leiden clustering explained

Porter, M. A., Onnela, J.-P. & Mucha, P. J. partition_type : Optional [ Type [ MutableVertexPartition ]] (default: None) Type of partition to use. Clustering algorithms look for similarities or dissimilarities among data points so that similar ones can be grouped together. How many iterations of the Leiden clustering algorithm to perform. 9, the Leiden algorithm also performs better than the Louvain algorithm in terms of the quality of the partitions that are obtained. Soc. 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 However, the Louvain algorithm does not consider this possibility, since it considers only individual node movements. Faster unfolding of communities: Speeding up the Louvain algorithm. In the most difficult case (=0.9), Louvain requires almost 2.5 days, while Leiden needs fewer than 10 minutes. Disconnected community. It is a directed graph if the adjacency matrix is not symmetric. Then the Leiden algorithm can be run on the adjacency matrix. However, modularity suffers from a difficult problem known as the resolution limit (Fortunato and Barthlemy 2007). Percentage of communities found by the Louvain algorithm that are either disconnected or badly connected compared to percentage of badly connected communities found by the Leiden algorithm. to use Codespaces. J. Assoc. The algorithm is described in pseudo-code in AlgorithmA.2 in SectionA of the Supplementary Information. Phys. It partitions the data space and identifies the sub-spaces using the Apriori principle. Fortunato, S. & Barthlemy, M. Resolution Limit in Community Detection. From Louvain to Leiden: guaranteeing well-connected communities, $$ {\mathcal H} =\frac{1}{2m}\,{\sum }_{c}({e}_{c}-{\rm{\gamma }}\frac{{K}_{c}^{2}}{2m}),$$, $$ {\mathcal H} ={\sum }_{c}[{e}_{c}-\gamma (\begin{array}{c}{n}_{c}\\ 2\end{array})],$$, https://doi.org/10.1038/s41598-019-41695-z. 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. 2(b). The algorithm continues to move nodes in the rest of the network. Node optimality is also guaranteed after a stable iteration of the Louvain algorithm. We show that this algorithm has a major defect that largely went unnoticed until now: the Louvain algorithm may yield arbitrarily badly connected communities. After the first iteration of the Louvain algorithm, some partition has been obtained. (We implemented both algorithms in Java, available from https://github.com/CWTSLeiden/networkanalysis and deposited at Zenodo23. 2008. Traag, V. A., Van Dooren, P. & Nesterov, Y. For lower values of , the correct partition is easy to find and Leiden is only about twice as fast as Louvain. E Stat. MathSciNet In this section, we analyse and compare the performance of the two algorithms in practice. In an experiment containing a mixture of cell types, each cluster might correspond to a different cell type. Complex brain networks: graph theoretical analysis of structural and functional systems. Lancichinetti, A., Fortunato, S. & Radicchi, F. Benchmark graphs for testing community detection algorithms. The Web of Science network is the most difficult one. We provide the full definitions of the properties as well as the mathematical proofs in SectionD of the Supplementary Information. Brandes, U. et al. This will compute the Leiden clusters and add them to the Seurat Object Class. We can guarantee a number of properties of the partitions found by the Leiden algorithm at various stages of the iterative process. & Girvan, M. Finding and evaluating community structure in networks. Moreover, when no more nodes can be moved, the algorithm will aggregate the network. Not. Besides being pervasive, the problem is also sizeable. J. This is not the case when nodes are greedily merged with the community that yields the largest increase in the quality function. Nonlin. Phys. 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. ADS First, we created a specified number of nodes and we assigned each node to a community. This commit does not belong to any branch on this repository, and may belong to a fork outside of the repository. If nothing happens, download GitHub Desktop and try again. The refined partition \({{\mathscr{P}}}_{{\rm{refined}}}\) is obtained as follows. More subtle problems may occur as well, causing Louvain to find communities that are connected, but only in a very weak sense. The idea of the refinement phase in the Leiden algorithm is to identify a partition \({{\mathscr{P}}}_{{\rm{refined}}}\) that is a refinement of \({\mathscr{P}}\). A community size of 50 nodes was used for the results presented below, but larger community sizes yielded qualitatively similar results. Random moving is a very simple adjustment to Louvain local moving proposed in 2015 (Traag 2015). This may have serious consequences for analyses based on the resulting partitions. When node 0 is moved to a different community, the red community becomes internally disconnected, as shown in (b). Clearly, it would be better to split up the community. 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. 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. 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 . However, for higher values of , Leiden becomes orders of magnitude faster than Louvain, reaching 10100 times faster runtimes for the largest networks. Rotta, R. & Noack, A. Multilevel local search algorithms for modularity clustering. 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. ADS Furthermore, by relying on a fast local move approach, the Leiden algorithm runs faster than the Louvain algorithm. E 74, 016110, https://doi.org/10.1103/PhysRevE.74.016110 (2006). As discussed earlier, the Louvain algorithm does not guarantee connectivity. http://arxiv.org/abs/1810.08473. Besides the relative flexibility of the implementation, it also scales well, and can be run on graphs of millions of nodes (as long as they can fit in memory). Speed and quality for the first 10 iterations of the Louvain and the Leiden algorithm for six empirical networks. Leiden is faster than Louvain especially for larger networks. One may expect that other nodes in the old community will then also be moved to other communities. Google Scholar. The Leiden algorithm starts from a singleton partition (a). A. Cite this article. 