specific neuron clusters involved in nictation control.
• Transportation Networks: By examining transportation reachability networks using two-hop motifs, they identified hubs and essential interconnecting airports. Remarkably, the primary spectral coordinate from this analysis showed a strong correlation with city populations, while the secondary spectral coordinate captured geographical insights.

The paper outlines the computational feasibility of their approach. Even for networks with up to two billion edges, the motif adjacency matrix and subsequent clustering computations were shown to be practical.

Implications and Future Directions

This framework is groundbreaking as it unifies motif analysis and network partitioning—two fundamental tools in network science. The discovery of higher-order organizational structures provides new dimensions to understand complex systems which were previously hidden when focusing solely on lower-order patterns.

Practical Implications:

• Biological Networks: Unveiling functional modules based on network motifs can advance our understanding of biological processes and gene regulation mechanisms.
• Infrastructure Networks: Identifying critical hubs and resilient paths in transportation networks can influence urban planning and emergency response strategies.

Theoretical Implications:

• Network Theory: The connection between higher-order motif clustering and spectral graph theory opens new research avenues for understanding the mathematical properties of networks.
• Algorithmic Developments: Further enhancements in the efficiency of the underlying computation, especially for larger motifs and more complex networks, could lead to more robust and scalable analysis.

Conclusion

The research by Benson, Gleich, and Leskovec extends the frontier of network science by shifting the focus to higher-order connectivity patterns. By leveraging motifs, the proposed framework offers unprecedented insights into the modular organization of complex networks. Future research could extend this framework to dynamic networks and explore its applicability in real-time analysis. The implications of this work are far-reaching across various scientific and engineering disciplines, paving the way for more nuanced and comprehensive network analyses.