(2) and m is the number of edges. This is not too difficult to explain. In subsequent iterations, the percentage of disconnected communities remains fairly stable. Phys. 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. In this paper, we show that the Louvain algorithm has a major problem, for both modularity and CPM. However, if communities are badly connected, this may lead to incorrect attributions of shared functionality. In this section, we analyse and compare the performance of the two algorithms in practice. import leidenalg as la import igraph as ig Example output. & Moore, C. Finding community structure in very large networks. 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. First iteration runtime for empirical networks. Get the most important science stories of the day, free in your inbox. Source Code (2018). CAS Rev. Additionally, we implemented a Python package, available from https://github.com/vtraag/leidenalg and deposited at Zenodo24). Ph.D. thesis, (University of Oxford, 2016). Louvain pruning keeps track of a list of nodes that have the potential to change communities, and only revisits nodes in this list, which is much smaller than the total number of nodes. 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. Hence, the problem of Louvain outlined above is independent from the issue of the resolution limit. This contrasts to benchmark networks, for which Leiden often converges after a few iterations. Importantly, the number of communities discovered is related only to the difference in edge density, and not the total number of nodes in the community. In the worst case, almost a quarter of the communities are badly connected. Leiden keeps finding better partitions for empirical networks also after the first 10 iterations of the algorithm. This is very similar to what the smart local moving algorithm does. This can be a shared nearest neighbours matrix derived from a graph object. (We ensured that modularity optimisation for the subnetwork was fully consistent with modularity optimisation for the whole network13) The Leiden algorithm was run until a stable iteration was obtained. 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. Agglomerative Clustering: Also known as bottom-up approach or hierarchical agglomerative clustering (HAC). Hence, the community remains disconnected, unless it is merged with another community that happens to act as a bridge. 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. The Leiden algorithm guarantees all communities to be connected, but it may yield badly connected communities. Technol. Note that the object for Seurat version 3 has changed. Practical Application of K-Means Clustering to Stock Data - Medium 10008, 6, https://doi.org/10.1088/1742-5468/2008/10/P10008 (2008). 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 The percentage of disconnected communities is more limited, usually around 1%. Node mergers that cause the quality function to decrease are not considered. Furthermore, by relying on a fast local move approach, the Leiden algorithm runs faster than the Louvain algorithm. Leiden consists of the following steps: The refinement step allows badly connected communities to be split before creating the aggregate network. In our experimental analysis, we observe that up to 25% of the communities are badly connected and up to 16% are disconnected. When a disconnected community has become a node in an aggregate network, there are no more possibilities to split up the community. & Arenas, A. Louvain quickly converges to a partition and is then unable to make further improvements. An alternative quality function is the Constant Potts Model (CPM)13, which overcomes some limitations of modularity. HiCBin: binning metagenomic contigs and recovering metagenome-assembled Inf. Each point corresponds to a certain iteration of an algorithm, with results averaged over 10 experiments. 6 show that Leiden outperforms Louvain in terms of both computational time and quality of the partitions. In the local moving phase, individual nodes are moved to the community that yields the largest increase in the quality function. Leiden is the most recent major development in this space, and highlighted a flaw in the original Louvain algorithm (Traag, Waltman, and Eck 2018). J. Speed and quality for the first 10 iterations of the Louvain and the Leiden algorithm for benchmark networks (n=106 and n=107). 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. How many iterations of the Leiden clustering algorithm to perform. 