Expander Graphs Research Articles

Turán Problems and Shadows III: Expansions of Graphs

The expansion $G^+$ of a graph $G$ is the 3-uniform hypergraph obtained from $G$ by enlarging each edge of $G$ with a new vertex disjoint from $V(G)$ such that distinct edges are enlarged by distinct vertices. Let ${ex}_3(n,F)$ denote the maximum number of edges in a 3-uniform hypergraph with $n$ vertices not containing any copy of a 3-uniform hypergraph $F$. The study of ${ex}_3(n,G^+)$ includes some well-researched problems, including the case that $F$ consists of $k$ disjoint edges, $G$ is a triangle, $G$ is a path or cycle, and $G$ is a tree. In this paper we initiate a broader study of the behavior of ${ex}_3(n,G^+)$. Specifically, we show $ {ex}_3(n,K_{s,t}^+) = \Theta(n^{3 - 3/s})$ whenever $t > (s - 1)!$ and $s \geq 3$. One of the main open problems is to determine for which graphs $G$ the quantity ${ex}_3(n,G^+)$ is quadratic in $n$. We show that this occurs when $G$ is any bipartite graph with Turán number $o(n^{\varphi})$ where $\varphi = \frac{1 + \sqrt{5}}{2}$, and in particular this shows ${ex}_3(n,G^+) = O(n^2)$ when $G$ is the three-dimensional cube graph.

Making the Long Code Shorter

The long code is a central tool in hardness of approximation, especially in questions related to the Unique Games Conjecture. We construct a new code that is exponentially more efficient, but can still be used in many of these applications. Using the new code we obtain exponential improvements over several known results, including the following: (1) For any $\varepsilon>0$, we show the existence of an $n$-vertex graph $G$ where every set of $o(n)$ vertices has expansion $1-\varepsilon$, but $G$'s adjacency matrix has more than $\exp(\log^{\delta}n)$ eigenvalues larger than $1-\varepsilon$, where $\delta$ depends only on $\varepsilon$. This answers an open question of Arora, Barak, and Steurer [Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, 2010, pp. 563--572], who asked whether one can improve over the noise graph on the Boolean hypercube that has ${\rm poly}(\log n)$ such eigenvalues. (2) A gadget that reduces Unique Games instances with linear constraints modulo $K$ into instances with alphabet $k$ with a blowup of $k^{{\rm polylog}(K)}$, improving over the previously known gadget with blowup of $k^{\Omega(K)}$. (3) An $n$-variable integrality gap for Unique Games that survives $\exp({\rm poly}(\log\log n))$ rounds of the semidefinite programming version of the Sherali--Adams hierarchy, improving on the previously known bound of ${\rm poly}(\log\log n)$. We show a connection between the local testability of linear codes and Small-Set Expansion in certain related Cayley graphs and use this connection to derandomize the noise graph on the Boolean hypercube.

Data Gathering with Compressive Sensing in Wireless Sensor Networks: A Random Walk Based Approach

In this paper, we study the problem of data gathering with compressive sensing (CS) in wireless sensor networks (WSNs). Unlike the conventional approaches, which require uniform sampling in the traditional CS theory, we propose a random walk algorithm for data gathering in WSNs. However, such an approach will conform to path constraints in networks and result in the non-uniform selection of measurements. It is still unknown whether such a non-uniform method can be used for CS to recover sparse signals in WSNs. In this paper, from the perspectives of CS theory and graph theory, we provide mathematical foundations to allow random measurements to be collected in a random walk based manner. We find that the random matrix constructed from our random walk algorithm can satisfy the expansion property of expander graphs. The theoretical analysis shows that a k-sparse signal can be recovered using `1 minimization decoding algorithm when it takes m = O(k log(n=k)) independent random walks with the length of each walk t = O(n=k) in a random geometric network with n nodes. We also carry out simulations to demonstrate the effectiveness of the proposed scheme. Simulation results show that our proposed scheme can significantly reduce communication cost compared to the conventional schemes using dense random projections and sparse random projections, indicating that our scheme can be a more practical alternative for data gathering applications in WSNs.

The calculation of expectation values in Gaussian random tensor theory via meanders

A difficult problem in the theory of random tensors is to calculate the expectation values of polynomials in the tensor entries, even in the large N limit and in a Gaussian distribution. Here we address this challenge for tensors of rank 4, focusing on a family of polynomials labeled by permutations, which generalize in a precise sense the single-trace invariants of random matrix models. rough Wick’s theorem, we show that the Feynman graph expansion of the expectation values of those polynomials enumerates meandric systems whose lower arch conguration is obtained from the upper arch conguration by a permutation on half of the arch feet. Our main theorem reduces the calculation of expectation values to those of polynomials labeled by stabilized-interval-free permutations (SIF) which are proved to enumerate irreducible meandric systems. is together with explicit calculations of expectation values associated to SIF permutations allows to exactly evaluate large N expectation values beyond the so-called melonic polynomials for the rst time.

