J. Communities in Networks. Nevertheless, when CPM is used as the quality function, the Louvain algorithm may still find arbitrarily badly connected communities. Leiden consists of the following steps: The refinement step allows badly connected communities to be split before creating the aggregate network. J. This is not too difficult to explain. The Leiden algorithm has been specifically designed to address the problem of badly connected communities. However, if communities are badly connected, this may lead to incorrect attributions of shared functionality. 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 this case we know the answer is exactly 10. 2. Faster Unfolding of Communities: Speeding up the Louvain Algorithm. Phys. Nonlin. Nature 433, 895900, https://doi.org/10.1038/nature03288 (2005). In an experiment containing a mixture of cell types, each cluster might correspond to a different cell type. The value of the resolution parameter was determined based on the so-called mixing parameter 13. The Louvain algorithm starts from a singleton partition in which each node is in its own community (a). Community detection is an important task in the analysis of complex networks. The numerical details of the example can be found in SectionB of the Supplementary Information. Importantly, the problem of disconnected communities is not just a theoretical curiosity. 2008. To address this important shortcoming, we introduce a new algorithm that is faster, finds better partitions and provides explicit guarantees and bounds. However, focussing only on disconnected communities masks the more fundamental issue: Louvain finds arbitrarily badly connected communities. We therefore require a more principled solution, which we will introduce in the next section. Introduction The Louvain method is an algorithm to detect communities in large networks. 81 (4 Pt 2): 046114. http://dx.doi.org/10.1103/PhysRevE.81.046114. Importantly, mergers are performed only within each community of the partition \({\mathscr{P}}\). This will compute the Leiden clusters and add them to the Seurat Object Class. Node optimality is also guaranteed after a stable iteration of the Louvain algorithm. This way of defining the expected number of edges is based on the so-called configuration model. However, in the case of the Web of Science network, more than 5% of the communities are disconnected in the first iteration. Article Each of these can be used as an objective function for graph-based community detection methods, with our goal being to maximize this value. 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. We can guarantee a number of properties of the partitions found by the Leiden algorithm at various stages of the iterative process. 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. 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. Elect. Performance of modularity maximization in practical contexts. Scientific Reports (Sci Rep) Empirical networks show a much richer and more complex structure. Clustering is a machine learning technique in which similar data points are grouped into the same cluster based on their attributes. Rev. Due to the resolution limit, modularity may cause smaller communities to be clustered into larger communities. As such, we scored leiden-clustering popularity level to be Limited. Communities may even be disconnected. A community is subset optimal if all subsets of the community are locally optimally assigned. If nothing happens, download GitHub Desktop and try again. Finding communities in large networks is far from trivial: algorithms need to be fast, but they also need to provide high-quality results. The corresponding results are presented in the Supplementary Fig. The R implementation of Leiden can be run directly on the snn igraph object in Seurat. Theory Exp. However, the Louvain algorithm does not consider this possibility, since it considers only individual node movements. J. Stat. Crucially, however, the percentage of badly connected communities decreases with each iteration of the Leiden algorithm. Int. However, for higher values of , Leiden becomes orders of magnitude faster than Louvain, reaching 10100 times faster runtimes for the largest networks. MathSciNet 8, 207218, https://doi.org/10.17706/IJCEE.2016.8.3.207-218 (2016). To use Leiden with the Seurat pipeline for a Seurat Object object that has an SNN computed (for example with Seurat::FindClusters with save.SNN = TRUE). The percentage of disconnected communities is more limited, usually around 1%. 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 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. 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 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. The 'devtools' package will be used to install 'leiden' and the dependancies (igraph and reticulate). Publishers note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. One may expect that other nodes in the old community will then also be moved to other communities. V.A.T. Cite this article. The docs are here. Nonlin. In the case of modularity, communities may have significant substructure both because of the resolution limit and because of the shortcomings of Louvain. Fortunato, S. & Barthlemy, M. Resolution Limit in Community Detection. The solution provided by Leiden is based on the smart local moving algorithm. In that case, nodes 16 are all locally optimally assigned, despite the fact that their community has become disconnected. We demonstrate the performance of the Leiden algorithm for several benchmark and real-world networks. Use Git or checkout with SVN using the web URL. Because the percentage of disconnected communities in the first iteration of the Louvain algorithm usually seems to be relatively low, the problem may have escaped attention from users of the algorithm. 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. PubMedGoogle Scholar. This problem is different from the well-known issue of the resolution limit of modularity14. At this point, it is guaranteed that each individual node is optimally assigned. 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. By creating the aggregate network based on \({{\mathscr{P}}}_{{\rm{refined}}}\) rather than P, the Leiden algorithm has more room for identifying high-quality partitions. However, modularity suffers from a difficult problem known as the resolution limit (Fortunato and Barthlemy 2007). To find an optimal grouping of cells into communities, we need some way of evaluating different partitions in the graph. These are the same networks that were also studied in an earlier paper introducing the smart local move algorithm15. The nodes that are more interconnected have been partitioned into separate clusters. Newman, M. E. J. These nodes can be approximately identified based on whether neighbouring nodes have changed communities. Perhaps surprisingly, iterating the algorithm aggravates the problem, even though it does increase the quality function. In fact, for the Web of Science and Web UK networks, Fig. An aggregate. We prove that the new algorithm is guaranteed to produce partitions in which all communities are internally connected. 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. 