The Spatial Evolution of Particles Diffusing in the Presence of Randomly Placed Traps

The evolution of a particle undergoing a continuous-time random walk in the presence of randomly placed imperfectly absorbing traps is studied. At long times, the spatial probability distribution becomes strongly localized in a sequence of trap-free regions. The subsequent intermittent transfer of the survival probability from small trap-free regions to larger trap-free regions is described as a time-directed variable range hopping among localized eigenstates in the Lifshitz tail. An asymptotic expression for the configurational average of the spatial distribution of surviving particles is obtained based on this description. The distribution is an exponential function of distance which expands superdiffusively, with the mean-square displacement increasing with time as t2/ln(2D+4/D)(t) in D dimensions

Limits on Relief through Constrained Exchange on Random Graphs

Agents are represented by nodes on a random graph (e.g., small world or truncated power law). Each agent is endowed with a zero-mean random value that may be either positive or negative. All agents attempt to find relief, i.e., to reduce the magnitude of that initial value, to zero if possible, through exchanges. The exchange occurs only between agents that are linked, a constraint that turns out to dominate the results. The exchange process continues until a Pareto equilibrium is achieved. Only 40%-90% of the agents achieved relief on small world graphs with mean degree between 2 and 40. Even fewer agents achieved relief on scale-free like graphs with a truncated power law degree distribution. The rate at which relief grew with increasing degree was slow, only at most logarithmic for all of the graphs considered; viewed in reverse, relief is resilient to the removal of links.Comment: 8 pages, 2 figures, 22 references Changes include name change for Lory A. Ellebracht (formerly Cooperstock, e-mail address stays the same), elimination of contractions and additional references. We also note that our results are less surprising in view of other work now cite

Tolerating the Community Detection Resolution Limit with Edge Weighting

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 Barth\'{e}lemy have proved that modularity maximization algorithms for community detection may fail to resolve communities with fewer than $\sqrt{L/2}$ 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 Barth\'{e}lemy'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 $\sqrt{W \epsilon/2}$ total edge weight, where $W$ is the total edge weight in the network and $\epsilon$ is the maximum weight of an inter-community edge. If $\epsilon$ 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 $\epsilon$, 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.Comment: revision with 8 pages 3 figures 2 table

Multidimensional geometric aspects of the solid-liquid transition in simple substances

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