Characteristic exponents of complex networks

We present a novel way to characterize the structure of complex networks by studying the statistical properties of the trajectories of random walks over them. We consider time series corresponding to different properties of the nodes visited by the walkers. We show that the analysis of the fluctuations of these time series allows to define a set of characteristic exponents which capture the local and global organization of a network. This approach provides a way of solving two classical problems in network science, namely the systematic classification of networks, and the identification of the salient properties of growing networks. The results contribute to the construction of a unifying framework for the investigation of the structure and dynamics of complex systems.Comment: 6 pages, 5 figures, 1 tabl

Clusters in randomly-coloured spatial networks

The behaviour and functioning of a variety of complex physical and biological systems depend on the spatial organisation of their constituent units, and on the presence and formation of clusters of functionally similar or related individuals. Here we study the properties of clusters in spatially-embedded networks where nodes are coloured according to a given colouring process. This characterisation will allow us to use spatial networks with uniformly-coloured nodes as a null-model against which the importance, relevance, and significance of clusters of related units in a given real-world system can be assessed. We show that even a uniform and uncorrelated random colouring process can generate coloured clusters of substantial size and interesting shapes, which can be distinguished by using some simple dynamical measures, like the average time needed for a random walk to escape from the cluster. We provide a mean-field approach to study the properties of those clusters in large two-dimensional lattices, and we show that the analytical treatment agrees very well with the numerical results.Comment: 21 pages, 11 figure

Non-parametric resampling of random walks for spectral network clustering

Parametric resampling schemes have been recently introduced in complex network analysis with the aim of assessing the statistical significance of graph clustering and the robustness of community partitions. We propose here a method to replicate structural features of complex networks based on the non-parametric resampling of the transition matrix associated with an unbiased random walk on the graph. We test this bootstrapping technique on synthetic and real-world modular networks and we show that the ensemble of replicates obtained through resampling can be used to improve the performance of standard spectral algorithms for community detection.Comment: 5 pages, 2 figure

Social and place-focused communities in location-based online social networks

Thanks to widely available, cheap Internet access and the ubiquity of smartphones, millions of people around the world now use online location-based social networking services. Understanding the structural properties of these systems and their dependence upon users' habits and mobility has many potential applications, including resource recommendation and link prediction. Here, we construct and characterise social and place-focused graphs by using longitudinal information about declared social relationships and about users' visits to physical places collected from a popular online location-based social service. We show that although the social and place-focused graphs are constructed from the same data set, they have quite different structural properties. We find that the social and location-focused graphs have different global and meso-scale structure, and in particular that social and place-focused communities have negligible overlap. Consequently, group inference based on community detection performed on the social graph alone fails to isolate place-focused groups, even though these do exist in the network. By studying the evolution of tie structure within communities, we show that the time period over which location data are aggregated has a substantial impact on the stability of place-focused communities, and that information about place-based groups may be more useful for user-centric applications than that obtained from the analysis of social communities alone.Comment: 11 pages, 5 figure