Statistical inference on graphs

Abstract

The problem of graph inference, or graph reconstruction, is to predict the presence or absence of edges between a set of given points known to form the vertices of a graph. Motivated by various applications including communication networks and systems biology, we propose a general model for studying the problem of graph inference in a supervised learning framework. In our setting, both the graph vertices and edges are assumed to be random, with a probability distribution that possibly depends on the size of the graph. We show that the problem can be transformed into one where we can use statistical learning methods based on empirical minimization of natural estimates of the reconstruction risk. Convex risk minimization methods are also studied to provide a theoretical framework for reconstruction algorithms based on boosting and support vector machines. Our approach is illustrated on simulated graphs.

On the ERM Principle With Networked Data

“…We now have a look at previous works that are closely related to our work, as shown in Table 1, and present the merits of our method. Biau and Bleakley (2006), Clémenc ¸on, Lugosi, and Vayatis (2008) and Papa, Bellet, and Clémenc ¸on (2016) deal with the case when the graph is complete, i.e., the target value of every pair of vertices is known. In this case, Clémenc ¸on, Lugosi, and Vayatis (2008) formulate the "low-noise" condition for the ranking problem and demonstrate that this condition can lead to tighter risk bounds by the moment inequality for U -processes.…”

Optimal link prediction with matrix logistic regression

“…Given observations X i , X j about nodes i and j of a network, we aim to understand how these two explanatory variables are related to the probability of connection between the two corresponding vertices, such that P(Y (i,j) = 1) = f (X i , X j ), by estimating f within a high-dimensional class based on logistic regression. Besides this high-dimensional parametric modelling, various fully non-parametric statistical frameworks were exploited in the literature, see, for example, Gao et al (2015); Wolfe and Olhede (2013), for graphon estimation, Biau and Bleakly (2008); Papa et al (2016) for graph reconstruction and Bickel and Chen (2009) for modularity analysis.…”

Robust estimation of U-statistics

“…Motivated by numerous applications, the theory of U -statistics and U -processes has received considerable attention in the past decades. U -statistics appear naturally in ranking (Clémençon et al, 2008), clustering (Clémençon, 2014) and learning on graphs (Biau and Bleakley, 2006) or as components of higher-order terms in expansions of smooth statistics, see, for example, Robins et al (2009). The general setting may be described as follows.…”