Towards Resistance Sparsifiers

We study resistance sparsification of graphs, in which the goal is to find a sparse subgraph (with reweighted edges) that approximately preserves the effective resistances between every pair of nodes. We show that every dense regular expander admits a $(1+\epsilon)$-resistance sparsifier of size $\tilde O(n/\epsilon)$, and conjecture this bound holds for all graphs on $n$ nodes. In comparison, spectral sparsification is a strictly stronger notion and requires $\Omega(n/\epsilon^2)$ edges even on the complete graph. Our approach leads to the following structural question on graphs: Does every dense regular expander contain a sparse regular expander as a subgraph? Our main technical contribution, which may of independent interest, is a positive answer to this question in a certain setting of parameters. Combining this with a recent result of von Luxburg, Radl, and Hein~(JMLR, 2014) leads to the aforementioned resistance sparsifiers

Exponentially Improving the Complexity of Simulating the Weisfeiler-Lehman Test with Graph Neural Networks

Recent work shows that the expressive power of Graph Neural Networks (GNNs) in distinguishing non-isomorphic graphs is exactly the same as that of the Weisfeiler-Lehman (WL) graph test. In particular, they show that the WL test can be simulated by GNNs. However, those simulations involve neural networks for the 'combine' function of size polynomial or even exponential in the number of graph nodes $n$, as well as feature vectors of length linear in $n$. We present an improved simulation of the WL test on GNNs with \emph{exponentially} lower complexity. In particular, the neural network implementing the combine function in each node has only a polylogarithmic number of parameters in $n$, and the feature vectors exchanged by the nodes of GNN consists of only $O(\log n)$ bits. We also give logarithmic lower bounds for the feature vector length and the size of the neural networks, showing the (near)-optimality of our construction.Comment: 22 pages,5 figures, accepted at NeurIPS 202

Complete genome sequence of an Israeli isolate of Xanthomonas hortorum pv. pelargonii strain 305 and novel type III effectors identified in Xanthomonas

Xanthomonas hortorum pv. pelargonii is the causative agent of bacterial blight in geranium ornamental plants, the most threatening bacterial disease of this plant worldwide. Xanthomonas fragariae is the causative agent of angular leaf spot in strawberries, where it poses a significant threat to the strawberry industry. Both pathogens rely on the type III secretion system and the translocation of effector proteins into the plant cells for their pathogenicity. Effectidor is a freely available web server we have previously developed for the prediction of type III effectors in bacterial genomes. Following a complete genome sequencing and assembly of an Israeli isolate of Xanthomonas hortorum pv. pelargonii - strain 305, we used Effectidor to predict effector encoding genes both in this newly sequenced genome, and in X. fragariae strain Fap21, and validated its predictions experimentally. Four and two genes in X. hortorum and X. fragariae, respectively, contained an active translocation signal that allowed the translocation of the reporter AvrBs2 that induced the hypersensitive response in pepper leaves, and are thus considered validated novel effectors. These newly validated effectors are XopBB, XopBC, XopBD, XopBE, XopBF, and XopBG