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30
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 Cesàro averages. Nonlinear spectral gaps of graphs are also shown to behave sub-multiplicatively under zigzag products. These results yield a ..."
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Cited by 15 (4 self)
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Nonlinear spectral gaps with respect to uniformly convex normed spaces are shown to satisfy a spectral calculus inequality that establishes their decay along Cesàro 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.
Resilient asymptotic consensus in robust networks
- IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS
, 2013
"... This paper addresses the problem of resilient innetwork consensus in the presence of misbehaving nodes. Secure and fault-tolerant consensus algorithms typically assume knowledge of nonlocal information; however, this assumption is not suitable for large-scale dynamic networks. To remedy this, we foc ..."
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Cited by 13 (6 self)
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This paper addresses the problem of resilient innetwork consensus in the presence of misbehaving nodes. Secure and fault-tolerant consensus algorithms typically assume knowledge of nonlocal information; however, this assumption is not suitable for large-scale dynamic networks. To remedy this, we focus on local strategies that provide resilience to faults and compromised nodes. We design a consensus protocol based on local information that is resilient to worst-case security breaches, assuming the compromised nodes have full knowledge of the network and the intentions of the other nodes. We provide necessary and sufficient conditions for the normal nodes to reach asymptotic consensus despite the influence of the misbehaving nodes under different threat assumptions. We show that traditional metrics such as connectivity are not adequate to characterize the behavior of such algorithms, and develop a novel graph-theoretic property referred to as network robustness. Network robustness formalizes the notion of redundancy of direct information exchange between subsets of nodes in the network, and is a fundamental property for analyzing the behavior of certain distributed algorithms that use only local information.
FROM APOLLONIUS TO ZAREMBA: LOCAL-GLOBAL PHENOMENA IN THIN ORBITS
"... Abstract. We discuss a number of natural problems in arithmetic, arising in completely unrelated settings, which turn out to have a common formulation involving “thin ” orbits. These include the local-global problem for integral Apollonian gaskets and Zaremba’s Conjecture on finite continued fractio ..."
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Cited by 6 (2 self)
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Abstract. We discuss a number of natural problems in arithmetic, arising in completely unrelated settings, which turn out to have a common formulation involving “thin ” orbits. These include the local-global problem for integral Apollonian gaskets and Zaremba’s Conjecture on finite continued fractions with absolutely bounded partial quotients. Though these problems could have been posed by the ancient Greeks, recent progress comes from a pleasant synthesis of modern techniques from a variety of fields, including harmonic analysis, algebra, geometry, combinatorics, and dynamics. We describe the problems, partial progress, and some of the tools alluded to above.
A Product Theorem in Free Groups
, 2007
"... commutingelementsthenjA If A is a nite subset of aAfree AjgroupjAj2 with at least two non-thesameconclusionholdsinanarbitraryvirtuallyfreegroup,unless (logjAj)O(1). Moregenerally, Ageneratesavirtuallycyclicsubgroup. timatingthenumberofcollisionsinmultipleproductsA1 Thecentralpartoftheproofofthisresu ..."
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Cited by 4 (0 self)
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commutingelementsthenjA If A is a nite subset of aAfree AjgroupjAj2 with at least two non-thesameconclusionholdsinanarbitraryvirtuallyfreegroup,unless (logjAj)O(1). Moregenerally, Ageneratesavirtuallycyclicsubgroup. timatingthenumberofcollisionsinmultipleproductsA1 Thecentralpartoftheproofofthisresultiscarriedonbyes-
SHORT COLLISION SEARCH IN ARBITRARY SL2 HASH FUNCTIONS HOMOMORPHIC
"... Abstract. We study homomorphic hash functions into SL2(q), the 2 × 2 matrices with determinant 1 over the field with q elements. Modulo a well supported number theoretic hypothesis, which holds in particular for all concrete homomorphisms proposed thus far, we prove that a random homomorphism is at ..."
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Cited by 2 (0 self)
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Abstract. We study homomorphic hash functions into SL2(q), the 2 × 2 matrices with determinant 1 over the field with q elements. Modulo a well supported number theoretic hypothesis, which holds in particular for all concrete homomorphisms proposed thus far, we prove that a random homomorphism is at least as secure as any concrete homomorphism. For a family of homomorphisms containing several concrete proposals in the literature, we prove that collisions of length O(log q) can be found in running time O ( √ q). For general homomorphisms we offer an algorithm that, heuristically and according to experiments, in running time O ( √ q) finds collisions of length O(log q) for q even, and length O(log 2 q / log log q) for arbitrary q. For any conceivable practical scenario, our algorithms are substantially faster than all earlier algorithms and produce much shorter collisions. Let {0, 1} ∗ 1.
Expansion of buildinglike complexes
, 1407
"... Abstract Following Gromov, the coboundary expansion of building-like complexes is studied. In particular, it is shown that for any n ≥ 1, there exists a constant (n) > 0 such that for any 0 ≤ k < n the k-th coboundary expansion constant of any n-dimensional spherical building is at least (n). ..."
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Cited by 2 (1 self)
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Abstract Following Gromov, the coboundary expansion of building-like complexes is studied. In particular, it is shown that for any n ≥ 1, there exists a constant (n) > 0 such that for any 0 ≤ k < n the k-th coboundary expansion constant of any n-dimensional spherical building is at least (n). MSC: 55U10 , 51E24
HIGHER DIMENSIONAL DISCRETE CHEEGER INEQUALITIES
- JOURNAL OF COMPUTATIONAL GEOMETRY
, 2015
"... For graphs there exists a strong connection between spectral and combinatorial expansion properties. This is expressed, e.g., by the discrete Cheeger inequality, the lower bound of which states that λ(G) ≤ h(G), where λ(G) is the second smallest eigenvalue of the Laplacian of a graph G and h(G) i ..."
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Cited by 2 (1 self)
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For graphs there exists a strong connection between spectral and combinatorial expansion properties. This is expressed, e.g., by the discrete Cheeger inequality, the lower bound of which states that λ(G) ≤ h(G), where λ(G) is the second smallest eigenvalue of the Laplacian of a graph G and h(G) is the Cheeger constant measuring the edge expansion of G. We are interested in generalizations of expansion properties to finite simplicial complexes of higher dimension (or uniform hypergraphs). Whereas higher dimensional Laplacians were introduced already in 1945 by Eck-mann, the generalization of edge expansion to simplicial complexes is not straightforward. Recently, a topologically motivated notion analogous to edge expansion that is based on Z2-cohomology was introduced by Gromov and independently by Linial, Meshulam and Wallach. It is known that for this generalization there is no direct higher dimensional analogue of the lower bound of the Cheeger inequality. A different, combinatorially motivated generalization of the Cheeger constant, de-noted by h(X), was studied by Parzanchevski, Rosenthal and Tessler. They showed that indeed λ(X) ≤ h(X), where λ(X) is the smallest non-trivial eigenvalue of the ((k − 1)-dimensional upper) Laplacian, for the case of k-dimensional simplicial complexes X with complete (k − 1)-skeleton. Whether this inequality also holds for k-dimensional complexes with non-complete (k−1)-skeleton has been an open question. We give two proofs of the inequality for arbitrary complexes. The proofs differ strongly in the methods and structures employed, and each allows for a different kind of additional strengthening of the original result.
