Exploring Complex Graphs by Random Walks

We present an algorithm to grow a graph with scale-free structure of {\it in-} and {\it out-links} and variable wiring diagram in the class of the world-wide Web. We then explore the graph by intentional random walks using local next-near-neighbor search algorithm to navigate through the graph. The topological properties such as betweenness are determined by an ensemble of independent walkers and efficiency of the search is compared on three different graph topologies. In addition we simulate interacting random walks which are created by given rate and navigated in parallel, representing transport with queueing of information packets on the graph.Comment: Latex, 4 figure

Time distribution and loss of scaling in granular flow

Two cellular automata models with directed mass flow and internal time scales are studied by numerical simulations. Relaxation rules are a combination of probabilistic critical height (probability of toppling $p$) and deterministic critical slope processes with internal correlation time $t_c$ equal to the avalanche lifetime, in Model A, and $t_c\equiv 1$, in Model B. In both cases nonuniversal scaling properties of avalanche distributions are found for $p\ge p^\star$, where $p^\star$ is related to directed percolation threshold in $d=3$. Distributions of avalanche durations for $p\ge p^\star$ are studied in detail, exhibiting multifractal scaling behavior in model A, and finite size scaling behavior in model B, and scaling exponents are determined as a function of $p$. At $p=p^\star$ a phase transition to noncritical steady state occurs. Due to difference in the relaxation mechanisms, avalanche statistics at $p^\star$ approaches the parity conserving universality class in Model A, and the mean-field universality class in Model B. We also estimate roughness exponent at the transition

Convergence and Convergence Rate of Stochastic Gradient Search in the Case of Multiple and Non-Isolated Extrema

The asymptotic behavior of stochastic gradient algorithms is studied. Relying on results from differential geometry (Lojasiewicz gradient inequality), the single limit-point convergence of the algorithm iterates is demonstrated and relatively tight bounds on the convergence rate are derived. In sharp contrast to the existing asymptotic results, the new results presented here allow the objective function to have multiple and non-isolated minima. The new results also offer new insights into the asymptotic properties of several classes of recursive algorithms which are routinely used in engineering, statistics, machine learning and operations research