65,951 research outputs found
Propagation Kernels
We introduce propagation kernels, a general graph-kernel framework for
efficiently measuring the similarity of structured data. Propagation kernels
are based on monitoring how information spreads through a set of given graphs.
They leverage early-stage distributions from propagation schemes such as random
walks to capture structural information encoded in node labels, attributes, and
edge information. This has two benefits. First, off-the-shelf propagation
schemes can be used to naturally construct kernels for many graph types,
including labeled, partially labeled, unlabeled, directed, and attributed
graphs. Second, by leveraging existing efficient and informative propagation
schemes, propagation kernels can be considerably faster than state-of-the-art
approaches without sacrificing predictive performance. We will also show that
if the graphs at hand have a regular structure, for instance when modeling
image or video data, one can exploit this regularity to scale the kernel
computation to large databases of graphs with thousands of nodes. We support
our contributions by exhaustive experiments on a number of real-world graphs
from a variety of application domains
Tight Bounds on Information Dissemination in Sparse Mobile Networks
Motivated by the growing interest in mobile systems, we study the dynamics of
information dissemination between agents moving independently on a plane.
Formally, we consider mobile agents performing independent random walks on
an -node grid. At time , each agent is located at a random node of the
grid and one agent has a rumor. The spread of the rumor is governed by a
dynamic communication graph process , where two agents are
connected by an edge in iff their distance at time is within their
transmission radius . Modeling the physical reality that the speed of radio
transmission is much faster than the motion of the agents, we assume that the
rumor can travel throughout a connected component of before the graph is
altered by the motion. We study the broadcast time of the system, which
is the time it takes for all agents to know the rumor. We focus on the sparse
case (below the percolation point ) where, with high
probability, no connected component in has more than a logarithmic number
of agents and the broadcast time is dominated by the time it takes for many
independent random walks to meet each other. Quite surprisingly, we show that
for a system below the percolation point the broadcast time does not depend on
the relation between the mobility speed and the transmission radius. In fact,
we prove that for any , even
when the transmission range is significantly larger than the mobility range in
one step, giving a tight characterization up to logarithmic factors. Our result
complements a recent result of Peres et al. (SODA 2011) who showed that above
the percolation point the broadcast time is polylogarithmic in .Comment: 19 pages; we rewrote Lemma 4, fixing a claim which was not fully
justified in the first version of the draf
Coined quantum walks on percolation graphs
Quantum walks, both discrete (coined) and continuous time, form the basis of
several quantum algorithms and have been used to model processes such as
transport in spin chains and quantum chemistry. The enhanced spreading and
mixing properties of quantum walks compared with their classical counterparts
have been well-studied on regular structures and also shown to be sensitive to
defects and imperfections in the lattice. As a simple example of a disordered
system, we consider percolation lattices, in which edges or sites are randomly
missing, interrupting the progress of the quantum walk. We use numerical
simulation to study the properties of coined quantum walks on these percolation
lattices in one and two dimensions. In one dimension (the line) we introduce a
simple notion of quantum tunneling and determine how this affects the
properties of the quantum walk as it spreads. On two-dimensional percolation
lattices, we show how the spreading rate varies from linear in the number of
steps down to zero, as the percolation probability decreases to the critical
point. This provides an example of fractional scaling in quantum walk dynamics.Comment: 25 pages, 14 figures; v2 expanded and improved presentation after
referee comments, added extra figur
Decoherence in quantum walks - a review
The development of quantum walks in the context of quantum computation, as
generalisations of random walk techniques, led rapidly to several new quantum
algorithms. These all follow unitary quantum evolution, apart from the final
measurement. Since logical qubits in a quantum computer must be protected from
decoherence by error correction, there is no need to consider decoherence at
the level of algorithms. Nonetheless, enlarging the range of quantum dynamics
to include non-unitary evolution provides a wider range of possibilities for
tuning the properties of quantum walks. For example, small amounts of
decoherence in a quantum walk on the line can produce more uniform spreading (a
top-hat distribution), without losing the quantum speed up. This paper reviews
the work on decoherence, and more generally on non-unitary evolution, in
quantum walks and suggests what future questions might prove interesting to
pursue in this area.Comment: 52 pages, invited review, v2 & v3 updates to include significant work
since first posted and corrections from comments received; some non-trivial
typos fixed. Comments now limited to changes that can be applied at proof
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