238 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
Generalized Graphon Process: Convergence of Graph Frequencies in Stretched Cut Distance
Graphons have traditionally served as limit objects for dense graph
sequences, with the cut distance serving as the metric for convergence.
However, sparse graph sequences converge to the trivial graphon under the
conventional definition of cut distance, which make this framework inadequate
for many practical applications. In this paper, we utilize the concepts of
generalized graphons and stretched cut distance to describe the convergence of
sparse graph sequences. Specifically, we consider a random graph process
generated from a generalized graphon. This random graph process converges to
the generalized graphon in stretched cut distance. We use this random graph
process to model the growing sparse graph, and prove the convergence of the
adjacency matrices' eigenvalues. We supplement our findings with experimental
validation. Our results indicate the possibility of transfer learning between
sparse graphs
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