Encoding Robust Representation for Graph Generation

Generative networks have made it possible to generate meaningful signals such as images and texts from simple noise. Recently, generative methods based on GAN and VAE were developed for graphs and graph signals. However, the mathematical properties of these methods are unclear, and training good generative models is difficult. This work proposes a graph generation model that uses a recent adaptation of Mallat's scattering transform to graphs. The proposed model is naturally composed of an encoder and a decoder. The encoder is a Gaussianized graph scattering transform, which is robust to signal and graph manipulation. The decoder is a simple fully connected network that is adapted to specific tasks, such as link prediction, signal generation on graphs and full graph and signal generation. The training of our proposed system is efficient since it is only applied to the decoder and the hardware requirements are moderate. Numerical results demonstrate state-of-the-art performance of the proposed system for both link prediction and graph and signal generation.Comment: 9 pages, 7 figures, 6 table

A generative model for protein contact networks

In this paper we present a generative model for protein contact networks. The soundness of the proposed model is investigated by focusing primarily on mesoscopic properties elaborated from the spectra of the graph Laplacian. To complement the analysis, we study also classical topological descriptors, such as statistics of the shortest paths and the important feature of modularity. Our experiments show that the proposed model results in a considerable improvement with respect to two suitably chosen generative mechanisms, mimicking with better approximation real protein contact networks in terms of diffusion properties elaborated from the Laplacian spectra. However, as well as the other considered models, it does not reproduce with sufficient accuracy the shortest paths structure. To compensate this drawback, we designed a second step involving a targeted edge reconfiguration process. The ensemble of reconfigured networks denotes improvements that are statistically significant. As a byproduct of our study, we demonstrate that modularity, a well-known property of proteins, does not entirely explain the actual network architecture characterizing protein contact networks. In fact, we conclude that modularity, intended as a quantification of an underlying community structure, should be considered as an emergent property of the structural organization of proteins. Interestingly, such a property is suitably optimized in protein contact networks together with the feature of path efficiency.Comment: 18 pages, 67 reference

Spectral Detection on Sparse Hypergraphs

We consider the problem of the assignment of nodes into communities from a set of hyperedges, where every hyperedge is a noisy observation of the community assignment of the adjacent nodes. We focus in particular on the sparse regime where the number of edges is of the same order as the number of vertices. We propose a spectral method based on a generalization of the non-backtracking Hashimoto matrix into hypergraphs. We analyze its performance on a planted generative model and compare it with other spectral methods and with Bayesian belief propagation (which was conjectured to be asymptotically optimal for this model). We conclude that the proposed spectral method detects communities whenever belief propagation does, while having the important advantages to be simpler, entirely nonparametric, and to be able to learn the rule according to which the hyperedges were generated without prior information.Comment: 8 pages, 5 figure

Neural 3D Morphable Models: Spiral Convolutional Networks for 3D Shape Representation Learning and Generation

Generative models for 3D geometric data arise in many important applications in 3D computer vision and graphics. In this paper, we focus on 3D deformable shapes that share a common topological structure, such as human faces and bodies. Morphable Models and their variants, despite their linear formulation, have been widely used for shape representation, while most of the recently proposed nonlinear approaches resort to intermediate representations, such as 3D voxel grids or 2D views. In this work, we introduce a novel graph convolutional operator, acting directly on the 3D mesh, that explicitly models the inductive bias of the fixed underlying graph. This is achieved by enforcing consistent local orderings of the vertices of the graph, through the spiral operator, thus breaking the permutation invariance property that is adopted by all the prior work on Graph Neural Networks. Our operator comes by construction with desirable properties (anisotropic, topology-aware, lightweight, easy-to-optimise), and by using it as a building block for traditional deep generative architectures, we demonstrate state-of-the-art results on a variety of 3D shape datasets compared to the linear Morphable Model and other graph convolutional operators.Comment: to appear at ICCV 201

Metrics for Graph Comparison: A Practitioner's Guide

Comparison of graph structure is a ubiquitous task in data analysis and machine learning, with diverse applications in fields such as neuroscience, cyber security, social network analysis, and bioinformatics, among others. Discovery and comparison of structures such as modular communities, rich clubs, hubs, and trees in data in these fields yields insight into the generative mechanisms and functional properties of the graph. Often, two graphs are compared via a pairwise distance measure, with a small distance indicating structural similarity and vice versa. Common choices include spectral distances (also known as $\lambda$ distances) and distances based on node affinities. However, there has of yet been no comparative study of the efficacy of these distance measures in discerning between common graph topologies and different structural scales. In this work, we compare commonly used graph metrics and distance measures, and demonstrate their ability to discern between common topological features found in both random graph models and empirical datasets. We put forward a multi-scale picture of graph structure, in which the effect of global and local structure upon the distance measures is considered. We make recommendations on the applicability of different distance measures to empirical graph data problem based on this multi-scale view. Finally, we introduce the Python library NetComp which implements the graph distances used in this work