##Great paper! The method uses random projections of the column vector of the matrix $S=(\alpha_0 I+\alpha_1 A+\alpha_2 A^2+...+\alpha_q A^q)$. It is shown that such a random projection is a heuristics to minimize a relevant cost function (equation (2)) with some guarantees (Theorem 1). As only the products of vectors $R$ with this matrix is needed, explicitly forming the (potentially dense) matrix $S$ is avoided: only the vectors $A^p\cdot R$ are computed. The running time is $O(n\cdot d^2+ m\cdot q\cdot d)$ ($m$ number of edges, $n$ number of nodes, $d$ dimensions of embedding and $q$ order of the embedding). The operations involved are very simple leading to a very fast practical algorithm. It is shown in the experiments that these simple ideas lead to the most scalable embedding method and that the method is still accurate for some machine learning tasks such as classification, link prediction, and network reconstruction. The executable in MATLAB is publicly available here: https://github.com/ZW-ZHANG/RandNE An alternative open source implementation in C is available here: https://github.com/maxdan94/RandNE ### Concerns regarding Theorem 3 "Theorem 3. The time complexity of Algorithm 3 is linear with respect to the number of changed nodes and the number of changed edges respectively." In the proof, the running time proven is in $O(M'\cdot R(q))$ (where $M'$ is the number of new or disappeared edges), the "$R(q)$" is the $q$-step neighborhood of an edge. We can have $R(q)=O(N)$ for some graphs especially in real-world graphs where the average distance between two nodes is very small (say less than 4) https://research.fb.com/three-and-a-half-degrees-of-separation/. The theorem is thus not correct without stating further assumptions. In addition, in the experiments, the running time of the suggested algorithm for dynamic graphs (Algorithm 3) is not compared to the static algorithm (Algorithm 1) re-run from scratch after each update. It is not clear to me if the gain in time is significant. ### I do not understand 5, C, 1 "Network Reconstruction: The experimental setting is similar to moderate-scale networks (Section V-B2), i.e. we rank pairs of nodes according to their inner product similarities and reconstruct the network. We report the AUC scores in Table V." There are approximately $\approx 10^{16}$ such pairs of nodes. This number of pairs of nodes is too large, I do not understand how the pairs can be ranked and the AUC computed. ### Parametters tuning The method needs many parameters: $d$, $q$, $a_0$, $a_1$, ..., $a_q$. Whyle $d$ and $q$ can be arbitrary fixed to say $d=128$ and $q=3$ (as done in the paper). It is not clear what default values would fit for the $a_k$'s. In the paper, the $a_k$'s are found using grid search. Note that the range of values and the steps are not specified in addition the optimal values are not given. This hurts reproducibility. What if grid search is not possible (e.g. no train set)? Which default value would make sense? ### Grid search and unfair comparison In the experiments, the suggested embedding is shown to lead to better results for some tasks (such as network reconstruction, classification and link prediction) compared to other embedding methods. However, the suggested method has parameters ($a_0$, $a_1$, $a_2$ and $a_3$ since $q$ is set to $3$). These parameters are tuned using grid search, while some of the parameters of the other methods seem to be kept to default. This may not be a very fair comparison. ### Real billion-scale networks? I read "We also report the exact running time in Table VII. It shows that RandNE can learn all the node embeddings of WeChat within 7 hours with 16 normal servers, which is promising for real billion-scale networks." I do not understand what is meant. WeChat has 250M nodes and 4.8G edges. The number of edges is thus in the order of billions. "Real billion-scale networks" means more than 1G nodes and not edges? Another graph with 1G edges is available here: http://snap.stanford.edu/data/com-Friendster.html In addition, contrary to WeChat, nodes have some labels that could be used to test the relevance of the embedding. Graphs with billion nodes are available here: http://law.di.unimi.it/datasets.php Unfortunately, they are directed... Another one is here: https://sparse.tamu.edu/Sybrandt/MOLIERE_2016 undirected, but weighted. A twitter graph with 25G edges can be obtained upon request: https://hal.inria.fr/hal-00948889/en/ ###Typos and minors: - "as an theoretical guarantee" - "an one-vs-all logistic regression" - This relevant WWW2018 paper is not cited and the associated method is not used for comparisons: https://papers-gamma.link/paper/48
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