Due to the fact that there is no fixed infrastructure or centralized management in ad hoc wireless networks, a Connected Dominating Set (CDS) of the graph representing the network is widely used as the virtual backbone of the network. Constructing a minimum CDS is NP-hard. In this paper, we propose a completely localized distributed algorithm which is r-CDS to construct a CDS with constant performance ratio. Our algorithm is better at maintenance since there is no need to build a tree or select a leader which are the common methods used in most of the distributed but serialized algorithms. We also compare our algorithm with other localized algorithms. The theoretical and simulation results show that our algorithm has a better performance ratio and constructs a CDS with smaller size in most cases.

a b s t r a c tIn this paper, we investigate the positive influence dominating set (PIDS) which has applications in social networks. We prove that PIDS is APX-hard and propose a greedy algorithm with an approximation ratio of H(δ) where H is the harmonic function and δ is the maximum vertex degree of the graph representing a social network.

Abstract-Since there is no fixed infrastructure or centralized management in wireless ad hoc networks, a Connected Dominating Set (CDS) has been proposed to serve as a virtual backbone. The CDS of a graph representing a network has a significant impact on the efficient design of routing protocols in wireless networks. This problem has been studied extensively in Unit Disk Graphs (UDG), in which all nodes have the same transmission ranges. However, in practice, the transmission ranges of all nodes are not necessarily equal. In this paper, we model a network as a disk graph and introduce the CDS problem in disk graphs. We present two efficient approximation algorithms to obtain a minimum CDS. The performance ratio of these algorithms is constant if the ratio of the maximum transmission range over the minimum transmission range in the network is bounded. These algorithms can be implemented as distributed algorithms. Furthermore, we show a size relationship between a maximal independent set and a CDS as well as a bound of the maximum number of independent neighbors of a node in disk graphs. The theoretical analysis and simulation results are also presented to verify our approaches.

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Abstract. Online social network has developed significantly in recent years as a medium of communicating, sharing and disseminating information and spreading influence. Most of current research has been on understanding the property of online social network and utilizing it to spread information and ideas. In this paper, we explored the problem of how to utilize online social networks to help alleviate social problems in the physical world, for example, the drinking, smoking, and drug related problems. We proposed a Positive Influence Dominating Set (PIDS) selection algorithm and analyzed its effect on a real online social network data set through simulations. By comparing the size and the average positive degree of PIDS with those of a 1-dominating set, we found that by strategically choosing 26% more people into the PIDS to participate in the intervention program, the average positive degree increases by approximately 3.3 times. In terms of the application, this result implies that by moderately increasing the participation related cost, the probability of positive influencing the whole community through the intervention program is significantly higher. We also discovered that a power law graph has empirically larger dominating sets (both the PIDS and 1-dominating set) than a random graph does.