In this paper, we show that the Louvain algorithm has a major problem, for both modularity and CPM. leiden_clsutering is distributed under a BSD 3-Clause License (see LICENSE). The constant Potts model might give better communities in some cases, as it is not subject to the resolution limit. Soft Matter Phys. E 92, 032801, https://doi.org/10.1103/PhysRevE.92.032801 (2015). Knowl. In the first iteration, Leiden is roughly 220 times faster than Louvain. The algorithm moves individual nodes from one community to another to find a partition (b), which is then refined (c). In this iterative scheme, Louvain provides two guarantees: (1) no communities can be merged and (2) no nodes can be moved. For this network, Leiden requires over 750 iterations on average to reach a stable iteration. This is well illustrated by figure 2 in the Leiden paper: When a community becomes disconnected like this, there is no way for Louvain to easily split it into two separate communities. On Modularity Clustering. It was found to be one of the fastest and best performing algorithms in comparative analyses11,12, and it is one of the most-cited works in the community detection literature. 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 partition to create an initial partition for the aggregate network. MathSciNet Scaling of benchmark results for network size. The Leiden algorithm starts from a singleton partition (a). performed the experimental analysis. Similarly, in citation networks, such as the Web of Science network, nodes in a community are usually considered to share a common topic26,27. Narrow scope for resolution-limit-free community detection. (We implemented both algorithms in Java, available from https://github.com/CWTSLeiden/networkanalysis and deposited at Zenodo23. That is, no subset can be moved to a different community. As can be seen in Fig. We start by initialising a queue with all nodes in the network. Modules smaller than the minimum size may not be resolved through modularity optimization, even in the extreme case where they are only connected to the rest of the network through a single edge. Again, if communities are badly connected, this may lead to incorrect inferences of topics, which will affect bibliometric analyses relying on the inferred topics. Two ways of doing this are graph modularity (Newman and Girvan 2004) and the constant Potts model (Ronhovde and Nussinov 2010). Modularity is used most commonly, but is subject to the resolution limit. It means that there are no individual nodes that can be moved to a different community. The corresponding results are presented in the Supplementary Fig. Reichardt, J. In the refinement phase, nodes are not necessarily greedily merged with the community that yields the largest increase in the quality function. They identified an inefficiency in the Louvain algorithm: computes modularity gain for all neighbouring nodes per loop in local moving phase, even though many of these nodes will not have moved. Ozaki, Naoto, Hiroshi Tezuka, and Mary Inaba. From Louvain to Leiden: Guaranteeing Well-Connected Communities, October. J. contrastive-sc works best on datasets with fewer clusters when using the KMeans clustering and conversely for Leiden. Introduction The Louvain method is an algorithm to detect communities in large networks. 8, the Leiden algorithm is significantly faster than the Louvain algorithm also in empirical networks. 104 (1): 3641. USA 104, 36, https://doi.org/10.1073/pnas.0605965104 (2007). By submitting a comment you agree to abide by our Terms and Community Guidelines. Algorithmics 16, 2.1, https://doi.org/10.1145/1963190.1970376 (2011). The Leiden community detection algorithm outperforms other clustering methods. Each point corresponds to a certain iteration of an algorithm, with results averaged over 10 experiments. An aggregate network (d) is created based on the refined partition, using the non-refined partition to create an initial partition for the aggregate network. In fact, when we keep iterating the Leiden algorithm, it will converge to a partition for which it is guaranteed that: A community is uniformly -dense if there are no subsets of the community that can be separated from the community. The authors show that the total computational time for Louvain depends a lot on the number of phase one loops (loops during the first local moving stage). When a sufficient number of neighbours of node 0 have formed a community in the rest of the network, it may be optimal to move node 0 to this community, thus creating the situation depicted in Fig. Ozaki, N., Tezuka, H. & Inaba, M. A Simple Acceleration Method for the Louvain Algorithm. The algorithm continues to move nodes in the rest of the network. The speed difference is especially large for larger networks. Even though clustering can be applied to networks, it is a broader field in unsupervised machine learning which deals with multiple attribute types. Ronhovde, Peter, and Zohar Nussinov. This should be the first preference when choosing an algorithm. Besides being pervasive, the problem is also sizeable. This makes sense, because after phase one the total size of the graph should be significantly reduced. Communities may even be disconnected. Ph.D. thesis, (University of Oxford, 2016). Note that if Leiden finds subcommunities, splitting up the