# The modularity of random graphs on the hyperbolic plane

@article{Chellig2021TheMO, title={The modularity of random graphs on the hyperbolic plane}, author={Jordan Chellig and Nikolaos Fountoulakis and Fiona Skerman}, journal={J. Complex Networks}, year={2021}, volume={10} }

Modularity is a quantity which has been introduced in the context of complex networks in order to quantify how close a network is to an ideal modular network in which the nodes form small interconnected communities that are joined together with relatively few edges. In this paper, we consider this quantity on a recent probabilistic model of complex networks introduced by Krioukov et al. (Phys. Rev. E 2010). This model views a complex network as an expression of hidden hierarchies, encapsulated…

## References

SHOWING 1-10 OF 27 REFERENCES

The probability of connectivity in a hyperbolic model of complex networks

- Computer Science, MathematicsRandom Struct. Algorithms
- 2016

This paper considers a model for complex networks that was introduced by Krioukov et al. (2010) and focuses on the probability that the graph is connected, which shows the following results.

Typical distances in a geometric model for complex networks

- Computer Science, MathematicsInternet Math.
- 2017

A further property is shown: the distance between two uniformly chosen vertices that belong to the same component is doubly logarithmic in $N$, i.e., the graph is an ~\emph{ultra-small world}.

On the Largest Component of a Hyperbolic Model of Complex Networks

- Mathematics, Computer ScienceElectron. J. Comb.
- 2015

The present paper focuses on the evolution of the component structure of the random graph, and shows that for $\alpha > 1$ and $\nu$ arbitrary, with high probability, as the number of vertices grows, the largest component of therandom graph has sublinear order.

The diameter of KPKVB random graphs

- MathematicsAdvances in Applied Probability
- 2019

Abstract We consider a random graph model that was recently proposed as a model for complex networks by Krioukov et al. (2010). In this model, nodes are chosen randomly inside a disk in the…

Random Hyperbolic Graphs: Degree Sequence and Clustering

- Mathematics, Computer ScienceArXiv
- 2012

This work initiates the rigorous study of random hyperbolic graphs and confirms rigorously that the degree sequence follows a power-law distribution with controllable exponent and that the clustering is nonvanishing.

Hyperbolic Geometry of Complex Networks

- Mathematics, MedicinePhysical review. E, Statistical, nonlinear, and soft matter physics
- 2010

It is shown that targeted transport processes without global topology knowledge are maximally efficient, according to all efficiency measures, in networks with strongest heterogeneity and clustering, and that this efficiency is remarkably robust with respect to even catastrophic disturbances and damages to the network structure.

Scale-free network clustering in hyperbolic and other random graphs

- Computer Science, PhysicsJournal of Physics A: Mathematical and Theoretical
- 2019

A variational principle is introduced to explain how vertices tend to cluster in triangles as a function of their degrees and it is shown that clustering in the hyperbolic model is non-vanishing and self-averaging, so that a single random graph sample is a good representation in the large-network limit.

Statistical mechanics of complex networks

- Computer Science, PhysicsArXiv
- 2001

A simple model based on these two principles was able to reproduce the power-law degree distribution of real networks, indicating a heterogeneous topology in which the majority of the nodes have a small degree, but there is a significant fraction of highly connected nodes that play an important role in the connectivity of the network.

Finding community structure in networks using the eigenvectors of matrices.

- Mathematics, MedicinePhysical review. E, Statistical, nonlinear, and soft matter physics
- 2006

A modularity matrix plays a role in community detection similar to that played by the graph Laplacian in graph partitioning calculations, and a spectral measure of bipartite structure in networks and a centrality measure that identifies vertices that occupy central positions within the communities to which they belong are proposed.

Network Clustering via Maximizing Modularity: Approximation Algorithms and Theoretical Limits

- Mathematics, Computer Science2015 IEEE International Conference on Data Mining
- 2015

This paper proposes the first additive approximation algorithm for modularity clustering with a constant factor, and provides a rigorous proof that a CS with modularity arbitrary close to maximum modularity QOPT might bear no similarity to the optimal CS ofmaximum modularity.