Here we introduce an approach to analyse the main statistical features of the interwoven sets of overlapping communities making a step towards the uncovering of the modular structure of complex systems. After defining a set of new characteristic quantities for the statistics of communities, we apply an efficient technique to explore overlapping communities on a large scale. We find that overlaps are significant, and the distributions we introduce reveal universal features of networks. Our studies of collaboration, word association, and protein interaction graphs demonstrate that the web of communities has non-trivial correlations and specific scaling properties.

Supplementary data are available on Bioinformatics online.

We show that the formation of membrane tubes (or membrane tethers), which is a crucial step in many biological processes, is highly non-trivial and involves first order shape transitions. The force exerted by an emerging tube is a non-monotonic function of its length. We point out that tubes attract each other, which eventually leads to their coalescence. We also show that detached tubes behave like semiflexible filaments with a rather short persistence length. We suggest that these properties play an important role in the formation and structure of tubular organelles.

Many natural and social systems develop complex networks that are usually modeled as random graphs. The eigenvalue spectrum of these graphs provides information about their structural properties. While the semicircle law is known to describe the spectral densities of uncorrelated random graphs, much less is known about the spectra of real-world graphs, describing such complex systems as the Internet, metabolic pathways, networks of power stations, scientific collaborations, or movie actors, which are inherently correlated and usually very sparse. An important limitation in addressing the spectra of these systems is that the numerical determination of the spectra for systems with more than a few thousand nodes is prohibitively time and memory consuming. Making use of recent advances in algorithms for spectral characterization, here we develop methods to determine the eigenvalues of networks comparable in size to real systems, obtaining several surprising results on the spectra of adjacency matrices corresponding to models of real-world graphs. We find that when the number of links grows as the number of nodes, the spectral density of uncorrelated random matrices does not converge to the semicircle law. Furthermore, the spectra of real-world graphs have specific features, depending on the details of the corresponding models. In particular, scale-free graphs develop a trianglelike spectral density with a power-law tail, while small-world graphs have a complex spectral density consisting of several sharp peaks. These and further results indicate that the spectra of correlated graphs represent a practical tool for graph classification and can provide useful insight into the relevant structural properties of real networks.

The notion of k-clique percolation in random graphs is introduced, where k is the size of the complete subgraphs whose large scale organizations are analytically and numerically investigated. For the Erdős-Rényi graph of N vertices we obtain that the percolation transition of k-cliques takes place when the probability of two vertices being connected by an edge reaches the threshold pc(k) = [(k − 1)N ] −1/(k−1) . At the transition point the scaling of the giant component with N is highly non-trivial and depends on k. We discuss why clique percolation is a novel and efficient approach to the identification of overlapping communities in large real networks.PACS numbers: 02.10. Ox, 89.75.Hc, 05.70.Fh, There has been a quickly growing interest in networks, since they can represent the structure of a wide class of complex systems occurring from the level of cells to society. Data obtained on real networks show that the corresponding graphs exhibit unexpected non-trivial properties, e.g., anomalous degree distributions, diameter, spreading phenomena, clustering coefficient, and correlations [1,2,3,4,5]. Very recently great attention has been paid to the local structural units of networks. Small and well defined subgraphs have been introduced as "motifs" [6]. Their distribution and clustering properties [6,7,8] can be used to interpret global features as well. Somewhat larger units, made up of vertices that are more densely connected to each other than to the rest of the network, are often referred to as communities [9,10,11,12,13,14,15,16], and have been considered to be the essential structural units of real networks. They have no obvious definition, and most of the recent methods for their identification rely on dividing the network into smaller pieces. The biggest drawback of these methods is that they do not allow for overlapping communities, although overlaps are generally assumed to be crucial features of communities. In this Letter we lay down the fundamentals of a kind of percolation phenomenon on graphs, which can also be used as an effective and deterministic method for uniquely identifying overlapping communities in large real networks [17].Meanwhile, the various aspects of the classical Erdős-Rényi (ER) uncorrelated random graph [18] remain still of great interest since such a graph can serve both as a test bed for checking all sorts of new ideas concerning complex networks in general, and as a prototype to which all other random graphs can be compared. Perhaps the most conspicuous early result on the ER graphs was related to the percolation transition taking place at p = p c ≡ 1/N , where p is the probability that two vertices are connected by an edge and N is the total number of vertices in the graph. The appearance of a giant component, which is also referred to as the percolating component, results in a dramatic change in the overall topological features of the graph and has been in the center of interest for other networks as well.In this Letter we address the general question of subgraph percolation i...

