Expander Graphs Research Articles

Lifts, derandomization, and diameters of Schreier graphs of Mealy automata

It is known that random 2-lifts of graphs give rise to expander graphs. We present a new conjectured derandomization of this construction based on certain Mealy automata. We verify that these graphs have polylogarithmic diameter, and present a class of automata for which the same is true. However, we also show that some automata in this class do not give rise to expander graphs.

Expander ℓ0-decoding

We introduce two new algorithms, Serial- ℓ 0 and Parallel- ℓ 0 for solving a large underdetermined linear system of equations y = A x ∈ R m when it is known that x ∈ R n has at most k < m nonzero entries and that A is the adjacency matrix of an unbalanced left d -regular expander graph. The matrices in this class are sparse and allow a highly efficient implementation. A number of algorithms have been designed to work exclusively under this setting, composing the branch of combinatorial compressed-sensing (CCS). Serial- ℓ 0 and Parallel- ℓ 0 iteratively minimise ‖ y − A x ˆ ‖ 0 by successfully combining two desirable features of previous CCS algorithms: the coordinate selection strategy of ER [1] for updating x ˆ , and the parallel updating mechanism of SMP [2] . We are able to link these elements and guarantee convergence in O ( d n log ⁡ k ) operations by introducing a randomised scaling of columns in A , with scaling chosen independent of the measured vector. We also empirically observe that the recovery properties of Serial- ℓ 0 and Parallel- ℓ 0 degrade gracefully as the signal is allowed to have its non-zero values chosen adversarially. Moreover, we observe Serial- ℓ 0 and Parallel- ℓ 0 to be able to solve large scale problems with a larger fraction of nonzeros than other algorithms when the number of measurements is substantially less than the signal length; in particular, they are able to reliably solve for a k -sparse vector x ∈ R n from m expander measurements with n / m = 10 3 and k / m up to four times greater than what is achievable by ℓ 1 -regularisation from dense Gaussian measurements. Additionally, due to their low computational complexity, Serial- ℓ 0 and Parallel- ℓ 0 are observed to be able to solve large problems sizes in substantially less time than other algorithms for compressed sensing. In particular, Parallel- ℓ 0 is structured to take advantage of massively parallel architectures.

Efficient robust secret sharing from expander graphs

Threshold secret sharing allows a dealer to share a secret among n players so that any coalition of t players learns nothing about the secret, but any t+1 players can reconstruct the secret in its entirety. Robust secret sharing (RSS) provides the additional guarantee that even if t malicious players mangle their shares, they cannot cause the honest players to reconstruct an incorrect secret. In this work, we construct a simple RSS protocol for $t = \left ({ \frac {1}{2} - \epsilon }\right )n$ that achieves logarithmic overhead in terms of share size and simultaneously allows efficient reconstruction. Our shares size increases by an additive term of $\mathcal {O}(\kappa + \log n)$ , and reconstruction succeeds except with probability at most 2−κ . Previous efficient RSS protocols like that of Rabin and Ben-Or (STOC ’89) and Cevallos et al. (Eurocrypt ’12) use MACs to allow each player to check the shares of each other player in the protocol. These checks provide robustness, but require significant overhead in share size. Our construction identifies the n players as nodes in an expander graph, each player only checks its neighbors in the expander graph.

For-All Sparse Recovery in Near-Optimal Time

An approximate sparse recovery system in ℓ 1 norm consists of parameters k , ϵ, N ; an m -by- N measurement Φ; and a recovery algorithm R . Given a vector, x , the system approximates x by xˆ = R (Φ x ), which must satisfy ‖ xˆ- x ‖ 1 ≤ (1+ϵ)‖ x - x k ‖ 1 . We consider the “for all” model, in which a single matrix Φ, possibly “constructed” non-explicitly using the probabilistic method, is used for all signals x . The best existing sublinear algorithm by Porat and Strauss [2012] uses O (ϵ −3 k log ( N / k )) measurements and runs in time O ( k 1 − α N α ) for any constant α &gt; 0. In this article, we improve the number of measurements to O (ϵ − 2 k log ( N / k )), matching the best existing upper bound (attained by super-linear algorithms), and the runtime to O ( k 1+β poly(log N ,1/ϵ)), with a modest restriction that k ⩽ N 1 − α and ϵ ⩽ (log k /log N ) γ for any constants α, β, γ &gt; 0. When k ⩽ log c N for some c &gt; 0, the runtime is reduced to O ( k poly( N ,1/ϵ)). With no restrictions on ϵ, we have an approximation recovery system with m = O ( k /ϵlog ( N / k )((log N /log k ) γ + 1/ϵ)) measurements. The overall architecture of this algorithm is similar to that of Porat and Strauss [2012] in that we repeatedly use a weak recovery system (with varying parameters) to obtain a top-level recovery algorithm. The weak recovery system consists of a two-layer hashing procedure (or with two unbalanced expanders for a deterministic algorithm). The algorithmic innovation is a novel encoding procedure that is reminiscent of network coding and that reflects the structure of the hashing stages. The idea is to encode the signal position index i by associating it with a unique message m i , which will be encoded to a longer message m ′ i (in contrast to Porat and Strauss [2012] in which the encoding is simply the identity). Portions of the message m ′ i correspond to repetitions of the hashing, and we use a regular expander graph to encode the linkages among these portions. The decoding or recovery algorithm consists of recovering the portions of the longer messages m ′ i and then decoding to the original messages m i , all the while ensuring that corruptions can be detected and/or corrected. The recovery algorithm is similar to list recovery introduced in Indyk et al. [2010] and used in Gilbert et al. [2013]. In our algorithm, the messages { m i } are independent of the hashing, which enables us to obtain a better result.

