Universal features of network topology

Recent studies have revealed characteristic general features in the topology of real-world networks. We investigate the universality of mechanisms that result in the power-law behaviour of many real-world networks, paying particular attention to the Barabasi-Albert process of preferential attachment as the most successful. We introduce a variation on this theme where at each time step either a new vertex and edge is added to the network or a new edge is created between two existing vertices. This process retains a power-law degree distribution, while other variations destroy it. We also introduce alternative models which favour connections to vertices with high degree but by a different mechanism and find that one of the models displays behaviour that is compatible with a power-law degree distribution

Dynamic scaling and universality in evolution of fluctuating random networks

We found that models of evolving random networks exhibit dynamic scaling similar to scaling of growing surfaces. It is demonstrated by numerical simulations of two variants of the model in which nodes are added as well as removed [Phys. Rev. Lett. 83, 5587 (1999)]. The averaged size and connectivity of the network increase as power-laws in early times but later saturate. Saturated values and times of saturation change with paramaters controlling the local evolution of the network topology. Both saturated values and times of saturation obey also power-law dependences on controlling parameters. Scaling exponents are calculated and universal features are discussed.Comment: 7 pages, 6 figures, Europhysics Letters for

From Network Structure to Dynamics and Back Again: Relating dynamical stability and connection topology in biological complex systems

The recent discovery of universal principles underlying many complex networks occurring across a wide range of length scales in the biological world has spurred physicists in trying to understand such features using techniques from statistical physics and non-linear dynamics. In this paper, we look at a few examples of biological networks to see how similar questions can come up in very different contexts. We review some of our recent work that looks at how network structure (e.g., its connection topology) can dictate the nature of its dynamics, and conversely, how dynamical considerations constrain the network structure. We also see how networks occurring in nature can evolve to modular configurations as a result of simultaneously trying to satisfy multiple structural and dynamical constraints. The resulting optimal networks possess hubs and have heterogeneous degree distribution similar to those seen in biological systems.Comment: 15 pages, 6 figures, to appear in Proceedings of "Dynamics On and Of Complex Networks", ECSS'07 Satellite Workshop, Dresden, Oct 1-5, 200

Fundamental statistical features and self-similar properties of tagged networks

We investigate the fundamental statistical features of tagged (or annotated) networks having a rich variety of attributes associated with their nodes. Tags (attributes, annotations, properties, features, etc.) provide essential information about the entity represented by a given node, thus, taking them into account represents a significant step towards a more complete description of the structure of large complex systems. Our main goal here is to uncover the relations between the statistical properties of the node tags and those of the graph topology. In order to better characterise the networks with tagged nodes, we introduce a number of new notions, including tag-assortativity (relating link probability to node similarity), and new quantities, such as node uniqueness (measuring how rarely the tags of a node occur in the network) and tag-assortativity exponent. We apply our approach to three large networks representing very different domains of complex systems. A number of the tag related quantities display analogous behaviour (e.g., the networks we studied are tag-assortative, indicating possible universal aspects of tags versus topology), while some other features, such as the distribution of the node uniqueness, show variability from network to network allowing for pin-pointing large scale specific features of real-world complex networks. We also find that for each network the topology and the tag distribution are scale invariant, and this self-similar property of the networks can be well characterised by the tag-assortativity exponent, which is specific to each system

Classification of scale-free networks

While the emergence of a power law degree distribution in complex networks is intriguing, the degree exponent is not universal. Here we show that the betweenness centrality displays a power-law distribution with an exponent \eta which is robust and use it to classify the scale-free networks. We have observed two universality classes with \eta \approx 2.2(1) and 2.0, respectively. Real world networks for the former are the protein interaction networks, the metabolic networks for eukaryotes and bacteria, and the co-authorship network, and those for the latter one are the Internet, the world-wide web, and the metabolic networks for archaea. Distinct features of the mass-distance relation, generic topology of geodesics and resilience under attack of the two classes are identified. Various model networks also belong to either of the two classes while their degree exponents are tunable.Comment: 6 Pages, 6 Figures, 1 tabl