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Constructing expander graphs by 2-lifts and discrepancy vs. spectral gap [article]

Yonatan Bilu, Nathan Linial
2004 arXiv   pre-print
We present a new explicit construction for expander graphs with nearly optimal spectral gap. The construction is based on a series of 2-lift operations. Let G be a graph on n vertices.  ...  A 2-lift of G is a graph H on 2n vertices, with a covering map π:H → G. It is not hard to see that all eigenvalues of G are also eigenvalues of H. In addition, H has n "new" eigenvalues.  ...  We thank Eran Ofek for suggesting that Corollary 5.1 might be used to bound the second eigenvalue of random d-regular graphs, and Avi Wigderson for Lemma 4.3.  ... 
arXiv:math/0312022v3 fatcat:iwd7mobg6ve5nbr5vnimcfdexe

Lifts, Discrepancy and Nearly Optimal Spectral Gap*

Yonatan Bilu, Nathan Linial
2006 Combinatorica  
We present a new explicit construction for expander graphs with nearly optimal spectral gap. The construction is based on a series of 2-lift operations. Let G be a graph on n vertices.  ...  A 2-lift of G is a graph H on 2n vertices, with a covering map π : H → G. It is not hard to see that all eigenvalues of G are also eigenvalues of H. In addition, H has n "new" eigenvalues.  ...  An explicit construction of Quasi-Ramanujan graphs For the purpose of constructing expanders, it is enough to prove a weaker version of Theorem 3.1 -that every expander graph has a 2-lift with small spectral  ... 
doi:10.1007/s00493-006-0029-7 fatcat:jokn73oimvfm5on5ycewvow4si

Explicit Abelian Lifts and Quantum LDPC Codes [article]

Fernando Granha Jeronimo, Tushant Mittal, Ryan O'Donnell, Pedro Paredes, Madhur Tulsiani
2021 arXiv   pre-print
For every (transitive) abelian group H ⩽Sym(ℓ), constant degree d ≥ 3 and ϵ > 0, we construct explicit d-regular expander graphs G obtained from an (H,ℓ)-lift of a (suitable) base n-vertex expander G_0  ...  In this work, we show the following explicit constructions of expanders obtained via abelian lifts.  ...  lifted graph is also expanding.  ... 
arXiv:2112.01647v1 fatcat:4wevb2ggnnaspknxyuvrauvnke

Xpander

Asaf Valadarsky, Michael Dinitz, Michael Schapira
2015 Proceedings of the 14th ACM Workshop on Hot Topics in Networks - HotNets-XIV  
We show, through a combination of theoretical analyses, extensive simulations, and experiments with a network emulator, that any "expander" network topology (as indeed are random graphs) comes with these  ...  show that datacenter designs with random network topologies outperform more sophisticated designs, achieving near-optimal throughput and bisection bandwidth, high resiliency to failures, incremental expandability  ...  This work was supported by NSF awards CCF-1464239 and CCF-1535887, ISF grant 420/12, the Israel Ministry of Science, the Israeli Center for Research Excellence in Algorithms, and a MSR PhD Scholarship.  ... 
doi:10.1145/2834050.2834059 dblp:conf/hotnets/ValadarskyDS15 fatcat:ey2743h75zavfpxs6eczel2tzi

Locally testable codes via high-dimensional expanders [article]

Yotam Dikstein, Irit Dinur, Prahladh Harsha, Noga Ron-Zewi
2020 arXiv   pre-print
This work unifies and generalizes the known results on testability of the Hadamard, Reed-Muller and lifted codes on the Subspace Complex, all of which are proved via local self correction.  ...  Informally, our result states that if this graph is part of a high-dimensional expander then the local testability of the code follows from the local testability of the base code.  ...  This is analogous to the celebrated expander codes [SS96] in which distance of the base codes gets lifted to distance of the entire code, assuming expansion of the underlying Tanner graph.  ... 
arXiv:2005.01045v1 fatcat:yda5uvuejvhzjlajluvwrc2rz4

Non-expander Cayley Graphs of Simple Groups

Gábor Somlai
2015 Communications in Algebra  
We construct a sequence of groups G n , and explicit sets of generators Y n ⊂ G n , such that all generating sets have bounded size, and the associated Cayley graphs are all expanders.  ...  We prove that direct products of such groups are expanding even with highly correlated tuples of generators.  ...  Acknowledgments We are grateful to Alex Lubotzky for many insightful conversations, in part supplying the group theoretic tools we ended up needing for the proof.  ... 
doi:10.1080/00927872.2013.865041 fatcat:yzundq3l7fgxhhr3bfgyjitp4a

Iterative Construction of Cayley Expander Graphs

Eyal Rozenman, Aner Shalev, Avi Wigderson
2006 Theory of Computing  
We construct a sequence of groups G n , and explicit sets of generators Y n ⊂ G n , such that all generating sets have bounded size, and the associated Cayley graphs are all expanders.  ...  We prove that direct products of such groups are expanding even with highly correlated tuples of generators.  ...  Acknowledgments We are grateful to Alex Lubotzky for many insightful conversations, in part supplying the group theoretic tools we ended up needing for the proof.  ... 
doi:10.4086/toc.2006.v002a005 dblp:journals/toc/RozenmanSW06 fatcat:wzjdosx6onf6zj3h4sbtduext4

Explicit Expanding Expanders [article]

