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. The algorithm may yield arbitrarily badly connected communities, over and above the well-known issue of the resolution limit14. The constant Potts model tries to maximize the number of internal edges in a community, while simultaneously trying to keep community sizes small, and the constant parameter balances these two characteristics. Rep. 6, 30750, https://doi.org/10.1038/srep30750 (2016). Inf. Acad. Hence, no further improvements can be made after a stable iteration of the Louvain algorithm. In single-cell biology we often use graph-based community detection methods to do this, as these methods are unsupervised, scale well, and usually give good results. As shown in Fig. It identifies the clusters by calculating the densities of the cells. Nevertheless, when CPM is used as the quality function, the Louvain algorithm may still find arbitrarily badly connected communities. CAS A community is subpartition -dense if it can be partitioned into two parts such that: (1) the two parts are well connected to each other; (2) neither part can be separated from its community; and (3) each part is also subpartition -dense itself. In the Louvain algorithm, an aggregate network is created based on the partition \({\mathscr{P}}\) resulting from the local moving phase. E 81, 046106, https://doi.org/10.1103/PhysRevE.81.046106 (2010). First, we show that the Louvain algorithm finds disconnected communities, and more generally, badly connected communities in the empirical networks. 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. 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. Trying to fix the problem by simply considering the connected components of communities19,20,21 is unsatisfactory because it addresses only the most extreme case and does not resolve the more fundamental problem. 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. Class wrapper based on scanpy to use the Leiden algorithm to directly cluster your data matrix with a scikit-learn flavor. However, the initial partition for the aggregate network is based on P, just like in the Louvain algorithm. ADS Nevertheless, depending on the relative strengths of the different connections, these nodes may still be optimally assigned to their current community. On the other hand, Leiden keeps finding better partitions, especially for higher values of , for which it is more difficult to identify good partitions. In this stage we essentially collapse communities down into a single representative node, creating a new simplified graph. Furthermore, if all communities in a partition are uniformly -dense, the quality of the partition is not too far from optimal, as shown in SectionE of the Supplementary Information. The algorithm is described in pseudo-code in AlgorithmA.2 in SectionA of the Supplementary Information. Requirements Developed using: scanpy v1.7.2 sklearn v0.23.2 umap v0.4.6 numpy v1.19.2 leidenalg Installation pip pip install leiden_clustering local Large network community detection by fast label propagation, Representative community divisions of networks, Gausss law for networks directly reveals community boundaries, A Regularized Stochastic Block Model for the robust community detection in complex networks, Community Detection in Complex Networks via Clique Conductance, A generalised significance test for individual communities in networks, Community Detection on Networkswith Ricci Flow, https://github.com/CWTSLeiden/networkanalysis, https://doi.org/10.1016/j.physrep.2009.11.002, https://doi.org/10.1103/PhysRevE.69.026113, https://doi.org/10.1103/PhysRevE.74.016110, https://doi.org/10.1103/PhysRevE.70.066111, https://doi.org/10.1103/PhysRevE.72.027104, https://doi.org/10.1103/PhysRevE.74.036104, https://doi.org/10.1088/1742-5468/2008/10/P10008, https://doi.org/10.1103/PhysRevE.80.056117, https://doi.org/10.1103/PhysRevE.84.016114, https://doi.org/10.1140/epjb/e2013-40829-0, https://doi.org/10.17706/IJCEE.2016.8.3.207-218, https://doi.org/10.1103/PhysRevE.92.032801, https://doi.org/10.1103/PhysRevE.76.036106, https://doi.org/10.1103/PhysRevE.78.046110, https://doi.org/10.1103/PhysRevE.81.046106, http://creativecommons.org/licenses/by/4.0/, A robust and accurate single-cell data trajectory inference method using ensemble pseudotime, Batch alignment of single-cell transcriptomics data using deep metric learning, ViralCC retrieves complete viral genomes and virus-host pairs from metagenomic Hi-C data, Community detection in brain connectomes with hybrid quantum computing. Note that communities found by the Leiden algorithm are guaranteed to be connected. We then remove the first node from the front of the queue and we determine whether the quality function can be increased by moving this node from its current community to a different one. We thank Lovro Subelj for his comments on an earlier version of this paper. Below, the quality of a partition is reported as \(\frac{ {\mathcal H} }{2m}\), where H is defined in Eq. Scaling of benchmark results for network size. In contrast, Leiden keeps finding better partitions in each iteration. & Fortunato, S. Community detection algorithms: A comparative analysis. A structure that is more informative than the unstructured set of clusters returned by flat clustering. E 72, 027104, https://doi.org/10.1103/PhysRevE.72.027104 (2005). Modularity is used most commonly, but is subject to the resolution limit. Phys. Knowl. MathSciNet This contrasts with the Leiden algorithm. We used modularity with a resolution parameter of =1 for the experiments. Speed of the first iteration of the Louvain and the Leiden algorithm for six empirical networks. The property of -separation is also guaranteed by the Louvain algorithm. For larger networks and higher values of , Louvain is much slower than Leiden. CAS and L.W. In addition, we prove that the algorithm converges to an asymptotically stable partition in which all subsets of all communities are locally optimally assigned. Are you sure you want to create this branch? Rev. Source Code (2018). AMS 56, 10821097 (2009). The quality improvement realised by the Leiden algorithm relative to the Louvain algorithm is larger for empirical networks than for benchmark networks. For example, the red community in (b) is refined into two subcommunities in (c), which after aggregation become two separate nodes in (d), both belonging to the same community. This will compute the Leiden clusters and add them to the Seurat Object Class. Table2 provides an overview of the six networks. The two phases are repeated until the quality function cannot be increased further. where nc is the number of nodes in community c. The interpretation of the resolution parameter is quite straightforward. Nodes 16 have connections only within this community, whereas node 0 also has many external connections. http://dx.doi.org/10.1073/pnas.0605965104. Leiden keeps finding better partitions for empirical networks also after the first 10 iterations of the algorithm. As such, we scored leiden-clustering popularity level to be Limited. Note that if Leiden finds subcommunities, splitting up the community is guaranteed to increase modularity. Phys. Louvain pruning is another improvement to Louvain proposed in 2016, and can reduce the computational time by as much as 90% while finding communities that are almost as good as Louvain (Ozaki, Tezuka, and Inaba 2016). Then, in order . Resolution Limit in Community Detection. Proc. volume9, Articlenumber:5233 (2019) The resulting clusters are shown as colors on the 3D model (top) and t -SNE embedding . The Leiden algorithm is typically iterated: the output of one iteration is used as the input for the next iteration. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Lancichinetti, A., Fortunato, S. & Radicchi, F. Benchmark graphs for testing community detection algorithms. PubMed Am. For a full specification of the fast local move procedure, we refer to the pseudo-code of the Leiden algorithm in AlgorithmA.2 in SectionA of the Supplementary Information. We denote by ec the actual number of edges in community c. The expected number of edges can be expressed as \(\frac{{K}_{c}^{2}}{2m}\), where Kc is the sum of the degrees of the nodes in community c and m is the total number of edges in the network. The algorithm then moves individual nodes in the aggregate network (e). Other networks show an almost tenfold increase in the percentage of disconnected communities. The corresponding results are presented in the Supplementary Fig. 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. 10, for the IMDB and Amazon networks, Leiden reaches a stable iteration relatively quickly, presumably because these networks have a fairly simple community structure. The authors act as bibliometric consultants to CWTS B.V., which makes use of community detection algorithms in commercial products and services. Phys. Brandes, U. et al. Cluster cells using Louvain/Leiden community detection Description. Default behaviour is calling cluster_leiden in igraph with Modularity (for undirected graphs) and CPM cost functions. The numerical details of the example can be found in SectionB of the Supplementary Information. Importantly, the output of the local moving stage will depend on the order that the nodes are considered in. After running local moving, we end up with a set of communities where we cant increase the objective function (eg, modularity) by moving any node to any neighboring community. 81 (4 Pt 2): 046114. http://dx.doi.org/10.1103/PhysRevE.81.046114. Hence, in general, Louvain may find arbitrarily badly connected communities. The difference in computational time is especially pronounced for larger networks, with Leiden being up to 20 times faster than Louvain in empirical networks. Community detection in complex networks using extremal optimization. We find that the Leiden algorithm commonly finds partitions of higher quality in less time. Ozaki, Naoto, Hiroshi Tezuka, and Mary Inaba. Google Scholar. