leiden clustering explained

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. Nonlin. The idea of the refinement phase in the Leiden algorithm is to identify a partition \({{\mathscr{P}}}_{{\rm{refined}}}\) that is a refinement of \({\mathscr{P}}\). J. Comput. Finding communities in large networks is far from trivial: algorithms need to be fast, but they also need to provide high-quality results. This is very similar to what the smart local moving algorithm does. Modularity optimization. E Stat. E 92, 032801, https://doi.org/10.1103/PhysRevE.92.032801 (2015). Rev. The Louvain algorithm10 is very simple and elegant. We thank Lovro Subelj for his comments on an earlier version of this paper. Leiden is both faster than Louvain and finds better partitions. In this post Ive mainly focused on the optimisation methods for community detection, rather than the different objective functions that can be used. In the first step of the next iteration, Louvain will again move individual nodes in the network. An iteration of the Leiden algorithm in which the partition does not change is called a stable iteration. For example, after four iterations, the Web UK network has 8% disconnected communities, but twice as many badly connected communities. Ozaki, N., Tezuka, H. & Inaba, M. A Simple Acceleration Method for the Louvain Algorithm. For example: If you do not have root access, you can use pip install --user or pip install --prefix to install these in your user directory (which you have write permissions for) and ensure that this directory is in your PATH so that Python can find it. The solution provided by Leiden is based on the smart local moving algorithm. In the fast local move procedure in the Leiden algorithm, only nodes whose neighbourhood has changed are visited. However, the Louvain algorithm does not consider this possibility, since it considers only individual node movements. Acad. It partitions the data space and identifies the sub-spaces using the Apriori principle. V.A.T. Figure6 presents total runtime versus quality for all iterations of the Louvain and the Leiden algorithm. Blondel, V D, J L Guillaume, and R Lambiotte. 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 case, refinement does not change the partition (f). We used modularity with a resolution parameter of =1 for the experiments. 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. 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. 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). ADS Agglomerative clustering is a bottom-up approach. 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. You will not need much Python to use it. 81 (4 Pt 2): 046114. http://dx.doi.org/10.1103/PhysRevE.81.046114. 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. Finally, we compare the performance of the algorithms on the empirical networks. Blondel, V. D., Guillaume, J.-L., Lambiotte, R. & Lefebvre, E. Fast unfolding of communities in large networks. This continues until the queue is empty. We also suggested that the Leiden algorithm is faster than the Louvain algorithm, because of the fast local move approach. How many iterations of the Leiden clustering algorithm to perform. E 74, 016110, https://doi.org/10.1103/PhysRevE.74.016110 (2006). Phys. 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 Community detection can then be performed using this graph. We find that the Leiden algorithm is faster than the Louvain algorithm and uncovers better partitions, in addition to providing 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. 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. Sci Rep 9, 5233 (2019). The increase in the percentage of disconnected communities is relatively limited for the Live Journal and Web of Science networks. Raghavan, U., Albert, R. & Kumara, S. Near linear time algorithm to detect community structures in large-scale networks. 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. In the local moving phase, individual nodes are moved to the community that yields the largest increase in the quality function. First, we created a specified number of nodes and we assigned each node to a community. Furthermore, if all communities in a partition are uniformly -dense, the quality of the partition is not too far from optimal, as shown in SectionE of the Supplementary Information. It maximizes a modularity score for each community, where the modularity quantifies the quality of an assignment of nodes to communities. Rev. Waltman, Ludo, and Nees Jan van Eck. Learn more. Communities may even be disconnected. & Bornholdt, S. Statistical mechanics of community detection. J. Stat. Performance of modularity maximization in practical contexts. J. Publishers note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 2015. Use Git or checkout with SVN using the web URL. In practical applications, the Leiden algorithm convincingly outperforms the Louvain algorithm, both in terms of speed and in terms of quality of the results, as shown by the experimental analysis presented in this paper. This way of defining the expected number of edges is based on the so-called configuration model. The algorithm may yield arbitrarily badly connected communities, over and above the well-known issue of the resolution limit14. Badly connected communities. Resolution Limit in Community Detection. Proc. Each point corresponds to a certain iteration of an algorithm, with results averaged over 10 experiments. where nc is the number of nodes in community c. The interpretation of the resolution parameter is quite straightforward. Empirical networks show a much richer and more complex structure. 