On the Expansion of Group-Based Lifts

Abstract

A k-lift of an n-vertex base graph G is a graph H on n × k vertices, where each vertex v of G is replaced by k vertices v 1 , · · · , v k and each edge (u, v) in G is replaced by a matching representing a bijection π uv so that the edges of H are of the form (u i , v πuv(i) ). Lifts have been studied as a means to efficiently construct expanders. In this work, we study lifts obtained from groups and group actions. We derive the spectrum of such lifts via the representation theory principles of the underlying group. Our main results are: * Expander graphs have spawned research in pure and applied mathematics during the last several years, with several applications to multiple fields including complexity theory, the design of robust computer networks, the design of error-correcting codes, de-randomization of randomized algorithms, compressed sensing and the study of metric embeddings. For a comprehensive survey of expander graphs see [Sar06,HLW06].Informally, an expander is a graph where every small subset of the vertices has a relatively large edge boundary. Most applications are concerned with sparse d-regular graphs G, where the largest eigenvalue of the adjacency matrix A G is d. In case of a bipartite graph, the largest and smallest eigenvalues of A G are d and −d, which are referred to as trivial eigenvalues. The expansion of the graph is related to the difference between d and λ, the first largest (in absolute value) non-trivial eigenvalue of A G . Roughly, the smaller λ is, the better the graph expansion. The Alon-Boppana bound ([Nil91]) states that λ ≥ 2 √ d − 1 − o(1) for non-bipartite graphs, thus graphs with λ ≤ 2 √ d − 1 are optimal expanders and are called Ramanujan. A simple probabilistic argument can show the existence of infinite families of expander graphs [Pin73]. However, constructing such infinite families explicitly has proven to be a challenging and important task. It is easy to construct Ramanujan graphs with a small number of vertices: d-regular complete graphs and complete bipartite graphs are Ramanujan. The challenge is to construct an infinite family of d-regular graphs that are all Ramanujan, which was first achieved by Lubotzky, Phillips and Sarnak [LPS88] and Margulis [Mar88]. They built Ramanujan graphs from Cayley graphs. All of their graphs are regular, have degrees p + 1 where p is a prime, and their proofs rely on deep number theoretic facts. In two recent breakthrough papers, Marcus, Spielman, and Srivastava showed the existence of bipartite Ramanujan graphs of all degrees [MSS13, MSS15]. However their results do not provide an efficient algorithm to construct those graphs. A striking result of Friedman [Fri08] and a slightly weaker but more general result of Puder [Pud13], shows that almost every d-regular graph on n vertices is very close to being Ramanujan i.e. for every ǫ > 0, asymptotically almost surely, λ < 2 √ d − 1 + ǫ. It is still unknown whether the event that a random dregular graph is exactly Ramanujan happens with constant probability. Despite the large body of wo...

Asymptotically Good Quantum and Locally Testable Classical LDPC Codes

“…However, the situation is not that bad if we apply a large shift ℓ-lift only once. As it was shown in Theorem 1.2 from [42], if the base graph Γ has good spectral expansion properties, then by using random shifts the obtained graph Γ also has good expansion properties, even when the lift size ℓ is very large. In [13], such graphs Γ were used to construct quasi-cyclic expander codes of very large lift size ℓ such that the corresponding parity-check matrix H and its transpose H * have good expansion properties.…”

Covers, Orientations and Factors

“…Csikvári [6] combined graph covers with graph limit theory to prove the so-called Lower Matching Conjecture of Friedland, Krop and Markström [7]. The properties of random lifts are also widely studied, see for instance the papers [1] and [8].…”

Explicit Abelian Lifts and Quantum LDPC Codes

“…When one restricts H to be abelian, Agarwal et al [ACKM19] showed that random (Z ℓ , ℓ)-lifts (also known as shift lifts) are expanding. Motivated by the applications of these lifts to codes, we obtain explicit constructions of expanding abelian lifts, for a wide range of lift sizes.…”

“…Panteleev and Kalachev use the aforementioned randomized construction of abelian lifted expanders by Agarwal et al [ACKM19], where each edge of the base graph is a associated with an element in Z ℓ sampled uniformly. When ℓ is in the exponential regime they obtain quantum LDPC codes with almost linear distance, i.e., Ω(N/ log(N)).…”

“…In this section, we give a simpler proof of a weaker result similar to one in [ACKM19] which says that if the lifts were picked independently and uniformly at random, then the lifted graph is also expanding. In place of the random signings, we will use the signings generated from random walks as described in the earlier section.…”