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate. Note that the object for Seurat version 3 has changed. Faster unfolding of communities: Speeding up the Louvain algorithm. Rev. You will not need much Python to use it. Google Scholar. Use Git or checkout with SVN using the web URL. Article Higher resolutions lead to more communities and lower resolutions lead to fewer communities, similarly to the resolution parameter for modularity. To address this problem, we introduce the Leiden algorithm. After a stable iteration of the Leiden algorithm, it is guaranteed that: All nodes are locally optimally assigned. These steps are repeated until no further improvements can be made. CAS To install the development version: The current release on CRAN can be installed with: First set up a compatible adjacency matrix: An adjacency matrix is any binary matrix representing links between nodes (column and row names). We abbreviate the leidenalg package as la and the igraph package as ig in all Python code throughout this documentation. The count of badly connected communities also included disconnected communities. As can be seen in the figure, Louvain quickly reaches a state in which it is unable to find better partitions. The minimum resolvable community size depends on the total size of the network and the degree of interconnectedness of the modules. This is not too difficult to explain. A smart local moving algorithm for large-scale modularity-based community detection. Leiden consists of the following steps: The refinement step allows badly connected communities to be split before creating the aggregate network. In fact, for the Web of Science and Web UK networks, Fig. The aggregate network is created based on the partition \({{\mathscr{P}}}_{{\rm{refined}}}\). Slider with three articles shown per slide. o CLIQUE (Clustering in Quest): - CLIQUE is a combination of density-based and grid-based clustering algorithm. 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. Eng. http://iopscience.iop.org/article/10.1088/1742-5468/2008/10/P10008/meta. At each iteration all clusters are guaranteed to be connected and well-separated. Modularity optimization. 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. import leidenalg as la import igraph as ig Example output. One of the most popular algorithms to optimise modularity is the so-called Louvain algorithm10, named after the location of its authors. It is good at identifying small clusters. According to CPM, it is better to split into two communities when the link density between the communities is lower than the constant. To overcome the problem of arbitrarily badly connected communities, we introduced a new algorithm, which we refer to as the Leiden algorithm. An aggregate. Phys. Cite this article. Introduction The Louvain method is an algorithm to detect communities in large networks. As can be seen in Fig. In subsequent iterations, the percentage of disconnected communities remains fairly stable. 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. b, The elephant graph (in a) is clustered using the Leiden clustering algorithm 51 (resolution r = 0.5). Speed of the first iteration of the Louvain and the Leiden algorithm for benchmark networks with increasingly difficult partitions (n=107). Google Scholar. The inspiration for this method of community detection is the optimization of modularity as the algorithm progresses. Article CAS In practice, this means that small clusters can hide inside larger clusters, making their identification difficult. Newman, M. E. J. This way of defining the expected number of edges is based on the so-called configuration model. Community detection can then be performed using this graph. Note that this code is designed for Seurat version 2 releases. The value of the resolution parameter was determined based on the so-called mixing parameter 13. CPM does not suffer from this issue13. From Louvain to Leiden: Guaranteeing Well-Connected Communities, October. The random component also makes the algorithm more explorative, which might help to find better community structures. With one exception (=0.2 and n=107), all results in Fig. Later iterations of the Louvain algorithm are very fast, but this is only because the partition remains the same. Each of these can be used as an objective function for graph-based community detection methods, with our goal being to maximize this value. 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. There is an entire Leiden package in R-cran here Communities were all of equal size. Additionally, we implemented a Python package, available from https://github.com/vtraag/leidenalg and deposited at Zenodo24). Finding community structure in networks using the eigenvectors of matrices. The Leiden algorithm is clearly faster than the Louvain algorithm. Scientific Reports (Sci Rep) The high percentage of badly connected communities attests to this. 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. The parameter functions as a sort of threshold: communities should have a density of at least , while the density between communities should be lower than . The R implementation of Leiden can be run directly on the snn igraph object in Seurat. We provide the full definitions of the properties as well as the mathematical proofs in SectionD of the Supplementary Information. The increase in the percentage of disconnected communities is relatively limited for the Live Journal and Web of Science networks. The algorithm moves individual nodes from one community to another to find a partition (b). Waltman, L. & van Eck, N. J. After the refinement phase is concluded, communities in \({\mathscr{P}}\) often will have been split into multiple communities in \({{\mathscr{P}}}_{{\rm{refined}}}\), but not always. Directed Undirected Homogeneous Heterogeneous Weighted 1. Scaling of benchmark results for difficulty of the partition. In short, the problem of badly connected communities has important practical consequences. However, if communities