Statistical mechanics of topological phase transitions in networks

We provide a phenomenological theory for topological transitions in restructuring networks. In this statistical mechanical approach energy is assigned to the different network topologies and temperature is used as a quantity referring to the level of noise during the rewiring of the edges. The associated microscopic dynamics satisfies the detailed balance condition and is equivalent to a lattice gas model on the edge-dual graph of a fully connected network. In our studies -- based on an exact enumeration method, Monte-Carlo simulations, and theoretical considerations -- we find a rich variety of topological phase transitions when the temperature is varied. These transitions signal singular changes in the essential features of the global structure of the network. Depending on the energy function chosen, the observed transitions can be best monitored using the order parameters Phi_s=s_{max}/M, i.e., the size of the largest connected component divided by the number of edges, or Phi_k=k_{max}/M, the largest degree in the network divided by the number of edges. If, for example the energy is chosen to be E=-s_{max}, the observed transition is analogous to the percolation phase transition of random graphs. For this choice of the energy, the phase-diagram in the [,T] plane is constructed. Single vertex energies of the form E=sum_i f(k_i), where k_i is the degree of vertex i, are also studied. Depending on the form of f(k_i), first order and continuous phase transitions can be observed. In case of f(k_i)=-(k_i+c)ln(k_i), the transition is continuous, and at the critical temperature scale-free graphs can be recovered.Comment: 12 pages, 12 figures, minor changes, added a new refernce, to appear in PR

Ontologies and tag-statistics

Due to the increasing popularity of collaborative tagging systems, the research on tagged networks, hypergraphs, ontologies, folksonomies and other related concepts is becoming an important interdisciplinary topic with great actuality and relevance for practical applications. In most collaborative tagging systems the tagging by the users is completely "flat", while in some cases they are allowed to define a shallow hierarchy for their own tags. However, usually no overall hierarchical organisation of the tags is given, and one of the interesting challenges of this area is to provide an algorithm generating the ontology of the tags from the available data. In contrast, there are also other type of tagged networks available for research, where the tags are already organised into a directed acyclic graph (DAG), encapsulating the "is a sub-category of" type of hierarchy between each other. In this paper we study how this DAG affects the statistical distribution of tags on the nodes marked by the tags in various real networks. We analyse the relation between the tag-frequency and the position of the tag in the DAG in two large sub-networks of the English Wikipedia and a protein-protein interaction network. We also study the tag co-occurrence statistics by introducing a 2d tag-distance distribution preserving both the difference in the levels and the absolute distance in the DAG for the co-occurring pairs of tags. Our most interesting finding is that the local relevance of tags in the DAG, (i.e., their rank or significance as characterised by, e.g., the length of the branches starting from them) is much more important than their global distance from the root. Furthermore, we also introduce a simple tagging model based on random walks on the DAG, capable of reproducing the main statistical features of tag co-occurrence.Comment: Submitted to New Journal of Physic