Traag, V. A., Van Dooren, P. & Nesterov, Y. Our analysis is based on modularity with resolution parameter =1. http://dx.doi.org/10.1073/pnas.0605965104. Community detection in complex networks using extremal optimization. We provide the full definitions of the properties as well as the mathematical proofs in SectionD of the Supplementary Information. It implies uniform -density and all the other above-mentioned properties. Proc. For each community, modularity measures the number of edges within the community and the number of edges going outside the community, and gives a value between -1 and +1. leiden_clustering Description Class wrapper based on scanpy to use the Leiden algorithm to directly cluster your data matrix with a scikit-learn flavor. Waltman, L. & van Eck, N. J. Please Cite this article. The horizontal axis indicates the cumulative time taken to obtain the quality indicated on the vertical axis. J. This continues until the queue is empty. Soft Matter Phys. Directed Undirected Homogeneous Heterogeneous Weighted 1. Figure3 provides an illustration of the algorithm. However, the Louvain algorithm does not consider this possibility, since it considers only individual node movements. HiCBin: binning metagenomic contigs and recovering metagenome-assembled For each network, Table2 reports the maximal modularity obtained using the Louvain and the Leiden algorithm. In addition, we prove that, when the Leiden algorithm is applied iteratively, it converges to a partition in which all subsets of all communities are locally optimally assigned. Value. After a stable iteration of the Leiden algorithm, it is guaranteed that: All nodes are locally optimally assigned. scanpy.tl.leiden Scanpy 1.9.3 documentation - Read the Docs The Leiden algorithm guarantees all communities to be connected, but it may yield badly connected communities. Finding and Evaluating Community Structure in Networks. Phys. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/. First calculate k-nearest neighbors and construct the SNN graph. The constant Potts model might give better communities in some cases, as it is not subject to the resolution limit. However, for higher values of , Leiden becomes orders of magnitude faster than Louvain, reaching 10100 times faster runtimes for the largest networks. The corresponding results are presented in the Supplementary Fig. It is a directed graph if the adjacency matrix is not symmetric. It was originally developed for modularity optimization, although the same method can be applied to optimize CPM. GitHub - vtraag/leidenalg: Implementation of the Leiden algorithm for As can be seen in the figure, Louvain quickly reaches a state in which it is unable to find better partitions. 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). Guimer, R. & Nunes Amaral, L. A. Functional cartography of complex metabolic networks. Uniform -density means that no matter how a community is partitioned into two parts, the two parts will always be well connected to each other. We thank Lovro Subelj for his comments on an earlier version of this paper. For example, nodes in a community in biological or neurological networks are often assumed to share similar functions or behaviour25. 8, the Leiden algorithm is significantly faster than the Louvain algorithm also in empirical networks. Community detection is often used to understand the structure of large and complex networks. PubMedGoogle Scholar. When iterating Louvain, the quality of the partitions will keep increasing until the algorithm is unable to make any further improvements. Clauset, A., Newman, M. E. J. The algorithm moves individual nodes from one community to another to find a partition (b). The thick edges in Fig. Are you sure you want to create this branch? 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. However, in the case of the Web of Science network, more than 5% of the communities are disconnected in the first iteration. Knowl. On the other hand, Leiden keeps finding better partitions, especially for higher values of , for which it is more difficult to identify good partitions. 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. Using the fast local move procedure, the first visit to all nodes in a network in the Leiden algorithm is the same as in the Louvain algorithm. In fact, by implementing the refinement phase in the right way, several attractive guarantees can be given for partitions produced by the Leiden algorithm. Hierarchical Clustering Explained - Towards Data Science Once aggregation is complete we restart the local moving phase, and continue to iterate until everything converges down to one node. Technol. 