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. Besides being pervasive, the problem is also sizeable. J. Comput. 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. However, as shown in this paper, the Louvain algorithm has a major shortcoming: the algorithm yields communities that may be arbitrarily badly connected. A number of iterations of the Leiden algorithm can be performed before the Louvain algorithm has finished its first iteration. The Leiden algorithm is clearly faster than the Louvain algorithm. Later iterations of the Louvain algorithm only aggravate the problem of disconnected communities, even though the quality function (i.e. For each community in a partition that was uncovered by the Louvain algorithm, we determined whether it is internally connected or not. bioRxiv, https://doi.org/10.1101/208819 (2018). 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). leidenalg. 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 iterative scheme, Louvain provides two guarantees: (1) no communities can be merged and (2) no nodes can be moved. 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. Waltman, L. & van Eck, N. J. Weights for edges an also be passed to the leiden algorithm either as a separate vector or weights or a weighted adjacency matrix. We show that this algorithm has a major defect that largely went unnoticed until now: the Louvain algorithm may yield arbitrarily badly connected communities. U. S. A. E Stat. Community detection is often used to understand the structure of large and complex networks. 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. Communities were all of equal size. Lancichinetti, A., Fortunato, S. & Radicchi, F. Benchmark graphs for testing community detection algorithms. To address this problem, we introduce the Leiden algorithm. 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. V. A. Traag. In many complex networks, nodes cluster and form relatively dense groupsoften called communities1,2. On Modularity Clustering. 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. PubMed E 76, 036106, https://doi.org/10.1103/PhysRevE.76.036106 (2007). Sci. Note that Leiden clustering directly clusters the neighborhood graph of cells, which we already computed in the previous section. Source Code (2018). Traag, V. A., Van Dooren, P. & Nesterov, Y. E 80, 056117, https://doi.org/10.1103/PhysRevE.80.056117 (2009). Cluster cells using Louvain/Leiden community detection Description. 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. 2(b). Is modularity with a resolution parameter equivalent to leidenalg.RBConfigurationVertexPartition? 68, 984998, https://doi.org/10.1002/asi.23734 (2017). Louvain quickly converges to a partition and is then unable to make further improvements. reviewed the manuscript. Google Scholar. Rev. 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. The Leiden algorithm is considerably more complex than the Louvain algorithm. https://leidenalg.readthedocs.io/en/latest/reference.html. We then created a certain number of edges such that a specified average degree \(\langle k\rangle \) was obtained. 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. In the initial stage of Louvain (when all nodes belong to their own community), nearly any move will result in a modularity gain, and it doesnt matter too much which move is chosen. Rev. That is, no subset can be moved to a different community. 2018. To ensure readability of the paper to the broadest possible audience, we have chosen to relegate all technical details to the Supplementary Information. 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. 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. In practice, this means that small clusters can hide inside larger clusters, making their identification difficult. How many iterations of the Leiden clustering algorithm to perform. Many Git commands accept both tag and branch names, so creating this branch may cause unexpected behavior. 8 (3): 207. https://pdfs.semanticscholar.org/4ea9/74f0fadb57a0b1ec35cbc5b3eb28e9b966d8.pdf. performed the experimental analysis. In this case, refinement does not change the partition (f). The phase one loop can be greatly accelerated by finding the nodes that have the potential to change community and only revisit those nodes. 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 The quality improvement realised by the Leiden algorithm relative to the Louvain algorithm is larger for empirical networks than for benchmark networks. CAS The percentage of badly connected communities is less affected by the number of iterations of the Louvain algorithm. In this section, we analyse and compare the performance of the two algorithms in practice. Rotta, R. & Noack, A. Multilevel local search algorithms for modularity clustering. sign in 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. 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. http://dx.doi.org/10.1073/pnas.0605965104. This is very similar to what the smart local moving algorithm does. As can be seen in Fig. Article In this post Ive mainly focused on the optimisation methods for community detection, rather than the different objective functions that can be used. This is similar to ideas proposed recently as pruning16 and in a slightly different form as prioritisation17. Use the Previous and Next buttons to navigate the slides or the slide controller buttons at the end to navigate through each slide. Additionally, we implemented a Python package, available from https://github.com/vtraag/leidenalg and deposited at Zenodo24). You signed in with another tab or window. It does not guarantee that modularity cant be increased by moving nodes between communities. Am. 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. Once aggregation is complete we restart the local moving phase, and continue to iterate until everything converges down to one node. Electr. Ozaki, Naoto, Hiroshi Tezuka, and Mary Inaba. Traag, V. A., Waltman, L. & van Eck, N. J. networkanalysis. Figure6 presents total runtime versus quality for all iterations of the Louvain and the Leiden algorithm. MATH It was originally developed for modularity optimization, although the same method can be applied to optimize CPM. Usually, the Louvain algorithm starts from a singleton partition, in which each node is in its own community. In a stable iteration, the partition is guaranteed to be node optimal and subpartition -dense. Another important difference between the Leiden algorithm and the Louvain algorithm is the implementation of the local moving phase. The Leiden algorithm starts from a singleton partition (a). Sci Rep 9, 5233 (2019). Random moving can result in some huge speedups, since Louvain spends about 95% of its time computing the modularity gain from moving nodes. Phys. running Leiden clustering finished: found 16 clusters and added 'leiden_1.0', the cluster labels (adata.obs, categorical) (0:00:00) running Leiden clustering finished: found 12 clusters and added 'leiden_0.6', the cluster labels (adata.obs, categorical) (0:00:00) running Leiden clustering finished: found 9 clusters and added 'leiden_0.4', the 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.
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