598,562 research outputs found
A metric on directed graphs and Markov chains based on hitting probabilities
The shortest-path, commute time, and diffusion distances on undirected graphs
have been widely employed in applications such as dimensionality reduction,
link prediction, and trip planning. Increasingly, there is interest in using
asymmetric structure of data derived from Markov chains and directed graphs,
but few metrics are specifically adapted to this task. We introduce a metric on
the state space of any ergodic, finite-state, time-homogeneous Markov chain
and, in particular, on any Markov chain derived from a directed graph. Our
construction is based on hitting probabilities, with nearness in the metric
space related to the transfer of random walkers from one node to another at
stationarity. Notably, our metric is insensitive to shortest and average walk
distances, thus giving new information compared to existing metrics. We use
possible degeneracies in the metric to develop an interesting structural theory
of directed graphs and explore a related quotienting procedure. Our metric can
be computed in time, where is the number of states, and in
examples we scale up to nodes and edges on a desktop
computer. In several examples, we explore the nature of the metric, compare it
to alternative methods, and demonstrate its utility for weak recovery of
community structure in dense graphs, visualization, structure recovering,
dynamics exploration, and multiscale cluster detection.Comment: 26 pages, 9 figures, for associated code, visit
https://github.com/zboyd2/hitting_probabilities_metric, accepted at SIAM J.
Math. Data Sc
Explicit Space-Time Codes Achieving The Diversity-Multiplexing Gain Tradeoff
A recent result of Zheng and Tse states that over a quasi-static channel,
there exists a fundamental tradeoff, referred to as the diversity-multiplexing
gain (D-MG) tradeoff, between the spatial multiplexing gain and the diversity
gain that can be simultaneously achieved by a space-time (ST) block code. This
tradeoff is precisely known in the case of i.i.d. Rayleigh-fading, for T>=
n_t+n_r-1 where T is the number of time slots over which coding takes place and
n_t,n_r are the number of transmit and receive antennas respectively. For T <
n_t+n_r-1, only upper and lower bounds on the D-MG tradeoff are available.
In this paper, we present a complete solution to the problem of explicitly
constructing D-MG optimal ST codes, i.e., codes that achieve the D-MG tradeoff
for any number of receive antennas. We do this by showing that for the square
minimum-delay case when T=n_t=n, cyclic-division-algebra (CDA) based ST codes
having the non-vanishing determinant property are D-MG optimal. While
constructions of such codes were previously known for restricted values of n,
we provide here a construction for such codes that is valid for all n.
For the rectangular, T > n_t case, we present two general techniques for
building D-MG-optimal rectangular ST codes from their square counterparts. A
byproduct of our results establishes that the D-MG tradeoff for all T>= n_t is
the same as that previously known to hold for T >= n_t + n_r -1.Comment: Revised submission to IEEE Transactions on Information Theor
An Algebraic Coding Scheme for Wireless Relay Networks With Multiple-Antenna Nodes
We consider the problem of coding over a half-duplex wireless relay network where both the transmitter and the receiver have respectively several transmit and receive antennas, whereas each relay is a small device with only a single antenna. Since, in this scenario, requiring the relays to decode results in severe rate hits, we propose a full rate strategy where the relays do a simple operation before forwarding the signal, based on the idea of distributed space-time coding. Our scheme relies on division algebras, an algebraic object which allows the design of fully diverse matrices. The code construction is applicable to systems with any number of transmit/receive antennas and relays, and has better performance than random code constructions, with much less encoding complexity. Finally, the robustness of the proposed distributed space-time codes to node failures is considered
Space Frequency Codes from Spherical Codes
A new design method for high rate, fully diverse ('spherical') space
frequency codes for MIMO-OFDM systems is proposed, which works for arbitrary
numbers of antennas and subcarriers. The construction exploits a differential
geometric connection between spherical codes and space time codes. The former
are well studied e.g. in the context of optimal sequence design in CDMA
systems, while the latter serve as basic building blocks for space frequency
codes. In addition a decoding algorithm with moderate complexity is presented.
This is achieved by a lattice based construction of spherical codes, which
permits lattice decoding algorithms and thus offers a substantial reduction of
complexity.Comment: 5 pages. Final version for the 2005 IEEE International Symposium on
Information Theor
Error-Correcting Codes for Automatic Control
Systems with automatic feedback control may consist of several remote devices, connected only by unreliable communication channels. It is necessary in these conditions to have a method for accurate, real-time state estimation in the presence of channel noise. This problem is addressed, for the case of polynomial-growth-rate state spaces, through a new type of error-correcting code that is online and computationally efficient. This solution establishes a constructive analog, for some applications in estimation and control, of the Shannon coding theorem
STBCs from Representation of Extended Clifford Algebras
A set of sufficient conditions to construct -real symbol Maximum
Likelihood (ML) decodable STBCs have recently been provided by Karmakar et al.
STBCs satisfying these sufficient conditions were named as Clifford Unitary
Weight (CUW) codes. In this paper, the maximal rate (as measured in complex
symbols per channel use) of CUW codes for is
obtained using tools from representation theory. Two algebraic constructions of
codes achieving this maximal rate are also provided. One of the constructions
is obtained using linear representation of finite groups whereas the other
construction is based on the concept of right module algebra over
non-commutative rings. To the knowledge of the authors, this is the first paper
in which matrices over non-commutative rings is used to construct STBCs. An
algebraic explanation is provided for the 'ABBA' construction first proposed by
Tirkkonen et al and the tensor product construction proposed by Karmakar et al.
Furthermore, it is established that the 4 transmit antenna STBC originally
proposed by Tirkkonen et al based on the ABBA construction is actually a single
complex symbol ML decodable code if the design variables are permuted and
signal sets of appropriate dimensions are chosen.Comment: 5 pages, no figures, To appear in Proceedings of IEEE ISIT 2007,
Nice, Franc
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