Graph Machine Learning

​​My thoughts on Kelly's conjecture. A Kelly's conjecture (a.k.a. reconstruction conjecture) states that for a graph with n vertices, a deck with n graphs on (n-1) vertices can reconstruct exactly a graph. In other words, pick a graph, remove one vertex (with all its outgoing edges), and put to the set. Then start with the original graph, remove another vertex, and put it to the set. Repeat for all vertices. Finally, you will have a set of n graphs with n-1 vertices, which is called a deck. Now, the conjecture states that deck uniquely determines the original graph, hence, it's a complete graph invariant, i.e. two graphs have the same deck if and only if, they are isomorphic. This conjecture has been known since 1957 and many results have been achieved. For example, it's known that regular graphs or trees can be reconstructed uniquely with a deck, but directed graphs are not. For general undirected case it's still not clear. There is even an ArXiv paper from 2017 that claimed to resolve the conjecture, but after communication with the authors, they admit there is a mistake in the proof. This idea has been in graph representation too. In particular, there is a graphlet kernel that decomposes a graph into a set of small graphlets/motifs. This is quite expressive kernel, i.e. it can recognize a lot of graphs, but whether it recognizes all of the graphs is still a question. For a long time, I believed that if Kelly's conjecture is true (which is quite likely), then not only a deck of graphs on (n-1) vertices is unique, but also of smaller graphs. The assumption was that you can in turn decompose a (n-1) graph into a set of (n-2) deck and move deeper to any number of vertices that you want. But yesterday I found that there are non-isomorphic graphs that have the same decomposition on smaller graphs. In particular, if you take a cycle of 4k+2 vertices as G1 and two cycles of 2k+1 as G2, then a decomposition into k-size graphlets will be the same. For example, for k=3, we have graphlets, i.e. triplets which are connected in all possible ways. In this case, G1 is a cycle on 10 vertices and G2 is two cycles on 5 vertices. The number of closed triangles would be 0 for G1 and G2, the number of 2 connected points and one disconnected would be 60 for G1 and G2, and so on. In the end, the decomposition would be the same, while the graphs are different. Moreover, in recent paper IJCAI '18 it's shown that graphlet kernel cannot identify some planar from non-planar, bijective from non-bijective, and connected from disconnected graphs. This is quite interesting as it shows the limitations of graphlet kernel and in general graphlet decomposition.

Fresh picks from ArXiv This week is accepted papers to CVPR and WebConf, submissions to ICML, 130-page survey on knowledge graphs and algorithms for rainbow vertex coloring 🌈 CVPR 20 * Point-GNN: Graph Neural Network for 3D Object Detection in a Point Cloud * Bundle Adjustment on a Graph Processor WebConf 20 * Just SLaQ When You Approximate: Accurate Spectral Distances for Web-Scale Graphs * Heterogeneous Graph Transformer * Learning to Hash with Graph Neural Networks for Recommender Systems ICML 20 * Neural Enhanced Belief Propagation on Factor Graphs by group of Max Welling Survey * A Survey on The Expressive Power of Graph Neural Networks by Ryoma Sato * A Survey on Deep Hashing Methods * Knowledge Graphs * Knowledge Graphs and Knowledge Networks: The Story in Brief Graph Theory * Properties of Erdős-Rényi Graphs * Algorithms for the rainbow vertex coloring problem on graph classes * Direct Product Primality Testing of Graphs is GI-hard

AISTATS 2020 stats 1490 number of submissions 423 accepted papers 28% acceptance rate 15 graph papers (4% of accepted)

AISTATS 20 will take place in Italy, as was planned. https://twitter.com/Aistats2020/status/1236321207559036928?s=20

Benchmarking Graph Neural Networks This is a recent submission to ICML that compares existing GNN on multiple data sets. There was already an accepted paper at ICLR 2020 so the two works largely overlap (and I guess that would decrease the chances for ICML submissions to be accepted). Good points: * The authors propose 6 new data sets with tens of thousands graphs. * They provide learned lessons: 1. GNNs are better than MLP for their proposed data sets. 2. Adding weights for each edge helps. 3. Residual connections and normalization helps. Bad points: * The data sets are still small in terms of the size of the graphs, ~100 of nodes. * No metrics beyond accuracy.

Bridging the Gap between Spatial and Spectral Domains: A Survey on Graph Neural Networks A new short survey on GNNs, submission to IJCAI survey track. * It formulates popular GNNs in the same context, i.e. Z = f(G)X, where Z is matrix of embeddings, f(G) is GNN function, and X is matrix of features. This helps to compare different GNNs by purely looking at f(G) function. * It provides examples for different local aggregation schemes, order of neighborhoods, and directions on the edges. * It discusses common GNNs from spectral perspective.