1 I am using the leiden algorithm implementation in iGraph, and noticed that when I repeat clustering using the same resolution parameter, I get different results. Modularity is a popular objective function used with the Louvain method for community detection. Below we offer an intuitive explanation of these properties. See the documentation on the leidenalg Python module for more information: https://leidenalg.readthedocs.io/en/latest/reference.html. First, we show that the Louvain algorithm finds disconnected communities, and more generally, badly connected communities in the empirical networks. igraph R manual pages Are you sure you want to create this branch? Note that if Leiden finds subcommunities, splitting up the community is guaranteed to increase modularity. the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Later iterations of the Louvain algorithm only aggravate the problem of disconnected communities, even though the quality function (i.e. By moving these nodes, Louvain creates badly connected communities. The thick edges in Fig. Leiden now included in python-igraph #1053 - Github Blondel, V D, J L Guillaume, and R Lambiotte. contrastive-sc works best on datasets with fewer clusters when using the KMeans clustering and conversely for Leiden. MATH Nevertheless, depending on the relative strengths of the different connections, these nodes may still be optimally assigned to their current community. The community with which a node is merged is selected randomly18. Article Waltman, L. & van Eck, N. J. It implies uniform -density and all the other above-mentioned properties. The two phases are repeated until the quality function cannot be increased further. 2. https://doi.org/10.1038/s41598-019-41695-z. This problem is different from the well-known issue of the resolution limit of modularity14. We now show that the Louvain algorithm may find arbitrarily badly connected communities. http://arxiv.org/abs/1810.08473. 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. Conversely, if Leiden does not find subcommunities, there is no guarantee that modularity cannot be increased by splitting up the community. Instead, a node may be merged with any community for which the quality function increases. Is modularity with a resolution parameter equivalent to leidenalg.RBConfigurationVertexPartition? Guimer, R. & Nunes Amaral, L. A. Functional cartography of complex metabolic networks. Note that nodes can be revisited several times within a single iteration of the local moving stage, as the possible increase in modularity will change as other nodes are moved to different 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. The second iteration of Louvain shows a large increase in the percentage of disconnected communities. 10X10Xleiden - Soft Matter Phys. Int. This is not the case when nodes are greedily merged with the community that yields the largest increase in the quality function. & Girvan, M. Finding and evaluating community structure in networks. Four popular community detection algorithms are explained . 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. Our analysis is based on modularity with resolution parameter =1. J. Then the Leiden algorithm can be run on the adjacency matrix. Ozaki, Naoto, Hiroshi Tezuka, and Mary Inaba. Traag, V. A. Nonlin. Lancichinetti, A., Fortunato, S. & Radicchi, F. Benchmark graphs for testing community detection algorithms. Directed Undirected Homogeneous Heterogeneous Weighted 1. 7, whereas Louvain becomes much slower for more difficult partitions, Leiden is much less affected by the difficulty of the partition. In an experiment containing a mixture of cell types, each cluster might correspond to a different cell type. In fact, by implementing the refinement phase in the right way, several attractive guarantees can be given for partitions produced by the Leiden algorithm. It only implies that individual nodes are well connected to their community. Google Scholar. As discussed earlier, the Louvain algorithm does not guarantee connectivity. In this new situation, nodes 2, 3, 5 and 6 have only internal connections. 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. Neurosci. Klavans, R. & Boyack, K. W. Which Type of Citation Analysis Generates the Most Accurate Taxonomy of Scientific and Technical Knowledge? 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. partition_type : Optional [ Type [ MutableVertexPartition ]] (default: None) Type of partition to use. Segmentation & Clustering SPATA2 - GitHub Pages Communities in Networks. There are many different approaches and algorithms to perform clustering tasks. Clustering with the Leiden Algorithm in R That is, one part of such an internally disconnected community can reach another part only through a path going outside the community. Porter, M. A., Onnela, J.