community is guaranteed to increase modularity. It is good at identifying small clusters. Somewhat stronger guarantees can be obtained by iterating the algorithm, using the partition obtained in one iteration of the algorithm as starting point for the next iteration. Networks with high modularity have dense connections between the nodes within modules but sparse connections between nodes in different modules. Optimising modularity is NP-hard5, and consequentially many heuristic algorithms have been proposed, such as hierarchical agglomeration6, extremal optimisation7, simulated annealing4,8 and spectral9 algorithms. Finally, we demonstrate the excellent performance of the algorithm for several benchmark and real-world networks. Moreover, when no more nodes can be moved, the algorithm will aggregate the network. Modularity is a popular objective function used with the Louvain method for community detection. Rep. 6, 30750, https://doi.org/10.1038/srep30750 (2016). In the meantime, to ensure continued support, we are displaying the site without styles This will compute the Leiden clusters and add them to the Seurat Object Class. Speed of the first iteration of the Louvain and the Leiden algorithm for benchmark networks with increasingly difficult partitions (n=107). 1 and summarised in pseudo-code in AlgorithmA.1 in SectionA of the Supplementary Information. However, focussing only on disconnected communities masks the more fundamental issue: Louvain finds arbitrarily badly connected communities. Phys. Nevertheless, depending on the relative strengths of the different connections, these nodes may still be optimally assigned to their current community. partition_type : Optional [ Type [ MutableVertexPartition ]] (default: None) Type of partition to use. Rather than progress straight to the aggregation stage (as we would for the original Louvain), we next consider each community as a new sub-network and re-apply the local moving step within each community. We first applied the Scanpy pipeline, including its clustering method (Leiden clustering), on the PBMC dataset. This package requires the 'leidenalg' and 'igraph' modules for python (2) to be installed on your system. Clauset, A., Newman, M. E. J. Each of these can be used as an objective function for graph-based community detection methods, with our goal being to maximize this value. As discussed earlier, the Louvain algorithm does not guarantee connectivity. Here is some small debugging code I wrote to find this. Random moving can result in some huge speedups, since Louvain spends about 95% of its time computing the modularity gain from moving nodes. The images or other third party material in this article are included in the articles Creative Commons license, unless indicated otherwise in a credit line to the material. Louvain can also be quite slow, as it spends a lot of time revisiting nodes that may not have changed neighborhoods. These nodes can be approximately identified based on whether neighbouring nodes have changed communities. However, if communities are badly connected, this may lead to incorrect attributions of shared functionality. Note that communities found by the Leiden algorithm are guaranteed to be connected. Four popular community detection algorithms are explained . Sci. 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. running Leiden clustering finished: found 16 clusters and added 'leiden_1.0', the cluster labels (adata.obs, categorical) (0:00:00) running Leiden clustering finished: found 12 clusters and added 'leiden_0.6', the cluster labels (adata.obs, categorical) (0:00:00) running Leiden clustering finished: found 9 clusters and added 'leiden_0.4', the This step will involve reducing the dimensionality of our data into two dimensions using uniform manifold approximation (UMAP), allowing us to visualize our cell populations as they are binned into discrete populations using Leiden clustering. Scaling of benchmark results for difficulty of the partition. leiden_clustering Description Class wrapper based on scanpy to use the Leiden algorithm to directly cluster your data matrix with a scikit-learn flavor. http://arxiv.org/abs/1810.08473. In practice, this means that small clusters can hide inside larger clusters, making their identification difficult. In addition, we prove that the algorithm converges to an asymptotically stable partition in which all subsets of all communities are locally optimally assigned. 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. Zenodo, https://doi.org/10.5281/zenodo.1466831 https://github.com/CWTSLeiden/networkanalysis. In this case we can solve one of the hard problems for K-Means clustering - choosing the right k value, giving the number of clusters we are looking for. We applied the Louvain and the Leiden algorithm to exactly the same networks, using the same seed for the random number generator. Google Scholar. The quality improvement realised by the Leiden algorithm relative to the Louvain algorithm is larger for empirical networks than for benchmark networks. The differences are not very large, which is probably because both algorithms find partitions for which the quality is close to optimal, related to the issue of the degeneracy of quality functions29. If nothing happens, download Xcode and try again. Class wrapper based on scanpy to use the Leiden algorithm to directly cluster your data matrix with a scikit-learn flavor. The inspiration for this method of community detection is the optimization of modularity as the algorithm progresses. Besides the Louvain algorithm and the Leiden algorithm (see the "Methods" section), there are several widely-used network clustering algorithms, such as the Markov clustering algorithm [], Infomap algorithm [], and label propagation algorithm [].Markov clustering and Infomap algorithm are both based on flow . 