## 4 Results
### Application to the U.S. Road Network  
We now assess the CORE-clustering algorithms on the U.S. road network out of the 9 th DIMACS Implementation challenge ([[http://users.diag.uniromal.it/challenge9/](http://users.diag.uniromal.it/challenge9/)]([http://users.diag.uniromal.it/challenge9/](http://users.diag.uniromal.it/challenge9/)) (accessed on 19 February 2021)). Our goal here is to discuss the pertinence of the detected CORE-clusters, and not specifically their representative variables, on a large scale and straightforwardly interpretable dataset. The graph contains here \(2.4\times 10^{7}\) nodes, each of them representing a crossing of the U.S. road/streets network, and \(5.8\times 10^{7}\) arcs, representing the part of the roads/streets between two crossings. Interestingly, the distance between the crossings is also associated with each edge of the graph. To assess the CORE-clustering algorithms on the U.S. road network, we first transformed the distances between the crossings into weights that are higher and higher for increasingly closer crossings. A weight \(\max(\left(4\times 10^{3}-l\right)/\left(4\times 10^{3}\right),10^{-4})\) was specifically given to each edge, where \(l\) is its original length in feet. We show Figure 3, the CORE-clusters (Clusters represented using _MI Map Tools: Geo-Plotter:_[[https://mobisoftinfotech.com/tools/plot-multiple-points-on-map/](https://mobisoftinfotech.com/tools/plot-multiple-points-on-map/)]([https://mobisoftinfotech.com/tools/plot-multiple-points-on-map/](https://mobisoftinfotech.com/tools/plot-multiple-points-on-map/)) (accessed on 19 February 2021)) obtained using the greedy algorithm with \(\tau=5\times 10^{4}\) and \(\xi=10^{-3}\)This means that each CORE-cluster contains a road network of \(5\times 10^{4}\) to \(10^{5}\) crossings, for which one can travel from any crossing to another one by only using streets of less than 3996 feet (about 1.2 km) between two crossings. As expected, the clusters represented in Figure 3 correspond to the 18 largest urban areas in the U.S. Note that two of them are made of three CORE-clusters (New York City and Los Angeles), and two other ones (Miami and Chicago) are made of two CORE-clusters. This is due to an algorithmic choice, discussed in Sections 3.2.2 and 3.2.3, which leads to the detection of CORE-clusters containing \(\tau\) to \(2\tau-1\) nodes by using the proposed CORE-clustering algorithms. As discussed in Section 2.6, an interesting strategy if connected CORE-clusters are found consists in running again the CORE-clustering algorithms with higher values of \(\tau\) or \(\xi\), to potentially detect more robust CORE-clusters: By reproducing this test with \(\tau=10^{5}\) instead of \(\tau=5\times 10^{4}\), we now try to find CORE-clusters containing \(10^{5}\) to \(2\times 10^{5}\) crossings, only 5 urban areas are found: New York City (2 CORE-clusters), Los Angeles, Chicago, Miami and San Fransisco. Now, by using \(\tau=5\times 10^{4}\) and now \(\xi=0.5\), i.e., with distances between the crossings lower than about 2000 feet (about 0.6 km) instead of 3996 feet, only 3 urban areas are found: New York City (2 CORE-clusters), Los Angeles and Chicago. What is interesting here in terms of interpretability is that we have been able to select specific subparts of the U.S. road network by explicitly controling the size of the clusters or a specific level of density in the network. This notion of interpretability can be further discussed by observing the CORE-clusters obtained in the New York city urban area, as shown in Figure 3b-d. By using \(\tau=5\times 10^{4}\) and \(\xi=10^{-3}\), the three CORE-clusters split the densest parts of the New York City urban area into the New Jersey state, Long Island and the rest of New York state.