-P. & Mucha, P. J. We prove that the Leiden algorithm yields communities that are guaranteed to be connected. Internet Explorer). In the refinement phase, nodes are not necessarily greedily merged with the community that yields the largest increase in the quality function. Preprocessing and clustering 3k PBMCs Scanpy documentation As such, we scored leiden-clustering popularity level to be Limited. SPATA2 currently offers the functions findSeuratClusters (), findMonocleClusters () and findNearestNeighbourClusters () which are wrapper around widely used clustering algorithms. Class wrapper based on scanpy to use the Leiden algorithm to directly cluster your data matrix with a scikit-learn flavor. Contrastive self-supervised clustering of scRNA-seq data E 80, 056117, https://doi.org/10.1103/PhysRevE.80.056117 (2009). The value of the resolution parameter was determined based on the so-called mixing parameter 13. In practice, this means that small clusters can hide inside larger clusters, making their identification difficult. Fortunato, Santo, and Marc Barthlemy. As we will demonstrate in our experimental analysis, the problem occurs frequently in practice when using the Louvain algorithm. and L.W. b, The elephant graph (in a) is clustered using the Leiden clustering algorithm 51 (resolution r = 0.5). E 81, 046106, https://doi.org/10.1103/PhysRevE.81.046106 (2010). One may expect that other nodes in the old community will then also be moved to other communities. Phys. Google Scholar. Disconnected community. Nodes 06 are in the same community. 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. 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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. Phys. To do this we just sum all the edge weights between nodes of the corresponding communities to get a single weighted edge between them, and collapse each community down to a single new node. Sci. This algorithm provides a number of explicit guarantees. http://iopscience.iop.org/article/10.1088/1742-5468/2008/10/P10008/meta. Clearly, it would be better to split up the community. 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 addition, we prove that the algorithm converges to an asymptotically stable partition in which all subsets of all communities are locally optimally assigned. At each iteration all clusters are guaranteed to be connected and well-separated. Speed and quality for the first 10 iterations of the Louvain and the Leiden algorithm for six empirical networks. Not. Another important difference between the Leiden algorithm and the Louvain algorithm is the implementation of the local moving phase. 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. CPM does not suffer from this issue13. A tag already exists with the provided branch name. 81 (4 Pt 2): 046114. http://dx.doi.org/10.1103/PhysRevE.81.046114. 8, 207218, https://doi.org/10.17706/IJCEE.2016.8.3.207-218 (2016). In contrast, Leiden keeps finding better partitions in each iteration. Finding communities in large networks is far from trivial: algorithms need to be fast, but they also need to provide high-quality results. 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. 2.3. An overview of the various guarantees is presented in Table1. The Leiden algorithm is considerably more complex than the Louvain algorithm. It partitions the data space and identifies the sub-spaces using the Apriori principle. Thank you for visiting nature.com. Then, in order . The Louvain algorithm10 is very simple and elegant. Google Scholar. In the worst case, communities may even be disconnected, especially when running the algorithm iteratively. However, for higher values of , Leiden becomes orders of magnitude faster than Louvain, reaching 10100 times faster runtimes for the largest networks. In other words, modularity may hide smaller communities and may yield communities containing significant substructure. Speed and quality of the Louvain and the Leiden algorithm for benchmark networks of increasing size (two iterations). However, focussing only on disconnected communities masks the more fundamental issue: Louvain finds arbitrarily badly connected communities. Rev. However, nodes 16 are still locally optimally assigned, and therefore these nodes will stay in the red community. Initially, \({{\mathscr{P}}}_{{\rm{refined}}}\) is set to a singleton partition, in which each node is in its own community. Performance of modularity maximization in practical contexts. volume9, Articlenumber:5233 (2019) Modularity is a scale value between 0.5 (non-modular clustering) and 1 (fully modular clustering) that measures the relative density of edges inside communities with respect to edges outside communities. 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. 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. B 86 (11): 471. https://doi.org/10.1140/epjb/e2013-40829-0. Traag, V. A., Waltman, L. & van Eck, N. J. networkanalysis. The speed difference is especially large for larger networks. Importantly, mergers are performed only within each community of the partition \({\mathscr{P}}\). Leiden algorithm. The Leiden algorithm starts from a singleton MATH For those wanting to read more, I highly recommend starting with the Leiden paper (Traag, Waltman, and Eck 2018) or the smart local moving paper (Waltman and Eck 2013). Rep. 486, 75174, https://doi.org/10.1016/j.physrep.2009.11.002 (2010). J. Comput. Presumably, many of the badly connected communities in the first iteration of Louvain become disconnected in the second iteration. Figure4 shows how well it does compared to the Louvain algorithm. Good, B. H., De Montjoye, Y.