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. First iteration runtime for empirical networks. Note that the object for Seurat version 3 has changed. Use the Previous and Next buttons to navigate the slides or the slide controller buttons at the end to navigate through each slide. At this point, it is guaranteed that each individual node is optimally assigned. Both conda and PyPI have leiden clustering in Python which operates via iGraph. Google Scholar. 6 show that Leiden outperforms Louvain in terms of both computational time and quality of the partitions. Google Scholar. Arguments can be passed to the leidenalg implementation in Python: In particular, the resolution parameter can fine-tune the number of clusters to be detected. If nothing happens, download Xcode and try again. Nevertheless, when CPM is used as the quality function, the Louvain algorithm may still find arbitrarily badly connected communities. As can be seen in the figure, Louvain quickly reaches a state in which it is unable to find better partitions. A community is subset optimal if all subsets of the community are locally optimally assigned. For higher values of , Leiden finds better partitions than Louvain. In particular, benchmark networks have a rather simple structure. 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. b, The elephant graph (in a) is clustered using the Leiden clustering algorithm 51 (resolution r = 0.5). 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. For the Amazon, DBLP and Web UK networks, Louvain yields on average respectively 23%, 16% and 14% badly connected communities. 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. Zenodo, https://doi.org/10.5281/zenodo.1469357 https://github.com/vtraag/leidenalg. Instead, a node may be merged with any community for which the quality function increases. 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). A score of -1 means that there are no edges connecting nodes within the community, and they instead all connect nodes outside the community. Consider the partition shown in (a). wrote the manuscript. Rev. Proc. It states that there are no communities that can be merged. Speed and quality for the first 10 iterations of the Louvain and the Leiden algorithm for benchmark networks (n=106 and n=107). The leidenalg package facilitates community detection of networks and builds on the package igraph. 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. You are using a browser version with limited support for CSS. Traag, V. A. leidenalg 0.7.0. Analyses based on benchmark networks have only a limited value because these networks are not representative of empirical real-world networks. Communities were all of equal size. Nonlin. Article The Louvain algorithm starts from a singleton partition in which each node is in its own community (a). Lancichinetti, A. 2 represent stronger connections, while the other edges represent weaker connections. Phys. E 74, 036104, https://doi.org/10.1103/PhysRevE.74.036104 (2006). When a disconnected community has become a node in an aggregate network, there are no more possibilities to split up the community. 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. The Leiden algorithm is clearly faster than the Louvain algorithm. Basically, there are two types of hierarchical cluster analysis strategies - 1. Hence, the Leiden algorithm effectively addresses the problem of badly connected communities. These steps are repeated until no further improvements can be made. 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. Presumably, many of the badly connected communities in the first iteration of Louvain become disconnected in the second iteration. Speed and quality for the first 10 iterations of the Louvain and the Leiden algorithm for six empirical networks. In subsequent iterations, the percentage of disconnected communities remains fairly stable. In this section, we analyse and compare the performance of the two algorithms in practice. The Louvain algorithm guarantees that modularity cannot be increased by merging communities (it finds a locally optimal solution). . For both algorithms, 10 iterations were performed. To address this problem, we introduce the Leiden algorithm. However, for higher values of , Leiden becomes orders of magnitude faster than Louvain, reaching 10100 times faster runtimes for the largest networks. Rev. Note that Leiden clustering directly clusters the neighborhood graph