The relationship of the rich-club property and geometry in complex networks

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Dr. Gulyás András
Department of Telecommunications and Media Informatics

In the last few decades, network theory has become one of the fastest developing domains in science. The significant increase in the performance of computers and the tremendous amount of available data have helped bring forward this progress. The detailed analysis of real-world networks carried out in the recent past have revealed the stunning similarity of these systems. These similarities include, but are not limited to properties like scale-free degree-distribution, small diameter and high clustering coefficient. Many of the mechanisms responsible for these similarities have been understood through different network generating models.

However, real networks are not alike when regarding some other properties, most notably the rich-club property. The rich-club coefficient gives a measure of how densely the most connected nodes are linked to each other in the network, so its presence indicates that the most influential elements in the system heavily interact with each other. Contrary to the properties mentioned above, real systems have remarkably different rich-club coefficients e.g. the network of flights possesses a strong rich-core while the protein-protein interaction network misses any kind of such organization. This great variety of the rich-club coefficient could possibly account for the different functions and circumstances real networks have beyond the impressive similarities they share.

However, no network model has given an intuitive model that could help understand why the results concerning the rich-club property are so different for certain systems.

In this thesis I will give a brief overview of the most significant network properties and models and introduce a new model that generates networks with different rich-club coefficients in an intuitive way. To investigate the behavior of the model I implemented a simulation in Python. The results of the simulations will be analyzed in detail and supported with analytic calculations showing the key properties of the resulting networks.


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