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 74, 036104, https://doi.org/10.1103/PhysRevE.74.036104 (2006). 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. Louvain keeps visiting all nodes in a network until there are no more node movements that increase the quality function. Basically, there are two types of hierarchical cluster analysis strategies - 1. Ozaki, Naoto, Hiroshi Tezuka, and Mary Inaba. Introduction The Louvain method is an algorithm to detect communities in large networks. Many Git commands accept both tag and branch names, so creating this branch may cause unexpected behavior. After a stable iteration of the Leiden algorithm, it is guaranteed that: All nodes are locally optimally assigned. We now show that the Louvain algorithm may find arbitrarily badly connected communities. Local Resolution-Limit-Free Potts Model for Community Detection. Phys. Importantly, the output of the local moving stage will depend on the order that the nodes are considered in. The Louvain method for community detection is a popular way to discover communities from single-cell data. MathSciNet Subset optimality is the strongest guarantee that is provided by the Leiden algorithm. The thick edges in Fig. Natl. Moreover, Louvain has no mechanism for fixing these communities. E 78, 046110, https://doi.org/10.1103/PhysRevE.78.046110 (2008). Note that this code is . 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. Waltman, L. & van Eck, N. J. Traag, V. A., Waltman, L. & van Eck, N. J. networkanalysis. the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Below we offer an intuitive explanation of these properties. Leiden consists of the following steps: The refinement step allows badly connected communities to be split before creating the aggregate network. Natl. The larger the increase in the quality function, the more likely a community is to be selected. In the refinement phase, nodes are not necessarily greedily merged with the community that yields the largest increase in the quality function. Clauset, A., Newman, M. E. J. Crucially, however, the percentage of badly connected communities decreases with each iteration of the Leiden algorithm. Rev. 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. This can be a shared nearest neighbours matrix derived from a graph object. As can be seen in Fig. The corresponding results are presented in the Supplementary Fig. The algorithm moves individual nodes from one community to another to find a partition (b), which is then refined (c). The value of the resolution parameter was determined based on the so-called mixing parameter 13. We gratefully acknowledge computational facilities provided by the LIACS Data Science Lab Computing Facilities through Frank Takes. We used modularity with a resolution parameter of =1 for the experiments. Internet Explorer). It is good at identifying small clusters. leiden_clsutering is distributed under a BSD 3-Clause License (see LICENSE). An alternative quality function is the Constant Potts Model (CPM)13, which overcomes some limitations of modularity. 5, for lower values of the partition is well defined, and neither the Louvain nor the Leiden algorithm has a problem in determining the correct partition in only two iterations. Leiden keeps finding better partitions for empirical networks also after the first 10 iterations of the algorithm. 2004. This enables us to find cases where its beneficial to split a community. E 76, 036106, https://doi.org/10.1103/PhysRevE.76.036106 (2007). For example, nodes in a community in biological or neurological networks are often assumed to share similar functions or behaviour25. Positive values above 2 define the total number of iterations to perform, -1 has the algorithm run until it reaches its optimal clustering. All experiments were run on a computer with 64 Intel Xeon E5-4667v3 2GHz CPUs and 1TB internal memory. PubMed The community with which a node is merged is selected randomly18. In fact, by implementing the refinement phase in the right way, several attractive guarantees can be given for partitions produced by the Leiden algorithm. Article In our experimental analysis, we observe that up to 25% of the communities are badly connected and up to 16% are disconnected. The quality of such an asymptotically stable partition provides an upper bound on the quality of an optimal partition. Subpartition -density does not imply that individual nodes are locally optimally assigned. Note that Leiden clustering directly clusters the neighborhood graph of cells, which we already computed in the previous section. In the Louvain algorithm, an aggregate network is created based on the partition \({\mathscr{P}}\) resulting from the local moving phase. The horizontal axis indicates the cumulative time taken to obtain the quality indicated on the vertical axis. As the use of clustering is highly depending on the biological question it makes sense to use several approaches and algorithms. E Stat. Agglomerative clustering is a bottom-up approach. J. In doing so, Louvain keeps visiting nodes that cannot be moved to a different community. We prove that the new algorithm is guaranteed to produce partitions in which all communities are internally connected. Community detection is often used to understand the structure of large and complex networks. 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. Am. Note that if Leiden finds subcommunities, splitting up the community is guaranteed to increase modularity. Learn more. 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. 