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. We start by initialising a queue with all nodes in the network. Community detection is an important task in the analysis of complex networks. Modularity is used most commonly, but is subject to the resolution limit. We now consider the guarantees provided by the Leiden algorithm. We therefore require a more principled solution, which we will introduce in the next section. However, values of within a range of roughly [0.0005, 0.1] all provide reasonable results, thus allowing for some, but not too much randomness. In practice, this means that small clusters can hide inside larger clusters, making their identification difficult. This contrasts with optimisation algorithms such as simulated annealing, which do allow the quality function to decrease4,8. Inf. This represents the following graph structure. Nonlin. For the results reported below, the average degree was set to \(\langle k\rangle =10\). How many iterations of the Leiden clustering algorithm to perform. Sci Rep 9, 5233 (2019). Phys. A community is subset optimal if all subsets of the community are locally optimally assigned. Zenodo, https://doi.org/10.5281/zenodo.1466831 https://github.com/CWTSLeiden/networkanalysis. 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. Any sub-networks that are found are treated as different communities in the next aggregation step. Run the code above in your browser using DataCamp Workspace. Article Blondel, V. D., Guillaume, J.-L., Lambiotte, R. & Lefebvre, E. Fast unfolding of communities in large networks. For the Amazon and IMDB networks, the first iteration of the Leiden algorithm is only about 1.6 times faster than the first iteration of the Louvain algorithm. The Leiden algorithm also takes advantage of the idea of speeding up the local moving of nodes16,17 and the idea of moving nodes to random neighbours18. Louvain can also be quite slow, as it spends a lot of time revisiting nodes that may not have changed neighborhoods. It identifies the clusters by calculating the densities of the cells. We show that this algorithm has a major defect that largely went unnoticed until now: the Louvain algorithm may yield arbitrarily badly connected communities. Ronhovde, Peter, and Zohar Nussinov. In the first step of the next iteration, Louvain will again move individual nodes in the network. contrastive-sc works best on datasets with fewer clusters when using the KMeans clustering and conversely for Leiden. Newman, M. E. J. Acad. Louvain algorithm. In that case, nodes 16 are all locally optimally assigned, despite the fact that their community has become disconnected. Elect. 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. Agglomerative Clustering: Also known as bottom-up approach or hierarchical agglomerative clustering (HAC). When a disconnected community has become a node in an aggregate network, there are no more possibilities to split up the community. In the worst case, almost a quarter of the communities are badly connected. Modularity scores of +1 mean that all the edges in a community are connecting nodes within the community. The aggregate network is created based on the partition \({{\mathscr{P}}}_{{\rm{refined}}}\). They show that the original Louvain algorithm that can result in badly connected communities (even communities that are completely disconnected internally) and propose an alternative method, Leiden, that guarantees that communities are well connected. 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). What is Clustering and Different Types of Clustering Methods If material is not included in the articles Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. In the meantime, to ensure continued support, we are displaying the site without styles 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. Runtime versus quality for benchmark networks. E 80, 056117, https://doi.org/10.1103/PhysRevE.80.056117 (2009). Sci. Yang, Z., Algesheimer, R. & Tessone, C. J. One may expect that other nodes in the old community will then also be moved to other communities. USA 104, 36, https://doi.org/10.1073/pnas.0605965104 (2007). V. A. Traag. 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. These steps are repeated until the quality cannot be increased further. All authors conceived the algorithm and contributed to the source code. The property of -separation is also guaranteed by the Louvain algorithm. A tag already exists with the provided branch name. leidenalg. The above results shows that the problem of disconnected and badly connected communities is quite pervasive in practice. We here introduce the Leiden algorithm, which guarantees that communities are well connected. Rev. The nodes are added to the queue in a random order. Table2 provides an overview of the six networks. Good, B. H., De Montjoye, Y. We generated benchmark networks in the following way. Iterating the Louvain algorithm can therefore be seen