Int. We find that the Leiden algorithm is faster than the Louvain algorithm and uncovers better partitions, in addition to providing explicit guarantees. Additionally, we implemented a Python package, available from https://github.com/vtraag/leidenalg and deposited at Zenodo24). That is, no subset can be moved to a different community. Moreover, the deeper significance of the problem was not recognised: disconnected communities are merely the most extreme manifestation of the problem of arbitrarily badly connected communities. 8 (3): 207. https://pdfs.semanticscholar.org/4ea9/74f0fadb57a0b1ec35cbc5b3eb28e9b966d8.pdf. Rev. We can guarantee a number of properties of the partitions found by the Leiden algorithm at various stages of the iterative process. Scaling of benchmark results for difficulty of the partition. 8, 207218, https://doi.org/10.17706/IJCEE.2016.8.3.207-218 (2016). For each community in a partition that was uncovered by the Louvain algorithm, we determined whether it is internally connected or not. Weights for edges an also be passed to the leiden algorithm either as a separate vector or weights or a weighted adjacency matrix. volume9, Articlenumber:5233 (2019) Fortunato, S. Community detection in graphs. (We implemented both algorithms in Java, available from https://github.com/CWTSLeiden/networkanalysis and deposited at Zenodo23. In that case, nodes 16 are all locally optimally assigned, despite the fact that their community has become disconnected. Ph.D. thesis, (University of Oxford, 2016). Am. 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. Scaling of benchmark results for network size. Requirements Developed using: scanpy v1.7.2 sklearn v0.23.2 umap v0.4.6 numpy v1.19.2 leidenalg Installation pip pip install leiden_clustering local Modularity is a measure of the structure of networks or graphs which measures the strength of division of a network into modules (also called groups, clusters or communities). You are using a browser version with limited support for CSS. In the refinement phase, nodes are not necessarily greedily merged with the community that yields the largest increase in the quality function. Nonlin. In this iterative scheme, Louvain provides two guarantees: (1) no communities can be merged and (2) no nodes can be moved. In practice, this means that small clusters can hide inside larger clusters, making their identification difficult. Rev. The value of the resolution parameter was determined based on the so-called mixing parameter 13. In this paper, we show that the Louvain algorithm has a major problem, for both modularity and CPM. Later iterations of the Louvain algorithm are very fast, but this is only because the partition remains the same. E 70, 066111, https://doi.org/10.1103/PhysRevE.70.066111 (2004). For example, nodes in a community in biological or neurological networks are often assumed to share similar functions or behaviour25. PubMed Central We start by initialising a queue with all nodes in the network. V.A.T. Again, if communities are badly connected, this may lead to incorrect inferences of topics, which will affect bibliometric analyses relying on the inferred topics. Sci Rep 9, 5233 (2019). 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. Note that if Leiden finds subcommunities, splitting up the community is guaranteed to increase modularity. Here we can see partitions in the plotted results. To find an optimal grouping of cells into communities, we need some way of evaluating different partitions in the graph. 10, for the IMDB and Amazon networks, Leiden reaches a stable iteration relatively quickly, presumably because these networks have a fairly simple community structure. Rep. 486, 75174, https://doi.org/10.1016/j.physrep.2009.11.002 (2010). & Fortunato, S. Community detection algorithms: A comparative analysis. Raghavan, U., Albert, R. & Kumara, S. Near linear time algorithm to detect community structures in large-scale networks. Publishers note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This may have serious consequences for analyses based on the resulting partitions. Data Eng. Google Scholar. The numerical details of the example can be found in SectionB of the Supplementary Information. However, in the case of the Web of Science network, more than 5% of the communities are disconnected in the first iteration. leiden_clsutering is distributed under a BSD 3-Clause License (see LICENSE). Hence, no further improvements can be made after a stable iteration of the Louvain algorithm. Google Scholar. If nothing happens, download Xcode and try again. Badly connected communities. In particular, benchmark networks have a rather simple structure. The algorithm moves individual nodes from one community to another to find a partition (b), which is then refined (c). ISSN 2045-2322 (online). Leiden consists of the following steps: The refinement step allows badly connected communities to be split before creating the aggregate network. The constant Potts model might give better communities in some cases, as it is not subject to the resolution limit. By submitting a comment you agree to abide by our Terms and Community Guidelines. This will compute the Leiden clusters and add them to the Seurat Object Class. More subtle problems may occur as well, causing Louvain to find communities that are connected, but only in a very weak sense. The Louvain algorithm guarantees that modularity cannot be increased by merging communities (it finds a locally optimal solution). Phys. The algorithm is described in pseudo-code in AlgorithmA.2 in SectionA of the Supplementary Information. By moving these nodes, Louvain creates badly connected communities. & Moore, C. Finding community structure in very large networks. As can be seen in Fig. The Louvain algorithm is illustrated in Fig. Traag, V.A., Waltman, L. & van Eck, N.J. From Louvain to Leiden: guaranteeing well-connected communities. In other words, modularity may hide smaller communities and may yield communities containing significant substructure. Traag, V. A. leidenalg 0.7.0. A community size of 50 nodes was used for the results presented below, but larger community sizes yielded qualitatively similar results. 