4,778 research outputs found
Coding in Undirected Graphs Is Either Very Helpful or Not Helpful at All
While it is known that using network coding can significantly improve the throughput of directed networks, it is a notorious open problem whether coding yields any advantage over the multicommodity flow (MCF) rate in undirected networks. It was conjectured that the answer is no. In this paper we show that even a small advantage over MCF can be amplified to yield a near-maximum possible gap.
We prove that any undirected network with k source-sink pairs that exhibits a (1+epsilon) gap between its MCF rate and its network coding rate can be used to construct a family of graphs G\u27 whose gap is log(|G\u27|)^c for some constant c < 1. The resulting gap is close to the best currently known upper bound, log(|G\u27|), which follows from the connection between MCF and sparsest cuts.
Our construction relies on a gap-amplifying graph tensor product that, given two graphs G1,G2 with small gaps, creates another graph G with a gap that is equal to the product of the previous two, at the cost of increasing the size of the graph. We iterate this process to obtain a gap of log(|G\u27|)^c from any initial gap
Graph Signal Processing: Overview, Challenges and Applications
Research in Graph Signal Processing (GSP) aims to develop tools for
processing data defined on irregular graph domains. In this paper we first
provide an overview of core ideas in GSP and their connection to conventional
digital signal processing. We then summarize recent developments in developing
basic GSP tools, including methods for sampling, filtering or graph learning.
Next, we review progress in several application areas using GSP, including
processing and analysis of sensor network data, biological data, and
applications to image processing and machine learning. We finish by providing a
brief historical perspective to highlight how concepts recently developed in
GSP build on top of prior research in other areas.Comment: To appear, Proceedings of the IEE
Non-linear index coding outperforming the linear optimum
The following source coding problem was introduced by Birk and Kol: a sender
holds a word , and wishes to broadcast a codeword to
receivers, . The receiver is interested in , and has
prior \emph{side information} comprising some subset of the bits. This
corresponds to a directed graph on vertices, where is an edge iff
knows the bit . An \emph{index code} for is an encoding scheme
which enables each to always reconstruct , given his side
information. The minimal word length of an index code was studied by
Bar-Yossef, Birk, Jayram and Kol (FOCS 2006). They introduced a graph
parameter, \minrk_2(G), which completely characterizes the length of an
optimal \emph{linear} index code for . The authors of BBJK showed that in
various cases linear codes attain the optimal word length, and conjectured that
linear index coding is in fact \emph{always} optimal.
In this work, we disprove the main conjecture of BBJK in the following strong
sense: for any and sufficiently large , there is an
-vertex graph so that every linear index code for requires codewords
of length at least , and yet a non-linear index code for
has a word length of . This is achieved by an explicit
construction, which extends Alon's variant of the celebrated Ramsey
construction of Frankl and Wilson.
In addition, we study optimal index codes in various, less restricted,
natural models, and prove several related properties of the graph parameter
\minrk(G).Comment: 16 pages; Preliminary version appeared in FOCS 200
Preserving Link Privacy in Social Network Based Systems
A growing body of research leverages social network based trust relationships
to improve the functionality of the system. However, these systems expose
users' trust relationships, which is considered sensitive information in
today's society, to an adversary.
In this work, we make the following contributions. First, we propose an
algorithm that perturbs the structure of a social graph in order to provide
link privacy, at the cost of slight reduction in the utility of the social
graph. Second we define general metrics for characterizing the utility and
privacy of perturbed graphs. Third, we evaluate the utility and privacy of our
proposed algorithm using real world social graphs. Finally, we demonstrate the
applicability of our perturbation algorithm on a broad range of secure systems,
including Sybil defenses and secure routing.Comment: 16 pages, 15 figure
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