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On Non-localization of Eigenvectors of High Girth Graphs [article]

Shirshendu Ganguly, Nikhil Srivastava
2021 arXiv   pre-print
We prove improved bounds on how localized an eigenvector of a high girth regular graph can be, and present examples showing that these bounds are close to sharp.  ...  Our construction is probabilistic and involves gluing together a pair of trees while maintaining high girth as well as control on the eigenvectors and could be of independent interest.  ...  We would like to thank Assaf Naor and Mark Rudelson for helpful conversations, and MSRI and the Simons Institute for the Theory of Computing, where this work was partially carried out.  ... 
arXiv:1803.08038v2 fatcat:ti6chfql35dt7f6vqlaxbtmkni

High-girth near-Ramanujan graphs with localized eigenvectors [article]

Noga Alon and Shirshendu Ganguly and Nikhil Srivastava
2019 arXiv   pre-print
/√(2))√(d), and many eigenvectors fully localized on small sets of size O(|V|^α).  ...  high girth regular graphs.  ...  Pairing trees Our goal is to construct high girth almost-Ramanujan expanders with one or many localized eigenvectors.  ... 
arXiv:1908.03694v1 fatcat:e7j3ekevmfcfhabpgwva5ceqxa

On high-girth expander graphs with localized eigenvectors [article]

Shohei Satake
2021 arXiv   pre-print
The main purpose of this paper is to construct high-girth regular expander graphs with localized eigenvectors for general degrees, which is inspired by a recent work due to Alon, Ganguly and Srivastava  ...  Satake has been supported by Grant-in-Aid for JSPS Fellows 20J00469 of the Japan Society for the Promotion of Science.  ...  While the above construction is based on a probabilistic method, Alon, Ganguly and Srivastava [1] gave a deterministic construction of (d + 1)-regular graphs with large girth and localized eigenvectors  ... 
arXiv:2101.08648v1 fatcat:adl4u4xanvhntlpbfbuju7e74q

Many nodal domains in random regular graphs [article]

Shirshendu Ganguly, Theo McKenzie, Sidhanth Mohanty, Nikhil Srivastava
2021 arXiv   pre-print
We prove that for every constant α > 0, with high probability every eigenvector of the adjacency matrix of G with eigenvalue less than -2√(d-2)-α has Ω(n/polylog(n)) nodal domains.  ...  Let G be a random d-regular graph.  ...  The remainder of the proof focuses on the case where the eigenvector f is ℓ 2 -localized on a small set S ⊂ G.  ... 
arXiv:2109.11532v3 fatcat:rx2zm6pt65ap3ltzeshfugl6li

Expanders have a spanning Lipschitz subgraph with large girth [article]

Gabor Kun
2021 arXiv   pre-print
We show that every regular graph with good local expansion has a spanning Lipschitz subgraph with large girth and minimum degree.  ...  Our finite theorems are kind of converse to the theorem of Bourgain and Gamburd showing that large girth implies expansion for Cayley graphs of SL_2(F_p).  ...  The authors thank to Gábor Pete for encouraging this collaboration, to Elad Tzalik for his suggestions to improve the paper and to Merav Parter for her comments on possible applications.  ... 
arXiv:1303.4982v3 fatcat:k6beb7tx7nhfvkvz5zs7xrzwhy

Approximating the Spectrum of a Graph [article]

David Cohen-Steiner, Weihao Kong, Christian Sohler, Gregory Valiant
2017 arXiv   pre-print
We demonstrate the practical viability of our algorithm on 15 different real-world graphs from the Stanford Large Network Dataset Collection, including social networks, academic collaboration graphs, and  ...  This set of eigenvalues encapsulates many aspects of the structure of the graph, including the extent to which the graph posses community structures at multiple scales.  ...  We then prove that a certain class of spectral properties is testable for any class of high girth graphs, i.e. when the input graph is promised to have high girth.  ... 
arXiv:1712.01725v1 fatcat:gxoc2sd7tjguxmxohys3wy4z2m

Expander Graphs – Both Local and Global [article]

Michael Chapman, Nati Linial, Yuval Peled
2019 arXiv   pre-print
Here we construct two families of (a,b)-regular graphs that expand both locally and globally. We also analyze the possible local and global spectral gaps of (a,b)-regular graphs.  ...  Namely, all the graphs {G_v|v∈ V} should be expanders as well. While random regular graphs are expanders with high probability, they almost surely fail to expand locally.  ...  Our conversations with Irit took place during a special year on high dimensional combinatorics held in the Israel Institute of Advanced Studies.  ... 
arXiv:1812.11558v3 fatcat:emokkoiqozdk7nw7urv6xzae5m

Point Spectrum of Periodic Operators on Universal Covering Trees [article]

Jess Banks, Jorge Garza-Vargas, Satyaki Mukherjee
2020 arXiv   pre-print
For any multi-graph G with edge weights and vertex potential, and its universal covering tree 𝒯, we completely characterize the point spectrum of operators A_𝒯 on 𝒯 arising as pull-backs of local, self-adjoint  ...  This builds on work of Aomoto, and includes an alternative proof of the necessary condition for point spectrum he derived in (Aomoto, 1991).  ...  We also thank Barry Simon for his many helpful comments and suggestions on a previous version of this manuscript, and in particular for his guidance in showing the current version of eorem 3.4.  ... 
arXiv:2008.03318v2 fatcat:n5a766z7r5g6zhvd6oy7ho4oia

