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In wireless sensor networks (WSNs), location information plays an important role in many fundamental services which includes geographic routing, target tracking, location-based coverage, topology control, and others. One promising approach in sensor network localization is the determination of location based on hop counts. A critical priori of this approach that directly influences the accuracy of location estimation is the hop-distance relationship. However, most of the related works on the hop-distance relationship assume the unit-disk graph (UDG) model that is unrealistic in a practical scenario. In this paper, we formulate the hop-distance relationship for quasi-UDG model in WSNs where sensor nodes are randomly and independently deployed in a circular region based on a Poisson point process. Different from the UDG model, quasi-UDG model has the non-uniformity property for connectivity. We derive an approximated recursive expression for the probability of the hop count with a given geographic distance. The border effect and dependence problem are also taken into consideration. Furthermore, we give the expressions describing the distribution of distance with known hop counts for inner nodes and those suffered from the border effect where we discover the insignificance of the border effect. The analytical results are validated by simulations showing the accuracy of the employed approximation. Besides, we demonstrate the localization application of the formulated relationship and show the accuracy improvement in the WSN localization.

Wormhole attack is one of the most devastating threats for range-free localization in wireless sensor networks. To address this issue, we propose a robust localization scheme in wireless sensor networks against wormhole attack, called ConSetLoc, which neither complicated distance measuring devices nor extra complex encrypting algorithms is necessary. With the relationship between hop counts and geographic distance of sensor nodes in the deployment territory, we design a partition method of consistent anchors sets by the convex constraints in geometry which can reduce the effect of bad measurements on estimates,  and then present a filtering strategy for the candidate locations with these anchors sets. In addition, we conduct simulation experiments for performance evaluation and the results demonstrate the proposed ConSetLoc can estimate the locations for most of sensor nodes with good accuracy and stability when wormhole attack exists in the network