The Leiden algorithm starts from a singleton partition (a). Phys. Ronhovde, Peter, and Zohar Nussinov. The R implementation of Leiden can be run directly on the snn igraph object in Seurat. 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. There was a problem preparing your codespace, please try again. As shown in Fig. As discussed earlier, the Louvain algorithm does not guarantee connectivity. The current state of the art when it comes to graph-based community detection is Leiden, which incorporates about 10 years of algorithmic improvements to the original Louvain method. Computer Syst. All communities are subpartition -dense. However, it is also possible to start the algorithm from a different partition15. Presumably, many of the badly connected communities in the first iteration of Louvain become disconnected in the second iteration. Initially, \({{\mathscr{P}}}_{{\rm{refined}}}\) is set to a singleton partition, in which each node is in its own community. As shown in Fig. For the Amazon, DBLP and Web UK networks, Louvain yields on average respectively 23%, 16% and 14% badly connected communities. This commit does not belong to any branch on this repository, and may belong to a fork outside of the repository. & Bornholdt, S. Statistical mechanics of community detection. Based on this partition, an aggregate network is created (c). However, if communities are badly connected, this may lead to incorrect attributions of shared functionality. Performance of modularity maximization in practical contexts. Finding communities in large networks is far from trivial: algorithms need to be fast, but they also need to provide high-quality results. For example, for the Web of Science network, the first iteration takes about 110120 seconds, while subsequent iterations require about 40 seconds. It states that there are no communities that can be merged. In this stage we essentially collapse communities down into a single representative node, creating a new simplified graph. MATH The Leiden algorithm is considerably more complex than the Louvain algorithm. Analyses based on benchmark networks have only a limited value because these networks are not representative of empirical real-world networks. Then optimize the modularity function to determine clusters. Ozaki, N., Tezuka, H. & Inaba, M. A Simple Acceleration Method for the Louvain Algorithm. We find that the Leiden algorithm is faster than the Louvain algorithm and uncovers better partitions, in addition to providing explicit guarantees. It was originally developed for modularity optimization, although the same method can be applied to optimize CPM. SPATA2 currently offers the functions findSeuratClusters (), findMonocleClusters () and findNearestNeighbourClusters () which are wrapper around widely used clustering algorithms. 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). Nonetheless, some networks still show large differences. The constant Potts model might give better communities in some cases, as it is not subject to the resolution limit. Learn more. In the most difficult case (=0.9), Louvain requires almost 2.5 days, while Leiden needs fewer than 10 minutes. Data Eng. Phys. U. S. A. Fortunato, Santo, and Marc Barthlemy. Google Scholar. 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 . In other words, modularity may hide smaller communities and may yield communities containing significant substructure. leiden_clustering Description Class wrapper based on scanpy to use the Leiden algorithm to directly cluster your data matrix with a scikit-learn flavor. 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. 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. The solution provided by Leiden is based on the smart local moving algorithm. 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. 6 show that Leiden outperforms Louvain in terms of both computational time and quality of the partitions. Clustering is a machine learning technique in which similar data points are grouped into the same cluster based on their attributes. The percentage of badly connected communities is less affected by the number of iterations of the Louvain algorithm. Speed and quality of the Louvain and the Leiden algorithm for benchmark networks of increasing size (two iterations). 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). The algorithm continues to move nodes in the rest of the network. 4. partition_type : Optional [ Type [ MutableVertexPartition ]] (default: None) Type of partition to use. First iteration runtime for empirical networks. You will not need much Python to use it. Introduction The Louvain method is an algorithm to detect communities in large networks. Is modularity with a resolution parameter equivalent to leidenalg.RBConfigurationVertexPartition? The corresponding results are presented in the Supplementary Fig. Leiden consists of the following steps: Local moving of nodes Partition refinement Network aggregation The refinement step allows badly connected communities to be split before creating the aggregate network. To ensure readability of the paper to the broadest possible audience, we have chosen to relegate all technical details to the Supplementary Information. The leidenalg package facilitates community detection of networks and builds on the package igraph. Phys. Article 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. 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. 