Thermodynamics of Hyperbolic Graphs and the Physics of Complex Systems

Significant Achievements, Anticipated Outcomes and Future Work

We study complex networks in terms of their embeddings on hyperbolic manifolds, classified according to their genus. We observe that the structural properties of the network (degree distribution, clustering coefficient, etc.) are related to features of the manifold. The advantage of considering the graph embedding is two sided: 1) it provides a novel quantity to characterize network properties; 2) it gives a locally planar representation of the network. The latter property is useful in many respects. For example, it allows the application of powerful tools to generate ensembles of networks with specific complex properties by means of simple elementary moves acting on simpler “seed” networks. In this project we have focused on generation of foam-like tessellations of very high genus manifolds, typically containing up to 100,000 handles and 100,000 nodes. Work on construction of ordered networks has led to a number of initial reports (S. T. Hyde, S. Ramsden, T. Di Matteo and J. Longdell, "Ab-initio construction of some crystalline 3D euclidean networks", Solid State Sciences, 5, 35-45, (2003); S. T. Hyde and S. Ramsden, "Some novel three-dimensional euclidean crystalline networks derived from two-dimensional hyperbolic tilings", Eur Phys J B, 31, 273—284, (2003)). A preliminary application of these networks has been to model exchange processes within such systems, in order to explore possible underlying features of wealth distributions. We have demonstrated that distributions with power law tails can emerge from additive stochastic processes with interacting agents. In this case, we show that the network of connections among agents plays a crucial role. Indeed, the resulting wealth distribution is shaped directly by the degree distribution of the network; we choose a scale-free distribution, generated from our hyperbolic tiling algorithm finding a good qualitative agreement with the empirical data for the income distribution in Australia. This work, presented at the prestigious International School of Physics "Enrico Fermi"- The Physics of Complex Systems (New Advances and Perspectives), 1/07/03 - 11/07/03 Varenna, Italy, has been accepted for publication in Il Nuovo Cimento [T. Di Matteo, T. Aste and S. T. Hyde, "Exchanges in complex networks: income and wealth distributions", Nuovo Cimento (2004) in press; also available at the LANL arXiv (cond mat/0310544)].

Computational Techniques Used

We use a C code that generates network from a seed structure by means of elementary moves.

Publications, Awards and External Funding

External Funding and Awards

The ARC DP0344004: "The architecture of networks: Characterisation and visualisation of complex systems as fluctuating networks" is partially funding this project by providing a QEII fellowship to Di Matteo and with some funding for research.

Publications

S. T. Hyde, S. Ramsden, T. Di Matteo and J. Longdell, "Ab-initio construction of some crystalline 3D euclidean networks", Solid State Sciences, 5, 2003, 35-45.

S. T. Hyde and S. Ramsden, "Some novel three-dimensional euclidean crystalline networks derived from two-dimensional hyperbolic tilings", Eur Phys J B, 31, 2003, 273—284.

T. Di Matteo, T. Aste and S. T. Hyde, "Exchanges in complex networks: income and wealth distributions", Nuovo Cimento (2004) (accepted November 2003); available at the LANL arXiv (cond mat/0310544).