NeuroMLR: Robust & Reliable Route Recommendation on Road Networks 

1 Introduction and Related Work . Given historical trajectory data , we study the problem of predicting the most likely route from a source node to a destination node in a road network . This problem has two prominent applications : route recommendation and route recovery . Route recommendation is one of the core functionalities in GPS-aided mapping applications . They are routinely used in the cab and food-delivery industry [ 26 , 7 , 34 , 6 ] , as well as by common people through navigation systems when they are unfamiliar with their surroundings [ 30 ] . Route recovery , as the name suggests , focuses on recovering the actual traversed route from a partially observed GPS trajectory [ 9 ] . Due to various reasons , for instance , limiting the power consumption of GPS devices , trajectories are often recorded at low sampling rate . 1.1 Existing work . The simplest approach is to predict the shortest or the quickest path ( route ) between the source and destination . However , several studies have shown that human beings rarely travel in shortest paths [ 21 ] . Rather , the probability of a path being taken is a complex mixture of several latent factors such as road quality [ 5 , 22 ] , road scenery [ 21 , 14 ] , pollution levels [ 23 ] , presence of road tolls , etc . Furthermore , it has also been shown that human beings systematically misperceive travel time [ 20 ] . Modeling these complex factors is challenging . Consequently , a large body of work exists on predicting the most likely route [ 1 , 28 , 33 , 4 , 10 , 27 ] , with DEEPST [ 10 ] and CSSRNN [ 27 ] being the best performing algorithms . CSSRNN models the trajectory patterns through a Recurrent Neural Network and exploits the topological constraints presented by the road network . DEEPST learns representations for the trajectories , destination and traffic conditions using variational autoencoders . ∗denotes equal contribution 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . 1 2 5 10 20 50 100 . 1.2 Limitations of existing work . • Reachability : Existing techniques have primarily used recall and precision to measure accuracy of predictions [ 10 , 27 ] . Critically , these ignore whether the predicted trajectory actually reaches the destination . The deployability of any route recommendation algorithm is severely compromised if the recommended route does not reach the destination . Our experiments reveal that for both DEEPST and CSSRNN , a significant portion of recommended routes fail to reach the destination ( See Table . 2 ) . Further , the state of the art does not scale well to large road networks ( see Fig . 1c ) . The performance gap between CSSRNN and NEUROMLR becomes more prominent with increasing network size . • Inductive Learning : A good prediction model should be capable of making predictions on parts of the road network which were unseen or rarely seen during training . From Fig 1b we can observe that there is a significant percentage of nodes in the road network that remain unseen even if we use the entire training data . Both CSSRNN and DEEPST learn node embeddings in a transductive manner , limiting knowledge sharing across nodes . Hence , quality on unseen/lesser seen nodes suffers . DEEPST employs a clustering-inspired approach for sharing statistical strength across trips having similar destinations . However , this introduces a limitation that different destinations get mapped to identical representations , impairing reachability . • Prediction Accuracy : The accuracy of even the best performing techniques is often below 50 % [ 10 ] . Consequently , there is scope for improvement . 1.3 Contributions . • Decoupled route prediction : Our proposed problem formulation ( § 2 ) allows decoupling the problem into two independent subproblems of route search ( § 3 ) and predicting transition probabilities ( § 4 ) . This allows us to reduce the problem of finding the most likely route to that of identifying the shortest path in a modified road network , thus guaranteeing destination reachability ( § 3 ) . • Generalization capability : We propose an inductive learning method using a novel combination of Lipschitz embeddings with Graph Convolutional Networks . Lipschitz embeddings serve as a rich initialisation derived from the global road network structure , independent of historical data . GCNs , via message passing , further propagate information learnt during training , to unseen and rarely seen nodes . • Empirical Evaluation : Extensive experiments on five large , real datasets establish : ( 1 ) NEUROMLR is up to 100 % more accurate and 4 times faster , ( 2 ) the recommended route reaches destination with > 0.96 probability , which is up to 90 % better than the state of the art , and ( 3 ) NEUROMLR is dramatically more effective in generalizing to unseen data . 