Local correctability of expander codes

In this work, we present the first local-decoding algorithm for expander codes. This yields a new family of constant-rate codes that can recover from a constant fraction of errors in the codeword symbols, and where any symbol of the codeword can be recovered with high probability by reading Nε symbols from the corrupted codeword, where N is the block-length of the code.Expander codes, introduced by Sipser and Spielman, are formed from an expander graph G=(V,E) of degree d, and an inner code of block-length d over an alphabet Σ. Each edge of the expander graph is associated with a symbol in Σ. A string in ΣE will be a codeword if for each vertex in V, the symbols on the adjacent edges form a codeword in the inner code.We show that if the inner code has a smooth reconstruction algorithm in the noiseless setting, then the corresponding expander code has an efficient local-correction algorithm in the noisy setting. Instantiating our construction with inner codes based on finite geometries, we obtain novel locally decodable codes with rate approaching one. This provides an alternative to the multiplicity codes of Kopparty, Saraf and Yekhanin (STOC '11) and the lifted codes of Guo, Kopparty and Sudan (ITCS '13).

Expansion of random graphs: new proofs, new results

We present a new approach to showing that random graphs are nearly optimal expanders. This approach is based on recent deep results in combinatorial group theory. It applies to both regular and irregular random graphs. Let G be a random d-regular graph on n vertices, and let \lambda be the largest absolute value of a non-trivial eigenvalue of its adjacency matrix. It was conjectured by Alon [86'] that a random d-regular graph is almost Ramanujan, in the following sense: for every e>0, \lambda<2\sqrt{d-1} + e asymptotically almost surely. Friedman famously presented a proof of this conjecture in [08']. Here we suggest a new, substantially simpler proof of a nearly-optimal result: we show that a random d-regular graph satisfies \lambda < 2\sqrt{d-1} + 1 a.a.s. A main advantage of our approach is that it is applicable to a generalized conjecture: For d even, a d-regular graph on n vertices is an n-covering space of a bouquet of d/2 loops. More generally, fixing an arbitrary base graph H, we study the spectrum of G, a random n-covering of H. Let \lambda be the largest absolute value of a non-trivial eigenvalue of G. Extending Alon's conjecture to this more general model, Friedman [03'] conjectured that for every e>0, a.a.s. \lambda < \rho+e, where \rho is the spectral radius of the universal cover of H. When H is regular we get a bound of \rho+0.84, and for an arbitrary H, we prove a nearly optimal upper bound of \sqrt{3}\rho. This is a substantial improvement upon all known results (by Friedman, Linial-Puder, Lubetzky-Sudakov-Vu and Addario-Berry-Griffiths).

The commuting local Hamiltonian problem on locally expanding graphs is approximable in $$\mathsf{{NP}}$$ NP

The local Hamiltonian problem is famously complete for the class $$\mathsf{{QMA}}$$QMA, the quantum analogue of $$\mathsf{{NP}}$$NP. The complexity of its semiclassical version, in which the terms of the Hamiltonian are required to commute (the $$\mathsf{{CLH}}$$CLH problem), has attracted considerable attention recently due to its intriguing nature, as well as in relation to growing interest in the $$\mathsf{{qPCP}}$$qPCP conjecture. We show here that if the underlying bipartite interaction graph of the $$\mathsf{{CLH}}$$CLH instance is a good locally expanding graph, namely the expansion of any constant-size set is $$\varepsilon $$?-close to optimal, then approximating its ground energy to within additive factor $$O(\varepsilon )$$O(?) lies in $$\mathsf{{NP}}$$NP. The proof holds for $$k$$k-local Hamiltonians for any constant $$k$$k and any constant dimensionality of particles $$d$$d. We also show that the approximation problem of $$\mathsf{{CLH}}$$CLH on such good local expanders is $$\mathsf{{NP}}$$NP-hard. This implies that too good local expansion of the interaction graph constitutes an obstacle against quantum hardness of the approximation problem, though it retains its classical hardness. The result highlights new difficulties in trying to mimic classical proofs (in particular, Dinur's $$\mathsf{{PCP}}$$PCP proof) in an attempt to prove the quantum $$\mathsf{{PCP}}$$PCP conjecture. A related result was discovered recently independently by Brandao and Harrow, for $$2$$2-local general Hamiltonians, bounding the quantum hardness of the approximation problem on good expanders, though no $$\mathsf{{NP}}$$NP hardness is known in that case.