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. Communities in \({\mathscr{P}}\) may be split into multiple subcommunities in \({{\mathscr{P}}}_{{\rm{refined}}}\). This algorithm provides a number of explicit guarantees. 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. Default behaviour is calling cluster_leiden in igraph with Modularity (for undirected graphs) and CPM cost functions. This method tries to maximise the difference between the actual number of edges in a community and the expected number of such edges. The random component also makes the algorithm more explorative, which might help to find better community structures. Phys. conda install -c conda-forge leidenalg pip install leiden-clustering Used via. This is the crux of the Leiden paper, and the authors show that this exact problem happens frequently in practice. 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. ADS First, we show that the Louvain algorithm finds disconnected communities, and more generally, badly connected communities in the empirical networks. Even worse, the Amazon network has 5% disconnected communities, but 25% badly connected communities. Removing such a node from its old community disconnects the old community. Source Code (2018). N.J.v.E. MathSciNet However, so far this problem has never been studied for the Louvain algorithm. Fortunato, S. Community detection in graphs. leidenalg. One of the best-known methods for community detection is called modularity3. E 84, 016114, https://doi.org/10.1103/PhysRevE.84.016114 (2011). 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. Contrary to what might be expected, iterating the Louvain algorithm aggravates the problem of badly connected communities, as we will also see in our experimental analysis. USA 104, 36, https://doi.org/10.1073/pnas.0605965104 (2007). CAS Google Scholar. 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. B 86 (11): 471. https://doi.org/10.1140/epjb/e2013-40829-0. Rev. Thank you for visiting nature.com. When a sufficient number of neighbours of node 0 have formed a community in the rest of the network, it may be optimal to move node 0 to this community, thus creating the situation depicted in Fig. Bullmore, E. & Sporns, O. * (2018). CAS Initially, \({{\mathscr{P}}}_{{\rm{refined}}}\) is set to a singleton partition, in which each node is in its own community. sign in 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. Phys. The numerical details of the example can be found in SectionB of the Supplementary Information. Again, if communities are badly connected, this may lead to incorrect inferences of topics, which will affect bibliometric analyses relying on the inferred topics. Modularity is used most commonly, but is subject to the resolution limit. Hence, no further improvements can be made after a stable iteration of the Louvain algorithm. http://iopscience.iop.org/article/10.1088/1742-5468/2008/10/P10008/meta, http://dx.doi.org/10.1073/pnas.0605965104, http://dx.doi.org/10.1103/PhysRevE.69.026113, https://pdfs.semanticscholar.org/4ea9/74f0fadb57a0b1ec35cbc5b3eb28e9b966d8.pdf, http://dx.doi.org/10.1103/PhysRevE.81.046114, http://dx.doi.org/10.1103/PhysRevE.92.032801, https://doi.org/10.1140/epjb/e2013-40829-0, Assign each node to a different community. Use Git or checkout with SVN using the web URL. As can be seen in Fig. Lancichinetti, A. Scaling of benchmark results for difficulty of the partition. http://dx.doi.org/10.1073/pnas.0605965104. Number of iterations before the Leiden algorithm has reached a stable iteration for six empirical networks. The leidenalg package facilitates community detection of networks and builds on the package igraph. Rev. Rep. 6, 30750, https://doi.org/10.1038/srep30750 (2016). Google Scholar. We find that the Leiden algorithm is faster than the Louvain algorithm and uncovers better partitions, in addition to providing explicit guarantees. Finding communities in large networks is far from trivial: algorithms need to be fast, but they also need to provide high-quality results. 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. Anyone you share the following link with will be able to read this content: Sorry, a shareable link is not currently available for this article. This aspect of the Louvain algorithm can be used to give information about the hierarchical relationships between communities by tracking at which stage the nodes in the communities were aggregated. However, the initial partition for the aggregate network is based on P, just like in the Louvain algorithm. E 80, 056117, https://doi.org/10.1103/PhysRevE.80.056117 (2009). 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. In short, the problem of badly connected communities has important practical consequences. Sci. 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. ISSN 2045-2322 (online). Directed Undirected Homogeneous Heterogeneous Weighted 1. Speed of the first iteration of the Louvain and the Leiden algorithm for benchmark networks with increasingly difficult partitions (n=107). The R implementation of Leiden can be run directly on the snn igraph object in Seurat. A number of iterations of the Leiden algorithm can be performed before the Louvain algorithm has finished its first iteration. Badly connected communities. Good, B. H., De Montjoye, Y. The Louvain algorithm is illustrated in Fig. As we will demonstrate in our experimental analysis, the problem occurs frequently in practice when using the Louvain algorithm. ADS After a stable iteration of the Leiden algorithm, the algorithm may still be able to make further improvements in later iterations. and L.W. 9 shows that more than 10 iterations of the Leiden algorithm can be performed before the Louvain algorithm has finished its first iteration. Randomness in the selection of a community allows the partition space to be explored more broadly. 