Non-equilibrium fluctuations can drive vectorial transport along an anisotropic structure in an isothermal medium by biasing the effect of thermal noise (kBT). Mechanisms based on this principle are often called Brownian ratchets and have been invoked as a possible explanation for the operation of biomolecular motors and pumps. We discuss the thermodynamics and kinetics for the operation of microscopic ratchet motors under conditions relevant to biology, showing how energy provided by external fluctuations or a non-equilibrium chemical reaction can cause unidirectional motion or uphill pumping of a substance. Our analysis suggests that molecular pumps such as Na,K-ATPase and molecular motors such as kinesin and myosin may share a common underlying mechanism.

PACS numbers:A serious obstacle that impedes the application of low and high temperature superconductor (SC) devices is the presence of trapped flux [1,2]. Flux lines or vortices are induced by fields as small as the Earth's magnetic field. Once present, vortices dissipate energy and generate internal noise, limiting the operation of numerous superconducting devices [2,3]. Methods used to overcome this difficulty include the pinning of vortices by the incorporation of impurities and defects [4], the construction of flux dams [5], slots and holes [6] and magnetic shields [2,3] which block the penetration of new flux lines in the bulk of the SC or reduce the magnetic field in the immediate vicinity of the superconducting device. Naturally, the most desirable would be to remove the vortices from the bulk of the SC. There is no known phenomenon, however, that could form the basis for such a process. Here we show that the application of an ac current to a SC that is patterned with an asymmetric pinning potential can induce vortex motion whose direction is determined only by the asymmetry of the pattern. The mechanism responsible for this phenomenon is the so called ratchet effect [7][8][9], and its working principle applies to both low and high temperature SCs. As a first step here we demonstrate that with an appropriate choice of the pinning potential the ratchet effect can be used to remove vortices from low temperature SCs in the parameter range required for various applications.Consider a type II superconductor film of the geometry shown in Fig. 1, placed in an external magnetic field H. The superconductor is patterned with a pinning potential U (x, y) = U (x) which is periodic with period ℓ along the x direction, has an asymmetric shape within one period, and is translationally invariant along the y direction of the sample. The simplest example of an asymmetric periodic potential, obtained for example by varying the sample thickness, is the asymmetric sawtooth potential, shown in Fig. 1b. In the presence of a current with density J flowing along the y axis the vortices move with the velocitywhere f L = (J ×ĥ)Φ 0 d/c is the Lorentz force moving the vortices transverse to the current,ĥ is the unit vector pointing in the direction of the external magnetic field H, f u = − dU dxx is the force generated by the periodic potential, f vv is the repulsive vortex-vortex interaction, Φ 0 = 2.07 × 10 −7 G cm 2 is the flux quantum, η is the viscous drag coefficient, and d is the length of the vortices (i.e. the thickness of the sample).When a dc current flows along the positive y direction, the Lorentz force moves the vortices along the positive x direction with velocity v + . Reversing the current reverses the direction of the vortex velocity, but its magnitude, |v − |, due to the asymmetry of the potential, is different from v + . For the sawtooth potential shown in Fig. 1b the vortex velocity is higher when the vortex is driven to the right, than when it is driven to the left (v + > |v − |). As a consequence the applicati...

Tethers are nanocylinders of lipid bilayer membrane, arising in situations ranging from micromanipulation experiments on synthetic vesicles to the formation of dynamic tubular networks in the Golgi apparatus. Relying on the extensive theoretical and experimental works aimed to understand the physics of individual tethers formation, we addressed the problem of the interaction between two nanotubes. By using a combination of micropipette manipulation and optical tweezers, we quantitatively studied the process of coalescence that occurred when the separation distance between both vesicle-tether junctions became smaller than a threshold length. Our experiments, which were supported by an original theoretical analysis, demonstrated that the measurements of the tether force and angle between tethers at coalescence directly yield the bending rigidity, kappa, and the membrane tension, sigma, of the vesicles. Contrary to other methods used to probe the bending rigidity of vesicles, the proposed approach permits a direct measurement of kappa without requiring any control of the membrane tension. Finally, after validation of the method and proposal of possible applications, we experimentally investigated the dynamics of the coalescence process.