of cells, which we already computed in the previous section. Traag, Vincent, Ludo Waltman, and Nees Jan van Eck. IEEE Trans. Knowl. This contrasts to benchmark networks, for which Leiden often converges after a few iterations. * (2018). Phys. Optimising modularity is NP-hard5, and consequentially many heuristic algorithms have been proposed, such as hierarchical agglomeration6, extremal optimisation7, simulated annealing4,8 and spectral9 algorithms. Number of iterations until stability. The numerical details of the example can be found in SectionB of the Supplementary Information. While current approaches are successful in reducing the number of sequence alignments performed, the generated clusters are . Note that if Leiden finds subcommunities, splitting up the community is guaranteed to increase modularity. The thick edges in Fig. Acad. sign in This can be a shared nearest neighbours matrix derived from a graph object. 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. Each of these can be used as an objective function for graph-based community detection methods, with our goal being to maximize this value. 8 (3): 207. https://pdfs.semanticscholar.org/4ea9/74f0fadb57a0b1ec35cbc5b3eb28e9b966d8.pdf. In the first iteration, Leiden is roughly 220 times faster than Louvain. 2(b). In the local move procedure in the Leiden algorithm, only nodes whose neighborhood . The DBLP network is somewhat more challenging, requiring almost 80 iterations on average to reach a stable iteration. CPM has the advantage that it is not subject to the resolution limit. The property of -connectivity is a slightly stronger variant of ordinary connectivity. 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. The Leiden algorithm is considerably more complex than the Louvain algorithm. However, in the case of the Web of Science network, more than 5% of the communities are disconnected in the first iteration. Phys. If you cant use Leiden, choosing Smart Local Moving will likely give very similar results, but might be a bit slower as it doesnt include some of the simple speedups to Louvain like random moving and Louvain pruning. Soft Matter Phys. Weights for edges an also be passed to the leiden algorithm either as a separate vector or weights or a weighted adjacency matrix. In the case of modularity, communities may have significant substructure both because of the resolution limit and because of the shortcomings of Louvain. In this iterative scheme, Louvain provides two guarantees: (1) no communities can be merged and (2) no nodes can be moved. The quality improvement realised by the Leiden algorithm relative to the Louvain algorithm is larger for empirical networks than for benchmark networks. 10, for the IMDB and Amazon networks, Leiden reaches a stable iteration relatively quickly, presumably because these networks have a fairly simple community structure. By submitting a comment you agree to abide by our Terms and Community Guidelines. This is the crux of the Leiden paper, and the authors show that this exact problem happens frequently in practice. MATH 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. The Louvain algorithm is a simple and popular method for community detection (Blondel, Guillaume, and Lambiotte 2008). A structure that is more informative than the unstructured set of clusters returned by flat clustering. 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. Phys. Speed of the first iteration of the Louvain and the Leiden algorithm for six empirical networks. According to CPM, it is better to split into two communities when the link density between the communities is lower than the constant. We provide the full definitions of the properties as well as the mathematical proofs in SectionD of the Supplementary Information. performed the experimental analysis. Leiden consists of the following steps: The refinement step allows badly connected communities to be split before creating the aggregate network. CPM is defined as. However, so far this problem has never been studied for the Louvain algorithm. Natl. Traag, V. A., Van Dooren, P. & Nesterov, Y. As such, we scored leiden-clustering popularity level to be Limited. Louvain quickly converges to a partition and is then unable to make further improvements. In particular, we show that Louvain may identify communities that are internally disconnected. 10008, 6, https://doi.org/10.1088/1742-5468/2008/10/P10008 (2008). Traag, V.A., Waltman, L. & van Eck, N.J. From Louvain to Leiden: guaranteeing well-connected communities. In many complex networks, nodes cluster and form relatively dense groupsoften called communities1,2. In later stages, most neighbors will belong to the same community, and its very likely that the best move for the node is to the community that most of its neighbors already belong to. The second iteration of Louvain shows a large increase in the percentage of disconnected communities. V. A. Traag. 2004. The nodes are added to the queue in a random order.

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