Performance Evaluation of Distance-Hop Proportionality on Geometric Graph Models of Dense Sensor Networks

Wireless sensor networks can often be viewed in terms of a uniform deployment of a large number of nodes on a region in Euclidean space, e.g., the unit square. After deployment, the nodes self-organise into a mesh topology. In a dense, homogeneous deployment, a frequently used approximation is to take the hop distance between nodes to be proportional to the Euclidean distance between them. In this paper, we analyse the performance of this approximation. We show that nodes with a certain hop distance from a fixed anchor node lie within a certain annulus with probability approach- ing unity as the number of nodes n → ∞. We take a uniform, i.i.d. deployment of n nodes on a unit square, and consider the geometric graph on these nodes with radius r(n) = c q ln n n . We show that, for a given hop distance h of a node from a fixed anchor on the unit square,the Euclidean distance lies within [(1−ǫ)(h−1)r(n), hr(n)],for ǫ > 0, with probability approaching unity as n → ∞.This result shows that it is more likely to expect a node, with hop distance h from the anchor, to lie within this an- nulus centred at the anchor location, and of width roughly r(n), rather than close to a circle whose radius is exactly proportional to h. We show that if the radius r of the ge- ometric graph is fixed, the convergence of the probability is exponentially fast. Similar results hold for a randomised lattice deployment. We provide simulation results that il- lustrate the theory, and serve to show how large n needs to be for the asymptotics to be useful.

Hardness of approximation results for the problem of finding the stopping distance in Tanner graphs

Tanner Graph representation of linear block codes is widely used by iterative decoding algorithms for recovering data transmitted across a noisy communication channel from errors and erasures introduced by the channel. The stopping distance of a Tanner graph T for a binary linear block code C determines the number of erasures correctable using iterative decoding on the Tanner graph T when data is transmitted across a binary erasure channel using the code C. We show that the problem of finding the stopping distance of a Tanner graph is hard to approximate within any positive constant approximation ratio in polynomial time unless P = NP. It is also shown as a consequence that there can be no approximation algorithm for the problem achieving an approximation ratio of 2(log n)(1-epsilon) for any epsilon > 0 unless NP subset of DTIME(n(poly(log n))).

Theory and Algorithms for Hop-Count-Based Localization with Random Geometric Graph Models of Dense Sensor Networks

Wireless sensor networks can often be viewed in terms of a uniform deployment of a large number of nodes in a region of Euclidean space. Following deployment, the nodes self-organize into a mesh topology with a key aspect being self-localization. Having obtained a mesh topology in a dense, homogeneous deployment, a frequently used approximation is to take the hop distance between nodes to be proportional to the Euclidean distance between them. In this work, we analyze this approximation through two complementary analyses. We assume that the mesh topology is a random geometric graph on the nodes; and that some nodes are designated as anchors with known locations. First, we obtain high probability bounds on the Euclidean distances of all nodes that are h hops away from a fixed anchor node. In the second analysis, we provide a heuristic argument that leads to a direct approximation for the density function of the Euclidean distance between two nodes that are separated by a hop distance h. This approximation is shown, through simulation, to very closely match the true density function. Localization algorithms that draw upon the preceding analyses are then proposed and shown to perform better than some of the well-known algorithms present in the literature. Belief-propagation-based message-passing is then used to further enhance the performance of the proposed localization algorithms. To our knowledge, this is the first usage of message-passing for hop-count-based self-localization.

Complexity of distance paired-domination problem in graphs

Suppose G = (V, E) is a simple graph and k is a fixed positive integer. A subset D subset of V is a distance k-dominating set of G if for every u is an element of V. there exists a vertex v is an element of D such that d(G)(u, v) <= k, where d(G)(u, v) is the distance between u and v in G. A set D subset of V is a distance k-paired-dominating set of G if D is a distance k-dominating set and the induced subgraph GD] contains a perfect matching. Given a graph G = (V, E) and a fixed integer k > 0, the MIN DISTANCE k-PAIRED-DOM SET problem is to find a minimum cardinality distance k-paired-dominating set of G. In this paper, we show that the decision version of MIN DISTANCE k-PAIRED-DOM SET iS NP-complete for undirected path graphs. This strengthens the complexity of decision version Of MIN DISTANCE k-PAIRED-DOM SET problem in chordal graphs. We show that for a given graph G, unless NP subset of DTIME (n(0)((log) (log) (n)) MIN DISTANCE k-PAIRED-Dom SET problem cannot be approximated within a factor of (1 -epsilon ) In n for any epsilon > 0, where n is the number of vertices in G. We also show that MIN DISTANCE k-PAIRED-DOM SET problem is APX-complete for graphs with degree bounded by 3. On the positive side, we present a linear time algorithm to compute the minimum cardinality of a distance k-paired-dominating set of a strongly chordal graph G if a strong elimination ordering of G is provided. We show that for a given graph G, MIN DISTANCE k-PAIRED-DOM SET problem can be approximated with an approximation factor of 1 + In 2 + k . In(Delta(G)), where Delta(G) denotes the maximum degree of G. (C) 2012 Elsevier B.V All rights reserved.

