26,963 research outputs found
Planting trees in graphs, and finding them back
In this paper we study detection and reconstruction of planted structures in
Erd\H{o}s-R\'enyi random graphs. Motivated by a problem of communication
security, we focus on planted structures that consist in a tree graph. For
planted line graphs, we establish the following phase diagram. In a low density
region where the average degree of the initial graph is below some
critical value , detection and reconstruction go from impossible
to easy as the line length crosses some critical value ,
where is the number of nodes in the graph. In the high density region
, detection goes from impossible to easy as goes from
to , and reconstruction remains impossible so
long as . For -ary trees of varying depth and ,
we identify a low-density region , such that the following
holds. There is a threshold with the following properties.
Detection goes from feasible to impossible as crosses . We also show
that only partial reconstruction is feasible at best for . We
conjecture a similar picture to hold for -ary trees as for lines in the
high-density region , but confirm only the following part of
this picture: Detection is easy for -ary trees of size ,
while at best only partial reconstruction is feasible for -ary trees of any
size . These results are in contrast with the corresponding picture for
detection and reconstruction of {\em low rank} planted structures, such as
dense subgraphs and block communities: We observe a discrepancy between
detection and reconstruction, the latter being impossible for a wide range of
parameters where detection is easy. This property does not hold for previously
studied low rank planted structures
Broadcasting on Random Directed Acyclic Graphs
We study a generalization of the well-known model of broadcasting on trees.
Consider a directed acyclic graph (DAG) with a unique source vertex , and
suppose all other vertices have indegree . Let the vertices at
distance from be called layer . At layer , is given a random
bit. At layer , each vertex receives bits from its parents in
layer , which are transmitted along independent binary symmetric channel
edges, and combines them using a -ary Boolean processing function. The goal
is to reconstruct with probability of error bounded away from using
the values of all vertices at an arbitrarily deep layer. This question is
closely related to models of reliable computation and storage, and information
flow in biological networks.
In this paper, we analyze randomly constructed DAGs, for which we show that
broadcasting is only possible if the noise level is below a certain degree and
function dependent critical threshold. For , and random DAGs with
layer sizes and majority processing functions, we identify the
critical threshold. For , we establish a similar result for NAND
processing functions. We also prove a partial converse for odd
illustrating that the identified thresholds are impossible to improve by
selecting different processing functions if the decoder is restricted to using
a single vertex.
Finally, for any noise level, we construct explicit DAGs (using expander
graphs) with bounded degree and layer sizes admitting
reconstruction. In particular, we show that such DAGs can be generated in
deterministic quasi-polynomial time or randomized polylogarithmic time in the
depth. These results portray a doubly-exponential advantage for storing a bit
in DAGs compared to trees, where but layer sizes must grow exponentially
with depth in order to enable broadcasting.Comment: 33 pages, double column format. arXiv admin note: text overlap with
arXiv:1803.0752
Random sampling of bandlimited signals on graphs
We study the problem of sampling k-bandlimited signals on graphs. We propose
two sampling strategies that consist in selecting a small subset of nodes at
random. The first strategy is non-adaptive, i.e., independent of the graph
structure, and its performance depends on a parameter called the graph
coherence. On the contrary, the second strategy is adaptive but yields optimal
results. Indeed, no more than O(k log(k)) measurements are sufficient to ensure
an accurate and stable recovery of all k-bandlimited signals. This second
strategy is based on a careful choice of the sampling distribution, which can
be estimated quickly. Then, we propose a computationally efficient decoder to
reconstruct k-bandlimited signals from their samples. We prove that it yields
accurate reconstructions and that it is also stable to noise. Finally, we
conduct several experiments to test these techniques
Sampling of graph signals via randomized local aggregations
Sampling of signals defined over the nodes of a graph is one of the crucial
problems in graph signal processing. While in classical signal processing
sampling is a well defined operation, when we consider a graph signal many new
challenges arise and defining an efficient sampling strategy is not
straightforward. Recently, several works have addressed this problem. The most
common techniques select a subset of nodes to reconstruct the entire signal.
However, such methods often require the knowledge of the signal support and the
computation of the sparsity basis before sampling. Instead, in this paper we
propose a new approach to this issue. We introduce a novel technique that
combines localized sampling with compressed sensing. We first choose a subset
of nodes and then, for each node of the subset, we compute random linear
combinations of signal coefficients localized at the node itself and its
neighborhood. The proposed method provides theoretical guarantees in terms of
reconstruction and stability to noise for any graph and any orthonormal basis,
even when the support is not known.Comment: IEEE Transactions on Signal and Information Processing over Networks,
201
Topology Discovery of Sparse Random Graphs With Few Participants
We consider the task of topology discovery of sparse random graphs using
end-to-end random measurements (e.g., delay) between a subset of nodes,
referred to as the participants. The rest of the nodes are hidden, and do not
provide any information for topology discovery. We consider topology discovery
under two routing models: (a) the participants exchange messages along the
shortest paths and obtain end-to-end measurements, and (b) additionally, the
participants exchange messages along the second shortest path. For scenario
(a), our proposed algorithm results in a sub-linear edit-distance guarantee
using a sub-linear number of uniformly selected participants. For scenario (b),
we obtain a much stronger result, and show that we can achieve consistent
reconstruction when a sub-linear number of uniformly selected nodes
participate. This implies that accurate discovery of sparse random graphs is
tractable using an extremely small number of participants. We finally obtain a
lower bound on the number of participants required by any algorithm to
reconstruct the original random graph up to a given edit distance. We also
demonstrate that while consistent discovery is tractable for sparse random
graphs using a small number of participants, in general, there are graphs which
cannot be discovered by any algorithm even with a significant number of
participants, and with the availability of end-to-end information along all the
paths between the participants.Comment: A shorter version appears in ACM SIGMETRICS 2011. This version is
scheduled to appear in J. on Random Structures and Algorithm
Structured sampling and fast reconstruction of smooth graph signals
This work concerns sampling of smooth signals on arbitrary graphs. We first
study a structured sampling strategy for such smooth graph signals that
consists of a random selection of few pre-defined groups of nodes. The number
of groups to sample to stably embed the set of -bandlimited signals is
driven by a quantity called the \emph{group} graph cumulative coherence. For
some optimised sampling distributions, we show that sampling
groups is always sufficient to stably embed the set of -bandlimited signals
but that this number can be smaller -- down to -- depending on the
structure of the groups of nodes. Fast methods to approximate these sampling
distributions are detailed. Second, we consider -bandlimited signals that
are nearly piecewise constant over pre-defined groups of nodes. We show that it
is possible to speed up the reconstruction of such signals by reducing
drastically the dimension of the vectors to reconstruct. When combined with the
proposed structured sampling procedure, we prove that the method provides
stable and accurate reconstruction of the original signal. Finally, we present
numerical experiments that illustrate our theoretical results and, as an
example, show how to combine these methods for interactive object segmentation
in an image using superpixels
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