Opinion Impact Models and Opinion Consensus Methods in Ad Hoc Tactical Social Networks

Ad hoc social networks are special social networks, such as ad hoc tactical social networks, ad hoc firefighter social networks, and ad hoc vehicular social networks. The social networks possess both the properties of ad hoc network and social network. One of the challenge problems in ad hoc social networks is opinion impact and consensus, and the opinion impact plays a key role for information fusion anddecision support in ad hoc social networks. In this paper, consider the impact of physical and logical distance on the opinions of individuals or nodes in heterogeneous social networks; we present a general opinion impact model, discuss the local and global opinion impact models in detail, and point out the relationship between the local opinion impact model and the global opinion impact model. For understanding the opinion impact models easily, we use the general opinion impact model to ad hoc tactical social networks and discuss the opinion impact and opinion consensus for ad hoc tactical social networks in the end.


Introduction
Ad hoc social networks are special social networks [1].They possess both the properties of ad hoc networks and social networks, such as the properties of self-organization, decentralization, multihop communication, structure influence, opinion impact, and small world.The ad hoc social networks are applied in many fields, such as ad hoc tactical social networks [2], ad hoc firefighter social networks [3], and ad hoc vehicular social networks [4].Lewenstein et al. [5] presented a social impact theory.Individuals are assumed to exchange, compare, adjust, and influence each other's attitudes.The total impact   that the th individual experiences from his or her social environment is a function of the persuasive impact of those individuals who hold the opposite opinion to the th individual, relative to the supportive impact of those individuals who share the opinion.
In the paper [5,6], the dynamics of the opinion is governed by the following rule: where   denotes the connection between the nodes , ,   () is the impact of the node  on , and ℎ  () is the noise of the system.Considering the properties of ad hoc social networks, we improve the impact model as follows.
Huang et al. [6] assumed that in the social networks, each of the nodes holds one of the two opposite opinions denoted by 1, −1.To illustrate the social impact of the community, the initial distributions of the opinion are as follows: where   () denotes the opinion of the th node at the time step  and   (0) denotes the opinion of the th node at the time step 0. We know that in real ad hoc social networks, the opinion   () of  may not be 1 or −1 and may be a fuzzy opinion or fuzzy number, such as triangular fuzzy number.For example, use [−0.For ad hoc social networks, especially for scale-not-free social networks, the connection   may not be simply 1 or 0 but should be approximately inversely proportional to the physical distance or the hop number of communication between  and .In other words, the smaller the hop number is, the stronger the connection between  and  is.Considering the change of dynamic topology with the time in ad hoc social networks, the physical location or distance also change; in this paper, we assume that   is a function of time and   () = 1/(ℎ  () + 1), where ℎ  () is the hop number of communication from  to  at time .
Ad hoc networks are usually assumed to be homogeneous, where each node shares the same radio capacity.However, homogeneous ad hoc networks suffer from poor scalability.Recent research has demonstrated its bottle neck performance through both theoretical analysis and simulation experiments and test bed measurements.Xu et al. [7] presented a design methodology to build a hierarchical largescale ad hoc network using different types of radio capabilities at different layers.In such a structure, nodes are dynamically grouped into multihop clusters.Each group elects a clusterhead to be a backbone node (BN).Then, higher-level links are established to connect the BNs into a backbone network.The backbone nodes have stronger social power or impact than the others.So, the backbone nodes have stronger opinion impact on the others.Illuminated in [8,9], we use the social power factor   = (  )  to describe social power strength, where   is the centrality of node  and  is a parameter controlling the social diversity.So, for   () in (1), considering the properties of ad hoc social networks, it may be approximately proportion to social power factor   = (  )  and inversely proportional to the logical distance or level distance between node  and .So, we may define   () =   /(|level() − level()| + 1), where level(), level() denote the levels of the node ,  live in respectively.
By above discussion, (1) can be rewritten as follows: We make some remarks to easily understand (3).
Remark 1.In (3), physical distance is a Euclid distance between node  and .For easy calculation, usually the physical distance is measured by the hop number of communication between  and .In other words, the hop number is approximately the number of nodes which connect node  to node , and the logical distance in (3) is a level distance between node  and .For easy calculation, especially in hierarchical network with different levels, usually the logical distance is measured by different levels between node  and node .
Remark 2. The difference between the physical distance and logical distance is that the physical distance is only considering the communication distance and not considering the relationship between the nodes, but the logical distance is considering the relationship and the social power between the nodes especially in hierarchical network.
Remark 3. In (3), physical distance mainly reflects the influence between nodes in the same level, and logical distance mainly reflects the influence between nodes in different levels.The impact of logical distance in (3) acts as an amplifier to physical distance.The impact mainly emphasizes the influence of backbone nodes and social powerful nodes.
To the best of our knowledge, there are no any papers using physical distance and logical distance that discuss opinion impact model.
The rest of the paper is organized as follows.In Section 2, we present the local and global opinion impact models and point out that the local opinion impact model is part of the global opinion impact model.By applying the models to ad hoc tactical social networks, in Section 3, we discuss the opinion impact.In Section 4, we give the opinion consensus for ad hoc tactical social networks.And we conclude in Section 5.