Belief propagation for minimum weight many-to-one matchings in the random complete graph

In a complete bipartite graph with vertex sets of cardinalities n and n', assign random weights from exponential distribution with mean 1, independently to each edge. We show that, as n -> infinity, with n' = n/alpha] for any fixed alpha > 1, the minimum weight of many-to-one matchings converges to a constant (depending on alpha). Many-to-one matching arises as an optimization step in an algorithm for genome sequencing and as a measure of distance between finite sets. We prove that a belief propagation (BP) algorithm converges asymptotically to the optimal solution. We use the objective method of Aldous to prove our results. We build on previous works on minimum weight matching and minimum weight edge cover problems to extend the objective method and to further the applicability of belief propagation to random combinatorial optimization problems.

PERCOLATION ON THE INFORMATION-THEORETICALLY SECURE SIGNAL TO INTERFERENCE RATIO GRAPH

We consider a continuum percolation model consisting of two types of nodes, namely legitimate and eavesdropper nodes, distributed according to independent Poisson point processes in R-2 of intensities lambda and lambda(E), respectively. A directed edge from one legitimate node A to another legitimate node B exists provided that the strength of the signal transmitted from node A that is received at node B is higher than that received at any eavesdropper node. The strength of the signal received at a node from a legitimate node depends not only on the distance between these nodes, but also on the location of the other legitimate nodes and an interference suppression parameter gamma. The graph is said to percolate when there exists an infinitely connected component. We show that for any finite intensity lambda(E) of eavesdropper nodes, there exists a critical intensity lambda(c) < infinity such that for all lambda > lambda(c) the graph percolates for sufficiently small values of the interference parameter. Furthermore, for the subcritical regime, we show that there exists a lambda(0) such that for all lambda < lambda(0) <= lambda(c) a suitable graph defined over eavesdropper node connections percolates that precludes percolation in the graphs formed by the legitimate nodes.

Long range resonance energy transfer from a dye molecule to graphene has (distance)-4 dependence

In our previous report on resonance energy transfer from a dye molecule to graphene [J. Chem. Phys.129, 054703 (2008)], we had derived an expression for the rate of energy transfer from a dye to graphene. An integral in the expression for the rate was evaluated approximately. We found a Yuwaka-type dependence of the rate on the distance. We now present an exact evaluation of the integral involved, leading to very interesting results. For short distances (z < 20 A), the present rate and the previous rate are in good agreement. For larger distances, the rate is found to have a z(-4) dependence on the distance, exactly. Thus we predict that for the case of pyrene on graphene, it is possible to observe fluorescence quenching up to a distance of 300 A. This is in sharp contrast to the traditional fluorescence resonance energy transfer where the quenching is observable only up to 100 A.

Approximate potential-distance relationship for clays

The conventional procedure of determining the surface potential of clay platelet and the variation of potential with distance is lengthy and time consuming. Simplified graphical procedures using Gouy theory have been developed and presented. The new procedures are simple, accurate and very much less time consuming.

Modeling of a pressure modulated desalination system using bond graph methodology

A desalination system is a complex multi energy domain system comprising power/energy flow across several domains such as electrical, thermal, and hydraulic. The dynamic modeling of a desalination system that comprehensively addresses all these multi energy domains is not adequately addressed in the literature. This paper proposes to address the issue of modeling the various energy domains for the case of a single stage flash evaporation desalination system. This paper presents a detailed bond graph modeling of a desalination unit with seamless integration of the power flow across electrical, thermal, and hydraulic domains. The paper further proposes a performance index function that leads to the tracking of the optimal chamber pressure giving the optimal flow rate for a given unit of energy expended. The model has been validated in steady state conditions by simulation and experimentation.

