12,697 research outputs found
Latent Distance Estimation for Random Geometric Graphs
Random geometric graphs are a popular choice for a latent points generative
model for networks. Their definition is based on a sample of points
on the Euclidean sphere~ which
represents the latent positions of nodes of the network. The connection
probabilities between the nodes are determined by an unknown function (referred
to as the "link" function) evaluated at the distance between the latent points.
We introduce a spectral estimator of the pairwise distance between latent
points and we prove that its rate of convergence is the same as the
nonparametric estimation of a function on , up to a
logarithmic factor. In addition, we provide an efficient spectral algorithm to
compute this estimator without any knowledge on the nonparametric link
function. As a byproduct, our method can also consistently estimate the
dimension of the latent space
Consistency of Maximum Likelihood for Continuous-Space Network Models
Network analysis needs tools to infer distributions over graphs of arbitrary
size from a single graph. Assuming the distribution is generated by a
continuous latent space model which obeys certain natural symmetry and
smoothness properties, we establish three levels of consistency for
non-parametric maximum likelihood inference as the number of nodes grows: (i)
the estimated locations of all nodes converge in probability on their true
locations; (ii) the distribution over locations in the latent space converges
on the true distribution; and (iii) the distribution over graphs of arbitrary
size converges.Comment: 21 page
Projective, Sparse, and Learnable Latent Position Network Models
When modeling network data using a latent position model, it is typical to
assume that the nodes' positions are independently and identically distributed.
However, this assumption implies the average node degree grows linearly with
the number of nodes, which is inappropriate when the graph is thought to be
sparse. We propose an alternative assumption---that the latent positions are
generated according to a Poisson point process---and show that it is compatible
with various levels of sparsity. Unlike other notions of sparse latent position
models in the literature, our framework also defines a projective sequence of
probability models, thus ensuring consistency of statistical inference across
networks of different sizes. We establish conditions for consistent estimation
of the latent positions, and compare our results to existing frameworks for
modeling sparse networks.Comment: 51 pages, 2 figure
Graphs in machine learning: an introduction
Graphs are commonly used to characterise interactions between objects of
interest. Because they are based on a straightforward formalism, they are used
in many scientific fields from computer science to historical sciences. In this
paper, we give an introduction to some methods relying on graphs for learning.
This includes both unsupervised and supervised methods. Unsupervised learning
algorithms usually aim at visualising graphs in latent spaces and/or clustering
the nodes. Both focus on extracting knowledge from graph topologies. While most
existing techniques are only applicable to static graphs, where edges do not
evolve through time, recent developments have shown that they could be extended
to deal with evolving networks. In a supervised context, one generally aims at
inferring labels or numerical values attached to nodes using both the graph
and, when they are available, node characteristics. Balancing the two sources
of information can be challenging, especially as they can disagree locally or
globally. In both contexts, supervised and un-supervised, data can be
relational (augmented with one or several global graphs) as described above, or
graph valued. In this latter case, each object of interest is given as a full
graph (possibly completed by other characteristics). In this context, natural
tasks include graph clustering (as in producing clusters of graphs rather than
clusters of nodes in a single graph), graph classification, etc. 1 Real
networks One of the first practical studies on graphs can be dated back to the
original work of Moreno [51] in the 30s. Since then, there has been a growing
interest in graph analysis associated with strong developments in the modelling
and the processing of these data. Graphs are now used in many scientific
fields. In Biology [54, 2, 7], for instance, metabolic networks can describe
pathways of biochemical reactions [41], while in social sciences networks are
used to represent relation ties between actors [66, 56, 36, 34]. Other examples
include powergrids [71] and the web [75]. Recently, networks have also been
considered in other areas such as geography [22] and history [59, 39]. In
machine learning, networks are seen as powerful tools to model problems in
order to extract information from data and for prediction purposes. This is the
object of this paper. For more complete surveys, we refer to [28, 62, 49, 45].
In this section, we introduce notations and highlight properties shared by most
real networks. In Section 2, we then consider methods aiming at extracting
information from a unique network. We will particularly focus on clustering
methods where the goal is to find clusters of vertices. Finally, in Section 3,
techniques that take a series of networks into account, where each network i
From random walks to distances on unweighted graphs
Large unweighted directed graphs are commonly used to capture relations
between entities. A fundamental problem in the analysis of such networks is to
properly define the similarity or dissimilarity between any two vertices.
Despite the significance of this problem, statistical characterization of the
proposed metrics has been limited. We introduce and develop a class of
techniques for analyzing random walks on graphs using stochastic calculus.
Using these techniques we generalize results on the degeneracy of hitting times
and analyze a metric based on the Laplace transformed hitting time (LTHT). The
metric serves as a natural, provably well-behaved alternative to the expected
hitting time. We establish a general correspondence between hitting times of
the Brownian motion and analogous hitting times on the graph. We show that the
LTHT is consistent with respect to the underlying metric of a geometric graph,
preserves clustering tendency, and remains robust against random addition of
non-geometric edges. Tests on simulated and real-world data show that the LTHT
matches theoretical predictions and outperforms alternatives.Comment: To appear in NIPS 201
Learning the distribution of latent variables in paired comparison models with round-robin scheduling
Paired comparison data considered in this paper originate from the comparison
of a large number N of individuals in couples. The dataset is a collection of
results of contests between two individuals when each of them has faced n
opponents, where n is much larger than N. Individual are represented by
independent and identically distributed random parameters characterizing their
abilities.The paper studies the maximum likelihood estimator of the parameters
distribution. The analysis relies on the construction of a graphical model
encoding conditional dependencies of the observations which are the outcomes of
the first n contests each individual is involved in. This graphical model
allows to prove geometric loss of memory properties and deduce the asymptotic
behavior of the likelihood function. This paper sets the focus on graphical
models obtained from round-robin scheduling of these contests.Following a
classical construction in learning theory, the asymptotic likelihood is used to
measure performance of the maximum likelihood estimator. Risk bounds for this
estimator are finally obtained by sub-Gaussian deviation results for Markov
chains applied to the graphical model
Testing for high-dimensional geometry in random graphs
We study the problem of detecting the presence of an underlying
high-dimensional geometric structure in a random graph. Under the null
hypothesis, the observed graph is a realization of an Erd\H{o}s-R\'enyi random
graph . Under the alternative, the graph is generated from the
model, where each vertex corresponds to a latent independent random
vector uniformly distributed on the sphere , and two vertices
are connected if the corresponding latent vectors are close enough. In the
dense regime (i.e., is a constant), we propose a near-optimal and
computationally efficient testing procedure based on a new quantity which we
call signed triangles. The proof of the detection lower bound is based on a new
bound on the total variation distance between a Wishart matrix and an
appropriately normalized GOE matrix. In the sparse regime, we make a conjecture
for the optimal detection boundary. We conclude the paper with some preliminary
steps on the problem of estimating the dimension in .Comment: 28 pages; v2 contains minor change
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