22 research outputs found
A Consistent Histogram Estimator for Exchangeable Graph Models
Exchangeable graph models (ExGM) subsume a number of popular network models.
The mathematical object that characterizes an ExGM is termed a graphon. Finding
scalable estimators of graphons, provably consistent, remains an open issue. In
this paper, we propose a histogram estimator of a graphon that is provably
consistent and numerically efficient. The proposed estimator is based on a
sorting-and-smoothing (SAS) algorithm, which first sorts the empirical degree
of a graph, then smooths the sorted graph using total variation minimization.
The consistency of the SAS algorithm is proved by leveraging sparsity concepts
from compressed sensing.Comment: 28 pages, 5 figure
Variational Bayes model averaging for graphon functions and motif frequencies inference in W-graph models
W-graph refers to a general class of random graph models that can be seen as
a random graph limit. It is characterized by both its graphon function and its
motif frequencies. In this paper, relying on an existing variational Bayes
algorithm for the stochastic block models along with the corresponding weights
for model averaging, we derive an estimate of the graphon function as an
average of stochastic block models with increasing number of blocks. In the
same framework, we derive the variational posterior frequency of any motif. A
simulation study and an illustration on a social network complete our work
Degree-based goodness-of-fit tests for heterogeneous random graph models : independent and exchangeable cases
The degrees are a classical and relevant way to study the topology of a
network. They can be used to assess the goodness-of-fit for a given random
graph model. In this paper we introduce goodness-of-fit tests for two classes
of models. First, we consider the case of independent graph models such as the
heterogeneous Erd\"os-R\'enyi model in which the edges have different
connection probabilities. Second, we consider a generic model for exchangeable
random graphs called the W-graph. The stochastic block model and the expected
degree distribution model fall within this framework. We prove the asymptotic
normality of the degree mean square under these independent and exchangeable
models and derive formal tests. We study the power of the proposed tests and we
prove the asymptotic normality under specific sparsity regimes. The tests are
illustrated on real networks from social sciences and ecology, and their
performances are assessed via a simulation study
A nonparametric two-sample hypothesis testing problem for random dot product graphs
We consider the problem of testing whether two finite-dimensional random dot
product graphs have generating latent positions that are independently drawn
from the same distribution, or distributions that are related via scaling or
projection. We propose a test statistic that is a kernel-based function of the
adjacency spectral embedding for each graph. We obtain a limiting distribution
for our test statistic under the null and we show that our test procedure is
consistent across a broad range of alternatives.Comment: 24 pages, 1 figure