Self-similarity of temporal interaction networks arises from hyperbolic geometry with time-varying curvature (2409.07733v1)

Abstract: The self-similarity of complex systems has been studied intensely across different domains due to its potential applications in system modeling, complexity analysis, etc., as well as for deep theoretical interest. Existing studies rely on scale transformations conceptualized over either a definite geometric structure of the system (very often realized as length-scale transformations) or purely temporal scale transformations. However, many physical and social systems are observed as temporal interactions among agents without any definitive geometry. Yet, one can imagine the existence of an underlying notion of distance as the interactions are mostly localized. Analysing only the time-scale transformations over such systems would uncover only a limited aspect of the complexity. In this work, we propose a novel technique of scale transformation that dissects temporal interaction networks under spatio-temporal scales, namely, flow scales. Upon experimenting with multiple social and biological interaction networks, we find that many of them possess a finite fractal dimension under flow-scale transformation. Finally, we relate the emergence of flow-scale self-similarity to the latent geometry of such networks. We observe strong evidence that justifies the assumption of an underlying, variable-curvature hyperbolic geometry that induces self-similarity of temporal interaction networks. Our work bears implications for modeling temporal interaction networks at different scales and uncovering their latent geometric structures.

• The paper develops a novel spatio-temporal box-counting technique to measure self-similarity and fractal dimensions in temporal interaction networks.
• Empirical analysis shows real networks exhibit scale-invariance or finite fractal dimensions, providing evidence for latent hyperbolic geometry with time-varying curvature.
• Theoretical simulations indicate that achieving scale-invariance across spatial and temporal dimensions requires an exponentially increasing negative curvature over time.

Analysis of Self-Similarity in Temporal Interaction Networks Using Hyperbolic Geometry

The paper by Subhabrata Dutta, Dipankar Das, and Tanmoy Chakraborty offers an intriguing exploration into the field of self-similarity in temporal interaction networks through the lens of hyperbolic geometry with time-varying curvature. This paper makes several methodical claims regarding the nature of complex systems and proposes novel methods of analyzing them.

Overview of the Research

The authors identify the self-similarity of complex systems—a characteristic where systems retain their structure across different scales—as a significant trait for understanding complexity in diverse network systems. Traditional approaches often focus on either spatial or temporal scale transformations independently, which may not capture the full complexity inherent in temporal interaction networks where interactions lack definitive geometry.

This paper fills a gap by developing a technique that examines these networks through a combined spatial and temporal lens, described as "flow scales." The flow scale transformations enable the authors to derive the fractal dimensions of such networks, empirically concluding that latent hyperbolic spaces underlie these networks with variable negative curvature induced over time.

Methodology

Central to the methodology is crafting a spatio-temporal box-counting technique for temporal interaction networks. This involves segmenting the networks into static snapshots using predefined time windows, which are then subjected to box-counting to uncover their self-similar properties. The fractal dimension is calculated across varying flow scales, revealing distinct patterns of scale invariance. The authors showcase these properties using real-world datasets, encompassing social and biological interaction networks.

Furthermore, the research extends to simulating point-particle motions over hyperbolic space, where hyperbolic geometry is characterized by a constant or dynamically changing negative curvature. Results from these simulations facilitate understanding the connection between the observed scale invariance in real networks and possible latent geometric structures.

Findings and Implications

The paper's findings are twofold:

1. Real-World Observations: Among multiple interaction networks analyzed, some exhibit finite fractal dimensions under the proposed flow-scale transformations, while others exhibit scale-invariance, providing strong evidence for the hypothesis of underlying hyperbolic geometry with time-varying curvature.
2. Theoretical Implications: Through simulations, it is theorized that a structure with an exponentially increasing negative curvature over time is necessary for achieving scale-invariance across both temporal and spatial dimensions.

The implications of these findings are vast. Practically, this framework could offer enhanced predictive models for interaction networks by embedding them into suitable hyperbolic spaces. Theoretically, the paper enriches the discourse around the geometric interpretations of temporal network dynamics and suggests potential new pathways in analyzing network evolution and structure.

Future Directions

The paper posits future research avenues in utilizing these geometric insights for better network embedding, a task pivotal for network analysis and applications such as recommendation systems and social network analysis. Further work could expand upon the scale of datasets analyzed, incorporate real-time interaction data, and refine the proposed methods to account for noises inherent in empirical data.

In summary, this research advances our understanding of network self-similarity by innovatively applying hyperbolic geometry to temporal interactions, paving the way for a deeper understanding of complex systems in computational science.