Multi-scale Modularity in Complex Networks

We focus on the detection of communities in multi-scale networks, namely networks made of different levels of organization and in which modules exist at different scales. It is first shown that methods based on modularity are not appropriate to uncover modules in empirical networks, mainly because modularity optimization has an intrinsic bias towards partitions having a characteristic number of modules which might not be compatible with the modular organization of the system. We argue for the use of more flexible quality functions incorporating a resolution parameter that allows us to reveal the natural scales of the system. Different types of multi-resolution quality functions are described and unified by looking at the partitioning problem from a dynamical viewpoint. Finally, significant values of the resolution parameter are selected by using complementary measures of robustness of the uncovered partitions. The methods are illustrated on a benchmark and an empirical network.Comment: 8 pages, 3 figure

Unanimity Rule on networks

We introduce a model for innovation-, evolution- and opinion dynamics whose spreading is dictated by unanimity rules, i.e. a node will change its (binary) state only if all of its neighbours have the same corresponding state. It is shown that a transition takes place depending on the initial condition of the problem. In particular, a critical number of initially activated nodes is needed so that the whole system gets activated in the long-time limit. The influence of the degree distribution of the nodes is naturally taken into account. For simple network topologies we solve the model analytically, the cases of random, small-world and scale-free are studied in detail.Comment: 7 pages 4 fig

Efficient spares matrix multiplication scheme for the CYBER 203

This work has been directed toward the development of an efficient algorithm for performing this computation on the CYBER-203. The desire to provide software which gives the user the choice between the often conflicting goals of minimizing central processing (CPU) time or storage requirements has led to a diagonal-based algorithm in which one of three types of storage is selected for each diagonal. For each storage type, an initialization sub-routine estimates the CPU and storage requirements based upon results from previously performed numerical experimentation. These requirements are adjusted by weights provided by the user which reflect the relative importance the user places on the resources. The three storage types employed were chosen to be efficient on the CYBER-203 for diagonals which are sparse, moderately sparse, or dense; however, for many densities, no diagonal type is most efficient with respect to both resource requirements. The user-supplied weights dictate the choice

Collaborative tagging as a tripartite network

We describe online collaborative communities by tripartite networks, the nodes being persons, items and tags. We introduce projection methods in order to uncover the structures of the networks, i.e. communities of users, genre families... To do so, we focus on the correlations between the nodes, depending on their profiles, and use percolation techniques that consist in removing less correlated links and observing the shaping of disconnected islands. The structuring of the network is visualised by using a tree representation. The notion of diversity in the system is also discussed

Energy non-equipartition in multicomponent granular mixtures

We study non-equipartition of energy in granular fluids composed by an arbitrarily large number of components. We focus on a simple mean field model, based upon a Maxwell collision operator kernel, and predict the temperature ratios for the species. Moreover, we perform Direct Monte Carlo simulations in order to verify the predictions.Comment: submitted to PR

A Brownian particle having a fluctuating mass

We focus on the dynamics of a Brownian particle whose mass fluctuates. First we show that the behaviour is similar to that of a Brownian particle moving in a fluctuating medium, as studied by Beck [Phys. Rev. Lett. 87 (2001) 180601]. By performing numerical simulations of the Langevin equation, we check the theoretical predictions derived in the adiabatic limit, and study deviations outside this limit. We compare the mass velocity distribution with truncated Tsallis distributions [J. Stat. Phys. 52 (1988) 479] and find excellent agreement if the masses are chi- squared distributed. We also consider the diffusion of the Brownian particle by studying a Bernoulli random walk with fluctuating walk length in one dimension. We observe the time dependence of the position distribution kurtosis and find interesting behaviours. We point out a few physical cases where the mass fluctuation problem could be encountered as a first approximation for agglomeration- fracture non equilibrium processes.Comment: submitted to PR

Temporal Pattern of Online Communication Spike Trains in Spreading a Scientific Rumor: How Often, Who Interacts with Whom?

We study complex time series (spike trains) of online user communication while spreading messages about the discovery of the Higgs boson in Twitter. We focus on online social interactions among users such as retweet, mention, and reply, and construct different types of active (performing an action) and passive (receiving an action) spike trains for each user. The spike trains are analyzed by means of local variation, to quantify the temporal behavior of active and passive users, as a function of their activity and popularity. We show that the active spike trains are bursty, independently of their activation frequency. For passive spike trains, in contrast, the local variation of popular users presents uncorrelated (Poisson random) dynamics. We further characterize the correlations of the local variation in different interactions. We obtain high values of correlation, and thus consistent temporal behavior, between retweets and mentions, but only for popular users, indicating that creating online attention suggests an alignment in the dynamics of the two interactions.Comment: A statistical data analysis & data mining on Social Dynamic Behavior, 9 pages and 7 figure

On the genre-fication of Music: a percolation approach (long version)

In this paper, we analyze web-downloaded data on people sharing their music library. By attributing to each music group usual music genres (Rock, Pop...), and analysing correlations between music groups of different genres with percolation-idea based methods, we probe the reality of these subdivisions and construct a music genre cartography, with a tree representation. We also show the diversity of music genres with Shannon entropy arguments, and discuss an alternative objective way to classify music, that is based on the complex structure of the groups audience. Finally, a link is drawn with the theory of hidden variables in complex networks.Comment: 7 pages, 5 figures, submitted to the proceedings of the 3rd International Conference NEXT-SigmaPh

Uncovering collective listening habits and music genres in bipartite networks

In this paper, we analyze web-downloaded data on people sharing their music library, that we use as their individual musical signatures (IMS). The system is represented by a bipartite network, nodes being the music groups and the listeners. Music groups audience size behaves like a power law, but the individual music library size is an exponential with deviations at small values. In order to extract structures from the network, we focus on correlation matrices, that we filter by removing the least correlated links. This percolation idea-based method reveals the emergence of social communities and music genres, that are visualised by a branching representation. Evidence of collective listening habits that do not fit the neat usual genres defined by the music industry indicates an alternative way of classifying listeners/music groups. The structure of the network is also studied by a more refined method, based upon a random walk exploration of its properties. Finally, a personal identification - community imitation model (PICI) for growing bipartite networks is outlined, following Potts ingredients. Simulation results do reproduce quite well the empirical data.Comment: submitted to PR

Growing network with j-redirection

A model for growing information networks is introduced where nodes receive new links through j-redirection, i.e. the probability for a node to receive a link depends on the number of paths of length j arriving at this node. In detail, when a new node enters the network, it either connects to a randomly selected node, or to the j -ancestor of this selected node. The j -ancestor is found by following j links from the randomly selected node. The system is shown to undergo a transition to a phase where condensates develop. We also find analytical predictions for the height statistics and show numerically the non-trivial behaviour of the degree distribution.Comment: 7 page