Albert Díaz-Guilera

Comparing community structure identification

We compare recent approaches to community structure identification in terms of sensitivity and computational cost. The recently proposed modularity measure is revisited and the performance of the methods as applied to ad hoc networks with known community structure, is compared. We find that the most accurate methods tend to be more computationally expensive, and that both aspects need to be considered when choosing a method for practical purposes. The work is intended as an introduction as well as a proposal for a standard benchmark test of community detection methods.

Self-similar community structure in a network of human interactions.

We propose a procedure for analyzing and characterizing complex networks. We apply this to the social network as constructed from email communications within a medium sized university with about 1700 employees. Email networks provide an accurate and nonintrusive description of the flow of information within human organizations. Our results reveal the self-organization of the network into a state where the distribution of community sizes is self-similar. This suggests that a universal mechanism, responsible for emergence of scaling in other self-organized complex systems, as, for instance, river networks, could also be the underlying driving force in the formation and evolution of social networks.

Synchronization Reveals Topological Scales in Complex Networks

We study the relationship between topological scales and dynamic time scales in complex networks. The analysis is based on the full dynamics towards synchronization of a system of coupled oscillators. In the synchronization process, modular structures corresponding to well-defined communities of nodes emerge in different time scales, ordered in a hierarchical way. The analysis also provides a useful connection between synchronization dynamics, complex networks topology, and spectral graph analysis.

Optimal network topologies for local search with congestion

The problem of searchability in decentralized complex networks is of great importance in computer science, economy, and sociology. We present a formalism that is able to cope simultaneously with the problem of search and the congestion effects that arise when parallel searches are performed, and we obtain expressions for the average search cost both in the presence and the absence of congestion. This formalism is used to obtain optimal network structures for a system using a local search algorithm. It is found that only two classes of networks can be optimal: starlike configurations, when the number of parallel searches is small, and homogeneous-isotropic configurations, when it is large.

Diffusion dynamics on multiplex networks.

We study the time scales associated with diffusion processes that take place on multiplex networks, i.e., on a set of networks linked through interconnected layers. To this end, we propose the construction of a supra-laplacian matrix, which consists of a dimensional lifting of the laplacian matrix of each layer of the multiplex network. We use perturbative analysis to reveal analytically the structure of eigenvectors and eigenvalues of the complete network in terms of the spectral properties of the individual layers. The spectrum of the supra-laplacian allows us to understand the physics of diffusionlike processes on top of multiplex networks.

Communication in Networks with Hierarchical Branching

We present a simple model of communication in networks with hierarchical branching. We analyze the behavior of the model from the viewpoint of critical systems under different situations. For certain values of the parameters, a continuous phase transition between a sparse and a congested regime is observed and accurately described by an order parameter and the power spectra. At the critical point the behavior of the model is totally independent of the number of hierarchical levels. Also scaling properties are observed when the size of the system varies. The presence of noise in the communication is shown to break the transition. The analytical results are a useful guide to forecasting the main features of real networks.

Community analysis in social networks

Abstract.We present an empirical study of different social networks obtained from digital repositories. Our analysis reveals the community structure and provides a useful visualising technique. We investigate the scaling properties of the community size distribution, and find that all the networks exhibit power law scaling in the community size distributions with exponent either -0.5 or -1. Finally we find that the networks’ community structure is topologically self-similar using the Horton-Strahler index.

The effect of size heterogeneity on community identification in complex networks

Identifying community structure can be a potent tool in the analysis and understanding of the structure of complex networks. Up to now, methods for evaluating the performance of identification algorithms use ad-hoc networks with communities of equal size. We show that inhomogeneities in community sizes can and do affect the performance of algorithms considerably, and propose an alternative method which takes these factors into account. Furthermore, we propose a simple modification of the algorithm proposed by Newman for community detection (Phys. Rev. E 69 066133) which treats communities of different sizes on an equal footing, and show that it outperforms the original algorithm while retaining its speed.

Dynamical properties of model communication networks

We study the dynamical properties of a collection of models for communication processes, characterized by a single parameter xi representing the relation between information load of the nodes and its ability to deliver this information. The critical transition to congestion reported so far occurs only for xi=1. This case is well analyzed for different network topologies. We focus on the properties of the order parameter, the susceptibility, and the time correlations when approaching the critical point. For xi<1, no transition to congestion is observed but it remains a crossover from a low-density to a high-density state. For xi>1, the transition to congestion is discontinuous and congestion nuclei arise.

Synchronization processes in complex networks

We present an extended analysis, based on the dynamics towards synchronization of a system of coupled oscillators, of the hierarchy of communities in complex networks. In the synchronization process, different structures corresponding to well defined communities of nodes appear in a hierarchical way. The analysis also provides a useful connection between synchronization dynamics, complex networks topology and spectral graph analysis.