text: This paper presents a tractable analytical framework for the exact calculation of probability of node isolation and minimum node degree distribution when @math sensor nodes are independently and uniformly distributed inside a finite square region. The proposed framework can accurately account for the boundary effects by partitioning the square into subregions, based on the transmission range and the node location. We show that for each subregion, the probability that a random node falls inside a disk centered at an arbitrary node located in that subregion can be expressed analytically in closed-form. Using the results for the different subregions, we obtain the exact probability of node isolation and minimum node degree distribution that serves as an upper bound for the probability of @math -connectivity. Our theoretical framework is validated by comparison with the simulation results and shows that the minimum node degree distribution serves as a tight upper bound for the probability of @math -connectivity. The proposed framework provides a very useful tool to accurately account for the boundary effects in the design of finite wireless networks.
  @cite_18: A range assignment to the nodes in a wireless ad hoc network induces a topology in which there is an edge between two nodes if and only if both of them are within each other's transmission range. The critical transmission radius for k-connectivity is the smallest r such that if all nodes have the transmission radius r, the induced topology is k-connected. In this paper, we study the asymptotic critical transmission radius for k-connectivity in a wireless ad hoc network whose nodes are uniformly and independently distributed in a unit-area square or disk. We provide a precise asymptotic distribution of the critical transmission radius for k-connectivity. In addition, the critical neighbor number for k-connectivity is the smallest integer l such that if every node sets its transmission radius equal to the distance between itself and its l-th nearest neighbor, the induced (symmetric) topology is k-connected. Applying the critical transmission radius for k-connectivity, we can obtain an asymptotic almost sure upper bound on the critical neighbor number for k-connectivity.
  @cite_14: This paper investigates two fundamental characteristics of a wireless multi -hop network: its minimum node degree and its k--connectivity. Both topology attributes depend on the spatial distribution of the nodes and their transmission range. Using typical modeling assumptions :--- :a random uniform distribution of the nodes and a simple link model :--- :we derive an analytical expression that enables the determination of the required range r0 that creates, for a given node density ρ, an almost surely k--connected network. Equivalently, if the maximum r0 of the nodes is given, we can find out how many nodes are needed to cover a certain area with a k--connected network. We also investigate these questions by various simulations and thereby verify our analytical expressions. Finally, the impact of mobility is discussed.The results of this paper are of practical value for researchers in this area, e.g., if they set the parameters in a network--level simulation of a mobile ad hoc network or if they design a wireless sensor network.
  @cite_9: In this paper we present a novel mathematical approach to evaluate the degree of connectivity of a multi-sink wireless sensor network. We consider both unbounded and bounded domains, specifically squares and rectangles, and the impact of border effects is also shown. Our channel model takes random fluctuations into account, as well as a distance-dependent deterministic path-loss. In particular, we deal with randomly shaped wireless footprints, rather than with the less realistic (yet widely adopted) disk model, which is a special case of our channel model. Our performance metric is based on the probability that a randomly chosen sensor is not isolated, from which the probability that a certain amount of nodes are connected can be easily derived.
  @cite_3: We develop a cluster expansion for the probability of full connectivity of high density random networks in confined geometries. In contrast to percolation phenomena at lower densities, boundary effects, which have previously been largely neglected, are not only relevant but dominant. We derive general analytical formulas that show a persistence of universality in a different form to percolation theory, and provide numerical confirmation. We also demonstrate the simplicity of our approach in three simple but instructive examples and discuss the practical benefits of its application to different models.
  @cite_24: This paper presents a framework for the calculation of stochastic connectivity properties of wireless multihop networks. Assuming that n nodes, each node with transmission range r0, are distributed according to some spatial probability density function, we study the level of connectivity of the resulting network topology from three viewpoints. First, we analyze the number of neighbors of a given node. Second, we study the probability that there is a communication path between two given nodes. Third, we investigate the probability that the entire network is connected, i.e. each node can communicate with every other node via a multihop path. For the last-mentioned issue, we compute a tight approximation for the critical (r0 ,n )pairs that are required to keep the network connected with a probability close to one. In fact, the problem is solved for the general case of a k-connected network, accounting for the robustness against node failures. These issues are studied for uniformly distributed nodes (with and without ‘border effects’), Gaussian distributed nodes, and nodes that move according to the commonly used random waypoint mobility model. The results are of practical value for the design and simulation of wireless sensor and mobile ad hoc networks.
