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Weisfeiler–Leman and Graph Spectra [article]

Gaurav Rattan, Tim Seppelt
<span title="2022-05-02">2022</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
by the WeisfeilerLeman (WL) algorithm.  ...  We devise a hierarchy of spectral graph invariants, generalising the adjacency spectra and Laplacian spectra, which are commensurate in power with the hierarchy of combinatorial graph invariants generated  ...  On the combinatorial front, the Weisfeiler-Leman procedure is the classical method for generating polynomial-time computable graph invariants.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2103.02972v2">arXiv:2103.02972v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ipgpn6ttnzhrbn56ql5o5qylcu">fatcat:ipgpn6ttnzhrbn56ql5o5qylcu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20220504213038/https://arxiv.org/pdf/2103.02972v2.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/49/b8/49b8b6a868368c12d286350d827953b4e0eb3b69.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2103.02972v2" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Weisfeiler-Leman, Graph Spectra, and Random Walks

Gaurav Rattan, Tim Frederik Seppelt
<span title="">2021</span>
The Weisfeiler-Leman algorithm is a ubiquitous tool for the Graph Isomorphism Problem with various characterisations in e.g. descriptive complexity and convex optimisation.  ...  As an application to random walks, we show using Generalised Laplacians that the edge colours produced by 2-WL determine commute distances.  ...  Random walks on graphs constitute one of the best-studied random processes, with myriad applications to statistics and computing [21] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.18154/rwth-2022-01183">doi:10.18154/rwth-2022-01183</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/4aus5vlbn5eonp2ajenvg2df54">fatcat:4aus5vlbn5eonp2ajenvg2df54</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20220209213014/https://publications.rwth-aachen.de/record/840312/files/840312.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/3b/20/3b204b275d3b008d460f51589eb70b7b7f1c0c7e.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.18154/rwth-2022-01183"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Descriptive complexity of graph spectra

Anuj Dawar, Simone Severini, Octavio Zapata
<span title="">2019</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/bnojym2hjzcgnpa4wixi2axhnq" style="color: black;">Annals of Pure and Applied Logic</a> </i> &nbsp;
We also show that the class of graphs that are determined by their spectra is definable in partial fixedpoint logic with counting.  ...  We relate these properties to other algebraic and combinatorial problems. The spectrum of a graph G is the multi-set of eigenvalues of its adjacency matrix.  ...  If the partitions P k G and P k H of graphs G and H are the same multi-set of labels obtained by the k-dimensional Weisfeiler-Leman algorithm, we say that G and H are k-WL equivalent.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.apal.2019.04.005">doi:10.1016/j.apal.2019.04.005</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/wqugzlilvrctxpibf2zxurh6v4">fatcat:wqugzlilvrctxpibf2zxurh6v4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190428150839/http://discovery.ucl.ac.uk/1526418/1/1603.07030v5.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/ad/d2/add2f81b4d21cc3b4005286798d6e96fe06de120.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.apal.2019.04.005"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Weisfeiler and Leman Go Infinite: Spectral and Combinatorial Pre-Colorings [article]

Or Feldman, Amit Boyarski, Shai Feldman, Dani Kogan, Avi Mendelson, Chaim Baskin
<span title="2022-03-02">2022</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Two popular alternatives that offer a good trade-off between expressive power and computational efficiency are combinatorial (i.e., obtained via the Weisfeiler-Leman (WL) test) and spectral invariants.  ...  Graph isomorphism testing is usually approached via the comparison of graph invariants.  ...  Acknowledgements This research was partially supported by the Technion Hiroshi Fujiwara Cyber Security Research Center and the Israel National Cyber Directorate.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2201.13410v2">arXiv:2201.13410v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/nqvzk544szd4zhpdtikavrkj3a">fatcat:nqvzk544szd4zhpdtikavrkj3a</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20220519143814/https://arxiv.org/pdf/2201.13410v2.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/21/1a/211af952734e053c01cae09f8f0aaca3b1564bf1.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2201.13410v2" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Graph Neural Networks Exponentially Lose Expressive Power for Node Classification [article]

Kenta Oono, Taiji Suzuki
<span title="2021-01-06">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Our theory enables us to relate the expressive power of GCNs with the topological information of the underlying graphs inherent in the graph spectra.  ...  We show that when the Erdős – Rényi graph is sufficiently dense and large, a broad range of GCNs on it suffers from the "information loss" in the limit of infinite layers with high probability.  ...  ACKNOWLEDGMENTS We thank Katsuhiko Ishiguro for providing a part of code for the experiments, Kohei Hayashi and Haru Negami Oono for giving us feedback and comments on the draft, Keyulu Xu and anonymous  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1905.10947v5">arXiv:1905.10947v5</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7eartph5xfalnmabgyvssozsge">fatcat:7eartph5xfalnmabgyvssozsge</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20201024234143/https://arxiv.org/pdf/1905.10947v4.pdf" title="fulltext PDF download [not primary version]" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <span style="color: #f43e3e;">&#10033;</span> <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/9d/88/9d88efc0dfd94e326428a1ac0bfbd21a11bd514f.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1905.10947v5" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Minimal matrices in the Bruhat order for symmetric (0,1)-matrices

Henrique F. da Cruz, Rosário Fernandes, Susana Furtado
<span title="">2017</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/wsx3rzhpingfvewcn5nwhfkq3e" style="color: black;">Linear Algebra and its Applications</a> </i> &nbsp;
graph theory and combinatorics.  ...  We have received 150 abstracts for 20-minute contributed talks which will run in four parallel sessions and cover a large variety of areas within graph theory and combinatorics.  ...  The boundary-type sets are rather trivial for trees and are all subsets of the leaves. We provide a characterization of ∂(G), Ct(G), and Ext(G) for the family of MOPs.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.laa.2017.05.014">doi:10.1016/j.laa.2017.05.014</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zlcp5rb2sjeqvlyaajvf2nyoby">fatcat:zlcp5rb2sjeqvlyaajvf2nyoby</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20180722202039/https://repositorio-aberto.up.pt/bitstream/10216/110839/2/253702.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/f0/e6/f0e65a9287c0f827f2588c08b410d7b847e9e847.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.laa.2017.05.014"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Proceedings of the 2021 Joint Workshop of the German Research Training Groups in Computer Science. May 31–June 1, 2021

Joint Workshop Of The German Research Training Groups In Computer Science, Erlangen May 31–June 1, 2021, Felix Freiling
<span title="2021-05-17">2021</span>
Initiated in 1996 and run regularly since 2007, researchers of the German Research and Training Groups (RTGs) funded by the Deutsche Forschungsgemeinschaft (DFG) in the field of computer science meet annually  ...  The goal of these workshops is to foster an interchange of ideas and experiences in order to strengthen the connection within the German computer science community.  ...  The Weisfeiler-Leman Algorithm. The Weisfeiler-Leman (WL) algorithm [13] is a ubiquitous tool in the context of the Graph Isomorphism Problem.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.25593/opus4-fau-16426">doi:10.25593/opus4-fau-16426</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/isoyq5lzdffxrdqzvkmfsnjgau">fatcat:isoyq5lzdffxrdqzvkmfsnjgau</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20210611191851/https://opus4.kobv.de/opus4-fau/frontdoor/deliver/index/docId/16426/file/Dagstuhl_GRK2021.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/2c/a7/2ca7c2f5c235c865cc75c8c6e2be35e3c02afbe8.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.25593/opus4-fau-16426"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>