Locally Estimating Core Numbers

Graphs are a powerful way to model interactions and relationships in data from a wide variety of application domains. In this setting, entities represented by vertices at the "center" of the graph are often more important than those associated with vertices on the "fringes". For example, central nodes tend to be more critical in the spread of information or disease and play an important role in clustering/community formation. Identifying such "core" vertices has recently received additional attention in the context of {\em network experiments}, which analyze the response when a random subset of vertices are exposed to a treatment (e.g. inoculation, free product samples, etc). Specifically, the likelihood of having many central vertices in any exposure subset can have a significant impact on the experiment. We focus on using $k$-cores and core numbers to measure the extent to which a vertex is central in a graph. Existing algorithms for computing the core number of a vertex require the entire graph as input, an unrealistic scenario in many real world applications. Moreover, in the context of network experiments, the subgraph induced by the treated vertices is only known in a probabilistic sense. We introduce a new method for estimating the core number based only on the properties of the graph within a region of radius $\delta$ around the vertex, and prove an asymptotic error bound of our estimator on random graphs. Further, we empirically validate the accuracy of our estimator for small values of $\delta$ on a representative corpus of real data sets. Finally, we evaluate the impact of improved local estimation on an open problem in network experimentation posed by Ugander et al.Comment: Main paper body is identical to previous version (ICDM version). Appendix with additional data sets and enlarged figures has been added to the en

Distribution of the brown bear (Ursus arctos marsicanus) in the Central Apennines, Italy, 2005-2014

Despite its critical conservation status, no formal estimate of the Apennine brown bear (Ursus arctos marsicanus) distribution has ever been attempted, nor a coordinated effort to compile and verify all recent occurrences has ever been ensured. We used 48331 verified bear location data collected by qualified personnel from 20052014 in the central Apennines, Italy, to estimate the current distribution of Apennine brown bears. Data sources included telemetry relocations, scats and DNA-verified hair samples, sightings, indirect signs of presence, photos from camera traps, and damage to properties. Using a grid-based zonal analysis to transform raw data density, we applied ordinary kriging and estimated a 4923 km2 main bear distribution, encompassing the historical stronghold of the bear population, and including a smaller (1460 km2) area of stable occupancy of reproducing female bears. National and Regional Parks cover 38.8% of the main bear distribution, plus an additional 19.5% encompassed by the Natura 2000 network alone. Despite some methodological and sampling problems related to spatial and temporal variation in sampling effort at the landscape scale, our approach provides an approximation of the current bear distribution that is suited to frequently update the distribution map. Future monitoring of this bear population would benefit from estimating detectability across a range on environmental and sampling variables, and from intensifying the collection of bear presence data in the peripheral portions of the distribution

Policy Gradients for CVaR-Constrained MDPs

We study a risk-constrained version of the stochastic shortest path (SSP) problem, where the risk measure considered is Conditional Value-at-Risk (CVaR). We propose two algorithms that obtain a locally risk-optimal policy by employing four tools: stochastic approximation, mini batches, policy gradients and importance sampling. Both the algorithms incorporate a CVaR estimation procedure, along the lines of Bardou et al. [2009], which in turn is based on Rockafellar-Uryasev's representation for CVaR and utilize the likelihood ratio principle for estimating the gradient of the sum of one cost function (objective of the SSP) and the gradient of the CVaR of the sum of another cost function (in the constraint of SSP). The algorithms differ in the manner in which they approximate the CVaR estimates/necessary gradients - the first algorithm uses stochastic approximation, while the second employ mini-batches in the spirit of Monte Carlo methods. We establish asymptotic convergence of both the algorithms. Further, since estimating CVaR is related to rare-event simulation, we incorporate an importance sampling based variance reduction scheme into our proposed algorithms