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k-Neigh is a basic neighbor-based topology control protocol based on the construction of k-neighbor graph as logical communication graph. Several topological aspects of the constructed topology which are crucial to the performance of the protocol are not yet analytically investigated. Moreover, the problem of determining the maximum "Hello" interval preserving the connectivity with high probability has not been extensively addressed yet. Since execution of the protocol is a resource consuming task, this problem is of great importance on the performance of the protocol in sense of power consumption and topology control overhead. In this paper, first, several topological properties of the constructed topology for a static network, e.g. average logical degree and average final transmission range, are investigated analytically. Then, temporal properties of the dynamic topology in presence of mobility are studied based on two phenomena, one concerning a connectivity phase transition with time which is reported for the first time and the other one concerning the average degree of the logical communication graph at the time of starting the phase transition defined as disconnection degree. Using the obtained results one may tune the protocol in sense of "Hello" interval, maximum transmission power, or number of neighbors (k). Although the present work considers k-Neigh protocol, many interesting quantities are derived which may have application beyond the specific problem considered in this paper, e.g. the cdf, mean, and variance of the distance to the ith neighbor.