7,393 research outputs found
Largest sparse subgraphs of random graphs
For the Erd\H{o}s-R\'enyi random graph G(n,p), we give a precise asymptotic
formula for the size of a largest vertex subset in G(n,p) that induces a
subgraph with average degree at most t, provided that p = p(n) is not too small
and t = t(n) is not too large. In the case of fixed t and p, we find that this
value is asymptotically almost surely concentrated on at most two explicitly
given points. This generalises a result on the independence number of random
graphs. For both the upper and lower bounds, we rely on large deviations
inequalities for the binomial distribution.Comment: 15 page
Spectral and Dynamical Properties in Classes of Sparse Networks with Mesoscopic Inhomogeneities
We study structure, eigenvalue spectra and diffusion dynamics in a wide class
of networks with subgraphs (modules) at mesoscopic scale. The networks are
grown within the model with three parameters controlling the number of modules,
their internal structure as scale-free and correlated subgraphs, and the
topology of connecting network. Within the exhaustive spectral analysis for
both the adjacency matrix and the normalized Laplacian matrix we identify the
spectral properties which characterize the mesoscopic structure of sparse
cyclic graphs and trees. The minimally connected nodes, clustering, and the
average connectivity affect the central part of the spectrum. The number of
distinct modules leads to an extra peak at the lower part of the Laplacian
spectrum in cyclic graphs. Such a peak does not occur in the case of
topologically distinct tree-subgraphs connected on a tree. Whereas the
associated eigenvectors remain localized on the subgraphs both in trees and
cyclic graphs. We also find a characteristic pattern of periodic localization
along the chains on the tree for the eigenvector components associated with the
largest eigenvalue equal 2 of the Laplacian. We corroborate the results with
simulations of the random walk on several types of networks. Our results for
the distribution of return-time of the walk to the origin (autocorrelator)
agree well with recent analytical solution for trees, and it appear to be
independent on their mesoscopic and global structure. For the cyclic graphs we
find new results with twice larger stretching exponent of the tail of the
distribution, which is virtually independent on the size of cycles. The
modularity and clustering contribute to a power-law decay at short return
times
On rigidity, orientability and cores of random graphs with sliders
Suppose that you add rigid bars between points in the plane, and suppose that
a constant fraction of the points moves freely in the whole plane; the
remaining fraction is constrained to move on fixed lines called sliders. When
does a giant rigid cluster emerge? Under a genericity condition, the answer
only depends on the graph formed by the points (vertices) and the bars (edges).
We find for the random graph the threshold value of
for the appearance of a linear-sized rigid component as a function of ,
generalizing results of Kasiviswanathan et al. We show that this appearance of
a giant component undergoes a continuous transition for and a
discontinuous transition for . In our proofs, we introduce a
generalized notion of orientability interpolating between 1- and
2-orientability, of cores interpolating between 2-core and 3-core, and of
extended cores interpolating between 2+1-core and 3+2-core; we find the precise
expressions for the respective thresholds and the sizes of the different cores
above the threshold. In particular, this proves a conjecture of Kasiviswanathan
et al. about the size of the 3+2-core. We also derive some structural
properties of rigidity with sliders (matroid and decomposition into components)
which can be of independent interest.Comment: 32 pages, 1 figur
Combinatorial theorems relative to a random set
We describe recent advances in the study of random analogues of combinatorial
theorems.Comment: 26 pages. Submitted to Proceedings of the ICM 201
Syntactic Separation of Subset Satisfiability Problems
Variants of the Exponential Time Hypothesis (ETH) have been used to derive lower bounds on the time complexity for certain problems, so that the hardness results match long-standing algorithmic results. In this paper, we consider a syntactically defined class of problems, and give conditions for when problems in this class require strongly exponential time to approximate to within a factor of (1-epsilon) for some constant epsilon > 0, assuming the Gap Exponential Time Hypothesis (Gap-ETH), versus when they admit a PTAS. Our class includes a rich set of problems from additive combinatorics, computational geometry, and graph theory. Our hardness results also match the best known algorithmic results for these problems
Sparse Learning over Infinite Subgraph Features
We present a supervised-learning algorithm from graph data (a set of graphs)
for arbitrary twice-differentiable loss functions and sparse linear models over
all possible subgraph features. To date, it has been shown that under all
possible subgraph features, several types of sparse learning, such as Adaboost,
LPBoost, LARS/LASSO, and sparse PLS regression, can be performed. Particularly
emphasis is placed on simultaneous learning of relevant features from an
infinite set of candidates. We first generalize techniques used in all these
preceding studies to derive an unifying bounding technique for arbitrary
separable functions. We then carefully use this bounding to make block
coordinate gradient descent feasible over infinite subgraph features, resulting
in a fast converging algorithm that can solve a wider class of sparse learning
problems over graph data. We also empirically study the differences from the
existing approaches in convergence property, selected subgraph features, and
search-space sizes. We further discuss several unnoticed issues in sparse
learning over all possible subgraph features.Comment: 42 pages, 24 figures, 4 table
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