A tournament approach to pattern avoiding matrices

We consider the following Turan-type problem: given a fixed tournament H, what is the least integer t = t(n,H) so that adding t edges to any n-vertex tournament, results in a digraph containing a copy of H. Similarly, what is the least integer t = t(Tn,H) so that adding t edges to the n-vertex transitive tournament, results in a digraph containing a copy of H. Besides proving several results on these problems, our main contributions are the following: Pach and Tardos conjectured that if M is an acyclic 0/1 matrix, then any n × n matrix with n(log n)O(1) entries equal to 1 contains the pattern M. We show that this conjecture is equivalent to the assertion that t(Tn,H) = n(log n)O(1) if and only if H belongs to a certain (natural) family of tournaments. We propose an approach for determining if t(n,H) = n(log n)O(1). This approach combines expansion in sparse graphs, together with certain structural characterizations of H-free tournaments. Our result opens the door for using structural graph theoretic tools in order to settle the Pach–Tardos conjecture.

Hyperbolicity, degeneracy, and expansion of random intersection graphs

We establish the conditions under which several algorithmically exploitable structural features hold for random intersection graphs, a natural model for many real-world networks where edges correspond to shared attributes. Specifically, we fully characterize the degeneracy of random intersection graphs, and prove that the model asymptotically almost surely produces graphs with hyperbolicity at least log(n). Further, we prove that in the parametric regime where random intersection graphs are degenerate an even stronger notion of sparseness, so called bounded expansion, holds with high probability. We supplement our theoretical findings with experimental evaluations of the relevant statistics.

Unbalanced Expander Based Compressive Data Gathering in Clustered Wireless Sensor Networks

Conventional compressive sensing-based data gathering (CS-DG) algorithms require a large number of sensors for each compressive sensing measurement, thereby resulting in high energy consumption in clustered wireless sensor networks (WSNs). To solve this problem, we propose a novel energy-efficient CS-DG algorithm, which exploits the better reconstruction accuracy of the adjacency matrix of an unbalanced expander graph. In the proposed CS-DG algorithm, each measurement is the sum of a few sensory data, which are jointly determined by random sampling and random walks. Through theoretical analysis, we prove that the constructed M × N sparse binary sensing matrix is the adjacency matrix of a (k, ε) unbalanced expander graph when M = O (k log N/k) and t = O (N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> /(kq)) for WSNs with Nc clusters, where 0 ≤ q ≤ 1 and N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> > k. Simulation results show our proposed CS-DG has better performance than existing algorithms in terms of reconstruction accuracy and energy consumption. When hybrid energy-efficient distributed clustering algorithm is used, to achieve the same reconstruction accuracy, our proposed CS-DG can save energy by at least 27.8%.

A continuum of expanders

A regular equivalence between two graphs $\Gamma,\Gamma'$ is a pair of uniformly proper Lipschitz maps $V\Gamma\to V\Gamma'$ and $V\Gamma'\to V\Gamma$. Using separation profiles we prove that there are $2^{\aleph_0}$ regular equivalence classes of expander graphs, and of finitely generated groups with a representative which isometrically contains expanders.

Explicit Expanding Expanders

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. ...

From graphs to keyed quantum hash functions

We present two new constructions of quantum hash functions: the first based on expander graphs and the second based on extractor functions and estimate the amount of randomness that is needed to construct them. We also propose a keyed quantum hash function based on extractor function that can be used in quantum message authentication codes and assess its security in a limited attacker model.

