80 research outputs found
High Dimensional Expanders
Expander graphs have been, during the last five decades, the subject of a
most fruitful interaction between pure mathematics and computer science, with
influence and applications going both ways (cf. [Lub94], [HLW06], [Lub12] and
the references therein). In the last decade, a theory of "high dimensional
expanders" has begun to emerge. The goal of the current paper is to describe
some paths of this new area of study.Comment: Paper to be presented as a Plenary talk at ICM 201
On Expanders Graphs: Parameters and Applications
We give a new lower bound on the expansion coefficient of an edge-vertex
graph of a -regular graph. As a consequence, we obtain an improvement on the
lower bound on relative minimum distance of the expander codes constructed by
Sipser and Spielman. We also derive some improved results on the vertex
expansion of graphs that help us in improving the parameters of the expander
codes of Alon, Bruck, Naor, Naor, and Roth.Comment: Submitted to SIAM J. DAM on Feb. 1, 200
Entropy waves, the zig-zag graph product, and new constant-degree
The main contribution of this work is a new type of graph product, which we
call the {\it 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 ``entropy
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 products affords the constructive
interference of two such waves.
Subsequent work [ALW01], [MW01] relates the zig-zag product of graphs to the
standard semidirect product of groups, leading to new results and constructions
on expanding Cayley graphs.Comment: 31 pages, published versio
Spectra of lifted Ramanujan graphs
A random -lift of a base graph is its cover graph on the vertices
, where for each edge in there is an independent
uniform bijection , and has all edges of the form .
A main motivation for studying lifts is understanding Ramanujan graphs, and
namely whether typical covers of such a graph are also Ramanujan.
Let be a graph with largest eigenvalue and let be the
spectral radius of its universal cover. Friedman (2003) proved that every "new"
eigenvalue of a random lift of is with high
probability, and conjectured a bound of , which would be tight by
results of Lubotzky and Greenberg (1995). Linial and Puder (2008) improved
Friedman's bound to . For -regular graphs,
where and , this translates to a bound of
, compared to the conjectured .
Here we analyze the spectrum of a random -lift of a -regular graph
whose nontrivial eigenvalues are all at most in absolute value. We
show that with high probability the absolute value of every nontrivial
eigenvalue of the lift is . This result is
tight up to a logarithmic factor, and for it
substantially improves the above upper bounds of Friedman and of Linial and
Puder. In particular, it implies that a typical -lift of a Ramanujan graph
is nearly Ramanujan.Comment: 34 pages, 4 figure
Arithmetic Groups (Banff, Alberta, April 14-19, 2013)
We present detailed summaries of the talks that were given during a week-long
workshop on Arithmetic Groups at the Banff International Research Station in
April 2013. The vast majority of these reports are based on abstracts that were
kindly provided by the speakers. Video recordings of many of the lectures are
available online.Comment: 34 page
Optimal network topologies: Expanders, Cages, Ramanujan graphs, Entangled networks and all that
We report on some recent developments in the search for optimal network
topologies. First we review some basic concepts on spectral graph theory,
including adjacency and Laplacian matrices, and paying special attention to the
topological implications of having large spectral gaps. We also introduce
related concepts as ``expanders'', Ramanujan, and Cage graphs. Afterwards, we
discuss two different dynamical feautures of networks: synchronizability and
flow of random walkers and so that they are optimized if the corresponding
Laplacian matrix have a large spectral gap. From this, we show, by developing a
numerical optimization algorithm that maximum synchronizability and fast random
walk spreading are obtained for a particular type of extremely homogeneous
regular networks, with long loops and poor modular structure, that we call
entangled networks. These turn out to be related to Ramanujan and Cage graphs.