Fresh picks from ArXiv This week is full of CVPR and AISTATS 20 accepted papers, new surveys, more submissions to ICML and KDD, and new GNN models 📚 CVPR 20 * Unbiased Scene Graph Generation from Biased Training * Social-STGCNN: A Social Spatio-Temporal Graph Convolutional Neural Network for Human Trajectory Prediction * 4D Association Graph for Realtime Multi-person Motion Capture Using Multiple Video Cameras * Representations, Metrics and Statistics For Shape Analysis of Elastic Graphs * Say As You Wish: Fine-grained Control of Image Caption Generation with Abstract Scene Graphs * Fine-grained Video-Text Retrieval with Hierarchical Graph Reasoning * SketchGCN: Semantic Sketch Segmentation with Graph Convolutional Networks Survey * Bridging the Gap between Spatial and Spectral Domains: A Survey on Graph Neural Networks * Graph Neural Networks Meet Neural-Symbolic Computing: A Survey and Perspective * Adversarial Attacks and Defenses on Graphs: A Review and Empirical Study * Knowledge Graphs on the Web -- an Overview GNN * Infinitely Wide Graph Convolutional Networks: Semi-supervised Learning via Gaussian Processes * Can graph neural networks count substructures? by group of Joan Bruna * Heterogeneous Graph Neural Networks for Malicious Account Detection by group of Le Song AISTATS 20 * Permutation Invariant Graph Generation via Score-Based Generative Modeling KDD 20 * PM2.5-GNN: A Domain Knowledge Enhanced Graph Neural Network For PM2.5 Forecasting ICML 20 * Semi-supervised Anomaly Detection on Attributed Graphs * Inverse Graphics GAN: Learning to Generate 3D Shapes from Unstructured 2D Data * Permutohedral-GCN: Graph Convolutional Networks with Global Attention * Benchmarking Graph Neural Networks by group of Yoshua Bengio Graph Theory * Finding large matchings in 1-planar graphs of minimum degree 3 * Trapping problem on star-type graphs with applications * On Fast Computation of Directed Graph Laplacian Pseudo-Inverse

Comparing Stars: On Approximating Graph Edit Distance Good old VLDB'09 paper on the computation of graph edit distance. * It shows the exact distance computation is NP-hard; * It discusses connections with graph matching; * And it provides lower and upper bound algorithms.

Channel photo updated

Transformers are Graph Neural Networks It seems today everyone is talking about this blog post. It describes how NLP's Transformers are just part of a more general concept we call Graph Neural Networks. In Transformer, we have an encoder that for every word gives an embedding. The idea is very simple: let's the embedding of word A be a weighted average of embeddings of other words in a sentence. That's it. That's what everyone calls attention. Attention = weighted average. The weights are trained together with embeddings, you also add normalization tricks, positional encodings, MLP, and other deep learning mumbles, but this is like in any other model. What's novel is that you use weighted average, instead of aggregation word by word sequentially. In GNN we also have an encoder, this encoder just takes a graph in and outputs embeddings of nodes. The most popular way in GNN to do this is through message-passing mechanism, the idea of which is also simple: the embedding of node A is a function of embeddings of its neighbors. Two changes from Transformer: 1) We can have any function, instead of weighted average. 2) We aggregate over the neighbors, instead of all nodes. If you take weighted average function, i.e. attention, then you have Graph Attention Network. Surprise. You could also aggregate the nodes beyond your neighbors and I believe there are works that do this, but common assumption is that graph connections are already constructed in such a way that node is most influenced by the incoming edges and not by some distant nodes. So Transformers are not GNNs, at most Transformers are Graph Attention Networks. This is important if we want to draw some conclusions for NLP from GNN area. For example, Graph Attention Network is less powerful than many other GNNs such as GIN. It also does not have function approximation guarantees that I described in previous posts. But sure the connection between NLP and GML exists and is not fully explored. I bet you can leverage graphs when you analyze text or you can use GNN for better encoding. GML on the other hand benefited from NLP in many ways already: usually we first see some results in NLP (RNN, Transformers, BERT) and then see its adaptation in GML. So maybe it's the moment we'll see more contributions to NLP with graphs.

Understanding the Representation Power of Graph Neural Networks in Learning Graph Topology This is NeurIPS 2019 work by the group of Albert-László Barabási, — the one who invented a famous Barabási–Albert graph model. In this work, they study if GNN can compute the function that approximates graph moments, i.e. powers of adjacency matrix. The key result is that if GNN has number of layers more than the power of the matrix then it can learn a corresponding graph moment. For those who followed previous post, which describes a paper What graph neural networks cannot learn: depth vs width by Loukas,will see that Barabási's paper is a precursor and a partial result of Loukas' paper that states the condition on GNN for computability of any function (not just graph moments).

ML and NLP Publications in 2019 One of the reasons why I have this channel is to encourage people do more research in general. And looking at the recent analysis of the number of accepted papers by countries you can understand why. USA has almost as much publications last year as all other countries combined. In particular, it's 2.5x > China, 6x > UK, and 100x > Russia. The reasons are obvious, USA spends more budget on research from both government and industry, but also the culture of doing research is different: making a publication at top conferences is considered a big deal and a lot of efforts is put on this, — which can definitely adopted by other countries.