Estimating and Sampling Graphs with Multidimensional Random Walks

Estimating characteristics of large graphs via sampling is a vital part of the study of complex networks. Current sampling methods such as (independent) random vertex and random walks are useful but have drawbacks. Random vertex sampling may require too many resources (time, bandwidth, or money). Random walks, which normally require fewer resources per sample, can suffer from large estimation errors in the presence of disconnected or loosely connected graphs. In this work we propose a new $m$-dimensional random walk that uses $m$ dependent random walkers. We show that the proposed sampling method, which we call Frontier sampling, exhibits all of the nice sampling properties of a regular random walk. At the same time, our simulations over large real world graphs show that, in the presence of disconnected or loosely connected components, Frontier sampling exhibits lower estimation errors than regular random walks. We also show that Frontier sampling is more suitable than random vertex sampling to sample the tail of the degree distribution of the graph

Ornstein-Uhlenbeck limit for the velocity process of an NN-particle system interacting stochastically

An $N$-particle system with stochastic interactions is considered. Interactions are driven by a Brownian noise term and total energy conservation is imposed. The evolution of the system, in velocity space, is a diffusion on a $(3N-1)$-dimensional sphere with radius fixed by the total energy. In the $N\rightarrow\infty$ limit, a finite number of velocity components are shown to evolve independently and according to an Ornstein-Uhlenbeck process.Comment: 19 pages ; streamlined notations ; new section on many particles with momentum conservation ; new appendix on Kac syste

Multiple Random Walks to Uncover Short Paths in Power Law Networks

Consider the following routing problem in the context of a large scale network $G$, with particular interest paid to power law networks, although our results do not assume a particular degree distribution. A small number of nodes want to exchange messages and are looking for short paths on $G$. These nodes do not have access to the topology of $G$ but are allowed to crawl the network within a limited budget. Only crawlers whose sample paths cross are allowed to exchange topological information. In this work we study the use of random walks (RWs) to crawl $G$. We show that the ability of RWs to find short paths bears no relation to the paths that they take. Instead, it relies on two properties of RWs on power law networks: 1) RW's ability observe a sizable fraction of the network edges; and 2) an almost certainty that two distinct RW sample paths cross after a small percentage of the nodes have been visited. We show promising simulation results on several real world networks

Arithmetic of positive characteristic L-series values in Tate algebras

The second author has recently introduced a new class of L-series in the arithmetic theory of function fields over finite fields. We show that the value at one of these L-series encode arithmetic informations of certain Drinfeld modules defined over Tate algebras. This enables us to generalize Anderson's log-algebraicity Theorem and Taelman's Herbrand-Ribet Theorem.Comment: final versio

Bayesian Inference of Online Social Network Statistics via Lightweight Random Walk Crawls

Online social networks (OSN) contain extensive amount of information about the underlying society that is yet to be explored. One of the most feasible technique to fetch information from OSN, crawling through Application Programming Interface (API) requests, poses serious concerns over the the guarantees of the estimates. In this work, we focus on making reliable statistical inference with limited API crawls. Based on regenerative properties of the random walks, we propose an unbiased estimator for the aggregated sum of functions over edges and proved the connection between variance of the estimator and spectral gap. In order to facilitate Bayesian inference on the true value of the estimator, we derive the approximate posterior distribution of the estimate. Later the proposed ideas are validated with numerical experiments on inference problems in real-world networks