  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.