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. The algorithm then moves individual nodes in the aggregate network (e). The main ideas of our algorithm are explained in an intuitive way in the main text of the paper. For each community in a partition that was uncovered by the Louvain algorithm, we determined whether it is internally connected or not. An iteration of the Leiden algorithm in which the partition does not change is called a stable iteration. To find an optimal grouping of cells into communities, we need some way of evaluating different partitions in the graph. The inspiration for this method of community detection is the optimization of modularity as the algorithm progresses. In a stable iteration, the partition is guaranteed to be node optimal and subpartition -dense. For lower values of , the correct partition is easy to find and Leiden is only about twice as fast as Louvain. Communities were all of equal size. You are using a browser version with limited support for CSS. Modularity is used most commonly, but is subject to the resolution limit. Therefore, clustering algorithms look for similarities or dissimilarities among data points. As can be seen in Fig. PubMed Central However, the Louvain algorithm does not consider this possibility, since it considers only individual node movements. Indeed, the percentage of disconnected communities becomes more comparable to the percentage of badly connected communities in later iterations. Raghavan, U., Albert, R. & Kumara, S. Near linear time algorithm to detect community structures in large-scale networks. Knowl. However, the initial partition for the aggregate network is based on P, just like in the Louvain algorithm. For example, nodes in a community in biological or neurological networks are often assumed to share similar functions or behaviour25. We name our algorithm the Leiden algorithm, after the location of its authors. J. The minimum resolvable community size depends on the total size of the network and the degree of interconnectedness of the modules. Furthermore, by relying on a fast local move approach, the Leiden algorithm runs faster than the Louvain algorithm. The Leiden algorithm guarantees all communities to be connected, but it may yield badly connected communities. Cite this article. Leiden is both faster than Louvain and finds better partitions. 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. 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 ). Detecting communities in a network is therefore an important problem. In this case we know the answer is exactly 10. In addition, a node is merged with a community in \({{\mathscr{P}}}_{{\rm{refined}}}\) only if both are sufficiently well connected to their community in \({\mathscr{P}}\). With one exception (=0.2 and n=107), all results in Fig. Speed of the first iteration of the Louvain and the Leiden algorithm for six empirical networks. As can be seen in the figure, Louvain quickly reaches a state in which it is unable to find better partitions. In that case, nodes 16 are all locally optimally assigned, despite the fact that their community has become disconnected. Article modularity) increases. Cluster Determination Source: R/generics.R, R/clustering.R Identify clusters of cells by a shared nearest neighbor (SNN) modularity optimization based clustering algorithm. to use Codespaces. PubMedGoogle Scholar. Figure4 shows how well it does compared to the Louvain algorithm. The nodes that are more interconnected have been partitioned into separate clusters. This is very similar to what the smart local moving algorithm does. After each iteration of the Leiden algorithm, it is guaranteed that: In these properties, refers to the resolution parameter in the quality function that is optimised, which can be either modularity or CPM. Bae, S., Halperin, D., West, J. D., Rosvall, M. & Howe, B. Scalable and Efficient Flow-Based Community Detection for Large-Scale Graph Analysis. Unlike the Louvain algorithm, the Leiden algorithm uses a fast local move procedure in this phase. Acad. Rev. 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. In fact, for the Web of Science and Web UK networks, Fig. Importantly, the problem of disconnected communities is not just a theoretical curiosity. This way of defining the expected number of edges is based on the so-called configuration model. 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. We now consider the guarantees provided by the Leiden algorithm. In particular, it yields communities that are guaranteed to be connected. Phys. An aggregate. There is an entire Leiden package in R-cran here For each set of parameters, we repeated the experiment 10 times. When iterating Louvain, the quality of the partitions will keep increasing until the algorithm is unable to make any further improvements. In addition, we prove that the algorithm converges to an asymptotically stable partition in which all subsets of all communities are locally optimally assigned. The nodes are added to the queue in a random order. 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. The Louvain algorithm is illustrated in Fig. In the local move procedure in the Leiden algorithm, only nodes whose neighborhood . 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. 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. Brandes, U. et al. The Beginner's Guide to Dimensionality Reduction. 