  @cite_23: Many analytic results for the connectivity, coverage, and capacity of wireless networks have been reported for the case where the number of nodes, n, tends to infinity (large-scale networks). The majority of these results have not been extended for small or moderate values of n; whereas in many practical networks, n is not very large. In this article, we consider finite (small-scale) wireless sensor networks. We first show that previous asymptotic results provide poor approximations for such networks. We provide a set of differences between small-scale and large-scale analysis and propose a methodology for analysis of finite sensor networks. Furthermore, we consider two models for such networks: unreliable sensor grids and sensor networks with random node deployment. We provide easily computable expressions for bounds on the coverage and connectivity of these networks. With validation from simulations, we show that the derived analytic expressions give very good estimates of such quantities for finite sensor networks. Our investigation confirms the fact that small-scale networks possess unique characteristics different from their large-scale counterparts, necessitating the development of a new framework for their analysis and design.
  @cite_16: In wireless networks, knowledge of internode distances is essential for performance analysis and protocol design. When determining distance distributions in random networks, the underlying nodal arrangement is almost universally taken to be a stationary Poisson point process. While this may be a good approximation in some cases, there are also certain shortcomings to this model, such as the fact that, in practical networks, the number of nodes in disjoint areas is not independent. This paper considers a more-realistic network model where a known and fixed number of nodes are independently distributed in a given region and characterizes the distribution of the Euclidean internode distances. The key finding is that, when the nodes are uniformly randomly placed inside a ball of arbitrary dimensions, the probability density function (pdf) of the internode distances follows a generalized beta distribution. This result is applied to study wireless network characteristics such as energy consumption, interference, outage, and connectivity.
  @cite_20: Connectivity and capacity are two fundamental properties of wireless multihop networks. The scalability of these properties has been a primary concern for which asymptotic analysis is a useful tool. Three related but logically distinct network models are often considered in asymptotic analyses, viz. the dense network model, the extended network model, and the infinite network model, which consider respectively a network deployed in a fixed finite area with a sufficiently large node density, a network deployed in a sufficiently large area with a fixed node density, and a network deployed in R2 with a sufficiently large node density. The infinite network model originated from continuum percolation theory and asymptotic results obtained from the infinite network model have often been applied to the dense and extended networks. In this paper, through two case studies related to network connectivity on the expected number of isolated nodes and on the vanishing of components of finite order k > 1 respectively, we demonstrate some subtle but important differences between the infinite network model and the dense and extended network models. Therefore, extra scrutiny has to be used in order for the results obtained from the infinite network model to be applicable to the dense and extended network models. Asymptotic results are also obtained on the expected number of isolated nodes, the vanishingly small impact of the boundary effect on the number of isolated nodes, and the vanishing of components of finite order k > 1 in the dense and extended network models using a generic random connection model.
  @cite_11: Sensors typically have limited power of transmission. Thus desired connectivity probability (CP) between the sensors is critical in the given de- ployment area. This paper presents a novel approach for analyzing the lower bound of CP for sensor networks under uniformly distribution of the sensors in a given area.
summary: Since many practical multi-hop networks are formed by distributing a finite number of nodes in a finite area, there has been an increasing interest to model and determine the connectivity properties in finite multi-hop networks @cite_14 @cite_24 @cite_11 @cite_9 @cite_18 @cite_16 @cite_23 @cite_20 @cite_3 . This is also due to the fact, established earlier in @cite_14 @cite_24 and recently in @cite_23 , that the asymptotic connectivity results for large-scale networks provide an extremely poor approximation for finite wireless networks. This poor approximation is due to the boundary effects experienced by the nodes near the borders of the finite region over which the nodes are deployed. Since the nodes located close to the physical boundaries of the network have a limited coverage area, they have a greater probability of isolation. Therefore, the boundary effects play an important role in determining the overall network connectivity.