The complexity of biological, social, and engineering networks makes it desirable to find natural partitions into clusters (or communities) that can provide insight into the structure of the overall system and even act as simplified functional descriptions. Although methods for community detection abound, there is a lack of consensus on how to quantify and rank the quality of partitions. We introduce here the stability of a partition, a measure of its quality as a community structure based on the clustered autocovariance of a dynamic Markov process taking place on the network. Because the stability has an intrinsic dependence on time scales of the graph, it allows us to compare and rank partitions at each time and also to establish the time spans over which partitions are optimal. Hence the Markov time acts effectively as an intrinsic resolution parameter that establishes a hierarchy of increasingly coarser communities. Our dynamical definition provides a unifying framework for several standard partitioning measures: modularity and normalized cut size can be interpreted as one-step time measures, whereas Fiedler's spectral clustering emerges at long times. We apply our method to characterize the relevance of partitions over time for constructive and real networks, including hierarchical graphs and social networks, and use it to obtain reduced descriptions for atomic-level protein structures over different time scales.community structure | Markov chain | modularity | multiscale modelling | networks I n recent years, there has been a surge of interest in the analysis of networks as models of complex systems. The literature is extensive, spanning areas as diverse as gene regulation, protein interactions and metabolic pathways, neural science, social networks or engineering systems such as transportation networks and the internet, to name but a few (1, 2). The tools for network analysis are firmly grounded on results in graph theory, with an influx of concepts from statistical physics, dynamical systems, and stochastic processes (3). Due to the large-scale, complex nature of many systems under study, an appealing idea is to obtain relevant partitions of the network (also called clusterings or communities) that can reveal the underlying structure of the system and hence provide insight into its function. These partitions could potentially lead to reduced, more manageable representations of the original system (4, 5).The topic of community detection in graphs has a long history and multiple methods and heuristics have been proposed to partition graphs into communities or clusters. (See for instance ref. 6 and references therein for a recent survey.) However, the extensive list of partitioning methods comes with a parallel lack of theory or consensus on measures to quantify the goodness of a community structure. For instance, consider the simplest such measure: the cut size, i.e., the sum of the weights of edges that lie at the boundaries of different communities. As a general rule, good community structures should hav...

We report on the effect of modifying the molecule−electrode binding interface of an α,α‘-xylyl-dithiol molecular wire. We find that except for the length of the surface bond, the conductance is not affected by variations of the surface geometry. We also compare the conductance of different terminal atom−electrode metal combinations and find that the conductance is substantially larger when the wire is terminated by selenium rather than sulfur or oxygen. We also find that gold makes a better electrode than silver.

To study the electronic transport of molecular wire circuits, we present a time-independent scattering formalism which includes an ab initio description of the molecular electronic structure. This allows us to obtain the molecule–metal coupling description at the same level of theory. The conductance of junction α, α′ xylyl dithiol and benzene-1,4-dithiol between gold electrodes is obtained and compared with available experimental data. The conductance depends dramatically on the relative position of the Fermi energy of the metal with respect to the molecular levels. We obtain an estimate for the injecting energy of the electron onto the molecule by varying the distance between the molecule and the attached gold clusters. Contrary to the standard assumption, we find that the injecting energy lies close to the molecular highest occupied molecular orbital, rather than in the middle of the gap; it is just the work function of the bulk metal. Finally, the adequacy of the widely used extended Hückel method for conductance calculations is discussed.

In recent years, there has been a surge of interest in community detection algorithms for complex networks. A variety of computational heuristics, some with a long history, have been proposed for the identification of communities or, alternatively, of good graph partitions. In most cases, the algorithms maximize a particular objective function, thereby finding the ‘right’ split into communities. Although a thorough comparison of algorithms is still lacking, there has been an effort to design benchmarks, i.e., random graph models with known community structure against which algorithms can be evaluated. However, popular community detection methods and benchmarks normally assume an implicit notion of community based on clique-like subgraphs, a form of community structure that is not always characteristic of real networks. Specifically, networks that emerge from geometric constraints can have natural non clique-like substructures with large effective diameters, which can be interpreted as long-range communities. In this work, we show that long-range communities escape detection by popular methods, which are blinded by a restricted ‘field-of-view’ limit, an intrinsic upper scale on the communities they can detect. The field-of-view limit means that long-range communities tend to be overpartitioned. We show how by adopting a dynamical perspective towards community detection [1], [2], in which the evolution of a Markov process on the graph is used as a zooming lens over the structure of the network at all scales, one can detect both clique- or non clique-like communities without imposing an upper scale to the detection. Consequently, the performance of algorithms on inherently low-diameter, clique-like benchmarks may not always be indicative of equally good results in real networks with local, sparser connectivity. We illustrate our ideas with constructive examples and through the analysis of real-world networks from imaging, protein structures and the power grid, where a multiscale structure of non clique-like communities is revealed.

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Allostery is a fundamental mechanism of biological regulation, in which binding of a molecule at a distant location affects the active site of a protein. Allosteric sites provide targets to fine-tune protein activity, yet we lack computational methodologies to predict them. Here we present an efficient graph-theoretical framework to reveal allosteric interactions (atoms and communication pathways strongly coupled to the active site) without a priori information of their location. Using an atomistic graph with energy-weighted covalent and weak bonds, we define a bond-to-bond propensity quantifying the non-local effect of instantaneous bond fluctuations propagating through the protein. Significant interactions are then identified using quantile regression. We exemplify our method with three biologically important proteins: caspase-1, CheY, and h-Ras, correctly predicting key allosteric interactions, whose significance is additionally confirmed against a reference set of 100 proteins. The almost-linear scaling of our method renders it suitable for high-throughput searches for candidate allosteric sites.

Transport studies of molecular wire circuits require a description of the molecule and the leads. Here we focus on the molecule–lead interaction. We extend a time-independent scattering formalism to include a more realistic description of the interface. This allows us to obtain the conductance as a function of dimensionality of contact and of electrode, number of contacts, and geometry between molecule and interface. We study conductance in adlayers of molecules by considering transport through two identical wires. Implications for experiments are discussed.

By molecular wires, one generally means molecular structures that transmit a signal between two termini. We discuss some theoretical models and analysis for electronically conductive molecular wires in which a single molecule conducts charge between two electrodes. This situation resembles both intramolecular non‐adiabatic electron transfer, in which electronic tunneling between donor and acceptor is seen, and mesoscopic quantum transport. We discuss formal methods for predicting conductance in molecular wire circuits. The critical component that differs from the usual conductivity is the interface between electrode continuum and the discrete levels of the molecule. This can be described in several ways. We present an analysis based on the Bardeen tunneling formula. Specific problems (electron polarization, disorder, nuclear scattering, charge distribution) are discussed. Finally, the differing mechanisms expected for the conductance, ranging from ballistic tunneling to gated transfer, are outlined.