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The high percentage of badly connected communities attests to this. 20, 172188, https://doi.org/10.1109/TKDE.2007.190689 (2008). V.A.T. Although originally defined for modularity, the Louvain algorithm can also be used to optimise other quality functions. 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. They identified an inefficiency in the Louvain algorithm: computes modularity gain for all neighbouring nodes per loop in local moving phase, even though many of these nodes will not have moved. Leiden consists of the following steps: The refinement step allows badly connected communities to be split before creating the aggregate network. Good, B. H., De Montjoye, Y. Phys. Data 11, 130, https://doi.org/10.1145/2992785 (2017). ISSN 2045-2322 (online). This algorithm provides a number of explicit guarantees. 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. PubMed Rep. 486, 75174, https://doi.org/10.1016/j.physrep.2009.11.002 (2010). Scaling of benchmark results for network size. We then created a certain number of edges such that a specified average degree \(\langle k\rangle \) was obtained. CAS Many Git commands accept both tag and branch names, so creating this branch may cause unexpected behavior. We now consider the guarantees provided by the Leiden algorithm. E 74, 016110, https://doi.org/10.1103/PhysRevE.74.016110 (2006). These are the same networks that were also studied in an earlier paper introducing the smart local move algorithm15. 8 (3): 207. https://pdfs.semanticscholar.org/4ea9/74f0fadb57a0b1ec35cbc5b3eb28e9b966d8.pdf. Rather than progress straight to the aggregation stage (as we would for the original Louvain), we next consider each community as a new sub-network and re-apply the local moving step within each community. leidenalg. Leiden is both faster than Louvain and finds better partitions. Hence, in general, Louvain may find arbitrarily badly connected communities. The 'devtools' package will be used to install 'leiden' and the dependancies (igraph and reticulate). We also suggested that the Leiden algorithm is faster than the Louvain algorithm, because of the fast local move approach. We consider these ideas to represent the most promising directions in which the Louvain algorithm can be improved, even though we recognise that other improvements have been suggested as well22. This is the crux of the Leiden paper, and the authors show that this exact problem happens frequently in practice. E 74, 036104, https://doi.org/10.1103/PhysRevE.74.036104 (2006). Traag, V.A., Waltman, L. & van Eck, N.J. From Louvain to Leiden: guaranteeing well-connected communities. 2. Rev. Moreover, when no more nodes can be moved, the algorithm will aggregate the network. Table2 provides an overview of the six networks. See the documentation for these functions. MathSciNet We thank Lovro Subelj for his comments on an earlier version of this paper. 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. Acad. Raghavan, U., Albert, R. & Kumara, S. Near linear time algorithm to detect community structures in large-scale networks. Rev. 2016. Electr. Another important difference between the Leiden algorithm and the Louvain algorithm is the implementation of the local moving phase. The Louvain algorithm guarantees that modularity cannot be increased by merging communities (it finds a locally optimal solution). The property of -connectivity is a slightly stronger variant of ordinary connectivity. Hence, the community remains disconnected, unless it is merged with another community that happens to act as a bridge. The property of -separation is also guaranteed by the Louvain algorithm. Below, the quality of a partition is reported as \(\frac{ {\mathcal H} }{2m}\), where H is defined in Eq. To obtain Note that communities found by the Leiden algorithm are guaranteed to be connected. At some point, node 0 is considered for moving. 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. The reasoning behind this is that the best community to join will usually be the one that most of the nodes neighbors already belong to. 10, 186198, https://doi.org/10.1038/nrn2575 (2009). Is modularity with a resolution parameter equivalent to leidenalg.RBConfigurationVertexPartition? Finding community structure in networks using the eigenvectors of matrices. Based on project statistics from the GitHub repository for the PyPI package leiden-clustering, we found that it has been starred 1 times. Note that this code is designed for Seurat version 2 releases. Speed and quality for the first 10 iterations of the Louvain and the Leiden algorithm for benchmark networks (n=106 and n=107). Clauset, A., Newman, M. E. J. See the documentation on the leidenalg Python module for more information: https://leidenalg.readthedocs.io/en/latest/reference.html. Presumably, many of the badly connected communities in the first iteration of Louvain become disconnected in the second iteration. 2.3. Louvain quickly converges to a partition and is then unable to make further improvements. Importantly, the first iteration of the Leiden algorithm is the most computationally intensive one, and subsequent iterations are faster. Modularity scores of +1 mean that all the edges in a community are connecting nodes within the community. 