Entropy of eigenfunctions on quantum graphs [article]

Lionel Kameni, Roman Schubert
2014 arXiv   pre-print
The optimal bounds are obtained for expanders with large girth, the bounds are similar to the ones obtained by Anantharaman for eigenfunctions on manifolds of negative curvature, and are based on  ...  on a small set on the graph.  ...  Acknowledgements: This work was carried out using the computational facilities of the Advanced Computing Research Centre at the University of Bristol.  ... 
arXiv:1405.5871v1 fatcat:qtn6c3gczfd3tfyppjpxkc3ura

Optimal network topologies: expanders, cages, Ramanujan graphs, entangled networks and all that

Luca Donetti, Franco Neri, Miguel A Muñoz
2006 Journal of Statistical Mechanics: Theory and Experiment  
First we review some basic concepts on spectral graph theory, including adjacency and Laplacian matrices, and paying special attention to the topological implications of having large spectral gaps.  ...  We report on some recent developments in the search for optimal network topologies.  ...  We are especially thankful to P I Hurtado, our co-author in the paper where entangled networks were first introduced, for a very enjoyable collaboration and a critical reading of the manuscript.  ... 
doi:10.1088/1742-5468/2006/08/p08007 fatcat:qgkw5fy545fv5jrees7ht2k4mq

Spectrum preserving short cycle removal on regular graphs [article]

Pedro Paredes
2020 arXiv   pre-print
Using this method we can show two results involving high girth spectral expander graphs.  ...  First, we show that given d ≥ 3 and n, there exists an explicit distribution of d-regular Θ(n)-vertex graphs where with high probability its samples have girth Ω(log_d - 1 n) and are ϵ-near-Ramanujan;  ...  Acknowledgments I am very grateful to Ryan O'Donnell for numerous comments and suggestions, as well as very thorough feedback on an earlier draft of this paper.  ... 
arXiv:2002.07211v1 fatcat:gdqgpwop3fh43norf73zqyu2ee

Network Modularity is essential for evolution of cooperation under uncertainty

David A. Gianetto, Babak Heydari
2015 Scientific Reports  
This work demonstrates that community structure, represented by network modularity, among all the tested structural parameters, has the most significant impact on the emergence of cooperative behavior,  ...  We also show that increased community structure reduces the dispersion of trust and forgiveness, thereby reducing the network-level uncertainties for these two components; graph transitivity and degree  ...  Kia Dalili for his helpful suggestions that improved the convergence of the evolutionary model.  ... 
doi:10.1038/srep09340 pmid:25849737 pmcid:PMC4388161 fatcat:pbzwcrwwmzb2hpzbwsnqjt6qia

An Alon-Boppana theorem for powered graphs and generalized Ramanujan graphs [article]

Emmanuel Abbe, Peter Ralli
2020 arXiv   pre-print
In particular, we show that certain graphs that are not good expanders due to local irregularities, such as Erdos-Renyi random graphs, become almost Ramanujan once powered.  ...  This paper gives a generalization of the Alon-Boppana Theorem for the r-th power of graphs, including irregular graphs.  ...  More specifically, Erdős-Rényi (ER) random graphs with an expected degree d will have their top two eigenvalues of order log(n)/ log log(n), due to eigenvectors localized on high-degree nodes, and therefore  ... 
arXiv:2006.11248v1 fatcat:o2kwrj5cbvhvpjwuyewoqfbb5m

A factor-graph-based random walk, and its relevance for LP decoding analysis and Bethe entropy characterization

Pascal O. Vontobel
2010 2010 Information Theory and Applications Workshop (ITA)  
of constraint function nodes, and with no restrictions on the girth.  ...  Moreover, formulating our results in terms of normal factor graphs will facilitate the generalization of the geometrical results of the ADS paper to setups with non-uniform node degrees, with other types  ...  absolute certainty or with high probability.)  ... 
doi:10.1109/ita.2010.5454077 dblp:conf/ita/Vontobel10 fatcat:yohl3lf6mfcg5ivzh56qow62qy

Invariant Gaussian processes and independent sets on regular graphs of large girth

Endre Csóka, Balázs Gerencsér, Viktor Harangi, Bálint Virág
2014 Random structures & algorithms (Print)  
We prove that every 3-regular, n-vertex simple graph with sufficiently large girth contains an independent set of size at least 0.4361n. (The best known bound is 0.4352n.)  ...  Our method uses invariant Gaussian processes on the d-regular tree that satisfy the eigenvector equation at each vertex for a certain eigenvalue λ.  ...  Instead of working on finite graphs with large girth, it will be more convenient for us to consider the regular (infinite) tree and look for independent sets on this tree that are i.i.d. factors.  ... 
doi:10.1002/rsa.20547 fatcat:vbr7erpnxjdy5bhdfvguy23pye
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