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. Sign up for the Nature Briefing newsletter what matters in science, free to your inbox daily. However, modularity suffers from a difficult problem known as the resolution limit (Fortunato and Barthlemy 2007). Louvain can also be quite slow, as it spends a lot of time revisiting nodes that may not have changed neighborhoods. These steps are repeated until no further improvements can be made. Fortunato, S. Community detection in graphs. In this section, we analyse and compare the performance of the two algorithms in practice. Phys. Complex brain networks: graph theoretical analysis of structural and functional systems. Starting from the second iteration, Leiden outperformed Louvain in terms of the percentage of badly connected communities. We provide the full definitions of the properties as well as the mathematical proofs in SectionD of the Supplementary Information. 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. 7, whereas Louvain becomes much slower for more difficult partitions, Leiden is much less affected by the difficulty of the partition. 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. 2(b). In the first iteration, Leiden is roughly 220 times faster than Louvain. In fact, by implementing the refinement phase in the right way, several attractive guarantees can be given for partitions produced by the Leiden algorithm. Excluding node mergers that decrease the quality function makes the refinement phase more efficient. Additionally, we implemented a Python package, available from https://github.com/vtraag/leidenalg and deposited at Zenodo24). While current approaches are successful in reducing the number of sequence alignments performed, the generated clusters are . MathSciNet The corresponding results are presented in the Supplementary Fig. Article Any sub-networks that are found are treated as different communities in the next aggregation step. See the documentation for these functions. Agglomerative Clustering: Also known as bottom-up approach or hierarchical agglomerative clustering (HAC). This is not the case when nodes are greedily merged with the community that yields the largest increase in the quality function. Higher resolutions lead to more communities, while lower resolutions lead to fewer communities. We generated networks with n=103 to n=107 nodes. 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. Communities in \({\mathscr{P}}\) may be split into multiple subcommunities in \({{\mathscr{P}}}_{{\rm{refined}}}\). 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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. This represents the following graph structure. In many complex networks, nodes cluster and form relatively dense groupsoften called communities1,2. Finding and Evaluating Community Structure in Networks. Phys. Correspondence to We demonstrate the performance of the Leiden algorithm for several benchmark and real-world networks. It therefore does not guarantee -connectivity either. 68, 984998, https://doi.org/10.1002/asi.23734 (2017). MathSciNet CPM is defined as. In the worst case, communities may even be disconnected, especially when running the algorithm iteratively. The degree of randomness in the selection of a community is determined by a parameter >0. The authors act as bibliometric consultants to CWTS B.V., which makes use of community detection algorithms in commercial products and services. Number of iterations before the Leiden algorithm has reached a stable iteration for six empirical networks. These nodes are therefore optimally assigned to their current community. 2 represent stronger connections, while the other edges represent weaker connections. Modularity is a measure of the structure of networks or graphs which measures the strength of division of a network into modules (also called groups, clusters or communities). https://doi.org/10.1038/s41598-019-41695-z. Rotta, R. & Noack, A. Multilevel local search algorithms for modularity clustering. performed the experimental analysis. https://doi.org/10.1038/s41598-019-41695-z, DOI: https://doi.org/10.1038/s41598-019-41695-z. J. Assoc. Although originally defined for modularity, the Louvain algorithm can also be used to optimise other quality functions. It implies uniform -density and all the other above-mentioned properties. This contrasts with optimisation algorithms such as simulated annealing, which do allow the quality function to decrease4,8. 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. 69 (2 Pt 2): 026113. http://dx.doi.org/10.1103/PhysRevE.69.026113. We show that this algorithm has a major defect that largely went unnoticed until now: the Louvain algorithm may yield arbitrarily badly connected communities. As the use of clustering is highly depending on the biological question it makes sense to use several approaches and algorithms. Soft Matter Phys. 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. Such a modular structure is usually not known beforehand. Eng. Slider with three articles shown per slide. E 81, 046106, https://doi.org/10.1103/PhysRevE.81.046106 (2010). E 69, 026113, https://doi.org/10.1103/PhysRevE.69.026113 (2004). This package requires the 'leidenalg' and 'igraph' modules for python (2) to be installed on your system. Good, B. H., De Montjoye, Y. 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. That is, no subset can be moved to a different community. Subpartition -density does not imply that individual nodes are locally optimally assigned. 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. 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. Source Code (2018). 