2 Problem Formulation . Definition 1 ( Road Network ) . A road network is a directed graph G = ( V , E , δ , τt ) , where V is the set of nodes representing road intersections , E ⊆ V × V is the set of edges representing road segments , a distance function δ : E → R representing the length ( weight ) of each road segment , and function τt : E → R representing the average time taken to traverse each road segment at time t. We use the notation e = ( u , v ) to denote a road segment ( edge ) from node u to v and its length is denoted by δ ( e ) . Definition 2 ( Route ) . A route ( a.k.a path ) R ( s , d ) = { v1 , · · · , vk } corresponds to a simple path from the source node s = v1 to destination d = vk in the road network G , i.e. , path without cycles . Analogously , a route can also be expressed in terms of a sequence of edgesR ( s , d ) = { e1 , · · · , ek−1 } , where ei = ( vi , vi+1 ) . To denote a generic route between any arbitrary source and destination nodes , we useR instead of R ( s , d ) . We use the notation |R| to denote the number of edges inR and R.ei to refer to the ith edge inR . Furthermore , e ∈ R denotes thatR goes through edge e. The above notations are analogously extended from edges to nodes . We assume we have access to a dataset of historical trajectories . A trajectory is a sequence of GPS pings , made by a vehichle , which corresponds to a path in the road network . Each GPS ping is a tuple of the form 〈latitude , longitude , time〉 , which records a vehicle ’ s location at a particular time . Definition 3 ( Query ) . In a route recommendation query , the input is a tuple q : 〈s , d , t〉 , where s , d ∈ V are the source and destination nodes respectively , and t is the time at which the journey is to be taken . Problem : Most Likely Route : Given a road network G , a historical database D of trajectories , and query q : 〈s , d , t〉 , we would like to infer the mostly likely routeR∗ ( s , d ) based on the traffic patterns embodied in D. Formally , R∗ ( s , d ) = arg max ∀R ( s , d ) ∈G Pr ( R ( s , d ) | q ) ( 1 ) 2.1 Problem Characterization . The route recommendation problem can be framed as a path search problem on the graph corresponding to the road network . Mathematically , the probability of a routeR can be expressed in terms of its constituent edges . Pr ( R | q ) = |R|∏ i=1 Pr ( R.ei | R.e0 → R.ei−1 , s , d , t ) ( 2 ) Here , Pr ( R.ei | R.e0 → R.ei−1 , s , d , t ) represents the probability that routeR goes through edge R.ei given the path taken so far and the query parameters q : 〈s , d , t〉 Past studies have shown that human mobility patterns conform to the Markovian assumption [ 1 , 25 ] . Thus , Eq . 2 reduces to : Pr ( R | q ) = |R|∏ i=1 Pr ( R.ei | vi , d , t ) ( 3 ) With these simplifications , Eq . 1 reduces to : R∗ ( s , d ) = arg max ∀R∈G |R|∏ i=1 Pr ( R.ei | vi , d , t ) = arg min ∀R∈G |R|∑ i=1 − log ( Pr ( R.ei | vi , d , t ) ) ( 4 ) To summarize the above observations , there are two key challenges that we need to tackle : • Route Search : Given the conditional transition probability distribution , searching the route ( path ) minimizing Eq . 4 . •Modeling : Learning the conditional distribution that governs transitions in the road network , from the historical trajectory dataset D. Henceforth , while talking about transitions in the road network , we will use the notation curr to refer to the current location ( node ) , curr ∈ V . 3 Route Search . In this section , we assume that the transition probability Pr ( e|curr , d , t ) for any edge e ∈ E is known . Our algorithm to learn this distribution is discussed in § 4 . The Optimal Search : We first note that the negative log likelihood of a transition probability ( see Eq . 4 ) would be non-negative . Thus , we have an edge weight for each edge in the road network and our goal is to identify the path from the source to the destination that has the minimum cumulative weight . This computational task maps to the problem of finding the shortest path in a graph and can be solved using Dijkstra ’ s Algorithm . More importantly , the optimal path is guaranteed to reach the destination . The pseudocode for this search algorithm is provided in App . A. Computational Complexity : The complexity of the Dijkstra ’ s Algorithm is O ( |V| + |E| log ( |V| ) ) . This may be prohibitively expensive on large road networks where real-time predictions are desired . Greedy Approach : We start from the source node and greedily choose the transition with the highest probability till the destination is reached . Inaccurate estimation of transition probabilities may however divert us towards the wrong direction and we may never reach the destination . This may result in |V| iterations in the worst case . To handle cases of this nature , we terminate when either the destination is reached or the Haversine distance from the current node to the destination is significantly higher than the closest point in the current route to the destination . This idea is motivated from the fact that , in general , a vehicle progressively moves closer to the destination with each transition [ 31 ] . The pseudocode of the greedy approach can be found in Alg . 2 in App . B . Computational Complexity : At each node inR∗ , we evaluate each neighbor and select the one with highest likelihood . Hence , the complexity is O ( g|R∗| ) , where g is the average degree in G .

Summary: The paper introduces a route recommendation model based on graph neural networks with Lipschitz embedding initialization. The proposed model is compared with several baseline models and the evaluation shows that it outperforms baselines in terms of precision, recall, and reachability. Ablation study is done to demonstrate the importance of both Lipschitz embedding and GCN components. The model which uses inductive learning shows that it significantly outperforms baselines when the source/destination node is unpopular.