# Temporal Graphs and Temporal Network Characteristics for Bio-Inspired Networks during Optimization
## 4 Review of Related Work  
Temporal networks, also called time-ordered networks or dynamic networks, are general network structures containing timing information about the actions of and interactions between network members [63]. This added information can make analysis significantly more difficult, however the added complexity makes temporal networks natural choices for studying dynamic systems [1,2]. Temporal network analysis has been used to study a wide variety of topics, including anomalies in urban transportation, disease transmission, and video super-resolution [64, 65, 66]. Temporal Network Graphs (TNGs) can be constructed in several different ways. For our purposes here, it is convenient to think of them as a sequence of unweighted, static graphs, each with an accompanying adjacency matrix, defined in a standard manner for static networks, where each element can be written:  
\[a_{ij}(t)= \begin{cases}1,&\text{if node $i$ is connected to $j$ at time $t$}\\ 0,&\text{otherwise}\end{cases}  \tag{1}\]  
The TNG representation is powerful, as it allows us to utilize familiar static network theoretic techniques on graphs within the sequence, specifically for each step of the sequence in time evolution of the network. Understanding the interplay between the underlying dynamics and the temporal sampling rate is critical to most temporal network analysis [1, 2, 3]. Both the Voronoi-only approach and the GA + Voronoi operate at discrete time steps, giving us a natural starting point for sampling times. We then construct TNGs for all trials as the sequence of static graphs occurring at each time step. Figure 2: Snapshots of a WSN generated by (**upper**) the Voronoi-only algorithm and (**lower**) the GA-BISON algorithm in an obstacle-free environment, at different time steps. ND signifies noise level (chosen from \(0.01,0.05,0.1\)). See Equation (2). Temporal network theory has been explored as a means of identifying shortest time-respecting paths between nodes, making it useful in the development of routing protocols for WSNs [67; 68; 69]. It also provides statistical analysis on the average number of hops and members required from source to destination. Temporal network analysis can also reveal members that are involved in shortest paths, thus playing an important role in mediating the data flow in the network [9]. Moreover, temporal and centrality measures are combined in static WSN to decide their best placement based on their response to changes in the surrounding environment [15; 70]. Regarding TNG measures and statistics, we consider two broad categories: (a) static measures recorded for individual graphs within the time-ordered sequence, referred to here as time traces, and (b) temporal measures, constructed by looking at a TNG as a single object. Given the added complexity of TNGs, as well as the relative newness of the field, it has been difficult to establish a consensus around the use and definition of measures of type (b) [1; 3]. We chose to focus a bit more on measures of type (a), largely for the reasons stated above, but we also recognize that future work utilizing measures of type (b) has significant merit. This being said, we did extract the distribution function of the length connections from the TNGs in the case of our simulated physical situations. Overall, this type of work may also include utilizing aggregate graphs [71; 72], supra-centrality measures [73; 74], and phase transition analysis [75]. Static centrality measures are commonly used network measures for categorizing and ranking nodes within a static network, and there exists a large variety of possible measures that can be employed [76]. In WSNs, various centralities have been introduced to measure connectivity, clustering/localization, data flow and energy expenditure [67; 68; 69; 77; 78; 79]. Still other centrality measures are used to show the robustness of the network against errors, particularly in the face of node failure [80].