# Predicting the Aggregate Mobility of a Vehicle Fleet within a City Graph
## 7 Concluding Remarks and Future Work  
The motivation behind this paper is to study the capabilities and limitations of GNN models in the prediction of aggregate mobility of a vehicle fleet in a city using only historical aggregate mobility information. An evaluation scenario is proposed, where asset mobility is governed by a Markovian model that is formalized and described in detail. Three schemes are proposed and evaluated. Two of these approaches exploit the underlying  
Figure 1: Errors for each model and type of dataset. graph structure used to define vehicle mobility in a city. One of them further exploits the assumption of an underlying Markov model. The first method is based on the estimation of the transition probability matrix of a Markov model that is supposed to characterize the mobility behavior of each individual. This estimation-based method shows good performance in scenarios where the mobility time series are generated using MC models, such as the ones used in the paper; in these cases, the Markov-estimation-based method outperforms other prediction schemes. The second proposed method employs an MLP in a recurrent formulation to generate predictions for the next node occupation as a function of previous time series values. Even though this approach can capture Markovian behaviors, its performance is not competitive since it does not exploit the available graph knowledge. The third proposed prediction method is based on a graph neural network trained in a supervised manner. In general, this method performs much better than the proposed baseline prediction schemes, and in some configurations outperforms or competes with the first proposed method (GNN). To do so, the GNN model needs enough representation power to discriminate between different nodes and neighbors. Our results show that, in the types of scenarios proposed, this can be achieved by using a higher value of characteristics \(W\), or by including node identifiers in the feature vector. It should be noted that for the GNN models to achieve competitive performance, node identifiers need to be provided as part of the node feature vector. This is true even in scenarios with a uniform transition matrix. Our study highlights the effectiveness of the Markov model, partly due to our reliance on Markov processes for synthetic data generation. However, the current synthetic data approach may not reflect some of the intricate time series patterns inherent in complex traffic data. Urban traffic dynamics inherently exhibit repetitive patterns and deviations, since humans very often tend to follow pre-established routes, and/or variations of these. GNNs may excel at modeling complex time series patterns, but the absence of such nuances in our synthetic data generation suggests untapped potential in our GNN approach. Future research should incorporate synthetic models tuned with real-world traffic data. By capturing the complexities of urban movement patterns, we can fully leverage GNNs in traffic analysis. In addition, more information could be incorporated into both Markov-based and GNN models to address the problem of aggregate prediction in these new scenarios. Specifically, the GNN architecture presented in this paper could be improved in three different ways: by using different neighbor sampling strategies, by using a more nuanced aggregation function that better discriminates both temporally and by node, and by using more expressive node identification features. Conceptualization, J.F.S.-R., J.M. and P.J.Z. ; methodology, J.F.S.-R. and P.J.Z. ; software, J.F.S.-R., R.V.-R. and P.J.Z. ; validation, J.F.S.-R.; formal analysis, J.F.S.-R. and P.J.Z. ; investigation, J.F.S.-R. and P.J.Z. ; resources, J.F.S.-R. and P.J.Z. ; writing--original draft preparation, P.J.Z. ; writing--review and editing, J.F.S.-R., J.M. and P.J.Z. ; visualization, J.F.S.-R., supervision, J.F.S.-R., [PERSON] and [PERSON]; project administration, J.F.S.-R., [PERSON] and P.J.Z. ; funding acquisition, J.F.S.-R., [PERSON] and P.J.Z.