14,759 research outputs found
A Survey on Graph Kernels
Graph kernels have become an established and widely-used technique for
solving classification tasks on graphs. This survey gives a comprehensive
overview of techniques for kernel-based graph classification developed in the
past 15 years. We describe and categorize graph kernels based on properties
inherent to their design, such as the nature of their extracted graph features,
their method of computation and their applicability to problems in practice. In
an extensive experimental evaluation, we study the classification accuracy of a
large suite of graph kernels on established benchmarks as well as new datasets.
We compare the performance of popular kernels with several baseline methods and
study the effect of applying a Gaussian RBF kernel to the metric induced by a
graph kernel. In doing so, we find that simple baselines become competitive
after this transformation on some datasets. Moreover, we study the extent to
which existing graph kernels agree in their predictions (and prediction errors)
and obtain a data-driven categorization of kernels as result. Finally, based on
our experimental results, we derive a practitioner's guide to kernel-based
graph classification
Topological Constraints at the Theta Point: Closed Loops at Two Loops
We map the problem of self-avoiding random walks in a Theta solvent with a
chemical potential for writhe to the three-dimensional symmetric
U(N)-Chern-Simons theory as N goes to 0. We find a new scaling regime of
topologically constrained polymers, with critical exponents that depend on the
chemical potential for writhe, which gives way to a fluctuation-induced
first-order transition.Comment: 5 pages, RevTeX, typo
Distribution of sizes of erased loops for loop-erased random walks
We study the distribution of sizes of erased loops for loop-erased random
walks on regular and fractal lattices. We show that for arbitrary graphs the
probability of generating a loop of perimeter is expressible in
terms of the probability of forming a loop of perimeter when a
bond is added to a random spanning tree on the same graph by the simple
relation . On -dimensional hypercubical lattices,
varies as for large , where for , where
z is the fractal dimension of the loop-erased walks on the graph. On
recursively constructed fractals with this relation is modified
to , where is the hausdorff and
is the spectral dimension of the fractal.Comment: 4 pages, RevTex, 3 figure
Graph Kernels
We present a unified framework to study graph kernels, special cases of which include the random
walk (Gärtner et al., 2003; Borgwardt et al., 2005) and marginalized (Kashima et al., 2003, 2004;
Mahé et al., 2004) graph kernels. Through reduction to a Sylvester equation we improve the time
complexity of kernel computation between unlabeled graphs with n vertices from O(n^6) to O(n^3).
We find a spectral decomposition approach even more efficient when computing entire kernel matrices.
For labeled graphs we develop conjugate gradient and fixed-point methods that take O(dn^3)
time per iteration, where d is the size of the label set. By extending the necessary linear algebra to
Reproducing Kernel Hilbert Spaces (RKHS) we obtain the same result for d-dimensional edge kernels,
and O(n^4) in the infinite-dimensional case; on sparse graphs these algorithms only take O(n^2)
time per iteration in all cases. Experiments on graphs from bioinformatics and other application
domains show that these techniques can speed up computation of the kernel by an order of magnitude
or more. We also show that certain rational kernels (Cortes et al., 2002, 2003, 2004) when
specialized to graphs reduce to our random walk graph kernel. Finally, we relate our framework to
R-convolution kernels (Haussler, 1999) and provide a kernel that is close to the optimal assignment
kernel of Fröhlich et al. (2006) yet provably positive semi-definite
Space-time random walk loop measures
In this work, we investigate a novel setting of Markovian loop measures and
introduce a new class of loop measures called Bosonic loop measures. Namely, we
consider loop soups with varying intensity (chemical potential in
physics terms), and secondly, we study Markovian loop measures on graphs with
an additional "time" dimension leading to so-called space-time random walks and
their loop measures and Poisson point loop processes. Interesting phenomena
appear when the additional coordinate of the space-time process is on a
discrete torus with non-symmetric jump rates. The projection of these
space-time random walk loop measures onto the space dimensions is loop measures
on the spatial graph, and in the scaling limit of the discrete torus, these
loop measures converge to the so-called [Bosonic loop measures]. This provides
a natural probabilistic definition of [Bosonic loop measures]. These novel loop
measures have similarities with the standard Markovian loop measures only that
they give weights to loops of certain lengths, namely any length which is
multiple of a given length which serves as an additional
parameter. We complement our study with generalised versions of Dynkin's
isomorphism theorem (including a version for the whole complex field) as well
as Symanzik's moment formulae for complex Gaussian measures. Due to the lacking
symmetry of our space-time random walks, the distributions of the occupation
time fields are given in terms of complex Gaussian measures over complex-valued
random fields ([B92,BIS09]. Our space-time setting allows obtaining quantum
correlation functions as torus limits of space-time correlation functions.Comment: 3 figure
Thermodynamic formalism for dissipative quantum walks
We consider the dynamical properties of dissipative continuous-time quantum
walks on directed graphs. Using a large-deviation approach we construct a
thermodynamic formalism allowing us to define a dynamical order parameter, and
to identify transitions between dynamical regimes. For a particular class of
dissipative quantum walks we propose a quantum generalization of the the
classical PageRank vector, used to rank the importance of nodes in a directed
graph. We also provide an example where one can characterize the dynamical
transition from an effective classical random walk to a dissipative quantum
walk as a thermodynamic crossover between distinct dynamical regimes.Comment: 8 page
The Type-problem on the Average for random walks on graphs
When averages over all starting points are considered, the Type Problem for
the recurrence or transience of a simple random walk on an inhomogeneous
network in general differs from the usual "local" Type Problem. This difference
leads to a new classification of inhomogeneous discrete structures in terms of
{\it recurrence} and {\it transience} {\it on the average}, describing their
large scale topology from a "statistical" point of view. In this paper we
analyze this classification and the properties connected to it, showing how the
average behavior affects the thermodynamic properties of statistical models on
graphs.Comment: 10 pages, 3 figures. to appear on EPJ
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