## 2 Model and Method
### Dynamic Model of a Complex Network  
A Dynamic Complex Network (DCN) is made up of several nodes, which stand in for individual devices, and edges, which represent the connections or temporal interactions between them. Traditional static complex network models cannot adequately capture such dynamic evolution since the topology and device placements coevolve over time. The time-ordered network was suggested by the authors of [17] to transform a dynamic network into a static network with directed flows. The authors of [26] analyzed the uniformity of device behavior over time. We build upon their work and propose a dynamic network model that captures the evolving nature of a DCN. Our objective is to predict the behavioral trends of devices in the network and measure their QoS parameters. The authors of [26] rely on Shannon entropy to verify the uniformity of device behavior, whereas we go further by conducting regression studies to analyze not only uniformity but also the trend of node behavior and directly apply it to QoS metrics in opportunistic networks with low connection density. By considering the temporal sequence, length, and correlations between connections or devices happening at various moments in time, our model seeks to give a clear and thorough framework for understanding the evolutionary patterns of the network. Our model illustrates the evolution of interactions between devices in a DCN over a certain time period by using a series of snapshots. We shall outline the basic concepts of the dynamic network model in the parts below:  
Define a finite set of devices \(V\) (nodes) and a set of connections \(E\) (edges) between these devices. The connections between devices are assumed to take place over a time span \(T\). We use \(L\) to denote the duration of each spatial snapshot (or time window size), and \(F_{T}=\frac{T}{L}\) represents the number of spatial snapshots during the time span T. The dynamics of the network can be subsequently described by \(G_{T}=(V_{T},E_{T},M_{T})\),  
where  
\(V_{T}\subseteq V\) is the collection of all networked devices throughout the duration of \(T\). \(E_{T}\subseteq E\) is the set of edges that stands for connections between devices throughout the course of time T.  
\(M_{T}\) is a sequence of connectivity matrices that record contact events of devices during the time span \(T\). A discretized collection of static complex networks, \(G_{T}=\left\{G_{1},G_{2},\ldots,G_{F_{T}}\right\}\), can be used to simulate a DCN. In this model, the edges in each connection matrix are not binary as they are in the adjacency matrix of an unweighted graph. The connectivity matrix's edge weights, which range from 0 to \(F_{T}\), indicate how frequently points of contact occur. For the sake of clarity, we provide the following example in Figure 1, where the network's aggregated view is represented by \(G_{T}\). In this scenario, two devices are said to be connected if they have made contact within a time interval \(t_{i}\). The network snapshots are denoted as \(\left\{G_{1},G_{2},\ldots,G_{10}\right\}\). All information from both geographical and temporal data is included in this network's representation. Figure 2 displays the connection matrixes in order. The connectivity matrix is symmetric since each snapshot is an undirected graph with a connection between devices denoting the presence of a contact link in both directions. For instance, the weight of \(M_{AC}\) in \(M_{1}\) is 0, indicating that device \(A\) and device \(C\) did not make contact during time unit \(t_{1}\). One link between device \(A\) and device \(C\) was represented by the weight of \(M_{AC}\) in \(M_{8}\), which is 1, during the time unit \(t_{8}\). The costs of the routes between a source and a destination in the domain \(G_{T}\) can be represented as a function of delay, \(\left\{\delta_{ij}x_{ij}\right\}_{t_{ij}}\), if the objective is to minimize the message delivery delay, or as a function of load, \(\left\{\lambda x_{ij}\right\}_{t_{ij}}\), if the objective is to minimize the overhead.