3,990 research outputs found
Bordered Floer homology and the spectral sequence of a branched double cover II: the spectral sequences agree
Given a link in the three-sphere, Ozsv\'ath and Szab\'o showed that there is
a spectral sequence starting at the Khovanov homology of the link and
converging to the Heegaard Floer homology of its branched double cover. The aim
of this paper is to explicitly calculate this spectral sequence in terms of
bordered Floer homology. There are two primary ingredients in this computation:
an explicit calculation of bimodules associated to Dehn twists, and a general
pairing theorem for polygons. The previous part (arXiv:1011.0499) focuses on
computing the bimodules; this part focuses on the pairing theorem for polygons,
in order to prove that the spectral sequence constructed in the previous part
agrees with the one constructed by Ozsv\'ath and Szab\'o.Comment: 85 pages, 19 figures, v3: Version to appear in Journal of Topolog
A Novel Approach to Finding Near-Cliques: The Triangle-Densest Subgraph Problem
Many graph mining applications rely on detecting subgraphs which are
near-cliques. There exists a dichotomy between the results in the existing work
related to this problem: on the one hand the densest subgraph problem (DSP)
which maximizes the average degree over all subgraphs is solvable in polynomial
time but for many networks fails to find subgraphs which are near-cliques. On
the other hand, formulations that are geared towards finding near-cliques are
NP-hard and frequently inapproximable due to connections with the Maximum
Clique problem.
In this work, we propose a formulation which combines the best of both
worlds: it is solvable in polynomial time and finds near-cliques when the DSP
fails. Surprisingly, our formulation is a simple variation of the DSP.
Specifically, we define the triangle densest subgraph problem (TDSP): given
, find a subset of vertices such that , where is the number of triangles induced
by the set . We provide various exact and approximation algorithms which the
solve the TDSP efficiently. Furthermore, we show how our algorithms adapt to
the more general problem of maximizing the -clique average density. Finally,
we provide empirical evidence that the TDSP should be used whenever the output
of the DSP fails to output a near-clique.Comment: 42 page
The polytope of non-crossing graphs on a planar point set
For any finite set \A of points in , we define a
-dimensional simple polyhedron whose face poset is isomorphic to the
poset of ``non-crossing marked graphs'' with vertex set \A, where a marked
graph is defined as a geometric graph together with a subset of its vertices.
The poset of non-crossing graphs on \A appears as the complement of the star
of a face in that polyhedron.
The polyhedron has a unique maximal bounded face, of dimension
where is the number of points of \A in the interior of \conv(\A). The
vertices of this polytope are all the pseudo-triangulations of \A, and the
edges are flips of two types: the traditional diagonal flips (in
pseudo-triangulations) and the removal or insertion of a single edge.
As a by-product of our construction we prove that all pseudo-triangulations
are infinitesimally rigid graphs.Comment: 28 pages, 16 figures. Main change from v1 and v2: Introduction has
been reshape
Searching for network modules
When analyzing complex networks a key target is to uncover their modular
structure, which means searching for a family of modules, namely node subsets
spanning each a subnetwork more densely connected than the average. This work
proposes a novel type of objective function for graph clustering, in the form
of a multilinear polynomial whose coefficients are determined by network
topology. It may be thought of as a potential function, to be maximized, taking
its values on fuzzy clusterings or families of fuzzy subsets of nodes over
which every node distributes a unit membership. When suitably parametrized,
this potential is shown to attain its maximum when every node concentrates its
all unit membership on some module. The output thus is a partition, while the
original discrete optimization problem is turned into a continuous version
allowing to conceive alternative search strategies. The instance of the problem
being a pseudo-Boolean function assigning real-valued cluster scores to node
subsets, modularity maximization is employed to exemplify a so-called quadratic
form, in that the scores of singletons and pairs also fully determine the
scores of larger clusters, while the resulting multilinear polynomial potential
function has degree 2. After considering further quadratic instances, different
from modularity and obtained by interpreting network topology in alternative
manners, a greedy local-search strategy for the continuous framework is
analytically compared with an existing greedy agglomerative procedure for the
discrete case. Overlapping is finally discussed in terms of multiple runs, i.e.
several local searches with different initializations.Comment: 10 page
The geometry of flip graphs and mapping class groups
The space of topological decompositions into triangulations of a surface has
a natural graph structure where two triangulations share an edge if they are
related by a so-called flip. This space is a sort of combinatorial
Teichm\"uller space and is quasi-isometric to the underlying mapping class
group. We study this space in two main directions. We first show that strata
corresponding to triangulations containing a same multiarc are strongly convex
within the whole space and use this result to deduce properties about the
mapping class group. We then focus on the quotient of this space by the mapping
class group to obtain a type of combinatorial moduli space. In particular, we
are able to identity how the diameters of the resulting spaces grow in terms of
the complexity of the underlying surfaces.Comment: 46 pages, 23 figure
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