Improved spectral algorithm for the detection of network communities

We review and improve a recently introduced method for the detection of communities in complex networks. This method combines spectral properties of some matrices encoding the network topology, with well known hierarchical clustering techniques, and the use of the modularity parameter to quantify the goodness of any possible community subdivision. This provides one of the best available methods for the detection of community structures in complex systems.Comment: 4 pages, 1 fugure; to appear in the Proceedings of the 8th Granada Seminar - Computational and Statistical Physic

Network synchronization: Optimal and Pessimal Scale-Free Topologies

By employing a recently introduced optimization algorithm we explicitely design optimally synchronizable (unweighted) networks for any given scale-free degree distribution. We explore how the optimization process affects degree-degree correlations and observe a generic tendency towards disassortativity. Still, we show that there is not a one-to-one correspondence between synchronizability and disassortativity. On the other hand, we study the nature of optimally un-synchronizable networks, that is, networks whose topology minimizes the range of stability of the synchronous state. The resulting ``pessimal networks'' turn out to have a highly assortative string-like structure. We also derive a rigorous lower bound for the Laplacian eigenvalue ratio controlling synchronizability, which helps understanding the impact of degree correlations on network synchronizability.Comment: 11 pages, 4 figs, submitted to J. Phys. A (proceedings of Complex Networks 2007

Optimal network topologies: Expanders, Cages, Ramanujan graphs, Entangled networks and all that

We report on some recent developments in the search for optimal network topologies. First we review some basic concepts on spectral graph theory, including adjacency and Laplacian matrices, and paying special attention to the topological implications of having large spectral gaps. We also introduce related concepts as ``expanders'', Ramanujan, and Cage graphs. Afterwards, we discuss two different dynamical feautures of networks: synchronizability and flow of random walkers and so that they are optimized if the corresponding Laplacian matrix have a large spectral gap. From this, we show, by developing a numerical optimization algorithm that maximum synchronizability and fast random walk spreading are obtained for a particular type of extremely homogeneous regular networks, with long loops and poor modular structure, that we call entangled networks. These turn out to be related to Ramanujan and Cage graphs. We argue also that these graphs are very good finite-size approximations to Bethe lattices, and provide almost or almost optimal solutions to many other problems as, for instance, searchability in the presence of congestion or performance of neural networks. Finally, we study how these results are modified when studying dynamical processes controlled by a normalized (weighted and directed) dynamics; much more heterogeneous graphs are optimal in this case. Finally, a critical discussion of the limitations and possible extensions of this work is presented.Comment: 17 pages. 11 figures. Small corrections and a new reference. Accepted for pub. in JSTA

Network evolution towards optimal dynamical performance

Understanding the mutual interdependence between the behavior of dynamical processes on networks and the underlying topologies promises new insight for a large class of empirical networks. We present a generic approach to investigate this relationship which is applicable to a wide class of dynamics, namely to evolve networks using a performance measure based on the whole spectrum of the dynamics' time evolution operator. As an example, we consider the graph Laplacian describing diffusion processes, and we evolve the network structure such that a given sub-diffusive behavior emerges.Comment: 5 pages, 4 figure

A New Comparative Definition of Community and Corresponding Identifying Algorithm

In this paper, a new comparative definition for community in networks is proposed and the corresponding detecting algorithm is given. A community is defined as a set of nodes, which satisfy that each node's degree inside the community should not be smaller than the node's degree toward any other community. In the algorithm, the attractive force of a community to a node is defined as the connections between them. Then employing attractive force based self-organizing process, without any extra parameter, the best communities can be detected. Several artificial and real-world networks, including Zachary Karate club network and College football network are analyzed. The algorithm works well in detecting communities and it also gives a nice description for network division and group formation.Comment: 11 pages, 4 fihure

Vertex similarity in networks

We consider methods for quantifying the similarity of vertices in networks. We propose a measure of similarity based on the concept that two vertices are similar if their immediate neighbors in the network are themselves similar. This leads to a self-consistent matrix formulation of similarity that can be evaluated iteratively using only a knowledge of the adjacency matrix of the network. We test our similarity measure on computer-generated networks for which the expected results are known, and on a number of real-world networks

Network theory approach for data evaluation in the dynamic force spectroscopy of biomolecular interactions

Investigations of molecular bonds between single molecules and molecular complexes by the dynamic force spectroscopy are subject to large fluctuations at nanoscale and possible other aspecific binding, which mask the experimental output. Big efforts are devoted to develop methods for effective selection of the relevant experimental data, before taking the quantitative analysis of bond parameters. Here we present a methodology which is based on the application of graph theory. The force-distance curves corresponding to repeated pulling events are mapped onto their correlation network (mathematical graph). On these graphs the groups of similar curves appear as topological modules, which are identified using the spectral analysis of graphs. We demonstrate the approach by analyzing a large ensemble of the force-distance curves measured on: ssDNA-ssDNA, peptide-RNA (system from HIV1), and peptide-Au surface. Within our data sets the methodology systematically separates subgroups of curves which are related to different intermolecular interactions and to spatial arrangements in which the molecules are brought together and/or pulling speeds. This demonstrates the sensitivity of the method to the spatial degrees of freedom, suggesting potential applications in the case of large molecular complexes and situations with multiple binding sites