15 research outputs found
Inhomogeneous random graphs, isolated vertices, and Poisson approximation
Consider a graph on randomly scattered points in an arbitrary space, with two
points connected with probability . Suppose the number of
points is large but the mean number of isolated points is . We give
general criteria for the latter to be approximately Poisson distributed. More
generally, we consider the number of vertices of fixed degree, the number of
components of fixed order, and the number of edges. We use a general result on
Poisson approximation by Stein's method for a set of points selected from a
Poisson point process. This method also gives a good Poisson approximation for
U-statistics of a Poisson process.Comment: 31 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
Computationally efficient inference for latent position network models
Latent position models are widely used for the analysis of networks in a
variety of research fields. In fact, these models possess a number of desirable
theoretical properties, and are particularly easy to interpret. However,
statistical methodologies to fit these models generally incur a computational
cost which grows with the square of the number of nodes in the graph. This
makes the analysis of large social networks impractical. In this paper, we
propose a new method characterised by a linear computational complexity, which
can be used to fit latent position models on networks of several tens of
thousands nodes. Our approach relies on an approximation of the likelihood
function, where the amount of noise introduced by the approximation can be
arbitrarily reduced at the expense of computational efficiency. We establish
several theoretical results that show how the likelihood error propagates to
the invariant distribution of the Markov chain Monte Carlo sampler. In
particular, we demonstrate that one can achieve a substantial reduction in
computing time and still obtain a good estimate of the latent structure.
Finally, we propose applications of our method to simulated networks and to a
large coauthorships network, highlighting the usefulness of our approach.Comment: 39 pages, 10 figures, 1 tabl
A Dynamic Latent-Space Model for Asset Clustering
Periods of financial turmoil are not only characterized by higher correlation across assets but also by modifications in their overall clustering structure. In this work, we develop a dynamic Latent-Space mixture model for capturing changes in the clustering structure of financial assets at a fine scale. Through this model, we are able to project stocks onto a lower dimensional manifold and detect the presence of clusters. The infinite-mixture assumption ensures tractability in inference and accommodates cases in which the number of clusters is large. The Bayesian framework we rely on accounts for uncertainty in the parameters’ space and allows for the inclusion of prior knowledge. After having tested our model’s effectiveness and inference on a suitable synthetic dataset, we apply the model to the cross-correlation series of two reference stock indices. Our model correctly captures the presence of time-varying asset clustering. Moreover, we notice how assets’ latent coordinates may be related to relevant financial factors such as market capitalization and volatility. Finally, we find further evidence that the number of clusters seems to soar in periods of financial distress
Continuous Latent Position Models for Instantaneous Interactions
We create a framework to analyse the timing and frequency of instantaneous
interactions between pairs of entities. This type of interaction data is
especially common nowadays, and easily available. Examples of instantaneous
interactions include email networks, phone call networks and some common types
of technological and transportation networks. Our framework relies on a novel
extension of the latent position network model: we assume that the entities are
embedded in a latent Euclidean space, and that they move along individual
trajectories which are continuous over time. These trajectories are used to
characterize the timing and frequency of the pairwise interactions. We discuss
an inferential framework where we estimate the individual trajectories from the
observed interaction data, and propose applications on artificial and real
data.Comment: 33 page
A dynamic network model to measure exposure diversification in the Austrian interbank market
We propose a statistical model for weighted temporal networks capable of
measuring the level of heterogeneity in a financial system. Our model focuses
on the level of diversification of financial institutions; that is, whether
they are more inclined to distribute their assets equally among partners, or if
they rather concentrate their commitment towards a limited number of
institutions. Crucially, a Markov property is introduced to capture time
dependencies and to make our measures comparable across time. We apply the
model on an original dataset of Austrian interbank exposures. The temporal span
encompasses the onset and development of the financial crisis in 2008 as well
as the beginnings of European sovereign debt crisis in 2011. Our analysis
highlights an overall increasing trend for network homogeneity, whereby core
banks have a tendency to distribute their market exposures more equally across
their partners