Michael Dinitz, Michael Schapira, Asaf Valadarsky
2015 arXiv   pre-print
We present an explicit construction of d-regular expanders with expansion cost at most 5d/2, for any d≥ 6. Our construction leverages the notion of a "2-lift" of a graph.  ...  Deterministic constructions of expander graphs have been an important topic of research in computer science and mathematics, with many well-studied constructions of infinite families of expanders.  ...  While d-regular random graphs are, in fact, very good expanders [4, 5] , many applications require explicit, deterministic constructions of expanders. 1 Consequently, a rich body of literature in graph  ... 
arXiv:1507.01196v1 fatcat:v24pao2wincpnppgpuoyylphdu

Asymptotically Good Quantum and Locally Testable Classical LDPC Codes [article]

Pavel Panteleev, Gleb Kalachev
2022 arXiv   pre-print
We study classical and quantum LDPC codes of constant rate obtained by the lifted product construction over non-abelian groups.  ...  We show that the obtained families of quantum LDPC codes are asymptotically good, which proves the qLDPC conjecture.  ...  This work was supported by the Ministry of Science and Higher Education of the Russian Federation (Grant 075-15-2020-801).  ... 
arXiv:2111.03654v2 fatcat:mvotjdxcfvf4ze2rymlzq424ii

Cryptographic Hash Functions and Expander Graphs: The End of the Story? [chapter]

Christophe Petit, Jean-Jacques Quisquater
2016 Lecture Notes in Computer Science  
Lifting attacks : LPS LPS : G = PSL(2, F p ) and S as in LPS Ramanujan graphs Lift from PSL(2, F p ) to SL(2, Z[i]) Here lifts of generators SL(2, Z[i]) Very small subset, but well structured [LPS88]  ...  ⊃ G N = {1} If |G i |/|G i+1 | small or if DLP easy in |G i |/|G i+1 | ⇒ preimages of 1 ⇒ 2nd preimage attack Subgroup attacks Assume G = G 0 ⊃ G 1 ⊃ G 2 ... ⊃ G N = {1} If |G i |/|G i+1 | small or  ... 
doi:10.1007/978-3-662-49301-4_19 fatcat:xupxke37nbgjjldjkssvu5umui

Simplified and Improved Separations Between Regular and General Resolution by Lifting [chapter]

Marc Vinyals, Jan Elffers, Jan Johannsen, Jakob Nordström
2020 Lecture Notes in Computer Science  
We give a significantly simplified proof of the exponential separation between regular and general resolution of Alekhnovich et al. (2007) as a consequence of a general theorem lifting proof depth to regular  ...  This simpler proof then allows us to strengthen the separation further, and to construct families of theoretically very easy benchmarks that are surprisingly hard for SAT solvers in practice.  ...  We are most grateful to Robert Robere for the interesting discussions that served as the starting point for this project.  ... 
doi:10.1007/978-3-030-51825-7_14 fatcat:zyk7tftjajbijm5tins3wwo2iq

Distributed Averaging Via Lifted Markov Chains

Kyomin Jung, Devavrat Shah, Jinwoo Shin
2010 IEEE Transactions on Information Theory  
We call this novel transformation pseudo-lifting. We apply our method to graphs with geometry, or graphs with doubling dimension.  ...  Specifically, the convergence time of our algorithm (equivalently, the mixing time of our Markov chain) is proportional to the diameter of the network graph and hence optimal.  ...  and are far from being expanders.  ... 
doi:10.1109/tit.2009.2034777 fatcat:s46nkx2vdnhxzey2qeebmemjvq

Stretch 'n' cut: Method for observing and ungrouping complex virtual objects in 3D space using elastic band metaphor

Mai Otsuki, Asako Kimura, Fumihisa Shibata, Hideyuki Tamura
2014 2014 IEEE Symposium on 3D User Interfaces (3DUI)  
We propose a system which realizes gesture-based separation, and the observation of a group of parts from complex virtual objects in 3D space.  ...  One practical application of our system is for training purposes such as learning the structures of complex 3D object such as the human body or industrial products.  ...  PROPOSED METHOD 3.1 Selecting The PartsThat Are Shown As Expanded Parts In this study, the contact relationships of each part are used as a graph ( Figure 2 ).  ... 
doi:10.1109/3dui.2014.6798836 dblp:conf/3dui/OtsukiKST14 fatcat:wd6ytqapmbblhadx2ej34d4mx4

On designing heteroclinic networks from graphs

Peter Ashwin, Claire Postlethwaite
2013 Physica D : Non-linear phenomena  
As these networks are of great interest as dynamical models of biological and cognitive processes, it is useful to understand how particular graphs can be realised as robust heteroclinic networks that  ...  are attracting.  ...  Acknowledgments This work was started during discussions at the Mathematical Biosciences Institute (MBI) in Ohio in 2011 and continued during a visit of CMP to Exeter.  ... 
doi:10.1016/j.physd.2013.09.006 fatcat:7ybzwnpu5ra2vovlcmd4pq4t3a

Distributed Averaging via Lifted Markov Chains [article]

Kyomin Jung, Devavrat Shah, Jinwoo Shin
2009 arXiv   pre-print
We call this novel transformation pseudo-lifting. We apply our method to graphs with geometry, or graphs with doubling dimension.  ...  Specifically, the convergence time of our algorithm (equivalently, the mixing time of our Markov chain) is proportional to the diameter of the network graph and hence optimal.  ...  Of course, this will complicate the mixing time analysis drastically. This is where our method developed in the expander-based lifting will be readily useful.  ... 
arXiv:0908.4073v1 fatcat:qme34amqjbgmdnx6zckgp66q64
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