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 ). Basically, there are two types of hierarchical cluster analysis strategies - 1. Ph.D. thesis, (University of Oxford, 2016). We prove that the new algorithm is guaranteed to produce partitions in which all communities are internally connected. We start by initialising a queue with all nodes in the network. Modularity is used most commonly, but is subject to the resolution limit. In many complex networks, nodes cluster and form relatively dense groupsoften called communities1,2. After the first iteration of the Louvain algorithm, some partition has been obtained. Even though clustering can be applied to networks, it is a broader field in unsupervised machine learning which deals with multiple attribute types. 2015. Conversely, if Leiden does not find subcommunities, there is no guarantee that modularity cannot be increased by splitting up the community. Soft Matter Phys. the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in In the Louvain algorithm, an aggregate network is created based on the partition \({\mathscr{P}}\) resulting from the local moving phase. As can be seen in Fig. A community is subset optimal if all subsets of the community are locally optimally assigned. 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. Rev. Usually, the Louvain algorithm starts from a singleton partition, in which each node is in its own community. http://iopscience.iop.org/article/10.1088/1742-5468/2008/10/P10008/meta. Sci Rep 9, 5233 (2019). Communities were all of equal size. Resolution Limit in Community Detection. Proc. Scientific Reports (Sci Rep) modularity) increases. 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). After a stable iteration of the Leiden algorithm, the algorithm may still be able to make further improvements in later iterations. (2) and m is the number of edges. Consider the partition shown in (a). Please Data Eng. Even worse, the Amazon network has 5% disconnected communities, but 25% badly connected communities. 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. PubMed The algorithm continues to move nodes in the rest of the network. 1 and summarised in pseudo-code in AlgorithmA.1 in SectionA of the Supplementary Information. Nodes 13 should form a community and nodes 46 should form another community. A structure that is more informative than the unstructured set of clusters returned by flat clustering. 2018. Speed and quality for the first 10 iterations of the Louvain and the Leiden algorithm for six empirical networks. Article Four popular community detection algorithms are explained . Each point corresponds to a certain iteration of an algorithm, with results averaged over 10 experiments. bioRxiv, https://doi.org/10.1101/208819 (2018). Unsupervised clustering of cells is a common step in many single-cell expression workflows. 8, the Leiden algorithm is significantly faster than the Louvain algorithm also in empirical networks. In this way, Leiden implements the local moving phase more efficiently than Louvain. The percentage of disconnected communities even jumps to 16% for the DBLP network. It maximizes a modularity score for each community, where the modularity quantifies the quality of an assignment of nodes to communities. 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 fact, for the Web of Science and Web UK networks, Fig. For each network, we repeated the experiment 10 times. CPM is defined as. reviewed the manuscript. The algorithm then moves individual nodes in the aggregate network (e). Based on this partition, an aggregate network is created (c). 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. Correspondence to Our analysis is based on modularity with resolution parameter =1. Leiden keeps finding better partitions for empirical networks also after the first 10 iterations of the algorithm. 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. According to CPM, it is better to split into two communities when the link density between the communities is lower than the constant. Then the Leiden algorithm can be run on the adjacency matrix. 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). N.J.v.E. 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. Default behaviour is calling cluster_leiden in igraph with Modularity (for undirected graphs) and CPM cost functions. Bullmore, E. & Sporns, O. In the worst case, communities may even be disconnected, especially when running the algorithm iteratively. Am. E Stat. Biological sequence clustering is a complicated data clustering problem owing to the high computation costs incurred for pairwise sequence distance calculations through sequence alignments, as well as difficulties in determining parameters for deriving robust clusters. On the other hand, after node 0 has been moved to a different community, nodes 1 and 4 have not only internal but also external connections. This contrasts with the Leiden algorithm. On the other hand, Leiden keeps finding better partitions, especially for higher values of , for which it is more difficult to identify good partitions. Nevertheless, depending on the relative strengths of the different