We argue also that these graphs are very good finite-size approximations to
Bethe lattices, and provide almost or almost optimal solutions to many other
problems as, for instance, searchability in the presence of congestion or
performance of neural networks. Finally, we study how these results are
modified when studying dynamical processes controlled by a normalized (weighted
and directed) dynamics; much more heterogeneous graphs are optimal in this
case. Finally, a critical discussion of the limitations and possible extensions
of this work is presented.Comment: 17 pages. 11 figures. Small corrections and a new reference. Accepted
for pub. in JSTA
From Ramanujan Graphs to Ramanujan Complexes
Ramanujan graphs are graphs whose spectrum is bounded optimally. Such graphs
have found numerous applications in combinatorics and computer science. In
recent years, a high dimensional theory has emerged. In this paper these
developments are surveyed. After explaining their connection to the Ramanujan
conjecture we will present some old and new results with an emphasis on random
walks on these discrete objects and on the Euclidean spheres. The latter lead
to "golden gates" which are of importance in quantum computation.Comment: To appear in the proceedings of the Ramanujan Centenary Celebration
at the Royal Societ
Isoperimetric Inequalities in Simplicial Complexes
In graph theory there are intimate connections between the expansion
properties of a graph and the spectrum of its Laplacian. In this paper we
define a notion of combinatorial expansion for simplicial complexes of general
dimension, and prove that similar connections exist between the combinatorial
expansion of a complex, and the spectrum of the high dimensional Laplacian
defined by Eckmann. In particular, we present a Cheeger-type inequality, and a
high-dimensional Expander Mixing Lemma. As a corollary, using the work of Pach,
we obtain a connection between spectral properties of complexes and Gromov's
notion of geometric overlap. Using the work of Gunder and Wagner, we give an
estimate for the combinatorial expansion and geometric overlap of random
Linial-Meshulam complexes
Recommended from our members
Entropy Waves, The Zig-Zag Graph Product, and New Constant-Degree Expanders and Extractors
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 every 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 "entropy 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.
A variant of this product can be applied to extractors, giving the first explicit extractors whose seed length depends (poly)logarithmically on only the entropy deficiency of the source (rather than its length) and that extract almost all the entropy of high min-entropy sources. These high min-entropy extractors have several interesting applications, including the first constant-degree explicit expanders which beat the "eigenvalue bound."Engineering and Applied Science
Endoscopic Approach for Tissue Expansion for Different Cosmetic Lesions in Pediatric Age
Background/Purpose: The use of tissue expanders in plastic and reconstruction surgery is now well established for large defects in adults & children. Tissue expansion is one of the reconstructive surgeon's alternatives in providing optimal tissue replacement when skin shortage is a major problem. Predesigned plan about the criteria of tissue expansion should be implied before embarking on removal of a lesion. Endoscopic placement of tissue expanders has the benefit of reducing operative time,
major complication rate, time to full expansion and length of hospital stay compared to the open technique for tissue expander placement.
Materials & Methods: The study was carried on 15 cases for which 22 expanders were inserted .All cases were in the pediatric age (6-15 years) .Nine cases had melanocytic pigmented naevi, four cases had post-burn scars, one had pigmented lymphangiomatous lesion & the last case had post grafting scarring .Eight patients needed single insertion of the expander followed by definitive reconstruction (2 months later), 4 cases needed multiple expanders on the same session & 3cases needed sequential expansion. The process of expander placement was done through a remote incision using endoscopic approach, after the required inflation (usually 2 months) reconstructive surgery was carried on for flap designing.
Results: Twenty two expanders were inserted in 15 cases .operating time ranged from 50-70 min. (mean= 1 hour) in early
cases .Later on the time was shortened to a mean of 40 ± 5 min. The mean duration for completion of expansion to the required dimensions needed for the flap was 2 months ( + 2 weeks ) . Complication rate was 18%. They occurred in 4 out of
22 expanders (hematoma ,wound dehiscence and seroma around 2 expanders).
Conclusion: Tissue expansion in the pediatric population has its implication in different plastic problems. Endoscopic assisted expansion is a new trend in expander placement that has its role in decreasing complications related to insertion of expanders.
Index Word: Tissue expansion- Endoscopic- Pediatric
- …