as a double-edged sword: it improves the partition in some way, but degrades it in another way. 2016. cluster_cells: Cluster cells using Louvain/Leiden community detection Each of these can be used as an objective function for graph-based community detection methods, with our goal being to maximize this value. For higher values of , Leiden finds better partitions than Louvain. Leiden is both faster than Louvain and finds better partitions. Google Scholar. Hence, the problem of Louvain outlined above is independent from the issue of the resolution limit. Scaling of benchmark results for difficulty of the partition. While current approaches are successful in reducing the number of sequence alignments performed, the generated clusters are . There is an entire Leiden package in R-cran here (2) and m is the number of edges. sign in With one exception (=0.2 and n=107), all results in Fig. & Fortunato, S. Community detection algorithms: A comparative analysis. Fortunato, S. & Barthlemy, M. Resolution Limit in Community Detection. Ozaki, Naoto, Hiroshi Tezuka, and Mary Inaba. Centre for Science and Technology Studies, Leiden University, Leiden, The Netherlands, You can also search for this author in CAS In this case we know the answer is exactly 10. Phys. Article Is modularity with a resolution parameter equivalent to leidenalg.RBConfigurationVertexPartition? An overview of the various guarantees is presented in Table1. Learn more. modularity) increases. E Stat. Introduction leidenalg 0.9.2.dev0+gb530332.d20221214 documentation 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. E 84, 016114, https://doi.org/10.1103/PhysRevE.84.016114 (2011). However, focussing only on disconnected communities masks the more fundamental issue: Louvain finds arbitrarily badly connected communities. Work fast with our official CLI. Fast Unfolding of Communities in Large Networks. Journal of Statistical , January. This is very similar to what the smart local moving algorithm does. Soft Matter Phys. E 76, 036106, https://doi.org/10.1103/PhysRevE.76.036106 (2007). This is because Louvain only moves individual nodes at a time. The algorithm is described in pseudo-code in AlgorithmA.2 in SectionA of the Supplementary Information. The authors act as bibliometric consultants to CWTS B.V., which makes use of community detection algorithms in commercial products and services. ADS In addition, to analyse whether a community is badly connected, we ran the Leiden algorithm on the subnetwork consisting of all nodes belonging to the community. Hence, for lower values of , the difference in quality is negligible. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate. Hence, in general, Louvain may find arbitrarily badly connected communities. The algorithm then locally merges nodes in \({{\mathscr{P}}}_{{\rm{refined}}}\): nodes that are on their own in a community in \({{\mathscr{P}}}_{{\rm{refined}}}\) can be merged with a different community. CAS MathSciNet We conclude that the Leiden algorithm is strongly preferable to the Louvain algorithm. Duch, J. 104 (1): 3641. Below we offer an intuitive explanation of these properties. Phys. Traag, V. A., Waltman, L. & van Eck, N. J. networkanalysis. E 74, 036104, https://doi.org/10.1103/PhysRevE.74.036104 (2006). The property of -connectivity is a slightly stronger variant of ordinary connectivity. For all networks, Leiden identifies substantially better partitions than Louvain. J. Assoc. Performance of modularity maximization in practical contexts. The steps for agglomerative clustering are as follows: Phys. Leiden consists of the following steps: The refinement step allows badly connected communities to be split before creating the aggregate network. 2010. Here we can see partitions in the plotted results. Publishers note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. In terms of the percentage of badly connected communities in the first iteration, Leiden performs even worse than Louvain, as can be seen in Fig. 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. Community Detection Algorithms - Towards Data Science These nodes are therefore optimally assigned to their current community. 20, 172188, https://doi.org/10.1109/TKDE.2007.190689 (2008). o CLIQUE (Clustering in Quest): - CLIQUE is a combination of density-based and grid-based clustering algorithm. CPM has the advantage that it is not subject to the resolution limit. Arguments can be passed to the leidenalg implementation in Python: In particular, the resolution parameter can fine-tune the number of clusters to be detected. 