Local and Global Opinion Impact Models
By the general opinion impact models in Section 1, in the following, we discuss the local opinion impact models for nodes in the same level and global opinion impact models for nodes in different levels in detail.

The Local Opinion Impact Model.
In this subsection, we only consider local opinion consensus in the same level, and   () =   .Assume   nodes in the same level with the node , so (3) can be rewritten as If we only consider the neighbor nodes of , we have that ℎ  () = 1, and ( If we only consider the neighbor nodes of , and neighbors' neighbor of  we have In general, assume that the maximum hop number is   for node  in the same level, then ()   , (7) where    are the number of neighbors of .The physical distance between those neighbors and  is  hops,  = 1, 2, . . ., , and   1 +   2 + ⋅ ⋅ ⋅ +    =   .

The Global Opinion Impact Model.
For global opinion impact, consider the different levels; from (3), we get that where     is the number of neighbors of the node .The logical distance between those neighbors and  is   hops,  = 1, 2, . . ., , and   1 +   2 + ⋅ ⋅ ⋅ +    = .Here,   1 =   in local opinion impact model.So, we can see easily that the local opinion impact model is part of the global opinion impact model.

The Opinion Impact Models for Tactical Social Networks
In Section 2, we discussed generally the local and global opinion impact models.In this section, we apply the results in Section 2 to the special ad hoc social networks, ad hoc tactical social networks.Lihui Gu et al. [2] discussed the characteristics of tactical environment and showed the architecture of multilevel heterogeneous ad hoc wireless networks with Unmanned Aerial Vehicles (UAVs).The hierarchical infrastructure (Figure 1) reflected the three layers, including level 1: ground ad hoc wireless networks; Level 2: ground embedded mobile backbone networks; and level 3: aerial mobile backbone networks.
In Figure 1, in level 1, there are three groups or clusters.The largest group, denoted by  1 , includes eight soldiers; the other two groups, denoted by  2 and  3 , each includes five soldiers.In level 2, there are also three groups or clusters.The largest group includes three tanks, denoted by  4 ; the other two groups, denoted by  5 and  6 , each includes one tank.In level 3, there is only one group, denoted  7 , which includes three planes.So, the total number of nodes is  = 18 + 5 + 3 = 26.For simplicity, we neglect the impact of centrality and discuss all situations as follows.
When node  is in the group  1 , the impacting weight of  1 is between 1/104 and 1/78; node  is in  2 and  3 , the impacting weights are 1/156; node  is in  4 , the impacting weight is between 1/156 and 1/104; node  is in  5 and  6 , the impacting weight is 1/260; node  is in  7 , the impacting weight is between 1/312 and 1/234.  () .

Opinion Consensus for the Ad Hoc Tactical Social Networks
In this section, based on the opinion models in Section 3, we discuss the opinion consensus of soldiers in an ad hoc tactical social network for six situations in detail.
Situation 1.When  ∈  1 and  ∈  1 , from (11), we transform the opinion consensus into vectors consensus, every components in a vector is the impacting weights of groups (from one to seven), respectively.So, the opinion consensus in time  +