Self-similarity of complex networks and hidden metric spaces (0710.2092v2)

Abstract: We demonstrate that the self-similarity of some scale-free networks with respect to a simple degree-thresholding renormalization scheme finds a natural interpretation in the assumption that network nodes exist in hidden metric spaces. Clustering, i.e., cycles of length three, plays a crucial role in this framework as a topological reflection of the triangle inequality in the hidden geometry. We prove that a class of hidden variable models with underlying metric spaces are able to accurately reproduce the self-similarity properties that we measured in the real networks. Our findings indicate that hidden geometries underlying these real networks are a plausible explanation for their observed topologies and, in particular, for their self-similarity with respect to the degree-based renormalization.

• The paper introduces a hidden metric space model that explains self-similarity in scale-free networks through degree-thresholding renormalization.
• The methodology preserves key network properties like degree distributions, degree correlations, and clustering coefficients in renormalized subgraphs.
• The findings imply practical applications in network design and routing by leveraging the intrinsic geometric principles of complex systems.

Self-Similarity of Complex Networks and Hidden Metric Spaces

The paper "Self-similarity of complex networks and hidden metric spaces" by M. Angeles Serrano, Dmitri Krioukov, and Marián Boguñá introduces a novel perspective on understanding the intrinsic properties of scale-free networks by associating them with hidden metric spaces. This approach aims to explain various non-trivial characteristics of complex networks, such as self-similarity and clustering, which are not adequately addressed by existing theories.

Overview of Concepts and Methodologies

The authors propose that many complex networks, which are typically scale-free, can be better understood by assuming that their nodes reside in hidden metric spaces. In this framework, the proximity between nodes in the hidden space correlates with the likelihood of connectivity in the observable network. This assumption aligns with the self-similarity observed in these networks when a degree-thresholding renormalization procedure is applied—a method that is central to this paper.

The paper employs degree-thresholding renormalization to extract subgraphs from larger network structures based on node degree. This method maintains the core topological features such as degree distributions, degree-degree correlations, and clustering coefficients across the renormalization process. These features are found to be remarkably consistent, and they adhere to power-law distributions similar to those observed in natural and technological networks, such as the Internet's Autonomous System level and social networks like the Pretty Good Privacy (PGP) network.

Empirical Observations and Theoretical Modeling

Empirical analysis conducted on real-world networks, including the Border Gateway Protocol (BGP) and PGP, as well as their randomized versions, highlight that while degree distributions and correlations remain invariant, clustering can be disrupted through randomization. This indicates an essential role of the underlying metric space which respects the triangle inequality, leading to natural clustering in the network.

The theoretical model incorporates hidden variables corresponding to nodes' coordinates within a metric space, which influences connection probabilities based on hidden metric distances. This model accounts for the observed self-similar properties without relying on explicit spatial embeddings. The degree distributions conform to power laws, supporting the assertion that complex network topologies can emerge from simple geometric principles.

Implications and Future Directions

The paper's findings suggest that hidden metric spaces offer a plausible explanation for the geometrical properties of scale-free networks, particularly regarding their self-similarity under renormalization. This theory extends beyond offering descriptive insights, as it holds potential for practical applications such as designing more efficient network protocols and algorithms for routing and search, leveraging the inherent self-similarity and clustering of networks.

Future research could further explore the relationship between fractality and self-similarity, as highlighted by earlier works on network box-covering techniques. Additionally, examining the extent to which hidden metric spaces and their geometric constraints impact the dynamic processes on networks, such as information dissemination and synchronization, could lead to more comprehensive models of complex systems.

In summary, this paper presents a noteworthy advancement in the paper of complex networks by framing them within hidden metric spaces, offering a unified explanation for several observed topological phenomena, and opening avenues for both theoretical exploration and practical application.