connections, these nodes may still be optimally assigned to their current community. Guimer, R. & Nunes Amaral, L. A. Functional cartography of complex metabolic networks. In the first iteration, Leiden is roughly 220 times faster than Louvain. Internet Explorer). First iteration runtime for empirical networks. where nc is the number of nodes in community c. The interpretation of the resolution parameter is quite straightforward. & Bornholdt, S. Statistical mechanics of community detection. https://doi.org/10.1038/s41598-019-41695-z. Similarly, in citation networks, such as the Web of Science network, nodes in a community are usually considered to share a common topic26,27. The algorithm is described in pseudo-code in AlgorithmA.2 in SectionA of the Supplementary Information. A Simple Acceleration Method for the Louvain Algorithm. Int. Run the code above in your browser using DataCamp Workspace. This should be the first preference when choosing an algorithm. The resulting clusters are shown as colors on the 3D model (top) and t -SNE embedding . 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. PubMedGoogle Scholar. Rev. MATH When the Leiden algorithm found that a community could be split into multiple subcommunities, we counted the community as badly connected. Phys. Rotta, R. & Noack, A. Multilevel local search algorithms for modularity clustering. The quality of such an asymptotically stable partition provides an upper bound on the quality of an optimal partition. Knowl. Each point corresponds to a certain iteration of an algorithm, with results averaged over 10 experiments. We generated networks with n=103 to n=107 nodes. Importantly, the problem of disconnected communities is not just a theoretical curiosity. Rev. Yang, Z., Algesheimer, R. & Tessone, C. J. 10008, 6, https://doi.org/10.1088/1742-5468/2008/10/P10008 (2008). 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. In doing so, Louvain keeps visiting nodes that cannot be moved to a different community. Later iterations of the Louvain algorithm only aggravate the problem of disconnected communities, even though the quality function (i.e. In contrast, Leiden keeps finding better partitions in each iteration. This contrasts to benchmark networks, for which Leiden often converges after a few iterations. The corresponding results are presented in the Supplementary Fig. Phys. Waltman, L. & van Eck, N. J. There is an entire Leiden package in R-cran here However, modularity suffers from a difficult problem known as the resolution limit (Fortunato and Barthlemy 2007). At some point, the Louvain algorithm may end up in the community structure shown in Fig. However, the initial partition for the aggregate network is based on P, just like in the Louvain algorithm. The algorithm then moves individual nodes in the aggregate network (d). However, this is not necessarily the case, as the other nodes may still be sufficiently strongly connected to their community, despite the fact that the community has become disconnected. However, it is also possible to start the algorithm from a different partition15. Furthermore, by relying on a fast local move approach, the Leiden algorithm runs faster than the Louvain algorithm. E 69, 026113, https://doi.org/10.1103/PhysRevE.69.026113 (2004). In general, Leiden is both faster than Louvain and finds better partitions. To ensure readability of the paper to the broadest possible audience, we have chosen to relegate all technical details to the Supplementary Information. 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. MathSciNet After a stable iteration of the Leiden algorithm, it is guaranteed that: All nodes are locally optimally assigned. Google Scholar. The larger the increase in the quality function, the more likely a community is to be selected. Performance of modularity maximization in practical contexts. Natl. Google Scholar. Rev. This can be a shared nearest neighbours matrix derived from a graph object. The Louvain method for community detection is a popular way to discover communities from single-cell data. Hence, the complex structure of empirical networks creates an even stronger need for the use of the Leiden algorithm. As shown in Fig. Figure4 shows how well it does compared to the Louvain algorithm. A new methodology for constructing a publication-level classification system of science. Networks with high modularity have dense connections between the nodes within modules but sparse connections between nodes in different modules. (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. It only implies that individual nodes are well connected to their community. Theory Exp. It identifies the clusters by calculating the densities of the cells. 