Temporal complex networks modeling applied to vehicular ad-hoc networks

Abstract

VANETs solutions use aggregated graph representation to model the interaction among the vehicles and different aggregated complex network measures to quantify some topological characteristics. This modeling ignores the temporal interactions between the cars, causing loss of information or unrealistic behavior. This work proposes the use of both temporal graphs and temporal measures to model VANETs applications. To verify the viability of this model, we initially perform a comparative analysis between the temporal and aggregated modeling considering five different real datasets. This analysis shows that the aggregated model is inefficient in modeling the temporal aspects of networks. After that, we perform a network evaluation through a simulation by considering the impact of temporal modeling applied to the deployment of RSUs. First, we compare a solution based on our temporal modeling with a greedy algorithm based on an aggregated model to choose the positions of RSUs. In a scenario with 70 RSUs, we have 77% and 65% of coverage in the temporal and aggregated model (greedy algorithm), respectively. Second, we evaluate the use of aggregated and temporal measures applied as features in a genetic algorithm. The approach with temporal betweenness had the better result with 90% of the coverage area against 61% of aggregated one applied to the same scenario.

Introduction

A Vehicular ad hoc network (VANET) (Naboulsi and Fiore, 2017) consists of groups of moving or stationary vehicles connected by a wireless network. According to their communication radius, the communication occurs among the cars or through a fixed infrastructure along the roads. In both cases, the connection patterns between their elements have a dynamic behavior with the insertion and removal of links over time. It is usual to use a graph representation to model the interaction among the vehicles and different complex network measures to quantify some topological characteristics (Newman, 2010). These measures represent a mathematical and computational framework to understand better non-trivial topological features, like the dynamics of growing a network over time, and to enable the characterization, analysis, and modeling of the network topology based on different features like connectivity centrality, cycles, and distances.

Several VANETs applications and network infrastructure solutions use graph modeling and its measures as a static representation called aggregated modeling (Fiore and Härri, 2008, Diniz et al., 2020, Glacet et al., 2015). In this modeling, all quick contacts among the vehicles are permanent connections. For example, consider that we observed three vehicles $v_1$, $v_2$, and $v_3$ on a VANET for ten minutes at one-minute intervals and obtained the following sequence of contacts between them: (i) $v_1$ with $v_2$ only during minute 1, (ii) $v_1$ with $v_3$ between minutes 2 and 5, and (iii) $v_2$ with $v_3$ between minutes 6 and 10. In this case, the aggregated modeling produces a complete graph among the three vehicles, and, consequently, we calculate the aggregated measures directly over this aggregated graph. This modeling ignores the temporal interactions between the vehicles, causing loss of information or unrealistic behavior.

Some applications and infrastructure solutions adopt temporal modeling to represent the graph and preserve the temporal characteristics (Qiao et al., 2017). In this case, quick contacts among the vehicles generate connections in a time interval $t$, getting different graphs. Considering the previous example with three vehicles, the sequence of contacts in the temporal modeling produces temporal graphs with three disjoint edges $(v_1,v_2)$ at minute 1, $(v_1,v_3)$ during minutes 2 and 5 and $(v_2,v_3)$ during minutes 5 and 10. However, the solutions calculate the average aggregated measures for all graphs in time interval t, losing global time information about the network.

In this way, the novelty of our proposal is the use of temporal models in conjunction with temporal measures to model and evaluate VANETs. To the best of our knowledge, we are the only ones that use these measures in VANETs scenarios. We use the temporal measures proposed by Kim and Anderson (2012): Degree, Betweenness, and Closeness. These measures relate node’s position and its ability to disseminate information efficiently in the network. In addition, they preserve the temporal relationships of the network. We perform a comparative analysis between the complete temporal (graph and measures) and aggregated modeling based on these measures.

To quantify the impact of temporal modeling compared with the aggregated one, we consider four static analyses: i. Quantification of the number of vertices and edges; ii. Kolmogorov–Smirnov test (KS-test) (Massey, 1951); iii. Hellinger distance (Basu et al., 2010); and iv. Visual scatter plot behavior. In all cases, we apply the temporal model to follow real scenarios: Cologne dataset (Naboulsi and Fiore, 2013) representing the car traffic in an urban area of Cologne, Germany; Motorway M40 (Gramaglia et al., 2016) representing part of the intermediate layer of the Madrid city; Autovía A6 (Lébre et al., 2015) representing the Motorway that connects the city of A Corunã to the city of Madrid; Créteil 7 am–9 am and Creteil 5 pm–7 pm Lébre et al. (2015) representing Créteil, Val-de-Marne (94) in France. The results show that the aggregated model is inefficient in modeling the network’s temporal aspects.

We also perform a network evaluation through a simulation by considering the deployment of Road Side Units (RSU) application (Moura et al., 2018). In this evaluation, we use both temporal and aggregated models to extract the features used by each algorithm evaluated and compare them using the temporal and aggregated modeling. We use only the Cologne scenario to perform this evaluation because it is a large-scale dataset that comprises more than 250.000 vehicle routes with varied road traffic conditions. We first compare our strategy with a greedy algorithm to choose the RSUs’ positions. This algorithm uses a contact matrix $\mathcal{T}$ to perform the solution. Thus, we generate $\mathcal{T}$ based on aggregated and temporal graphs. In a scenario with 70 RSUs, we have 77% and 65% of coverage in the temporal and aggregated model, respectively. After that, we compare aggregated modeling against the temporal ones as features in the genetic algorithm proposed by Moura et al. (2018). This algorithm considers a preprocessing based on different centrality measures. Thus, we evaluate both aggregated and temporal measures in specifics scenarios. The approach with temporal betweenness had the best result with 90% of the coverage area against 61% of aggregated one applied to the same scenario. The evaluation showed that the temporal model is adequate because it truly captures the network behavior.

We organized the remainder of this paper as follows: Section 2 gives an overview of the main related work. Section 3 describes the temporal VANETs topology modeling. Section 4 shows the evaluations and experiments. Finally, Section 5 presents the conclusion and future work.