are badly connected, this may lead to incorrect attributions of shared functionality. Leiden is the most recent major development in this space, and highlighted a flaw in the original Louvain algorithm (Traag, Waltman, and Eck 2018). wrote the manuscript. In the case of modularity, communities may have significant substructure both because of the resolution limit and because of the shortcomings of Louvain. The Leiden algorithm starts from a singleton partition (a). Louvain algorithm. We first applied the Scanpy pipeline, including its clustering method (Leiden clustering), on the PBMC dataset. Sci Rep 9, 5233 (2019). The idea of the refinement phase in the Leiden algorithm is to identify a partition \({{\mathscr{P}}}_{{\rm{refined}}}\) that is a refinement of \({\mathscr{P}}\). Clauset, A., Newman, M. E. J. Based on this partition, an aggregate network is created (c). Subpartition -density is not guaranteed by the Louvain algorithm. The classic Louvain algorithm should be avoided due to the known problem with disconnected communities. An overview of the various guarantees is presented in Table1. 2(a). This problem is different from the well-known issue of the resolution limit of modularity14. MathSciNet Clustering is the task of grouping a set of objects with similar characteristics into one bucket and differentiating them from the rest of the group. Get the most important science stories of the day, free in your inbox. One of the best-known methods for community detection is called modularity3. The Louvain algorithm is a simple and popular method for community detection (Blondel, Guillaume, and Lambiotte 2008). The Louvain algorithm guarantees that modularity cannot be increased by merging communities (it finds a locally optimal solution). In the local moving phase, individual nodes are moved to the community that yields the largest increase in the quality function. Soft Matter Phys. Therefore, clustering algorithms look for similarities or dissimilarities among data points. It starts clustering by treating the individual data points as a single cluster then it is merged continuously based on similarity until it forms one big cluster containing all objects. 2004. Consider the partition shown in (a). To ensure readability of the paper to the broadest possible audience, we have chosen to relegate all technical details to the Supplementary Information. Ph.D. thesis, (University of Oxford, 2016). In the worst case, communities may even be disconnected, especially when running the algorithm iteratively. USA 104, 36, https://doi.org/10.1073/pnas.0605965104 (2007). Not. This contrasts with optimisation algorithms such as simulated annealing, which do allow the quality function to decrease4,8. In other words, modularity may hide smaller communities and may yield communities containing significant substructure. Therefore, by selecting a community based by choosing randomly from the neighbors, we choose the community to evaluate with probability proportional to the composition of the neighbors communities. PubMed Central Traag, Vincent, Ludo Waltman, and Nees Jan van Eck. The Louvain algorithm starts from a singleton partition in which each node is in its own community (a). We applied the Louvain and the Leiden algorithm to exactly the same networks, using the same seed for the random number generator. Neurosci. One of the most widely used algorithms is the Louvain algorithm10, which is reported to be among the fastest and best performing community detection algorithms11,12. Sci. Figure6 presents total runtime versus quality for all iterations of the Louvain and the Leiden algorithm. Furthermore, by relying on a fast local move approach, the Leiden algorithm runs faster than the Louvain algorithm. Phys. A. The Leiden community detection algorithm outperforms other clustering methods. https://doi.org/10.1038/s41598-019-41695-z, DOI: https://doi.org/10.1038/s41598-019-41695-z. Hence, the problem of Louvain outlined above is independent from the issue of the resolution limit. In the meantime, to ensure continued support, we are displaying the site without styles There was a problem preparing your codespace, please try again. Usually, the Louvain algorithm starts from a singleton partition, in which each node is in its own community. 2. For all networks, Leiden identifies substantially better partitions than Louvain. Ayan Sinha, David F. Gleich & Karthik Ramani, Marinka Zitnik, Rok Sosi & Jure Leskovec, Zhenqi Lu, Johan Wahlstrm & Arye Nehorai, Natalie Stanley, Roland Kwitt, Peter J. Mucha, Scientific Reports & Clauset, A. 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. Work fast with our official CLI. Moreover, Louvain has no mechanism for fixing these communities. A Smart Local Moving Algorithm for Large-Scale Modularity-Based Community Detection. Eur. The degree of randomness in the selection of a community is determined by a parameter >0. To use Leiden with the Seurat pipeline for a Seurat Object object that has an SNN computed (for example with Seurat::FindClusters with save.SNN = TRUE). The 'devtools' package will be used to install 'leiden' and the dependancies (igraph and reticulate). Starting from the second iteration, Leiden outperformed Louvain in terms of the percentage of badly connected communities.
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