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. Speed of the first iteration of the Louvain and the Leiden algorithm for benchmark networks with increasingly difficult partitions (n=107). The high percentage of badly connected communities attests to this. Newman, M. E. J. Modularity is given by. In the aggregation phase, an aggregate network is created based on the partition obtained in the local moving phase. leiden_clsutering is distributed under a BSD 3-Clause License (see LICENSE). Rev. Louvain quickly converges to a partition and is then unable to make further improvements. For the results reported below, the average degree was set to \(\langle k\rangle =10\). Removing such a node from its old community disconnects the old community. Rev. where nc is the number of nodes in community c. The interpretation of the resolution parameter is quite straightforward. 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). 2018. Instead, a node may be merged with any community for which the quality function increases. Use Git or checkout with SVN using the web URL. In this way, the constant acts as a resolution parameter, and setting the constant higher will result in fewer communities. This should be the first preference when choosing an algorithm. Importantly, the output of the local moving stage will depend on the order that the nodes are considered in. By submitting a comment you agree to abide by our Terms and Community Guidelines. Once no further increase in modularity is possible by moving any node to its neighboring community, we move to the second phase of the algorithm: aggregation. Community detection in complex networks using extremal optimization. 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. Importantly, mergers are performed only within each community of the partition \({\mathscr{P}}\). The Louvain method for community detection is a popular way to discover communities from single-cell data. & Moore, C. Finding community structure in very large networks. Nevertheless, depending on the relative strengths of the different connections, these nodes may still be optimally assigned to their current community. While smart local moving and multilevel refinement can improve the communities found, the next two improvements on Louvain that Ill discuss focus on the speed/efficiency of the algorithm. The resolution limit describes a limitation where there is a minimum community size able to be resolved by optimizing modularity (or other related functions). GitHub on Feb 15, 2020 Do you think the performance improvements will also be implemented in leidenalg? The above results shows that the problem of disconnected and badly connected communities is quite pervasive in practice. Louvain keeps visiting all nodes in a network until there are no more node movements that increase the quality function. In short, the problem of badly connected communities has important practical consequences. 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. Leiden consists of the following steps: The refinement step allows badly connected communities to be split before creating the aggregate network. Edges were created in such a way that an edge fell between two communities with a probability and within a community with a probability 1. One of the most popular algorithms for uncovering community structure is the so-called Louvain algorithm. A partition of clusters as a vector of integers Examples * (2018). A community size of 50 nodes was used for the results presented below, but larger community sizes yielded qualitatively similar results. Communities in Networks. MathSciNet Rev. 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. That is, one part of such an internally disconnected community can reach another part only through a path going outside the community. This makes sense, because after phase one the total size of the graph should be significantly reduced. 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. Hence, no further improvements can be made after a stable iteration of 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 partition to create an initial partition for the aggregate network. In this paper, we show that the Louvain algorithm has a major problem, for both modularity and CPM. Not. Speed and quality for the first 10 iterations of the Louvain and the Leiden algorithm for benchmark networks (n=106 and n=107). This is the crux of the Leiden paper, and the authors show that this exact problem happens frequently in practice. Traag, V A. reviewed the manuscript. We generated benchmark networks in the following way. The speed difference is especially large for larger networks. Two ways of doing this are graph modularity (Newman and Girvan 2004) and the constant Potts model (Ronhovde and Nussinov 2010). Note that if Leiden finds subcommunities, splitting up the community is guaranteed to increase modularity. B 86 (11): 471. https://doi.org/10.1140/epjb/e2013-40829-0. In fact, if we keep iterating the Leiden algorithm, it will converge to a partition without any badly connected communities, as discussed earlier. The horizontal axis indicates the cumulative time taken to obtain the quality indicated on the vertical axis. 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). This may have serious consequences for analyses based on the resulting partitions. Leiden keeps finding better partitions for empirical networks also after the first 10 iterations of the algorithm. A Comparative Analysis of Community Detection Algorithms on Artificial Networks. 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.