2(a). Google Scholar. For all networks, Leiden identifies substantially better partitions than Louvain. This represents the following graph structure. Article 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. Theory Exp. Indeed, the percentage of disconnected communities becomes more comparable to the percentage of badly connected communities in later iterations. The idea of the refinement phase in the Leiden algorithm is to identify a partition \({{\mathscr{P}}}_{{\rm{refined}}}\) that is a refinement of \({\mathscr{P}}\). After a stable iteration of the Leiden algorithm, it is guaranteed that: All nodes are locally optimally assigned. Empirical networks show a much richer and more complex structure. volume9, Articlenumber:5233 (2019) Runtime versus quality for benchmark networks. In the most difficult case (=0.9), Louvain requires almost 2.5 days, while Leiden needs fewer than 10 minutes. These steps are repeated until the quality cannot be increased further. Clustering algorithms look for similarities or dissimilarities among data points so that similar ones can be grouped together. First calculate k-nearest neighbors and construct the SNN graph. Our analysis is based on modularity with resolution parameter =1. The larger the increase in the quality function, the more likely a community is to be selected. The property of -separation is also guaranteed by the Louvain algorithm. In the Louvain algorithm, a node may be moved to a different community while it may have acted as a bridge between different components of its old community. The percentage of disconnected communities even jumps to 16% for the DBLP network. All authors conceived the algorithm and contributed to the source code. ACM Trans. Article At each iteration all clusters are guaranteed to be connected and well-separated. As shown in Fig. (2) and m is the number of edges. 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). Klavans, R. & Boyack, K. W. Which Type of Citation Analysis Generates the Most Accurate Taxonomy of Scientific and Technical Knowledge? In the worst case, communities may even be disconnected, especially when running the algorithm iteratively. The algorithm may yield arbitrarily badly connected communities, over and above the well-known issue of the resolution limit14. E 69, 026113, https://doi.org/10.1103/PhysRevE.69.026113 (2004). You signed in with another tab or window. Resolution Limit in Community Detection. Proc. PubMedGoogle Scholar. For higher values of , Leiden finds better partitions than Louvain. In our experimental analysis, we observe that up to 25% of the communities are badly connected and up to 16% are disconnected. Article The thick edges in Fig. 9, the Leiden algorithm also performs better than the Louvain algorithm in terms of the quality of the partitions that are obtained. This is because Louvain only moves individual nodes at a time. We typically reduce the dimensionality of the data first by running PCA, then construct a neighbor graph in the reduced space. Rev. DBSCAN Clustering Explained Detailed theorotical explanation and scikit-learn implementation Clustering is a way to group a set of data points in a way that similar data points are grouped together. ADS Thank you for visiting nature.com. We used modularity with a resolution parameter of =1 for the experiments. 6 show that Leiden outperforms Louvain in terms of both computational time and quality of the partitions. The horizontal axis indicates the cumulative time taken to obtain the quality indicated on the vertical axis. Learn more. All communities are subpartition -dense. 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. We find that the Leiden algorithm is faster than the Louvain algorithm and uncovers better partitions, in addition to providing explicit guarantees. Phys. The percentage of badly connected communities is less affected by the number of iterations of the Louvain algorithm. A new methodology for constructing a publication-level classification system of science. Nonlin. Traag, V.A., Waltman, L. & van Eck, N.J. From Louvain to Leiden: guaranteeing well-connected communities. In the case of the Louvain algorithm, after a stable iteration, all subsequent iterations will be stable as well. Louvain quickly converges to a partition and is then unable to make further improvements. It was originally developed for modularity optimization, although the same method can be applied to optimize CPM. This is similar to what we have seen for benchmark networks. Complex brain networks: graph theoretical analysis of structural and functional systems. This continues until the queue is empty. In doing so, Louvain keeps visiting nodes that cannot be moved to a different community. When iterating Louvain, the quality of the partitions will keep increasing until the algorithm is unable