10008, 6, https://doi.org/10.1088/1742-5468/2008/10/P10008 (2008). The two phases are repeated until the quality function cannot be increased further. Figure6 presents total runtime versus quality for all iterations of the Louvain and the Leiden algorithm. Clustering with the Leiden Algorithm in R This should be the first preference when choosing an algorithm. This is the crux of the Leiden paper, and the authors show that this exact problem happens frequently in practice. To address this important shortcoming, we introduce a new algorithm that is faster, finds better partitions and provides explicit guarantees and bounds. Many Git commands accept both tag and branch names, so creating this branch may cause unexpected behavior. Brandes, U. et al. To overcome the problem of arbitrarily badly connected communities, we introduced a new algorithm, which we refer to as the Leiden algorithm. Newman, M E J, and M Girvan. Introduction The Louvain method is an algorithm to detect communities in large networks. In the local moving phase, individual nodes are moved to the community that yields the largest increase in the quality function. Using UMAP for Clustering umap 0.5 documentation - Read the Docs 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). Clustering with the Leiden Algorithm in R This package allows calling the Leiden algorithm for clustering on an igraph object from R. See the Python and Java implementations for more details: https://github.com/CWTSLeiden/networkanalysis https://github.com/vtraag/leidenalg Install In contrast, Leiden keeps finding better partitions in each iteration. & Bornholdt, S. Statistical mechanics of community detection. to use Codespaces. MathSciNet As we prove in SectionC1 of the Supplementary Information, even when node mergers that decrease the quality function are excluded, the optimal partition of a set of nodes can still be uncovered. Rev. For the Amazon, DBLP and Web UK networks, Louvain yields on average respectively 23%, 16% and 14% badly connected communities. Later iterations of the Louvain algorithm are very fast, but this is only because the partition remains the same. Node mergers that cause the quality function to decrease are not considered. This makes sense, because after phase one the total size of the graph should be significantly reduced. In this way, Leiden implements the local moving phase more efficiently than Louvain. Starting from the second iteration, Leiden outperformed Louvain in terms of the percentage of badly connected communities. 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. Analyses based on benchmark networks have only a limited value because these networks are not representative of empirical real-world networks. MATH b, The elephant graph (in a) is clustered using the Leiden clustering algorithm 51 (resolution r = 0.5). In this iterative scheme, Louvain provides two guarantees: (1) no communities can be merged and (2) no nodes can be moved. Node optimality is also guaranteed after a stable iteration of the Louvain algorithm. Natl. 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 . The algorithm is run iteratively, using the partition identified in one iteration as starting point for the next iteration. Each point corresponds to a certain iteration of an algorithm, with results averaged over 10 experiments. leiden_clsutering is distributed under a BSD 3-Clause License (see LICENSE). We then created a certain number of edges such that a specified average degree \(\langle k\rangle \) was obtained. One of the most popular algorithms to optimise modularity is the so-called Louvain algorithm10, named after the location of its authors. Source Code (2018). Rep. 6, 30750, https://doi.org/10.1038/srep30750 (2016). 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). Modularity is given by. A new methodology for constructing a publication-level classification system of science. Note that the object for Seurat version 3 has changed. From Louvain to Leiden: guaranteeing well-connected communities - Nature The idea of the refinement phase in the Leiden algorithm is to identify a partition \({{\mathscr{P}}}_{{\rm{refined}}}\) that is a refinement of \({\mathscr{P}}\). The Leiden algorithm starts from a singleton partition (a). Article As discussed earlier, the Louvain algorithm does not guarantee connectivity. In particular, we show that Louvain may identify communities that are internally disconnected. ADS Scaling of benchmark results for network size. Luecken, M. D. Application of multi-resolution partitioning of interaction networks to the study of complex disease. Not. First iteration runtime for empirical networks. The Leiden algorithm has been specifically designed to address the problem of badly connected communities. 