Detecting communities with the multi-scale Louvain method: robustness test on the metropolitan area of Brussels

Abstract

Detecting communities in large networks has become a common practice in socio-spatial analyses and has led to the development of numerous dedicated mathematical algorithms. Nowadays, however, researchers face a deluge of data and algorithms, and great care must be taken regarding methodological questions such as the values of the parameters and the geographical characteristics of the data. We aim here at testing the sensitivity of multi-scale modularity optimized by the Louvain method to the value of the resolution parameter (introduced by Reichardt and Bornholdt (Phys Rev Lett 93(21):218701, 2004. https://doi.org/10.1103/PhysRevLett.93.218701) and controlling the size of the communities) and to a number of spatial issues such as the inclusion of internal loops and the delineation of the study area. We compare the community structures with those found by another well-known community detection algorithm (Infomap), and we further interpret the final results in terms of urban geography. Sensitivity analyses are conducted for commuting movements in and around Brussels. Results reveal slight effects of spatial issues (inclusion of the internal loops, definition of the study area) on the partition into job basins, while the resolution parameter plays a major role in the final results and their interpretation in terms of urban geography. Community detection methods seem to reveal a surprisingly strong spatial effect of commuting patterns: Similar partitions are obtained with different methods. This paper highlights the advantages and sensitivities of the multi-scale Louvain method and more particularly of defining communities of places. Despite these sensitivities, the method proves to be a valuable tool for geographers and planners.

Appendix

1.1 Extraction of the dominant community

With the multiple runs of the Louvain method, the nodes are characterized by a list of 1000 identification numbers (id) corresponding to the community number. It is important to underline that the allocation of nodes to communities is specific to each run of the Louvain method, implying that the ID of the communities is renumbered at each run. Hence, a community numbered “one” in the first run is not automatically labeled “one” in the other simulations; so we have to make the ID of the communities comparable through the runs of the Louvain method. The methodology is illustrated in Fig. 13.

First, each node is characterized by a list of 1000 elements corresponding to the communities where the nodes are allocated. We call this list the signature of the node and calculate the number of occurrences of each signature in the results. Second, the list of signatures is ordered by descending order of occurrence and we attribute to each specific id that we call the new community number. Third, the highly represented signature is used to renumber the communities detected. The renumbering starts with the nodes having the highest represented signature. The initial community numbers are modified with the new community number. Moreover, this renumbering is applied for the entire run and so, if the initial community number is still present, it is renumbered with the new community number too. This special case is illustrated in Fig. 13. In run 5, the nodes with the highest represented signature are nodes 1, 2, and 3 which belong to Community 1. In this run, we observe that a number of other nodes are initially classified within this community 1 (nodes 4 and 5). So, all these nodes (1, 2, 3, 4, and 5) are renumbered with the new community number 100. Once all the initial IDs of the communities have been modified by the new community number, the same procedure is applied to the second most frequent signature. This renumbering is applied until all the communities are renumbered. Finally, the dominant community (most frequent community) is extracted for each node and the stability of the nodes is calculated.

Fig. 13

Renumbering of the communities for five runs of the Louvain method