Another important difference between the Leiden algorithm and the Louvain algorithm is the implementation of the local moving phase. Finding and Evaluating Community Structure in Networks. Phys. As we will demonstrate in our experimental analysis, the problem occurs frequently in practice when using the Louvain algorithm. However, nodes 16 are still locally optimally assigned, and therefore these nodes will stay in the red community. CAS Later iterations of the Louvain algorithm are very fast, but this is only because the partition remains the same. Run the code above in your browser using DataCamp Workspace. Additionally, we implemented a Python package, available from https://github.com/vtraag/leidenalg and deposited at Zenodo24). Louvain keeps visiting all nodes in a network until there are no more node movements that increase the quality function. This is because Louvain only moves individual nodes at a time. An alternative quality function is the Constant Potts Model (CPM)13, which overcomes some limitations of modularity. This phenomenon can be explained by the documented tendency KMeans has to identify equal-sized , combined with the significant class imbalance associated with the datasets having more than 8 clusters (Table 1). However, so far this problem has never been studied for the Louvain algorithm. These nodes are therefore optimally assigned to their current community. Ozaki, N., Tezuka, H. & Inaba, M. A Simple Acceleration Method for the Louvain Algorithm. As can be seen in the figure, Louvain quickly reaches a state in which it is unable to find better partitions. For each network, we repeated the experiment 10 times. However, after all nodes have been visited once, Leiden visits only nodes whose neighbourhood has changed, whereas Louvain keeps visiting all nodes in the network. The difference in computational time is especially pronounced for larger networks, with Leiden being up to 20 times faster than Louvain in empirical networks. Am. sign in After a stable iteration of the Leiden algorithm, the algorithm may still be able to make further improvements in later iterations. Thank you for visiting nature.com. We then created a certain number of edges such that a specified average degree \(\langle k\rangle \) was obtained. In the most difficult case (=0.9), Louvain requires almost 2.5 days, while Leiden needs fewer than 10 minutes. As can be seen in Fig. Rev. 2007. In the first step of the next iteration, Louvain will again move individual nodes in the network. The smart local moving algorithm (Waltman and Eck 2013) identified another limitation in the original Louvain method: it isnt able to split communities once theyre merged, even when it may be very beneficial to do so. The nodes are added to the queue in a random order. Crucially, however, the percentage of badly connected communities decreases with each iteration of the Leiden algorithm. Hence, the problem of Louvain outlined above is independent from the issue of the resolution limit. Subpartition -density does not imply that individual nodes are locally optimally assigned. 4. CAS We generated benchmark networks in the following way. Complex brain networks: graph theoretical analysis of structural and functional systems. See the documentation for these functions. & Arenas, A. PubMed The two phases are repeated until the quality function cannot be increased further. At some point, the Louvain algorithm may end up in the community structure shown in Fig. A Simple Acceleration Method for the Louvain Algorithm. Int. This is not the case when nodes are greedily merged with the community that yields the largest increase in the quality function. Leiden is faster than Louvain especially for larger networks. Node mergers that cause the quality function to decrease are not considered. The Leiden algorithm provides several guarantees. This way of defining the expected number of edges is based on the so-called configuration model. This aspect of the Louvain algorithm can be used to give information about the hierarchical relationships between communities by tracking at which stage the nodes in the communities were aggregated. J. Exp. Fortunato, S. & Barthlemy, M. Resolution Limit in Community Detection. In practical applications, the Leiden algorithm convincingly outperforms the Louvain algorithm, both in terms of speed and in terms of quality of the results, as shown by the experimental analysis presented in this paper. The DBLP network is somewhat more challenging, requiring almost 80 iterations on average to reach a stable iteration. Rev. Blondel, V D, J L Guillaume, and R Lambiotte. In fact, although it may seem that the Louvain algorithm does a good job at finding high quality partitions, in its standard form the algorithm provides only one guarantee: the algorithm yields partitions for which it is guaranteed that no communities can be merged. An aggregate. When a disconnected community has become a node in an aggregate network, there are no more possibilities to split up the community. For lower values of , the correct partition is easy to find and Leiden is only about twice as fast as Louvain. Faster unfolding of communities: Speeding up the Louvain algorithm. The Leiden algorithm consists of three phases: (1) local moving of nodes, (2) refinement of the partition and (3) aggregation of the network based on the refined partition, using the non-refined. Electr. In all experiments reported here, we used a value of 0.01 for the parameter that determines the degree of randomness in the refinement phase of the Leiden algorithm. Modularity is used most commonly, but is subject to the resolution limit. Similarly, in citation networks, such as the Web of Science network, nodes in a community are usually considered to share a common topic26,27. 