The purpose of this crate is to provide embedding of directed or undirected graphs with positively weighted edges and possibly discrete labels attached to nodes.
For simple graphs, without data attached to nodes/labels, we provide 2 (rust) modules nodesketch and atp. A simple executable with a validation option based on link prediction is also provided.
The module gkernel is dedicated to graphs with discrete labels attached to nodes/edges. We use the petgraph crate for graph description.
The algorithm is based on an extension of the hashing strategy used in the module nodesketch.
In the undirected case, this module also computes a global embedding vector for the whole graph. It is still in an early version.
NodeSketch : Highly-Efficient Graph Embeddings via Recursive Sketching KDD 2019. [https://dl.acm.org/doi/10.1145/3292500.3330951]
D. Yang,P. Rosso,Bin-Li, P. Cudre-Mauroux.
It is based on multi hop neighbourhood identification via sensitive hashing.
Instead of using ICWS for hashing we use the more recent algorithm probminhash. See probminhash.
The algorithm associates a probability distribution on neighbours of each point depending on edge weights and distance to the point.
Then this distribution is hashed to build an embedding vector.
The distance between embedded vectors is the Jaccard distance so we get
a real distance on the embedding space, at least for the symetric embedding.
An extension of the paper is also implemented to get asymetric embedding for directed graph. The similarity is also based on the hash of sets (nodes going to or from) a given node but then the dissimilarity is no more a distance (no symetry and some discrepancy with the triangular inequality).
Asymetric Transitivity Preserving Graph Embedding 2016
M. Ou, P Cui, J. Pei, Z. Zhang and W. Zhu.
The objective is to provide an asymetric graph embedding and get estimate of the precision of the embedding in function of its dimension.
We use the Adamic-Adar matricial representation of the graph. (It must be noted that the ponderation of a node by the number of couples joined by it is called Resource Allocation in the Graph Kernel litterature).
The asymetric embedding is obtained from the left and right singular eigenvectors of the Adamic-Adar representation of the graph.
Source node are related to left singular vectors and target nodes to the right ones.
The similarity measure is the dot product, so it is not a norm.
The svd is approximated by randomization as described in Halko-Tropp 2011 as implemented in the annembed crate.
Asymetric embeddings compute 2 vectors for each node, once considered as a source, once considered as a target.
These vectors are computed either as left and right singular vectors of a proximity matrix in the case of the atp module. For the nodesketch and gkernel the embedding vectors are computed as the summary of node id's or labels attached to nodes flowing through the edges going into or from a node.
Small datasets are given in the Data subdirectory (with 7z compression) to run tests.
Larger datasets can be downloaded from the SNAP data collections https://snap.stanford.edu/data
Les miserables http://konect.cc/networks/moreno_lesmis les miserables co occurence of characters in Victor Hugo's novel 'Les Misérables'.
CA-GrQc.txt https://snap.stanford.edu/data/ca-GrQc.html
*p2p-Gnutella08.txt.gz https://snap.stanford.edu/data/p2p-Gnutella08.html
wiki-vote https://snap.stanford.edu/data/wiki-Vote.html 7115 nodes 103689 edges
Cora : http://konect.cc/networks/subelj_cora citation network 23166 nodes 91500 edges
These graphs were used in results see below.
Beware of the possible need to convert from Windows to Linux End Of Line, see the dos2unix utility.
Possibly some data can need to be converted from Tsv format to Csv, before being read by the program.
*youtube. Nodes: 1134890 Edges: 2987624 https://snap.stanford.edu/data/com-Youtube.html
*twitter as tested in Hope http://konect.cc/networks/munmun_twitter_social 465017 nodes 834797 edges
Embedding and link prediction evaluation for the above data sets are given in file resultats.md
For the embedding using the randomized svd, increasing the embedding dimension is interesting as far as the corresponding eigenvalues continue to decrease significantly.
The munmuntwittersocial graph shows that treating a directed graph as an undirected graph give significantly different results in terms of link prediction AUC.
An implementation of Generalized Svd comes as a by-product in module gsvd.
The crate provides three features, required by the annembed dependency, to specify which version of lapack you want to use.
For example compilation is done by :
cargo build --release --features="openblas-static" to link statically with openblas.
The choice of one feature is mandatory to provide required linear algebra library.
The Hope embedding relying on matrices computations limits the size of the graph to some hundred thousands nodes.
It is intrinsically asymetric in nature. It nevertheless gives access to the spectrum of Adamic Adar representing the graph and
so to the required dimension to get a valid embedding in $R^{n}$.
The Sketching embedding is much faster for large graphs but embeds in a space consisting in sequences of node id equipped with the Jaccard distance.
The embed module takes embedding and possibly validation commands in one directive.
The general syntax is :
embed filedescription [validationcommand --validationarguments] embeddingcommand --embedding_arguments
It is detailed in docs of the embed module. Use cargo doc --no-deps as usual.
Use the environment variable RUSTLOG gives access to some information at various level (debug, info, error) via the log and envlogger crates.