9, the Leiden algorithm also performs better than the Louvain algorithm in terms of the quality of the partitions that are obtained. Faster unfolding of communities: Speeding up the Louvain algorithm. CAS 7, whereas Louvain becomes much slower for more difficult partitions, Leiden is much less affected by the difficulty of the partition. Rev. First, we show that the Louvain algorithm finds disconnected communities, and more generally, badly connected communities in the empirical networks. MathSciNet In the fast local move procedure in the Leiden algorithm, only nodes whose neighbourhood has changed are visited. Zenodo, https://doi.org/10.5281/zenodo.1469357 https://github.com/vtraag/leidenalg. Complex brain networks: graph theoretical analysis of structural and functional systems. 92 (3): 032801. http://dx.doi.org/10.1103/PhysRevE.92.032801. A Simple Acceleration Method for the Louvain Algorithm. Int. 68, 984998, https://doi.org/10.1002/asi.23734 (2017). Slider with three articles shown per slide. We abbreviate the leidenalg package as la and the igraph package as ig in all Python code throughout this documentation. Cluster your data matrix with the Leiden algorithm. There was a problem preparing your codespace, please try again. Phys. If you cant use Leiden, choosing Smart Local Moving will likely give very similar results, but might be a bit slower as it doesnt include some of the simple speedups to Louvain like random moving and Louvain pruning. When node 0 is moved to a different community, the red community becomes internally disconnected, as shown in (b). We generated networks with n=103 to n=107 nodes. In that case, some optimal partitions cannot be found, as we show in SectionC2 of the Supplementary Information. Moreover, the deeper significance of the problem was not recognised: disconnected communities are merely the most extreme manifestation of the problem of arbitrarily badly connected communities. Speed and quality for the first 10 iterations of the Louvain and the Leiden algorithm for benchmark networks (n=106 and n=107). If nothing happens, download Xcode and try again. Phys. See the documentation for these functions. However, so far this problem has never been studied for the Louvain algorithm. The Louvain algorithm is a simple and popular method for community detection (Blondel, Guillaume, and Lambiotte 2008). The Leiden algorithm is considerably more complex than the Louvain algorithm. 8 (3): 207. https://pdfs.semanticscholar.org/4ea9/74f0fadb57a0b1ec35cbc5b3eb28e9b966d8.pdf. Modularity is a popular objective function used with the Louvain method for community detection. Inf. The algorithm optimises a quality function such as modularity or CPM in two elementary phases: (1) local moving of nodes; and (2) aggregation of the network. This is similar to ideas proposed recently as pruning16 and in a slightly different form as prioritisation17. Again, if communities are badly connected, this may lead to incorrect inferences of topics, which will affect bibliometric analyses relying on the inferred topics. Ozaki, N., Tezuka, H. & Inaba, M. A Simple Acceleration Method for the Louvain Algorithm. Number of iterations before the Leiden algorithm has reached a stable iteration for six empirical networks. partition_type : Optional [ Type [ MutableVertexPartition ]] (default: None) Type of partition to use. Source Code (2018). Discovering cell types using manifold learning and enhanced E 74, 016110, https://doi.org/10.1103/PhysRevE.74.016110 (2006). Discov. Leiden now included in python-igraph #1053 - Github Hence, the complex structure of empirical networks creates an even stronger need for the use of the Leiden algorithm. As can be seen in Fig. . Article All communities are subpartition -dense. I tracked the number of clusters post-clustering at each step. In a stable iteration, the partition is guaranteed to be node optimal and subpartition -dense. This method tries to maximise the difference between the actual number of edges in a community and the expected number of such edges. Theory Exp. https://leidenalg.readthedocs.io/en/latest/reference.html. The resulting clusters are shown as colors on the 3D model (top) and t -SNE embedding . A structure that is more informative than the unstructured set of clusters returned by flat clustering. A. You will not need much Python to use it. However, the initial partition for the aggregate network is based on P, just like in the Louvain algorithm. Sci. We prove that the Leiden algorithm yields communities that are guaranteed to be connected. The Louvain local moving phase consists of the following steps: This process is repeated for every node in the network until no further improvement in modularity is possible. E 81, 046106, https://doi.org/10.1103/PhysRevE.81.046106 (2010). Conversely, if Leiden does not find subcommunities, there is no guarantee that modularity cannot be increased by splitting up the community. A major goal of single-cell analysis is to study the cell-state heterogeneity within a sample by discovering groups within the population of cells.