Zig-zag Product Research Articles

Permutational Powers of a Graph

This paper introduces a new graph construction, the permutational power of a graph, whose adjacency matrix is obtained by the composition of a permutation matrix with the adjacency matrix of the graph. It is shown that this construction recovers the classical zig-zag product of graphs when the permutation is an involution, and it is in fact more general. We start by discussing necessary and sufficient conditions on the permutation and on the adjacency matrix of a graph to guarantee their composition to represent an adjacency matrix of a graph, then we focus our attention on the cases in which the permutational power does not reduce to a zig-zag product. We show that the cases of interest are those in which the adjacency matrix is singular. This leads us to frame our problem in the context of equitable partitions, obtained by identifying vertices having the same neighborhood. The families of cyclic and complete bipartite graphs are treated in details.

Mini-Workshop: Superexpanders and Their Coarse Geometry

It is a deep open problem whether all expanders are superexpanders. In fact, it was already a major challenge to prove the mere existence of superexpanders. However, by now, some classes of examples are known: Lafforgue’s expanders constructed as sequences of finite quotients of groups with strong Banach property (T), the examples coming from zigzag products due to Mendel and Naor, and the recent examples coming from group actions on compact manifolds. The methods which are used to construct superexpanders are typically functional analytic in nature, but also rely on arguments from geometry and combinatorics. Another important aspect of the study of superexpanders is their (coarse) geometry, in particular in order to distinguish them from each other. The aim of this workshop was to bring together researchers working on superexpanders and their coarse geometry from different perspectives, with the aim of sharing expertise and stimulating new research.

Zeta functions of finite Schreier graphs and their zig-zag products

We investigate the Ihara zeta functions of finite Schreier graphs [Formula: see text] of the Basilica group. We show that [Formula: see text] is two sheeted unramified normal covering of [Formula: see text], for all [Formula: see text] with Galois group [Formula: see text] In fact, for any [Formula: see text], the graph [Formula: see text] is [Formula: see text] sheeted unramified, non-normal covering of [Formula: see text]. In order to do this, we give the definition of the [Formula: see text] [Formula: see text] [Formula: see text] of Schreier graphs of Basilica groups. We also show the corresponding results in zig-zag product of Schreier graphs [Formula: see text] with a [Formula: see text]-cycle.

Symmetric LDPC codes and local testing

Coding theoretic and complexity theoretic considerations naturally lead to the question of generating symmetric, sparse, redundant linear systems. This paper provides a new way of construction with better parameters and new lower bounds. Low Density Parity Check (LDPC) codes are linear codes defined by short constraints (a property essential for local testing of a code). Some of the best (theoretically and practically) used codes are LDPC. Symmetric codes are those in which all coordinates look the same, namely there is some transitive group acting on the coordinates which preserves the code. Some of the most commonly used locally testable codes (especially in PCPs and other proof systems), including all low-degree codes, are symmetric. Requiring that a symmetric binary code of length n has large (linear or near-linear) distance seems to suggest a con ict between 1/rate and density (constraint length). In known constructions, if one is constant, then the other is almost the worst possible - n/poly(logn). Our main positive result simultaneously achieves symmetric low density, constant rate codes generated by a single constraint. We present an explicit construction of a symmetric and transitive binary code of length n, near-linear distance n/(log logn)2, of constant rate and with constraints of length (logn)4. The construction is in the spirit of Tanner codes, namely the codewords are indexed by the edges of a sparse regular expander graph. The main novelty is in our construction of a transitive (non Abelian!) group acting on these edges which preserves the code. Our construction is one instantiation of a framework we call Cayley Codes developed here, that may be viewed as extending zig-zag product to symmetric codes. Our main negative result is that the parameters obtained above cannot be significantly improved, as long as the acting group is solvable (like the one we use). More specifically, we show that in constant rate and linear distance codes (aka good codes) invariant under solvable groups, the density (length of generating constraints) cannot go down to a constant, and is bounded below by (log(Ω(l))n)(an Ω(l) iterated logarithm) if the group has a derived series of length l. This negative result precludes natural local tests with constantly many queries for such solvable good codes.

Self-similar groups and the zig-zag and replacement products of graphs

Every finitely generated self-similar group naturally produces an infinite sequence of finite d-regular graphs Γn. We construct self-similar groups, whose graphs Γn can be represented as an iterated zig-zag product and graph powering: Γn+1=Γnkz⃝Γ (k≥1). We also construct self-similar groups, whose graphs Γn can be represented as an iterated replacement product and graph powering: Γn+1=Γnkr⃝Γ (k≥1). This gives simple explicit examples of self-similar groups, whose graphs Γn form an expanding family, and examples of automaton groups, whose graphs Γn have linear diameters diam(Γn)=O(n) and bounded girth.

