We propose two novel algorithms for distributed and location-free boundary recognition in wireless sensor networks. Both approaches enable a node to decide autonomously whether it is a boundary node, based solely on connectivity information of a small neighborhood. This makes our algorithms highly applicable for dynamic networks where nodes can move or become inoperative. We compare our algorithms qualitatively and quantitatively with several previous approaches. In extensive simulations, we consider various models and scenarios. Although our algorithms use less information than most other approaches, they produce significantly better results. They are very robust against variations in node degree and do not rely on simplified assumptions of the communication model. Moreover, they are much easier to implement on real sensor nodes than most existing approaches.
try to recognize boundary nodes by low node degree or similar statistical properties. As long as nodes are smoothly distributed, this works quite well as boundary nodes usually have less neighbors than interior nodes. However, as soon as node degrees fluctuate noticeably, most statistical approaches produce many misclassifications. Besides, these algorithms often require unrealistic high average node degrees. Prominent statistical approaches are Fekete @cite_18 @cite_1 , and Bi @cite_17 .
@cite_18 @cite_1 @cite_17
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The classical setting of community detection consists of networks exhibiting a clustered structure. To more accurately model real systems we consider a class of networks (i) whose edges may carry labels and (ii) which may lack a clustered structure. Specifically we assume that nodes possess latent attributes drawn from a general compact space and edges between two nodes are randomly generated and labeled according to some unknown distribution as a function of their latent attributes. Our goal is then to infer the edge label distributions from a partially observed network. We propose a computationally efficient spectral algorithm and show it allows for asymptotically correct inference when the average node degree could be as low as logarithmic in the total number of nodes. Conversely, if the average node degree is below a specific constant threshold, we show that no algorithm can achieve better inference than guessing without using the observations. As a byproduct of our analysis, we show that our model provides a general procedure to construct random graph models with a spectrum asymptotic to a pre-specified eigenvalue distribution such as a power-law distribution. This research focuses on distributed and localized algorithms for precise boundary detection in 3D wireless networks. Our objectives are in two folds. First, we aim to identify the nodes on the boundaries of a 3D network, which serve as a key attribute that characterizes the network, especially in such geographic exploration tasks as terrain and underwater reconnaissance. Second, we construct locally planarized 2-manifold surfaces for inner and outer boundaries, in order to enable available graph theory tools to be applied on 3D surfaces, such as embedding, localization, partition, and greedy routing among many others. To achieve the first objective, we propose a Unit Ball Fitting (UBF) algorithm that discovers a set of potential boundary nodes, followed by a refinement algorithm, named Isolated Fragment Filtering (IFF), which removes isolated nodes that are misinterpreted as boundary nodes by UBF. Based on the identified boundary nodes, we develop an algorithm that constructs a locally planarized triangular mesh surface for each 3D boundary. Our proposed scheme is localized, requiring information within one-hop neighborhood only. Our simulation results demonstrate that the proposed algorithms can effectively identify boundary nodes and surfaces, even under high measurement errors. As far as we know, this is the first work for discovering boundary nodes and constructing boundary surfaces in 3D wireless networks. We study the problem of detecting a structured, low-rank signal matrix corrupted with additive Gaussian noise. This includes clustering in a Gaussian mixture model, sparse PCA, and submatrix localization. Each of these problems is conjectured to exhibit a sharp information-theoretic threshold, below which the signal is too weak for any algorithm to detect. We derive upper and lower bounds on these thresholds by applying the first and second moment methods to the likelihood ratio between these "planted models" and null models where the signal matrix is zero. Our bounds differ by at most a factor of root two when the rank is large (in the clustering and submatrix localization problems, when the number of clusters or blocks is large) or the signal matrix is very sparse. Moreover, our upper bounds show that for each of these problems there is a significant regime where reliable detection is information- theoretically possible but where known algorithms such as PCA fail completely, since the spectrum of the observed matrix is uninformative. This regime is analogous to the conjectured 'hard but detectable' regime for community detection in sparse graphs. We consider a random sparse graph with bounded average degree, in which a subset of vertices has higher connectivity than the background. In particular, the average degree inside this subset of vertices is larger than outside (but still bounded). Given a realization of such graph, we aim at identifying the hidden subset of vertices. This can be regarded as a model for the problem of finding a tightly knitted community in a social network, or a cluster in a relational dataset. In this paper we present two sets of contributions: (i) We use the cavity method from spin glass theory to derive an exact phase diagram for the reconstruction problem. In particular, as the difference in edge probability increases, the problem undergoes two phase transitions, a static phase transition and a dynamic one. (ii) We establish rigorous bounds on the dynamic phase transition and prove that, above a certain threshold, a local algorithm (belief propagation) correctly identify most of the hidden set. Below the same threshold no local algorithm can achieve this goal. However, in this regime the subset can be identified by exhaustive search. For small hidden sets and large average degree, the phase transition for local algorithms takes an intriguingly simple form. Local algorithms succeed with high probability for ( _ in - _ out > _ out e ) and fail for ( _ in - _ out < _ out e ) (with ( _ in ), ( _ out ) the average degrees inside and outside the community). We argue that spectral algorithms are also ineffective in the latter regime. It is an open problem whether any polynomial time algorithms might succeed for ( _ in - _ out < _ out e ).