Simplicial Closure and higher-order link prediction (1802.06916v2)

Abstract: Networks provide a powerful formalism for modeling complex systems by using a model of pairwise interactions. But much of the structure within these systems involves interactions that take place among more than two nodes at once; for example, communication within a group rather than person-to person, collaboration among a team rather than a pair of coauthors, or biological interaction between a set of molecules rather than just two. Such higher-order interactions are ubiquitous, but their empirical study has received limited attention, and little is known about possible organizational principles of such structures. Here we study the temporal evolution of 19 datasets with explicit accounting for higher-order interactions. We show that there is a rich variety of structure in our datasets but datasets from the same system types have consistent patterns of higher-order structure. Furthermore, we find that tie strength and edge density are competing positive indicators of higher-order organization, and these trends are consistent across interactions involving differing numbers of nodes. To systematically further the study of theories for such higher-order structures, we propose higher-order link prediction as a benchmark problem to assess models and algorithms that predict higher-order structure. We find a fundamental differences from traditional pairwise link prediction, with a greater role for local rather than long-range information in predicting the appearance of new interactions.

• The paper introduces simplicial closure, extending traditional triadic closure to model interactions among multiple nodes.
• The study analyzes 19 diverse datasets to uncover unique structural patterns in higher-order network interactions.
• The paper develops predictive models using local heuristics and supervised learning to accurately forecast multi-node link formation.

Overview of "Simplicial Closure and Higher-Order Link Prediction"

The paper "Simplicial Closure and Higher-Order Link Prediction" presents a comprehensive paper on higher-order interactions within networked systems, shifting the typical focus from dyadic interactions to more complex multi-node structures. This work highlights the ubiquity and significance of higher-order interactions in diverse domains by analyzing 19 datasets, seeking to uncover consistent structural features, and proposing methods to predict the evolution of these interactions.

A central theme of the paper is the notion of "simplicial closure," analogously extending the concept of triadic closure to multi-node interactions. Unlike the traditional pairwise link prediction that solely focuses on the potential formation of dyads, this research broadens the scope to include interactions that may form among three or more nodes simultaneously. The authors introduce higher-order link prediction as a benchmark problem for which they develop predictive models and algorithms.

Structural Analysis of Higher-Order Interactions

By examining datasets across various domains—such as academic coauthorship, tagging systems, email networks, and biological networks—the research identifies that these systems exhibit distinctive patterns of higher-order connectivity that cannot be reduced to or fully captured by dyadic interaction representations.

Key findings from the structural analysis include:

• Networks display a broad spectrum of patterns when analyzing higher-order interactions, with the fraction of open triangles vastly differing across domains.
• Increased tie strength and edge density consistently emerge as positive indicators for higher-order link formations, highlighting the role of cohesive substructures in facilitating multi-node interactions.
• Local network statistics are effective discriminators of the system domain, highlighting that higher-order features reflect organizing principles unique to the contextual setting.

Temporal Dynamics and Predictive Models

The temporal analysis shows that open triangles often transition to closed states, termed simplicial closure events, driven by rich local structures. This provides evidence that networks evolve in ways that preferentially create cohesive groups. Furthermore, the paper's exploration demonstrates that these organic formations are best modeled using local tightly-knit interactions rather than drawing solely from long-range connectivity cues.

The authors propose various algorithmic approaches to the higher-order link prediction problem, including:

• Employing simple local heuristics, such as generalized means of edge weights, which surprisingly outperform more complex path-based models traditionally used in dyadic link prediction tasks.
• A supervised learning predictor leveraging multiple network features, achieving high prediction precision, particularly in dense datasets with rich feature variability.

Implications and Future Directions

The introduction of higher-order link prediction as a framework serves as an avenue for further development in understanding the complexities of networked systems. This work illustrates that while existing pairwise models provide a foundational understanding, they overlook crucial aspects presented in higher-order interactions. The findings have significant implications for fields like sociology, biology, and computer science, where modeling group dynamics is essential.

Further research directions may explore:

• Developing more nuanced models to understand better the mechanistic underpinnings of higher-order interactions across broader contexts.
• Extending the higher-order predictions to incorporate dynamic network changes, such as node addition or deletion, thus offering a more holistic model of real-world systems.
• Integrating these models with computational topology techniques, potentially uncovering new dimensions of network evolution and topology.

This paper offers a well-rounded framework not only for deciphering the intrinsic organizational structures within complex networks but also for methodologically advancing the analysis and prediction of these intricate higher-order interactions.