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. Excluding node mergers that decrease the quality function makes the refinement phase more efficient. With one exception (=0.2 and n=107), all results in Fig. Runtime versus quality for empirical networks. In this paper, we show that the Louvain algorithm has a major problem, for both modularity and CPM. Value. Data Eng. Importantly, mergers are performed only within each community of the partition \({\mathscr{P}}\). The algorithm moves individual nodes from one community to another to find a partition (b), which is then refined (c). By moving these nodes, Louvain creates badly connected communities. Both conda and PyPI have leiden clustering in Python which operates via iGraph. 2007. Communities may even be disconnected. MATH In this post, I will cover one of the common approaches which is hierarchical clustering. 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. For example, for the Web of Science network, the first iteration takes about 110120 seconds, while subsequent iterations require about 40 seconds. The Leiden algorithm is considerably more complex than the Louvain algorithm. Newman, M. E. J. Acad. In the case of the Louvain algorithm, after a stable iteration, all subsequent iterations will be stable as well. The main ideas of our algorithm are explained in an intuitive way in the main text of the paper. The classic Louvain algorithm should be avoided due to the known problem with disconnected communities. The percentage of disconnected communities is more limited, usually around 1%. Technol. Due to the resolution limit, modularity may cause smaller communities to be clustered into larger communities. Algorithmics 16, 2.1, https://doi.org/10.1145/1963190.1970376 (2011). 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. In the previous section, we showed that the Leiden algorithm guarantees a number of properties of the partitions uncovered at different stages of the algorithm. 63, 23782392, https://doi.org/10.1002/asi.22748 (2012). 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). Phys. Discov. The algorithm is run iteratively, using the partition identified in one iteration as starting point for the next iteration. Community detection can then be performed using this graph. In particular, in an attempt to find better partitions, multiple consecutive iterations of the algorithm can be performed, using the partition identified in one iteration as starting point for the next iteration. We generated networks with n=103 to n=107 nodes. Sci. Inf. Clustering the neighborhood graph As with Seurat and many other frameworks, we recommend the Leiden graph-clustering method (community detection based on optimizing modularity) by Traag *et al. Luecken, M. D. Application of multi-resolution partitioning of interaction networks to the study of complex disease. It identifies the clusters by calculating the densities of the cells. This is very similar to what the smart local moving algorithm does. Slider with three articles shown per slide. The Louvain algorithm is a simple and popular method for community detection (Blondel, Guillaume, and Lambiotte 2008). However, focussing only on disconnected communities masks the more fundamental issue: Louvain finds arbitrarily badly connected communities. 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 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 solution provided by Leiden is based on the smart local moving algorithm. We also suggested that the Leiden algorithm is faster than the Louvain algorithm, because of the fast local move approach. One of the most popular algorithms for uncovering community structure is the so-called Louvain algorithm. The Leiden algorithm has been specifically designed to address the problem of badly connected communities. It only implies that individual nodes are well connected to their community. Such a modular structure is usually not known beforehand. For example, after four iterations, the Web UK network has 8% disconnected communities, but twice as many badly connected communities. See the documentation for these functions. To elucidate the problem, we consider the example illustrated in Fig. Hence, the complex structure of empirical networks creates an even stronger need for the use of the Leiden algorithm. Although originally defined for modularity, the Louvain algorithm can also be used to optimise other quality functions. Rev. The problem of disconnected communities has been observed before19,20, also in the context of the label propagation algorithm21. 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. At each iteration all clusters are guaranteed to be connected and well-separated. Phys. where nc is the number of nodes in community c. The interpretation of the resolution parameter is quite straightforward. If we move the node to a different community, we add to the rear of the queue all neighbours of the node that do not belong to the nodes new community and that are not yet in the queue. Unsupervised clustering of cells is a common step in many single-cell expression workflows. The second iteration of Louvain shows a large increase in the percentage of disconnected communities. In this case we know the answer is exactly 10. 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. The smart local moving algorithm (Waltman and Eck 2013) identified another limitation in the original Louvain method: it isnt able to split communities once theyre merged, even when it may be very beneficial to do so. Narrow scope for resolution-limit-free community detection. http://iopscience.iop.org/article/10.1088/1742-5468/2008/10/P10008/meta. We prove that the Leiden algorithm yields communities that are guaranteed to be connected. o CLIQUE (Clustering in Quest): - CLIQUE is a combination of density-based and grid-based clustering algorithm. This is because Louvain only moves individual nodes at a time. We therefore require a more principled solution, which we will introduce in the next section. Cluster your data matrix with the Leiden algorithm. 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.