On Compiling Structured CNFs to OBDDs

We present new results on the size of OBDD representations of structurally characterized classes of CNF formulas. First, we prove that variable convex formulas (that is, formulas with incidence graphs that are convex with respect to the set of variables) have polynomial OBDD size. Second, we prove an exponential lower bound on the OBDD size of a family of CNF formulas with incidence graphs of bounded degree. We obtain the first result by identifying a simple sufficient condition—which we call the few subterms property—for a class of CNF formulas to have polynomial OBDD size, and show that variable convex formulas satisfy this condition. To prove the second result, we exploit the combinatorial properties of expander graphs; this approach allows us to establish an exponential lower bound on the OBDD size of formulas satisfying strong syntactic restrictions.

A retrospective of knowledge graphs

Information on the Internet is fragmented and presented in different data sources, which makes automatic knowledge harvesting and understanding formidable for machines, and even for humans. Knowledge graphs have become prevalent in both of industry and academic circles these years, to be one of the most efficient and effective knowledge integration approaches. Techniques for knowledge graph construction can mine information from either structured, semi-structured, or even unstructured data sources, and finally integrate the information into knowledge, represented in a graph. Furthermore, knowledge graph is able to organize information in an easy-to-maintain, easy-to-understand and easy-to-use manner. In this paper, we give a summarization of techniques for constructing knowledge graphs. We review the existing knowledge graph systems developed by both academia and industry. We discuss in detail about the process of building knowledge graphs, and survey state-of-the-art techniques for automatic knowledge graph checking and expansion via logical inferring and reasoning. We also review the issues of graph data management by introducing the knowledge data models and graph databases, especially from a NoSQL point of view. Finally, we overview current knowledge graph systems and discuss the future research directions.

On the effective size of a non-Weyl graph

We show how to find the coefficient of the leading term of the resonance asymptotics using the method of pseudo-orbit expansion for quantum graphs which do not obey Weyl asymptotics. For a non-Weyl graph we develop a method to reduce the number of edges of a corresponding directed graph. Through this method we prove bounds on the above coefficient depending on the structure of the graph, for graphs with the same lengths of internal edges. We explicitly find the positions of the resolvent resonances.

Low-energy physics of thet−Jmodel ind=∞using extremely correlated Fermi liquid theory: Cutoff second-order equations

We present the results for the low energy properties of the infinite dimensional t-J model with $J=0$, using $O(\lambda^2)$ equations of the extremely correlated Fermi liquid formalism. The parameter $\lambda \in [0,1]$ is analogous to the inverse spin parameter $1/(2S)$ in quantum magnets. The present analytical scheme allows us to approach the physically most interesting regime near the Mott insulating state $n\lesssim 1$. It overcomes the limitation to low densities $n \lesssim .7$ of earlier calculations, by employing a variant of the skeleton graph expansion, and a high frequency cutoff that is essential for maintaining the known high-T entropy. The resulting quasiparticle weight $Z$, the low $\omega,T $ self energy and the resistivity are reported. These are quite close at all densities to the exact numerical results of the $U=\infty$ Hubbard model, obtained using the dynamical mean field theory. The present calculation offers the advantage of generalizing to finite $T$ rather easily, and allows the visualization of the loss of coherence of Fermi liquid quasiparticles by raising $T$. The present scheme is generalizable to finite dimensions and a non vanishing $J$.

A Generalized Alon-Boppana Bound and Weak Ramanujan Graphs

A basic eigenvalue bound due to Alon and Boppana holds only for regular graphs. In this paper we give a generalized Alon-Boppana bound for eigenvalues of graphs that are not required to be regular. We show that a graph $G$ with diameter $k$ and vertex set $V$, the smallest nontrivial eigenvalue $\lambda_1$ of the normalized Laplacian $\mathcal L$ satisfies$$ \lambda_1 \leq 1-\sigma \big(1- \frac c {k} \big)$$ for some constant $c$ where $\sigma = 2\sum_v d_v \sqrt{d_v-1}/\sum_v d_v^2 $ and $d_v$ denotes the degree of the vertex $v$.We consider weak Ramanujan graphs defined as graphs satisfying $ \lambda_1 \geq 1-\sigma$. We examine the vertex expansion and edge expansion of weak Ramanujan graphs and then use the expansion properties among other methods to derive the above Alon-Boppana bound. A corrigendum was added on the 3rd of November 2017.