Groups graded by root systems and property (T).

We establish property (T) for a large class of groups graded by root systems, including elementary Chevalley groups and Steinberg groups of rank at least 2 over finitely generated commutative rings with 1. We also construct a group with property (T) which surjects onto all finite simple groups of Lie type and rank at least two.

Logarithmically small minors and topological minors

Mader proved that for every integer $t$ there is a smallest real number $c(t)$ such that any graph with average degree at least $c(t)$ must contain a $K_t$-minor. Fiorini, Joret, Theis and Wood conjectured that any graph with $n$ vertices and average degree at least $c(t)+\epsilon$ must contain a $K_t$-minor consisting of at most $C(\epsilon,t)\log n$ vertices. Shapira and Sudakov subsequently proved that such a graph contains a $K_t$-minor consisting of at most $C(\epsilon,t)\log n \log\log n$ vertices. Here we build on their method using graph expansion to remove the $\log\log n$ factor and prove the conjecture. Mader also proved that for every integer $t$ there is a smallest real number $s(t)$ such that any graph with average degree larger than $s(t)$ must contain a $K_t$-topological minor. We prove that, for sufficiently large $t$, graphs with average degree at least $(1+\epsilon)s(t)$ contain a $K_t$-topological minor consisting of at most $C(\epsilon,t)\log n$ vertices. Finally, we show that, for sufficiently large $t$, graphs with average degree at least $(1+\epsilon)c(t)$ contain either a $K_t$-minor consisting of at most $C(\epsilon,t)$ vertices or a $K_t$-topological minor consisting of at most $C(\epsilon,t)\log n$ vertices.

Optimal Designs for Lasso and Dantzig Selector Using Expander Codes

We investigate the high-dimensional regression problem using adjacency matrices of unbalanced expander graphs. In this frame, we prove that the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -prediction error and ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -risk of the lasso, and the Dantzig selector are optimal up to an explicit multiplicative constant. Thus, we can estimate a high-dimensional target vector with an error term similar to the one obtained in a situation where one knows the support of the largest coordinates in advance. Moreover, we show that these design matrices have an explicit restricted eigenvalue. Precisely, they satisfy the restricted eigenvalue assumption and compatibility condition with an explicit constant. Eventually, we capitalize on the recent construction of unbalanced expander graphs due to Guruswami, Umans, and Vadhan, to provide a deterministic polynomial time construction of these design matrices.

Expansion Properties of Topology for Networking of Information in Cloud

Toward the progress in the era of globalization and ubiquity of sensors and devices, sharing and dissemination of information dominate todays networks. Content-centric networking, cloud services, and open connectivity form the main ingredients of the future Internet architecture. With the problem of information overload, the networking paradigm of cloud computing can benefit from transitioning to a network of information in which information is the main token of communication instead of physical address. Available methods may not be efficient in exploiting the semantics of information for content dissemination. Considering a content-centric approach, we intend to tackle this problem by using the expander graphs for an enhanced network coding scheme that takes an opportunistic strategy to utilize the spectral characteristics of the network topology to achieve a better solvability and reliability and lowering the processing cost for the entire system. By simulation and analytical evaluation, we compare our proposed method with an epidemic network coding based approach. Our evaluation examines the performance of our clustering method in the presence of different random topology models as well as examining the impact on the network coding technique.

Fibred coarse embedding into non-positively curved manifolds and higher index problem

Let X=⨆n=1∞Xn be the coarse disjoint union of a sequence of finite metric spaces with uniform bounded geometry. In this paper, we show that the coarse Novikov conjecture holds for X, if X admits a fibred coarse embedding into a simply connected complete Riemannian manifold of non-positive sectional curvature. This includes a large class of expander graphs with geometric property (T).

Short lists for shortest descriptions in short time

Is it possible to find a shortest description for a binary string? The well-known answer is no, Kolmogorov complexity is not computable. Faced with this barrier, one might instead seek a short list of candidates which includes a laconic description. Remarkably such approximations exist. This paper presents an efficient algorithm which generates a polynomial-size list containing an optimal description for a given input string. Along the way, we employ expander graphs and randomness dispersers to obtain an Explicit Online Matching Theorem for bipartite graphs and a refinement of Muchnik's Conditional Complexity Theorem. Our main result extends recent work by Bauwens, Mahklin, Vereshchagin, and Zimand.