9 shows that more than 10 iterations of the Leiden algorithm can be performed before the Louvain algorithm has finished its first iteration. We typically reduce the dimensionality of the data first by running PCA, then construct a neighbor graph in the reduced space. According to CPM, it is better to split into two communities when the link density between the communities is lower than the constant. Natl. The larger the increase in the quality function, the more likely a community is to be selected. By creating the aggregate network based on \({{\mathscr{P}}}_{{\rm{refined}}}\) rather than P, the Leiden algorithm has more room for identifying high-quality partitions. Random moving is a very simple adjustment to Louvain local moving proposed in 2015 (Traag 2015). Acad. leidenalg. Subset optimality is the strongest guarantee that is provided by the Leiden algorithm. To find an optimal grouping of cells into communities, we need some way of evaluating different partitions in the graph. As discussed earlier, the Louvain algorithm does not guarantee connectivity. The algorithm is described in pseudo-code in AlgorithmA.2 in SectionA of the Supplementary Information. Package 'leiden' October 13, 2022 Type Package Title R Implementation of Leiden Clustering Algorithm Version 0.4.3 Date 2022-09-10 Description Implements the 'Python leidenalg' module to be called in R. Enables clustering using the leiden algorithm for partition a graph into communities. The algorithm may yield arbitrarily badly connected communities, over and above the well-known issue of the resolution limit14. In the meantime, to ensure continued support, we are displaying the site without styles The constant Potts model (CPM), so called due to the use of a constant value in the Potts model, is an alternative objective function for community detection. Excluding node mergers that decrease the quality function makes the refinement phase more efficient. the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in In the fast local move procedure in the Leiden algorithm, only nodes whose neighbourhood has changed are visited. However, Leiden is more than 7 times faster for the Live Journal network, more than 11 times faster for the Web of Science network and more than 20 times faster for the Web UK network. Such a modular structure is usually not known beforehand. The Leiden algorithm is partly based on the previously introduced smart local move algorithm15, which itself can be seen as an improvement of the Louvain algorithm. Is modularity with a resolution parameter equivalent to leidenalg.RBConfigurationVertexPartition? Nonetheless, some networks still show large differences. The random component also makes the algorithm more explorative, which might help to find better community structures. 92 (3): 032801. http://dx.doi.org/10.1103/PhysRevE.92.032801. To obtain Good, B. H., De Montjoye, Y. The Louvain algorithm guarantees that modularity cannot be increased by merging communities (it finds a locally optimal solution). Nevertheless, when CPM is used as the quality function, the Louvain algorithm may still find arbitrarily badly connected communities. As such, we scored leiden-clustering popularity level to be Limited. Soc. The algorithm then moves individual nodes in the aggregate network (e). E 69, 026113, https://doi.org/10.1103/PhysRevE.69.026113 (2004). CAS Rather than evaluating the modularity gain for moving a node to each neighboring communities, we choose a neighboring node at random and evaluate whether there is a gain in modularity if we were to move the node to that neighbors community. J. This continues until the queue is empty. The Beginner's Guide to Dimensionality Reduction. Detecting communities in a network is therefore an important problem. It is a directed graph if the adjacency matrix is not symmetric. For each set of parameters, we repeated the experiment 10 times. Then optimize the modularity function to determine clusters. b, The elephant graph (in a) is clustered using the Leiden clustering algorithm 51 (resolution r = 0.5). Rev. Rev. Our analysis is based on modularity with resolution parameter =1. Porter, M. A., Onnela, J.-P. & Mucha, P. J. In the first iteration, Leiden is roughly 220 times faster than Louvain. Speed and quality for the first 10 iterations of the Louvain and the Leiden algorithm for six empirical networks. Google Scholar. The Leiden algorithm has been specifically designed to address the problem of badly connected communities. See the documentation on the leidenalg Python module for more information: https://leidenalg.readthedocs.io/en/latest/reference.html. where >0 is a resolution parameter4. contrastive-sc works best on datasets with fewer clusters when using the KMeans clustering and conversely for Leiden. Randomness in the selection of a community allows the partition space to be explored more broadly. The PyPI package leiden-clustering receives a total of 15 downloads a week. To study the scaling of the Louvain and the Leiden algorithm, we rely on a variant of a well-known approach for constructing benchmark networks28. With one exception (=0.2 and n=107), all results in Fig. In that case, nodes 16 are all locally optimally assigned, despite the fact that their community has become disconnected. Lancichinetti, A., Fortunato, S. & Radicchi, F. Benchmark graphs for testing community detection algorithms. In that case, some optimal partitions cannot be found, as we show in SectionC2 of the Supplementary Information. The quality of such an asymptotically stable partition provides an upper bound on the quality of an optimal partition. Moreover, when the algorithm is applied iteratively, it converges to a partition in which all subsets of all communities are guaranteed to be locally optimally assigned.