Bounds on the Parameters of Locally Recoverable Codes

A locally recoverable code (LRC code) is a code over a finite alphabet, such that every symbol in the encoding is a function of a small number of other symbols that form a recovering set. In this paper, we derive new finite-length and asymptotic bounds on the parameters of LRC codes. For LRC codes with a single recovering set for every coordinate, we derive an asymptotic Gilbert-Varshamov type bound for LRC codes and find the maximum attainable relative distance of asymptotically good LRC codes. Similar results are established for LRC codes with two disjoint recovering sets for every coordinate. For the case of multiple recovering sets (the availability problem), we derive a lower bound on the parameters using expander graph arguments. Finally, we also derive finite-length upper bounds on the rate and the distance of LRC codes with multiple recovering sets.

Spectra of the extended neighborhood corona and extended corona of two graphs

In this paper we define extended corona and extended neighborhood corona of two graphs $G_{1}$ and $G_{2}$, which are denoted by $G_{1}\bullet G_{2}$ and $G_{1}\ast G_{2}$ respectively. We compute their adjacency spectrum, Laplacian spectrum and signless Laplacian spectrum. As applications, we give methods to construct infinite families of integral graphs, Laplacian integral graphs and expander graphs from known ones.

Implementation, Scheduling, and Adaptation of Partial Expansion Graphs on Multicore Platforms

The dynamic nature of state-of-the-art multicore signal processing systems limits the ability of designers to derive accurate models for the targeted applications. Inaccurate assumptions in the model can lead to inefficient implementations and restrict the runtime re-configuration of these systems. On the other hand, dataflow models have provided powerful techniques to analyze and explore the design space for many classes of signal processing systems. In this context, we develop the Partial Expansion Graph (PEG) as an implementation model where existing dataflow graph analysis is augmented with dynamic adaptation, efficient parallelism utilization, and online reconfiguration based on the measured performance of the targeted applications. In this paper, we develop new methods for scheduling and mapping DSP systems using PEGs. Collectively, these methods tune the amount of data parallelism in the application graph and distribute data- and task-parallel instances over different cores while balancing the load across the available processing units. We enable online adaptation for PEG systems using low-overhead customizable solutions. We demonstrate the utility of our PEG-based scheduling and mapping algorithms through experiments on real application models and various synthetic graphs.

Isoperimetric Inequalities for Ramanujan Complexes and Topological Expanders

Expander graphs have been intensively studied in the last four decades (Hoory et al., Bull Am Math Soc, 43(4):439–562, 2006; Lubotzky, Bull Am Math Soc, 49:113–162, 2012). In recent years a high dimensional theory of expanders has emerged, and several variants have been studied. Among them stand out coboundary expansion and topological expansion. It is known that for every d there are unbounded degree simplicial complexes of dimension d with these properties. However, a major open problem, formulated by Gromov (Geom Funct Anal 20(2):416–526, 2010), is whether bounded degree high dimensional expanders exist for \({d \geq 2}\). We present an explicit construction of bounded degree complexes of dimension \({d = 2}\) which are topological expanders, thus answering Gromov’s question in the affirmative. Conditional on a conjecture of Serre on the congruence subgroup property, infinite sub-family of these give also a family of bounded degree coboundary expanders. The main technical tools are new isoperimetric inequalities for Ramanujan Complexes. We prove linear size bounds on \({\mathbb{F}_2}\) systolic invariants of these complexes, which seem to be the first linear\({\mathbb{F}_2}\) systolic bounds. The expansion results are deduced from these isoperimetric inequalities.

Compressive Sampling Photoacoustic Microscope System based on Low Rank Matrix Completion

Abstract Photoacoustic Microscopy (PAM) has developed into a powerful tool for deep tissue imaging with a better spatial resolution. But the data acquisition time in PAM is so long that it is a great challenge for real time imaging. In this paper, a new PAM data acquisition and image recovery method, called Compressive Sampling PAM System based on Low Rank Matrix Completion (CSLRM-PAM) is proposed to obtain a high-resolution PAM image with relatively low sampling rates. In order to successfully set up a CSLRM-PAM system, the two key problems which we need to keep focus on are design of the compressive sampling scheme and the corresponding image recovery algorithm. In this paper, two compressive sampling schemes based on expander graphs are proposed to replace the conventional point-by-point scanning scheme to implement fast data acquisition. Then, the low rank matrix completion is utilized to obtain high-resolution PAM image directly from the compressive sampling data. The effectiveness of the proposed scheme is validated using both numerical analysis and PAM experiments. In contrast with the conventional system, the proposed CSLRM-PAM system is able to dramatically decrease the total sampling points for a relatively high-resolution PAM image and to implement accelerated data acquisition.