Eigenvector Synchronization, Graph Rigidity and the Molecule Problem

The graph realization problem has received a great deal of attention in recent years, due to its importance in applications such as wireless sensor networks and structural biology. In this paper, we extend on previous work and propose the 3D-ASAP algorithm, for the graph realization problem in $\mathbb{R}^3$, given a sparse and noisy set of distance measurements. 3D-ASAP is a divide and conquer, non-incremental and non-iterative algorithm, which integrates local distance information into a global structure determination. Our approach starts with identifying, for every node, a subgraph of its 1-hop neighborhood graph, which can be accurately embedded in its own coordinate system. In the noise-free case, the computed coordinates of the sensors in each patch must agree with their global positioning up to some unknown rigid motion, that is, up to translation, rotation and possibly reflection. In other words, to every patch there corresponds an element of the Euclidean group Euc(3) of rigid transformations in $\mathbb{R}^3$, and the goal is to estimate the group elements that will properly align all the patches in a globally consistent way. Furthermore, 3D-ASAP successfully incorporates information specific to the molecule problem in structural biology, in particular information on known substructures and their orientation. In addition, we also propose 3D-SP-ASAP, a faster version of 3D-ASAP, which uses a spectral partitioning algorithm as a preprocessing step for dividing the initial graph into smaller subgraphs. Our extensive numerical simulations show that 3D-ASAP and 3D-SP-ASAP are very robust to high levels of noise in the measured distances and to sparse connectivity in the measurement graph, and compare favorably to similar state-of-the art localization algorithms.Comment: 49 pages, 8 figure

A hybrid localization approach in 3D wireless sensor network

Location information acquisition is crucial for many wireless sensor network (WSN) applications. While existing localization approaches mainly focus on 2D plane, the emerging 3D localization brings WSNs closer to reality with much enhanced accuracy. Two types of 3D localization algorithms are mainly used in localization application: the range-based localization and the range-free localization. The range-based localization algorithm has strict requirements on hardware and therefore is costly to implement in practice. The range-free localization algorithm reduces the hardware cost but at the expense of low localization accuracy. On addressing the shortage of both algorithms, in this paper, we develop a novel hybrid localization scheme, which utilizes the range-based attribute RSSI and the range-free attribute hopsize, to achieve accurate yet low-cost 3D localization. As anchor node deployment strategy plays an important role in improving the localization accuracy, an anchor node configuration scheme is also developed in this work by utilizing the MIS (maximal independent set) of a network. With proper anchor node configuration and propagation model selection, using simulations, we show that our proposed algorithm improves the localization accuracy by 38.9% compared with 3D DV-HOP and 52.7% compared with 3D centroid

Network Topology Mapping from Partial Virtual Coordinates and Graph Geodesics

For many important network types (e.g., sensor networks in complex harsh environments and social networks) physical coordinate systems (e.g., Cartesian), and physical distances (e.g., Euclidean), are either difficult to discern or inapplicable. Accordingly, coordinate systems and characterizations based on hop-distance measurements, such as Topology Preserving Maps (TPMs) and Virtual-Coordinate (VC) systems are attractive alternatives to Cartesian coordinates for many network algorithms. Herein, we present an approach to recover geometric and topological properties of a network with a small set of distance measurements. In particular, our approach is a combination of shortest path (often called geodesic) recovery concepts and low-rank matrix completion, generalized to the case of hop-distances in graphs. Results for sensor networks embedded in 2-D and 3-D spaces, as well as a social networks, indicates that the method can accurately capture the network connectivity with a small set of measurements. TPM generation can now also be based on various context appropriate measurements or VC systems, as long as they characterize different nodes by distances to small sets of random nodes (instead of a set of global anchors). The proposed method is a significant generalization that allows the topology to be extracted from a random set of graph shortest paths, making it applicable in contexts such as social networks where VC generation may not be possible.Comment: 17 pages, 9 figures. arXiv admin note: substantial text overlap with arXiv:1712.1006

Virtual and topological coordinate based routing, mobility tracking and prediction in 2D and 3D wireless sensor networks

