Communities of vertices within a giant network such as the World-Wide Web are likely to be vastly smaller than the network itself. However, Fortunato and Barthélemy have proved that modularity maximization algorithms for community detection may fail to resolve communities with fewer than p L/2 edges, where L is the number of edges in the entire network. This resolution limit leads modularity maximization algorithms to have notoriously poor accuracy on many real networks. Fortunato and Barthélemy's argument can be extended to networks with weighted edges as well, and we derive this corollary argument. We conclude that weighted modularity algorithms may fail to resolve communities with fewer than p W /2 total edge weight, where W is the total edge weight in the network and is the maximum weight of an inter-community edge. If is small, then small communities can be resolved.Given a weighted or unweighted network, we describe how to derive new edge weights in order to achieve a low , we modify the "CNM" community detection algorithm to maximize weighted modularity, and show that the resulting algorithm has greatly improved accuracy. In experiments with an emerging community standard benchmark, we find that our simple CNM variant is competitive with the most accurate community detection methods yet proposed.

We develop a quasi-chemical theory for the study of packing thermodynamics in dense liquids. The situation of hard-core interactions is addressed by considering the binding of solvent molecules to a precisely defined 'cavity' in order to assess the probability that the 'cavity' is entirely evacuated. The primitive quasi-chemical approximation corresponds to a extension of the Poisson distribution used as a default model in an information theory approach. This primitive quasi-chemical theory is in good qualitative agreement with the observations for the hard sphere fluid of occupancy distributions that are central to quasi-chemical theories but begins to be quantitatively erroneous for the equation of state in the dense liquid regime of ρd 3 >0.6. How the quasi-chemical approach can be iterated to treat correlation effects is addressed. Consideration of neglected correlation effects leads to a simple model for the form of those contributions neglected by the primitive quasi-chemical approximation. These considerations, supported by simulation observations, identify a 'break away' phenomena that requires special thermodynamic consideration for the zero (0) occupancy case as distinct from the rest of the distribution. A empirical treatment leads to a one parameter model occupancy distribution that accurately fits the hard sphere equation of state and observed distributions.

Any molecular system explores significantly different regions of the potential-energy hypersurface as the system is found, respectively, in the solid and liquid phases. We study in detail the multidimensional geometry of these different regions with molecular-dynamics calculations for 256 simple atoms in a fixed volume. The atomic interactions are chosen to represent the noble gases. The stable crystal for this model displays a face-centered cubic structure. We evaluate the local gradient and curvatures of the regions of the hypersurface sampled by the system for a wide range of temperatures. We observe that a significant fraction of the curvatures become negative in the region sampled by the system at temperatures even as low as one-fourth the melting temperature. Further, the curvature distribution changes dramatically with respect to temperature at the melting point. We also construct and evaluate a new distribution for the distance between the atoms in their instantaneous dynamical configurations and those in their corresponding "quenched" configuration (i.e., the configuration found at the corresponding potential-energy minimum). With the help of this new distribution, we conclude that the quenched configurations which are encountered during the melting process are structures which contain vacancy-interstitial defect pairs.

Comprehensive investigation of lithium ion complexation with 15N-labeled polyphosphazenes 15 N-poly[bis(2-(2-methoxyethoxy)ethoxy)phosphazene] (15 N-MEEP) and 15 N-poly-[((2-allylphenoxy)0.12(4-methoxyphenoxy)1.02(2-(2-methoxyethoxy)ethoxy)0.86)phosphazene] (15 N-HPP)was performed by NMR, IR, and Raman spectroscopies. Previous studies characterized the ionic transport through the polymer matrix in terms of “jumps” between neighboring polymer strands utilizing the electron lone pairs of the etherial oxygen nuclei with the nitrogen nuclei on the polyphosphazene backbone not involved. However, noteworthy changes were observed in the NMR, IR, and Raman spectra with the addition of lithium trifluoromethanesulfonate (LiOTf) to the polyphosphazenes. The data indicate that the preferred association for the lithium ion with the polymer is with the nitrogen nuclei, resulting in the formation of a “pocket” with the pendant groups folding around the backbone. NMR temperature-dependent spin−lattice relaxation (T 1) studies (13C, 31P, and 15N) indicate significant lithium ion interaction with the backbone nitrogen nuclei. These studies are in agreement with molecular dynamics simulations investigating lithium ion movement within the polyphosphazene matrix.

Glassy structures of water were generated by rapidly quenching configurations of 64 and 343 molecules of liquid water. The potential energy was then expanded through quadratic order around local minima generated this way and properties of the resulting harmonic system were calculated. The results were used to test the extent to which the structure of liquid water is similar to that of a harmonic aqueous glass. The radial distribution functions for the glass are remarkably similar to those of the liquid. The vibrational density of states for the glassy water exhibits a gap between 300 and 400 cm-1. The normal modes below 300 cm-1 correspond to molecular translations while the modes above 400 cm-1 are ascribed to molecular librations. Translational modes are almost entirely responsible for the broadening of oxygen-oxygen radial distribution function of the quenched configuration. They are also primarily responsible for the broadening of other radial distribution functions. Vibrational density of states leads to classical and quantum free energies for the harmonic system equal -9.62 +/- 0.12 and -8.89 +/- 0.12 kcal/mol, respectively, at T = 300 K. Both free energies were found to be insensitive to sample size and to the configurational differences between the quenched structures.