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Very nice paper! The main idea: if the distance between two embedded nodes has to be large then one should not waste energy in order to make this distance a bit smaller. Trying to make large distances a bit smaller is actually what Laplacian embedding is doing with its quadratic objective. One should rather focus on small distances and give up on the pairs of embedded nodes that are far away. ### Embedding complex networks: This method could be applied to social networks (or complex networks in general): social networks are known to have "weak ties" (roughly speaking, weak ties are links between people that do not have many friends in common) that may perturb laplacian embedding. ### Asymptotic complexity and scalability in practice: The suggested algorithm to solve the optimization does not seem very scalable. The asymptotic running time complexity is not given. ### Concerns about the optimization in dimension higher than 1. The example in 2-D. In equations (9-11), what if we change the constraints into: $||x||^2+||y||^2=1$, $e^Tx=0$ and $e^Ty=0$? In n-D, denoting $r_i$ the vector associated to node $i$, that would give $\sum_ir_i^2=1$ and $\sum_ir_i=0$. This seems to be a more relevant optimization to me: minimize the distance between nodes in 2-D without setting constraints for each dimension, but rather global constraints. If the objective is the one of the laplacian embedding, is it true that, in the optimal solution, $y$ will be proportional to $x$? ($x$ and $y$ will both be proportional to the eigenvector associated with the smallest eigenvalue of the Laplacian?) If so, is it still true if the objective is another one? Like the Cauchy embedding objective for instance? ### Others: - How to tune the parameter $\sigma$?
> - How to tune the parameter $σ$? I don't see where $σ$ is defined, did I miss something?
Note that $\sum_i r_i = 0$ means that vector $r$ is orthogonal to vector $v_0 = (1,1,\ldots\,1,1,1)$. The latter corresponds to the smallest eigen-value $\lambda_0 = 0$ of [Laplacian matrix](https://en.wikipedia.org/wiki/Laplacian_matrix). Adding constraint $r \perp v_0$ to the optimization problem drives us to seek the second smallest eigenpair. You may also read very related and well-written [lecture](https://ocw.mit.edu/courses/mathematics/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/lecture-notes/MIT18_409F09_scribe3.pdf) about Rayleigh quotients, min-max theorem (Courant–Fischer–Wey). Gives a good background in the field of eigen-based solutions for optimization problems. In 2016, during the preparation of [my talk](http://www.complexnetworks.fr/temporal-density-of-complex-networks-and-ego-community-dynamics/) titled "Temporal density of complex networks and ego-community dynamics" ([slides](https://kirgizov.link/talks/temporal-density-4-july-2016.pdf), french). I have read stuff about graph optimization problems. Below I present my personal (perhaps a little bit random selection) of related papers and books * [Four Cheeger-type Inequalities for Graph Partitioning Algorithms](https://www.math.ucsd.edu/~fan/wp/heaticcm.pdf), Fan Chung, 2007 * [Normalized cuts and image segmentation](https://people.eecs.berkeley.edu/~malik/papers/SM-ncut.pdf), Jianbo Shi and Jitendra Malik, 2000 * [Eigenvalues of graphs](https://web.cs.elte.hu/~lovasz/eigenvals-x.pdf), László Lovász, 2007 * [Spectres de graphes](https://www-fourier.ujf-grenoble.fr/~ycolver/All-Articles/98a.pdf) de Yves Colin de Verdière, 1998
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