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Register a new userNetwork estimation via graphon with node features
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Added by
Yi Su
Publication date
2018
fields
Mathematical Statistics
and research's language is
English
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Estimating the probabilities of linkages in a network has gained increasing interest in recent years. One popular model for network analysis is the exchangeable graph model (ExGM) characterized by a two-dimensional function known as a graphon. Estimating an underlying graphon becomes the key of such analysis. Several nonparametric estimation methods have been proposed, and some are provably consistent. However, if certain useful features of the nodes (e.g., age and schools in social network context) are available, none of these methods was designed to incorporate this source of information to help with the estimation. This paper develops a consistent graphon estimation method that integrates the information from both the adjacency matrix itself and node features. We show that properly leveraging the features can improve the estimation. A cross-validation method is proposed to automatically select the tuning parameter of the method.
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Networks describe a variety of interacting complex systems in social science, biology and information technology. Usually the nodes of real networks are identified not only by their connections but also by some other characteristics. Examples of characteristics of nodes can be age, gender or nationality of a person in a social network, the abundance of proteins in the cell taking part in a protein-interaction networks or the geographical position of airports that are connected by directed flights. Integrating the information on the connections of each node with the information about its characteristics is crucial to discriminating between the essential and negligible characteristics of nodes for the structure of the network. In this paper we propose a general indicator, based on entropy measures, to quantify the dependence of a networks structure on a given set of features. We apply this method to social networks of friendships in US schools, to the protein-interaction network of Saccharomyces cerevisiae and to the US airport network, showing that the proposed measure provides information which complements other known measures.
This paper develops a novel approach to density estimation on a network. We formulate nonparametric density estimation on a network as a nonparametric regression problem by binning. Nonparametric regression using local polynomial kernel-weighted least squares have been studied rigorously, and its asymptotic properties make it superior to kernel estimators such as the Nadaraya-Watson estimator. When applied to a network, the best estimator near a vertex depends on the amount of smoothness at the vertex. Often, there are no compelling reasons to assume that a density will be continuous or discontinuous at a vertex, hence a data driven approach is proposed. To estimate the density in a neighborhood of a vertex, we propose a two-step procedure. The first step of this pretest estimator fits a separate local polynomial regression on each edge using data only on that edge, and then tests for equality of the estimates at the vertex. If the null hypothesis is not rejected, then the second step re-estimates the regression function in a small neighborhood of the vertex, subject to a joint equality constraint. Since the derivative of the density may be discontinuous at the vertex, we propose a piecewise polynomial local regression estimate to model the change in slope. We study in detail the special case of local piecewise linear regression and derive the leading bias and variance terms using weighted least squares theory. We show that the proposed approach will remove the bias near a vertex that has been noted for existing methods, which typically do not allow for discontinuity at vertices. For a fixed network, the proposed method scales sub-linearly with sample size and it can be extended to regression and varying coefficient models on a network. We demonstrate the workings of the proposed model by simulation studies and apply it to a dendrite network data set.
Graph neural networks (GNNs) are learning architectures that rely on knowledge of the graph structure to generate meaningful representations of large-scale network data. GNN stability is thus important as in real-world scenarios there are typically uncertainties associated with the graph. We analyze GNN stability using kernel objects called graphons. Graphons are both limits of convergent graph sequences and generating models for deterministic and stochastic graphs. Building upon the theory of graphon signal processing, we define graphon neural networks and analyze their stability to graphon perturbations. We then extend this analysis by interpreting the graphon neural network as a generating model for GNNs on deterministic and stochastic graphs instantiated from the original and perturbed graphons. We observe that GNNs are stable to graphon perturbations with a stability bound that decreases asymptotically with the size of the graph. This asymptotic behavior is further demonstrated in an experiment of movie recommendation.
In this work, we propose to train a graph neural network via resampling from a graphon estimate obtained from the underlying network data. More specifically, the graphon or the link probability matrix of the underlying network is first obtained from which a new network will be resampled and used during the training process at each layer. Due to the uncertainty induced from the resampling, it helps mitigate the well-known issue of over-smoothing in a graph neural network (GNN) model. Our framework is general, computationally efficient, and conceptually simple. Another appealing feature of our method is that it requires minimal additional tuning during the training process. Extensive numerical results show that our approach is competitive with and in many cases outperform the other over-smoothing reducing GNN training methods.
Komlos [Komlos: Tiling Turan Theorems, Combinatorica, 2000] determined the asymptotically optimal minimum-degree condition for covering a given proportion of vertices of a host graph by vertex-disjoint copies of a fixed graph H, thus essentially extending the Hajnal-Szemeredi theorem which deals with the case when H is a clique. We give a proof of a graphon version of Komloss theorem. To prove this graphon version, and also to deduce from it the original statement about finite graphs, we use the machinery introduced in [Hladky, Hu, Piguet: Tilings in graphons, arXiv:1606.03113]. We further prove a stability version of Komloss theorem.
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