4 research outputs found
From Quasirandom graphs to Graph Limits and Graphlets
We generalize the notion of quasirandom which concerns a class of equivalent
properties that random graphs satisfy. We show that the convergence of a graph
sequence under the spectral distance is equivalent to the convergence using the
(normalized) cut distance. The resulting graph limit is called graphlets. We
then consider several families of graphlets and, in particular, we characterize
graphlets with low ranks for both dense and sparse graphs
Efficient polynomial-time approximation scheme for the genus of dense graphs
The main results of this paper provide an Efficient Polynomial-Time
Approximation Scheme (EPTAS) for approximating the genus (and non-orientable
genus) of dense graphs. By dense we mean that for
some fixed . While a constant factor approximation is trivial for
this class of graphs, approximations with factor arbitrarily close to 1 need a
sophisticated algorithm and complicated mathematical justification. More
precisely, we provide an algorithm that for a given (dense) graph of order
and given , returns an integer such that has an
embedding into a surface of genus , and this is -close to a
minimum genus embedding in the sense that the minimum genus of
satisfies: . The
running time of the algorithm is , where is
an explicit function. Next, we extend this algorithm to also output an
embedding (rotation system) whose genus is . This second algorithm is an
Efficient Polynomial-time Randomized Approximation Scheme (EPRAS) and runs in
time .Comment: 36 pages. An extended abstract of the preliminary version of this
paper appeared in FOCS 201
Linear embeddings of graphs and graph limits
Consider a random graph process where vertices are chosen from the interval
, and edges are chosen independently at random, but so that, for a given
vertex , the probability that there is an edge to a vertex decreases as
the distance between and increases. We call this a random graph with a
linear embedding. We define a new graph parameter , which aims to
measure the similarity of the graph to an instance of a random graph with a
linear embedding. For a graph , if and only if is a unit
interval graph, and thus a deterministic example of a graph with a linear
embedding. We show that the behaviour of is consistent with the
notion of convergence as defined in the theory of dense graph limits. In this
theory, graph sequences converge to a symmetric, measurable function on
. We define an operator which applies to graph limits, and
which assumes the value zero precisely for graph limits that have a linear
embedding. We show that, if a graph sequence converges to a function
, then converges as well. Moreover, there exists a
function arbitrarily close to under the box distance, so that
is arbitrarily close to .Comment: In pres
Harmonic analysis on graphs via Bratteli diagrams and path-space measures
The past decade has seen a flourishing of advances in harmonic analysis of
graphs. They lie at the crossroads of graph theory and such analytical tools as
graph Laplacians, Markov processes and associated boundaries, analysis of
path-space, harmonic analysis, dynamics, and tail-invariant measures. Motivated
by recent advances for the special case of Bratteli diagrams, our present focus
will be on those graph systems with the property that the sets of vertices
and edges admit discrete level structures. A choice of discrete levels
in turn leads to new and intriguing discrete-time random-walk models.
Our main extension (which greatly expands the earlier analysis of Bratteli
diagrams) is the case when the levels in the graph system under
consideration are now allowed to be standard measure spaces. Hence, in the
measure framework, we must deal with systems of transition probabilities, as
opposed to incidence matrices (for the traditional Bratteli diagrams).
The paper is divided into two parts, (i) the special case when the levels are
countable discrete systems, and (ii) the (non-atomic) measurable category,
i.e., when each level is a prescribed measure space with standard Borel
structure. The study of the two cases together is motivated in part by recent
new results on graph-limits. Our results depend on a new analysis of certain
duality systems for operators in Hilbert space; specifically, one dual system
of operator for each level. We prove new results in both cases, (i) and (ii);
and we further stress both similarities, and differences, between results and
techniques involved in the two cases.Comment: 73 pages, 3 figure