9 shows that more than 10 iterations of the Leiden algorithm can be performed before the Louvain algorithm has finished its first iteration. Are you sure you want to create this branch? The solution proposed in smart local moving is to alter how the local moving step in Louvain works. It partitions the data space and identifies the sub-spaces using the Apriori principle. In fact, when we keep iterating the Leiden algorithm, it will converge to a partition for which it is guaranteed that: A community is uniformly -dense if there are no subsets of the community that can be separated from the community. Moreover, Louvain has no mechanism for fixing these communities. One may expect that other nodes in the old community will then also be moved to other communities. We used modularity with a resolution parameter of =1 for the experiments. Eng. Nat. Runtime versus quality for empirical networks. The algorithm may yield arbitrarily badly connected communities, over and above the well-known issue of the resolution limit14. The main ideas of our algorithm are explained in an intuitive way in the main text of the paper. The algorithm moves individual nodes from one community to another to find a partition (b). As the problem of modularity optimization is NP-hard, we need heuristic methods to optimize modularity (or CPM). Narrow scope for resolution-limit-free community detection. Article The percentage of badly connected communities is less affected by the number of iterations of the Louvain algorithm. E 76, 036106, https://doi.org/10.1103/PhysRevE.76.036106 (2007). For larger networks and higher values of , Louvain is much slower than Leiden. Ozaki, Naoto, Hiroshi Tezuka, and Mary Inaba. The difference in computational time is especially pronounced for larger networks, with Leiden being up to 20 times faster than Louvain in empirical networks. 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. E Stat. Inf. To elucidate the problem, we consider the example illustrated in Fig. Note that this code is . Lancichinetti, A. S3. Number of iterations until stability. A partition of clusters as a vector of integers Examples 6 show that Leiden outperforms Louvain in terms of both computational time and quality of the partitions. The inspiration for this method of community detection is the optimization of modularity as the algorithm progresses. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate. To overcome the problem of arbitrarily badly connected communities, we introduced a new algorithm, which we refer to as the Leiden algorithm. 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. 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. We will use sklearns K-Means implementation looking for 10 clusters in the original 784 dimensional data. SPATA2 currently offers the functions findSeuratClusters (), findMonocleClusters () and findNearestNeighbourClusters () which are wrapper around widely used clustering algorithms. In the meantime, to ensure continued support, we are displaying the site without styles The Leiden algorithm provides several guarantees. 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. In fact, if we keep iterating the Leiden algorithm, it will converge to a partition without any badly connected communities, as discussed earlier. Waltman, Ludo, and Nees Jan van Eck. To install the development version: The current release on CRAN can be installed with: First set up a compatible adjacency matrix: An adjacency matrix is any binary matrix representing links between nodes (column and row names). Nonlin. Importantly, the output of the local moving stage will depend on the order that the nodes are considered in. 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. The numerical details of the example can be found in SectionB of the Supplementary Information. ADS http://dx.doi.org/10.1073/pnas.0605965104. The quality improvement realised by the Leiden algorithm relative to the Louvain algorithm is larger for empirical networks than for benchmark networks. In the local moving phase, individual nodes are moved to the community that yields the largest increase in the quality function. 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. 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. In fact, by implementing the refinement phase in the right way, several attractive guarantees can be given for partitions produced by the Leiden algorithm. Soft Matter Phys. CPM does not suffer from this issue13. https://leidenalg.readthedocs.io/en/latest/reference.html. The second iteration of Louvain shows a large increase in the percentage of disconnected communities. The Web of Science network is the most difficult one. Badly connected communities. However, in the case of the Web of Science network, more than 5% of the communities are disconnected in the first iteration. Source Code (2018). 9, the Leiden algorithm also performs better than the Louvain algorithm in terms of the quality of the partitions that are obtained. E 78, 046110, https://doi.org/10.1103/PhysRevE.78.046110 (2008). 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. One of the best-known methods for community detection is called modularity3. 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. J. From Louvain to Leiden: Guaranteeing Well-Connected Communities, October. Complex brain networks: graph theoretical analysis of structural and functional systems. Besides being pervasive, the problem is also sizeable. At this point, it is guaranteed that each individual node is optimally assigned. The leidenalg package facilitates community detection of networks and builds on the package igraph. We demonstrate the performance of the Leiden algorithm for several benchmark and real-world networks.