Constraints, MMSNP and expander relational structures

We give a poly-time construction for a combinatorial classic known as Sparse Incomparability Lemma, studied by Erdős, Lovász, Nešetřil, Rödl and others: We show that every Constraint Satisfaction Problem is poly-time equivalent to its restriction to structures with large girth. This implies that the complexity classes CSP and Monotone Monadic Strict NP introduced by Feder and Vardi are computationally equivalent. The technical novelty of the paper is a concept of expander relations and a new type of product for relational structures: a generalization of the zig-zag product, the twisted product.

Nonlinear spectral calculus and super-expanders

Nonlinear spectral gaps with respect to uniformly convex normed spaces are shown to satisfy a spectral calculus inequality that establishes their decay along Cesaro averages. Nonlinear spectral gaps of graphs are also shown to behave sub-multiplicatively under zigzag products. These results yield a combinatorial construction of super-expanders, i.e., a sequence of 3-regular graphs that does not admit a coarse embedding into any uniformly convex normed space.

A key-distribution mechanism for wireless sensor networks using Zig-Zag product

Secure communication in Wireless Sensor Networks (WSNs) necessitates utilisation of efficient key distribution mechanisms. Several works using probabilistic, deterministic, and hybrid methods have been conducted to address key distribution among nodes. In this paper, we propose a method to distribute key–chains utilising expander graphs based on the Zig–Zag product. Given a set of constraints such as network size, storage, and radio range, we are able to construct a resilient key distribution graph. The main advantage of the obtained method is attaining a shorter key–path length compared with previously proposed approaches. Simulation results demonstrate the efficiency of the proposed method.

A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag Product

Reingold, Vadhan, and Wigderson [Ann. of Math. (2), 155 (2002), pp. 157–187] introduced the graph zig-zag product. This product combines a large and a small graph into one, such that the resulting graph inherits its size from the large graph, its degree from the small graph, and its spectral gap from both. Using this product, they gave a fully explicit combinatorial construction of D-regular graphs having spectral gap $1-O(D^{-\frac{1}{3}})$. In the same paper, they posed the open problem of whether a similar graph product could be used to achieve the almost optimal spectral gap $1-O(D^{-\frac{1}{2}})$. In this paper we propose a generalization of the zig-zag product that combines a large graph and several small graphs. The new product gives a better relation between the degree and the spectral gap of the resulting graph. We use the new product to give a fully explicit combinatorial construction of D-regular graphs having spectral gap $1-D^{-\frac{1}{2}+o(1)}$.

ON CONSTRUCTION OF ALMOST-RAMANUJAN GRAPHS

O. Reingold et al. introduced the notion zig-zag product on two different graphs, and presented a fully explicit construction of d-regular expanders with the second largest eigenvalue O(d-1/3). In the same paper, they ask whether or not the similar technique can be used to construct expanders with the second largest eigenvalue O(d-1/2). Such graphs are called Ramanujan graphs. Recently, zig-zag product has been generalized by A. Ben-Aroya and A. Ta-Shma. Using this technique, they present a family of expanders with the second largest eigenvalue d-1/2 + o(1), what they call almost-Ramanujan graphs. However, their construction relies on local invertible functions and the dependence between the big graph and several small graphs, which makes the construction more complicated. In this paper, we shall give a generalized theorem of zig-zag product. Specifically, the zig-zag product of one "big" graph and several "small" graphs with the same size will be formalized. By choosing the big graph and several small graphs individually, we shall present a family of fully explicitly almost-Ramanujan graphs with locally invertible function waived.

Zig-zag and replacement product graphs and LDPC codes

It is known that the expansion property of a graph influences the performance of the corresponding code when decoded using iterative algorithms. Certain graph products may be used to obtain larger expander graphs from smaller ones. In particular, the zig-zag product and replacement product may be used to construct infinite families of constant degree expander graphs. This paper investigates the use of zig-zag and replacement product graphs for the construction of codes on graphs. A modification of the zig-zag product is also introduced, which can operate on two unbalanced biregular bipartite graphs, and a proof of the expansion property of this modified zig-zag product is presented.

Entropy Waves, the Zig-Zag Graph Product, and New Constant-Degree Expanders

The main contribution of this work is a new type of graph product, which we call the zig-zag product. Taking a product of a large graph with a small graph, the resulting graph inherits (roughly) its size from the large one, its degree from the small one, and its expansion properties from both! Iteration yields simple explicit constructions of constant-degree expanders of arbitrary size, starting from one constant-size expander. Crucial to our intuition (and simple analysis) of the properties of this graph product is the view of expanders as functions which act as wave propagators -they transform probability distributions in which entropy is concentrated in one area to distributions where that concentration is dissipated. In these terms, the graph product affords the constructive interference of two such waves.