# Temporal Graphs and Temporal Network Characteristics for Bio-Inspired Networks during Optimization
## 7 Centrality Measure  
The motivation of this work is driven by the need to understand and improve the network coverage in the context of the spatial, temporal and energy constraints. Specifically, implementations are often driven by  
1. cumulative distance traveled (CDT) by the nodes,
2. percent area coverage (PAC) that nodes manage to cover by the signal,
3. minimal time to achieve target coverage, and
4. optimized energy allowed for the network of drones. The analysis so far illustrates how uneven the roles of most nodes are during the deployment process. We also noted that as the deployment progresses, the nodes that have few links can quickly gather many links, indicating that much temporal switching occurs within the networks. With the goal of initial quantification of the role of individual nodes and the time-dependent role they play, we determined eigenvector centrality (\(EC\)) [76], one of the leading measures that help understands the evolving nature of temporal network graphs. We select eigenvector centrality as it is sufficiently general (typically defined as a generalization of Katz index [76]), applicable to smaller size networks (N \(\approx\) 30-100 in most our cases), and is readily obtainable from the adjacency matrices. It measures the importance of a node based on the importance of its (connected) neighbors. Comparing our values with [67] we notice that local centrality measures (our degree centrality measures and their betweenness centrality measures) provide high spike values  
Figure 5: Temporal evolution of the deviation from regularity for Voronoi-only and GA + Voronoi approaches under different environmental conditions. We note the need to use log scale for the time axis indicating an order of magnitude difference between the characteristic time scale for two methods. The measure \(\Delta_{\text{reg}}\) is defined in Equation (4), above. (**a**) Comparison of BISON only and GA+BISON performance with varied noise but without scatterers. (**b**) Comparison of BISON only and GA+BISON performance with varied noise, and in the presence of scaterrers. for nodes that are more connected to their neighbors than others, while global centralities (our eigenvector centrality measures and their closeness centrality measures) have a smooth distribution of values among the nodes. We also observed that the differences in the eigenvector centralities of the nodes in the network at different time steps revealed the uneven role of most of the nodes in the network during their deployment process, which is considered an important finding to be used later to keep track of the important nodes that can be relied on to transfer data packets among the network members. We present eigenvector centrality measurements for the various BISON cases in Figure 6 for the selected environmental cases. Nodes with early injection time immediately spike in importance, but drop off as deployment continues. By contrast, nodes deployed later spike early as well, and then maintain some non-zero importance throughout their lifetime. This initial surge is to be expected, as nodes are injected from the same corner, resulting in an area with a high node density. As spreading continues, early nodes migrate to distant regions, and thus have little opportunity to be deeply connected, while later nodes remain relatively close to this highly dense region. These EC findings also clearly support our previous insight that adding noise does, in fact, have a beneficial impact on the performance of Voronoi-only generated networks with regard to deployment speed. When comparing plots from simulations differing only in noise level, we consistently see that the presence of noise smooths the EC time trace curve, suggesting a more homogeneous distribution of the nodes. We see somewhat similar spiking behavior in the GA + Voronoi trials featured in Figure 7, although early deployment nodes are more likely to feature significant spikes in EC later in their lifetimes.