2013 Fall.Includes bibliographical references.A Virtual Coordinate System (VCS) for Wireless Sensor Networks (WSNs) characterizes each sensor node's location using the minimum number of hops to a specific set of sensor nodes called anchors. VCS does not require geographic localization hardware such as Global Positioning System (GPS), or localization algorithms based on Received Signal Strength Indication (RSSI) measurements. Topological Coordinates (TCs) are derived from Virtual Coordinates (VCs) of networks using Singular Value Decomposition (SVD). Topology Preserving Maps (TPMs) based on TCs contain 2D or 3D network topology and directional information that are lost in VCs. This thesis extends the scope of VC and TC based techniques to 3D sensor networks and networks with mobile nodes. Specifically, we apply existing Extreme Node Search (ENS) for anchor placement for 3D WSNs. 3D Geo-Logical Routing (3D-GLR), a routing algorithm for 3D sensor networks that alternates between VC and TC domains is evaluated. VC and TC based methods have hitherto been used only in static networks. We develop methods to use VCs in mobile networks, including the generation of coordinates, for mobile sensors without having to regenerate VCs every time the topology changes. 2D and 3D Topological Coordinate based Tracking and Prediction (2D-TCTP and 3D-TCTP) are novel algorithms developed for mobility tracking and prediction in sensor networks without the need of physical distance measurements. Most existing 2D sensor networking algorithms fail or perform poorly in 3D networks. Developing VC and TC based algorithms for 3D sensor networks is crucial to benefit from the scalability, adjustability and flexibility of VCs as well as to overcome the many disadvantages associated with geographic coordinate systems. Existing ENS algorithm for 2D sensor networks plays a key role in providing a good anchor placement and we continue to use ENS algorithm for anchor selection in 3D network. Additionally, we propose a comparison algorithm for ENS algorithm named Double-ENS algorithm which uses two independent pairs of initial anchors and thereby increases the coverage of ENS anchors in 3D networks, in order to further prove if anchor selection from original ENS algorithm is already optimal. Existing Geo-Logical Routing (GLR) algorithm demonstrates very good routing performance by switching between greedy forwarding in virtual and topological domains in 2D sensor networks. Proposed 3D-GLR extends the algorithm to 3D networks by replacing 2D TCs with 3D TCs in TC distance calculation. Simulation results show that the 3D-GLR algorithm with ENS anchor placement can significantly outperform current Geographic Coordinates (GCs) based 3D Greedy Distributed Spanning Tree Routing (3D-GDSTR) algorithm in various network environments. This demonstrates the effectiveness of ENS algorithm and 3D-GLR algorithm in 3D sensor networks. Tracking and communicating with mobile sensors has so far required the use of localization or geographic information. This thesis presents a novel approach to achieve tracking and communication without geographic information, thus significantly reducing the hardware cost and energy consumption. Mobility of sensors in WSNs is considered under two scenarios: dynamic deployment and continuous movement. An efficient VC generation scheme, which uses the average of neighboring sensors' VCs, is proposed for newly deployed sensors to get coordinates without flooding based VC generation. For the second scenario, a prediction and tracking algorithm called 2D-TCTP for continuously moving sensors is developed for 2D sensor networks. Predicted location of a mobile sensor at a future time is calculated based on current sampled velocity and direction in topological domain. The set of sensors inside an ellipse-shaped detection area around the predicted future location is alerted for the arrival of mobile sensor for communication or detection purposes. Using TPMs as a 2D guide map, tracking and prediction performances can be achieved similar to those based on GCs. A simple modification for TPMs generation is proposed, which considers radial information contained in the first principle component from SVD. This modification improves the compression or folding at the edges that has been observed in TPMs, and thus the accuracy of tracking. 3D-TCTP uses a detection area in the shape of a 3D sphere. 3D-TCTP simulation results are similar to 2D-TCTP and show competence comparable to the same algorithms based on GCs although without any 3D geographic information

DisLoc: A Convex Partitioning Based Approach for Distributed 3-D Localization in Wireless Sensor Networks

Accurate localization in wireless sensor networks (WSNs) is fundamental to many applications, such as geographic routing and position-aware data processing. This, however, is challenging in large scale 3-D WSNs due to the irregular topology, such as holes in the path, of the network. The irregular topology may cause overestimated Euclidean distance between nodes as the communication path is bent and accordingly introduces severe errors in 3-D WSN localization. As an effort towards the issue, this paper develops a distributed algorithm to achieve accurate 3-D WSN localization. Our proposal is composed of two steps, segmentation and joint localization. In specific, the entire network is first divided into several subnetworks by applying the approximate convex partitioning. A spatial convex node recognition mechanism is developed to assist the network segmentation, which relies on the connectivity information only. After that, each subnetwork is accurately localized by using the multidimensional scaling-based algorithm. The proposed localization algorithm also applies a new 3-D coordinate transformation algorithm, which helps reduce the errors introduced by coordinate integration between subnetworks and improve the localization accuracy. Using extensive simulations, we show that our proposal can effectively segment a complex 3-D sensor network and significantly improve the localization rate in comparison with existing solutions

Probability-based Distance Estimation Model for 3D DV-Hop Localization in WSNs

Localization is one of the pivotal issues in wireless sensor network applications. In 3D localization studies, most algorithms focus on enhancing the location prediction process, lacking theoretical derivation of the detection distance of an anchor node at the varying hops, engenders a localization performance bottleneck. To address this issue, we propose a probability-based average distance estimation (PADE) model that utilizes the probability distribution of node distances detected by an anchor node. The aim is to mathematically derive the average distances of nodes detected by an anchor node at different hops. First, we develop a probability-based maximum distance estimation (PMDE) model to calculate the upper bound of the distance detected by an anchor node. Then, we present the PADE model, which relies on the upper bound obtained of the distance by the PMDE model. Finally, the obtained average distance is used to construct a distance loss function, and it is embedded with the traditional distance loss function into a multi-objective genetic algorithm to predict the locations of unknown nodes. The experimental results demonstrate that the proposed method achieves state-of-the-art performance in random and multimodal distributed sensor networks. The average localization accuracy is improved by 3.49\%-12.66\% and 3.99%-22.34%, respectively