Computing personalized PageRank quickly by exploiting graph structures

We propose a new scalable algorithm that can compute Personalized PageRank (PPR) very quickly. The Power method is a state-of-the-art algorithm for computing exact PPR; however, it requires many iterations. Thus reducing the number of iterations is the main challenge. We achieve this by exploiting graph structures of web graphs and social networks. The convergence of our algorithm is very fast. In fact, it requires up to 7.5 times fewer iterations than the Power method and is up to five times faster in actual computation time. To the best of our knowledge, this is the first time to use graph structures explicitly to solve PPR quickly. Our contributions can be summarized as follows. 1. We provide an algorithm for computing a tree decomposition, which is more efficient and scalable than any previous algorithm. 2. Using the above algorithm, we can obtain a core-tree decomposition of any web graph and social network. This allows us to decompose a web graph and a social network into (1) the core , which behaves like an expander graph, and (2) a small tree-width graph, which behaves like a tree in an algorithmic sense. 3. We apply a direct method to the small tree-width graph to construct an LU decomposition. 4. Building on the LU decomposition and using it as pre-conditoner , we apply GMRES method (a state-of-the-art advanced iterative method) to compute PPR for whole web graphs and social networks.

Ramanujan complexes and high dimensional expanders

Expander graphs in general, and Ramanujan graphs in particular, have been of great interest in the last three decades with many applications in computer science, combinatorics and even pure mathematics. In these notes we describe various efforts made in recent years to generalize these notions from graphs to higher dimensional simplicial complexes.

Secure sets and their expansion in cubic graphs

Abstract Consider a graph whose vertices play the role of members of the opposing groups. The edge between two vertices means that these vertices may defend or attack each other. At one time, any attacker may attack only one vertex. Similarly, any defender fights for itself or helps exactly one of its neighbours. If we have a set of defenders that can repel any attack, then we say that the set is secure. Moreover, it is strong if it is also prepared for a raid of one additional foe who can strike anywhere. We show that almost any cubic graph of order n has a minimum strong secure set of cardinality less or equal to n/2 + 1. Moreover, we examine the possibility of an expansion of secure sets and strong secure sets.

Skeleton graph expansion of critical exponents in "cultural revolution" years

Kenneth Wilson's Nobel Prize winning breakthrough in the renormalization group theory of phase transition and critical phenomena almost overlapped with the violent "cultural revolution" years (1966–1976) in China. An unexpected chance in 1972 brought the author of these lines close to the Wilson–Fisher ϵ-expansion of critical exponents and eventually led to a joint paper with Lu Yu published entirely in Chinese without any English title and abstract. Even the original acknowledgment was deleted because of mentioning foreign names like Kenneth Wilson and Kerson Huang. In this article I will tell the 40-year old story as a much belated tribute to Kenneth Wilson and to reproduce the essence of our work in English. At the end, I give an elementary derivation of the Callan–Symanzik equation without referring to field theory.

Geometric complexity of embeddings in $${\mathbb{R}^{d}}$$ R d

Given a simplicial complex K, we consider several notions of geometric complexity of embeddings of K in a Euclidean space $${\mathbb{R}^d}$$ : thickness, distortion, and refinement complexity (the minimal number of simplices needed for a PL embedding). We show that any n-complex with N simplices which topologically embeds in $${\mathbb{R}^{2n}, n > 2}$$ , can be PL embedded in $${\mathbb{R}^{2n}}$$ with refinement complexity $${O(e^{N^{4+{\epsilon}}})}$$ . Families of simplicial n-complexes K are constructed such that any embedding of K into $${\mathbb{R}^{2n}}$$ has an exponential lower bound on thickness and refinement complexity as a function of the number of simplices of K. This contrasts embeddings in the stable range, $${K\subset \mathbb{R}^{2n+k}, k > 0}$$ , where all known bounds on geometric complexity functions are polynomial. In addition, we give a geometric argument for a bound on distortion of expander graphs in Euclidean spaces. Several related open problems are discussed, including questions about the growth rate of complexity functions of embeddings, and about the crossing number and the ropelength of classical links.

On the sizes of expander graphs and minimum distances of graph codes

We give lower bounds for the minimum distances of graph codes based on expander graphs. The bounds depend only on the second eigenvalue of the graph and the parameters of the component codes. We also give an upper bound on the size of a degree regular graph with given second eigenvalue.

Algebraic Cayley graphs over finite fields

A new algebraic Cayley graph is constructed using finite fields. It provides a more flexible source of expander graphs. Its connectedness, the number of connected components, and diameter bound are studied via Weil's estimate for character sums. Furthermore, we study the algorithmic problem of computing the number of connected components and establish a link to the integer factorization problem.