## 3 Social Relationship Measurement
### Definitions  
It is supposed that node \(i\) encounters node \(j\) at time \(t\), and node \(i\) can measure its social relationship \(S_{ij}(t)\) with node \(j\). \(S_{ij}(t)\) is the strength of the social relationship from node \(i\) to \(j\) at time \(t\), and \(E_{ij}(t)\) and \(H_{ij}(t)\) are explicit and implicit parts of \(S_{ij}(t)\), respectively. Then, \(S_{ij}(t)\) can be denoted as Equation (1), where \(W_{i}^{E}(t)\in[0,1]\) and \(W_{i}^{H}(t)\in[0,1]\) are used to adjust the weights of explicit and implicit social relationships for the strength of social relationships with \(W_{i}^{E}(t)+W_{i}^{H}(t)=1\). Depending on the frequency of node encounters, the weight functions \(W_{i}^{E}(t)\) and \(W_{i}^{H}(t)\) may vary over time:  
\[S_{ij}(t)=W_{i}^{E}(t)E_{ij}(t)+W_{i}^{H}(t)H_{ij}(t). \tag{1}\]  
In MONs, we suppose that each node has an encounter table, which stores node identification and encounter frequency during recent time \(T\). \(N_{i}(t)\) is the set of nodes encountered by node \(i\) during recent time \(T\) at time \(t\), and \(N_{ij}(t)=N_{i}(t)\cap N_{j}(t)\) is the set of nodes encountered by both nodes \(i\) and \(j\). We also say that \(N_{ij}(t)\) is the set of common friends of nodes \(i\) and \(j\) at time \(t\), or the circles of friends of node \(i\) and \(j\). In addition, \(f_{ij}(t)\) is the number of encounter times between node \(i\) and \(j\) during recent time \(T\) at time \(t\). Due to the symmetry of node encounters, we have \(N_{ij}(t)=N_{ji}(t)\) and \(f_{ij}(t)=f_{ji}(t)\). Observing social phenomena in our real lives, we find that if nodes \(i\) and \(j\) have social relationships with each other, they must meet one of the following two conditions: (1) they have direct contact with each other, and (2) they have a common circle of friends. In this paper, the two conditions above form Explicit Social Relationships (ExSRs) and Implicit Social Relationships (ImSRs), respectively. **Definition 1** (Strength of ExSR). : _The strength of ExSR (\(E_{ij}(t)\)) between nodes \(i\) and \(j\) at time \(t\) is denoted as Equation (2), where \(\mu\) is a constant integer and the threshold of encounter times. If the encounter times between nodes \(i\) and \(j\) is more than \(\mu\), the strength of ExSR will be equal to \(1\), which means a close ExSR between node \(i\) and \(j\). In general, \(\mu\) is an integer, set as 1.2 times the average encounters among all nodes. As shown in Equation (2), \(E_{ij}(t)\in[0,1]\) is determined by encounter times of nodes \(i\) and \(j\):_  
\[E_{ij}(t)=\frac{min(\mu,f_{ij}(t))}{\mu}. \tag{2}\]  
**Definition 2** (Strength of ImSR). : _The strength of ImSR (\(H_{ij}(t)\)) between nodes \(i\) and \(j\) at time \(t\) is denoted as Equations (3) and (4), where \(R_{ij}(t)\) is used to denote the encounter times between nodes \(i\) or \(j\) and their common circles of friends, and represents the indirect message delivery capability between nodes \(i\) and \(j\). Equation (3) has the same \(\mu\) as Equation (2). \(H_{ij}(t)\in[0,1]\) depends on the common circles of friends between node \(i\) and \(j\), and the bigger size and more frequent contact of common circles of friends will lead to a stronger ImSR with each other:_  
\[H_{ij}(t)=\frac{min(\mu,R_{ij}(t))}{\mu}, \tag{3}\]  
\[R_{ij}(t)=\sum_{k\in N_{ij}(t)}min(f_{ik}(t),f_{jk}(t)). \tag{4}\]  
As shown in Figure 1, nodes \(A\), \(B\) and \(C\) stand for [PERSON] and his parents, respectively, as is mentioned in Section 1, \(D\) is the mother of [PERSON]'s mother, and \(E\) is the mother of [PERSON]'s father. The encounter times between two nodes are marked with the solid lines. Supposing \(\mu=40\), according to Equations (3) and (4), we can get \(R_{DE}=min(10,10)+min(10,10)+min(10,10)=30\) and \(H_{DE}=\frac{30}{40}=0.75\) in Figure 0(a), and \(R_{DE}=min(8,10)+min(8,10)+min(8,10)=24\) and \(H_{DE}=\frac{24}{40}=0.6\) in Figure 0(b). Since \(D\) and \(E\) do not have direct contact at all, the communication between \(D\) and \(E\) should be relayed by \(A\), \(B\) or \(C\).