Bond graph toolbox for handling complex variable

Bond graph is an apt modelling tool for any system working across multiple energy domains. Power electronics system modelling is usually the study of the interplay of energy in the domains of electrical, mechanical, magnetic and thermal. The usefulness of bond graph modelling in power electronic field has been realised by researchers. Consequently in the last couple of decades, there has been a steadily increasing effort in developing simulation tools for bond graph modelling that are specially suited for power electronic study. For modelling rotating magnetic fields in electromagnetic machine models, a support for vector variables is essential. Unfortunately, all bond graph simulation tools presently provide support only for scalar variables. We propose an approach to provide complex variable and vector support to bond graph such that it will enable modelling of polyphase electromagnetic and spatial vector systems. We also introduced a rotary gyrator element and use it along with the switched junction for developing the complex/vector variable's toolbox. This approach is implemented by developing a complex S-function tool box in Simulink inside a MATLAB environment This choice has been made so as to synthesise the speed of S-function, the user friendliness of Simulink and the popularity of MATLAB.

Distance Relay Co-ordination Using Support Vector Machines in Power Transmission System

In this paper, a new approach to enhance the transmission system distance relay co-ordination is presented. The approach depends on the apparent impedance loci seen by the distance relay during all possible disturbances. In a distance relay, the impedance loci seen at the relay location is obtained by extensive transient stability studies. Support vector machines (SVMs), a class of patterns classifiers are used in discriminating zone settings (zone-1, zone-2 and zone-3) using the signals to be used by the relay. Studies on a sample 9-bus are presented for illustrating the proposed scheme.

Flow-graph techniques for epicyclic gear train analysis

Flow-graph techniques are applied in this article for the analysis of an epicyclic gear train. A gear system based on this is designed and constructed for use in Numerical Control Systems.

Acoustic emission source location and damage detection in a metallic structure using a graph-theory-based geodesic pproach

A geodesic-based approach using Lamb waves is proposed to locate the acoustic emission (AE) source and damage in an isotropic metallic structure. In the case of the AE (passive) technique, the elastic waves take the shortest path from the source to the sensor array distributed in the structure. The geodesics are computed on the meshed surface of the structure using graph theory based on Dijkstra's algorithm. By propagating the waves in reverse virtually from these sensors along the geodesic path and by locating the first intersection point of these waves, one can get the AE source location. The same approach is extended for detection of damage in a structure. The wave response matrix of the given sensor configuration for the healthy and the damaged structure is obtained experimentally. The healthy and damage response matrix is compared and their difference gives the information about the reflection of waves from the damage. These waves are backpropagated from the sensors and the above method is used to locate the damage by finding the point where intersection of geodesics occurs. In this work, the geodesic approach is shown to be suitable to obtain a practicable source location solution in a more general set-up on any arbitrary surface containing finite discontinuities. Experiments were conducted on aluminum specimens of simple and complex geometry to validate this new method.

Distance dependence of fluorescence resonance energy transfer

Deviations from the usual R (-6) dependence of the rate of fluorescence resonance energy transfer (FRET) on the distance between the donor and the acceptor have been a common scenario in the recent times. In this paper, we present a critical analysis of the distance dependence of FRET, and try to illustrate the non R (-6) type behaviour of the rate for the case of transfer from a localized electronic excitation on the donor, a dye molecule to three different energy acceptors with delocalized electronic excitations namely, graphene,two-dimensional semiconducting sheet and the case of such a semiconducting sheet rolled to obtain a nanotube. We use simple analytic models to understand the distance dependence in each case.

Relative electrical distance concept for evaluation of network reactive power and loss contributions in a deregulated system

With the liberalisation of electricity market it has become very important to determine the participants making use of the transmission network.Transmission line usage computation requires information of generator to load contributions and the path used by various generators to meet loads and losses. In this study relative electrical distance (RED) concept is used to compute reactive power contributions from various sources like generators, switchable volt-amperes reactive(VAR) sources and line charging susceptances that are scattered throughout the network, to meet the system demands. The transmission line charge susceptances contribution to the system reactive flows and its aid extended in reducing the reactive generation at the generator buses are discussed in this paper. Reactive power transmission cost evaluation is carried out in this study. The proposed approach is also compared with other approaches viz.,proportional sharing and modified Y-bus.Detailed case studies with base case and optimised results are carried out on a sample 8-bus system. IEEE 39-bus system and a practical 72-bus system, an equivalent of Indian Southern grid are also considered for illustration and results are discussed.