to make any further improvements. In short, the problem of badly connected communities has important practical consequences. It only implies that individual nodes are well connected to their community. First, we created a specified number of nodes and we assigned each node to a community. It states that there are no communities that can be merged. CPM has the advantage that it is not subject to the resolution limit. The docs are here. & Bornholdt, S. Statistical mechanics of community detection. The degree of randomness in the selection of a community is determined by a parameter >0. At this point, it is guaranteed that each individual node is optimally assigned. For example, nodes in a community in biological or neurological networks are often assumed to share similar functions or behaviour25. Below, the quality of a partition is reported as \(\frac{ {\mathcal H} }{2m}\), where H is defined in Eq. Percentage of communities found by the Louvain algorithm that are either disconnected or badly connected compared to percentage of badly connected communities found by the Leiden algorithm. Communities in Networks. ISSN 2045-2322 (online). Fortunato, Santo, and Marc Barthlemy. Use Git or checkout with SVN using the web URL. To find an optimal grouping of cells into communities, we need some way of evaluating different partitions in the graph. Furthermore, by relying on a fast local move approach, the Leiden algorithm runs faster than the Louvain algorithm. Soft Matter Phys. Although originally defined for modularity, the Louvain algorithm can also be used to optimise other quality functions. However, the Louvain algorithm does not consider this possibility, since it considers only individual node movements. In later stages, most neighbors will belong to the same community, and its very likely that the best move for the node is to the community that most of its neighbors already belong to. Run the code above in your browser using DataCamp Workspace. Rotta, R. & Noack, A. Multilevel local search algorithms for modularity clustering. Fast Unfolding of Communities in Large Networks. Journal of Statistical , January. It maximizes a modularity score for each community, where the modularity quantifies the quality of an assignment of nodes to communities. Hence, by counting the number of communities that have been split up, we obtained a lower bound on the number of communities that are badly connected. E 80, 056117, https://doi.org/10.1103/PhysRevE.80.056117 (2009). Newman, M. E. J. ADS Electr. The above results shows that the problem of disconnected and badly connected communities is quite pervasive in practice. Technol. CAS Mech. In the fast local move procedure in the Leiden algorithm, only nodes whose neighbourhood has changed are visited. Bullmore, E. & Sporns, O. Porter, M. A., Onnela, J.-P. & Mucha, P. J. Figure3 provides an illustration of the algorithm. For each community in a partition that was uncovered by the Louvain algorithm, we determined whether it is internally connected or not. Excluding node mergers that decrease the quality function makes the refinement phase more efficient. You are using a browser version with limited support for CSS. The new algorithm integrates several earlier improvements, incorporating a combination of smart local move15, fast local move16,17 and random neighbour move18. From Louvain to Leiden: guaranteeing well-connected communities, $$ {\mathcal H} =\frac{1}{2m}\,{\sum }_{c}({e}_{c}-{\rm{\gamma }}\frac{{K}_{c}^{2}}{2m}),$$, $$ {\mathcal H} ={\sum }_{c}[{e}_{c}-\gamma (\begin{array}{c}{n}_{c}\\ 2\end{array})],$$, https://doi.org/10.1038/s41598-019-41695-z. ADS The R implementation of Leiden can be run directly on the snn igraph object in Seurat. E 70, 066111, https://doi.org/10.1103/PhysRevE.70.066111 (2004). Nonlin. Hence, the Leiden algorithm effectively addresses the problem of badly connected communities. Later iterations of the Louvain algorithm only aggravate the problem of disconnected communities, even though the quality function (i.e. The property of -connectivity is a slightly stronger variant of ordinary connectivity. A community size of 50 nodes was used for the results presented below, but larger community sizes yielded qualitatively similar results. bioRxiv, https://doi.org/10.1101/208819 (2018). Note that this code is designed for Seurat version 2 releases. The problem of disconnected communities has been observed before19,20, also in the context of the label propagation algorithm21. Is modularity with a resolution parameter equivalent to leidenalg.RBConfigurationVertexPartition? 8, 207218, https://doi.org/10.17706/IJCEE.2016.8.3.207-218 (2016). Consider the partition shown in (a). The algorithm then moves individual nodes in the aggregate network (e). Rev. Hence, in general, Louvain may find arbitrarily badly connected communities. Article In the local moving phase, individual nodes are moved to the community that yields the largest increase in the quality function. Cite this article. Modularity scores of +1 mean that all the edges in a community are connecting nodes within the community. Hence, the problem of Louvain outlined above is independent from the issue of the resolution limit. Article Louvain keeps visiting all nodes in a network until there are no more node movements that increase the quality function.