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. This method tries to maximise the difference between the actual number of edges in a community and the expected number of such edges. The Leiden algorithm starts from a singleton partition (a). Knowl. Newman, M. E. J. Crucially, however, the percentage of badly connected communities decreases with each iteration of the Leiden algorithm. Rev. Source Code (2018). 4. Phys. Source Code (2018). We applied the Louvain and the Leiden algorithm to exactly the same networks, using the same seed for the random number generator. In this post Ive mainly focused on the optimisation methods for community detection, rather than the different objective functions that can be used. In this section, we analyse and compare the performance of the two algorithms in practice. In other words, communities are guaranteed to be well separated. In many complex networks, nodes cluster and form relatively dense groupsoften called communities1,2. 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. Here is some small debugging code I wrote to find this. 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 http://arxiv.org/abs/1810.08473. The Louvain algorithm is a simple and popular method for community detection (Blondel, Guillaume, and Lambiotte 2008). 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. However, so far this problem has never been studied for the Louvain algorithm. The leidenalg package facilitates community detection of networks and builds on the package igraph. Value. Rev. This will compute the Leiden clusters and add them to the Seurat Object Class. Clearly, it would be better to split up the community. 81 (4 Pt 2): 046114. http://dx.doi.org/10.1103/PhysRevE.81.046114. Two ways of doing this are graph modularity (Newman and Girvan 2004) and the constant Potts model (Ronhovde and Nussinov 2010). Finding community structure in networks using the eigenvectors of matrices. Bae, S., Halperin, D., West, J. D., Rosvall, M. & Howe, B. Scalable and Efficient Flow-Based Community Detection for Large-Scale Graph Analysis. Rep. 6, 30750, https://doi.org/10.1038/srep30750 (2016). It partitions the data space and identifies the sub-spaces using the Apriori principle. This makes sense, because after phase one the total size of the graph should be significantly reduced. The phase one loop can be greatly accelerated by finding the nodes that have the potential to change community and only revisit those nodes. Note that this code is . Guimer, R. & Nunes Amaral, L. A. Functional cartography of complex metabolic networks. 2(b). Phys. It identifies the clusters by calculating the densities of the cells. One of the most popular algorithms for uncovering community structure is the so-called Louvain algorithm. Communities may even be disconnected. Newman, M. E. J. For example, for the Web of Science network, the first iteration takes about 110120 seconds, while subsequent iterations require about 40 seconds. Rev. Speed and quality for the first 10 iterations of the Louvain and the Leiden algorithm for benchmark networks (n=106 and n=107). The property of -connectivity is a slightly stronger variant of ordinary connectivity. Each point corresponds to a certain iteration of an algorithm, with results averaged over 10 experiments. The 'devtools' package will be used to install 'leiden' and the dependancies (igraph and reticulate). Nevertheless, when CPM is used as the quality function, the Louvain algorithm may still find arbitrarily badly connected communities. reviewed the manuscript. 63, 23782392, https://doi.org/10.1002/asi.22748 (2012). Rev. partition_type : Optional [ Type [ MutableVertexPartition ]] (default: None) Type of partition to use. 7, whereas Louvain becomes much slower for more difficult partitions, Leiden is much less affected by the difficulty of the partition. The algorithm then moves individual nodes in the aggregate network (d). For larger networks and higher values of , Louvain is much slower than Leiden. Natl. A Smart Local Moving Algorithm for Large-Scale Modularity-Based Community Detection. Eur. This algorithm provides a number of explicit guarantees. Leiden algorithm. MATH Fortunato, S. & Barthlemy, M. Resolution Limit in Community Detection. Faster unfolding of communities: Speeding up the Louvain algorithm. It is good at identifying small clusters. Is modularity with a resolution parameter equivalent to leidenalg.RBConfigurationVertexPartition? Communities may even be internally disconnected. Speed and quality for the first 10 iterations of the Louvain and the Leiden algorithm for six empirical networks. Disconnected community. Unsupervised clustering of cells is a common step in many single-cell expression workflows. We now compare how the Leiden and the Louvain algorithm perform for the six empirical networks listed in Table2. 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. Technol. With one exception (=0.2 and n=107), all results in Fig. Lancichinetti, A. Finally, we compare the performance of the algorithms on the empirical networks. Sign up for the Nature Briefing newsletter what matters in science, free to your inbox daily. 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. * (2018). Community detection in complex networks using extremal optimization. Louvain pruning is another improvement to Louvain proposed in 2016, and can reduce the computational time by as much as 90% while finding communities that are almost as good as Louvain (Ozaki, Tezuka, and Inaba 2016). The corresponding results are presented in the Supplementary Fig. Rotta, R. & Noack, A. Multilevel local search algorithms for modularity clustering. Scientific Reports (Sci Rep) 69 (2 Pt 2): 026113. http://dx.doi.org/10.1103/PhysRevE.69.026113. This package requires the 'leidenalg' and 'igraph' modules for python (2) to be installed on your system. 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. Perhaps surprisingly, iterating the algorithm aggravates the problem, even though it does increase the quality function. 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. In the Louvain algorithm, an aggregate network is created based on the partition \({\mathscr{P}}\) resulting from the local moving phase. J. Assoc. The R implementation of Leiden can be run directly on the snn igraph object in Seurat. 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. 2(a). In general, Leiden is both faster than Louvain and finds better partitions. 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 also takes advantage of the idea of speeding up the local moving of nodes16,17 and the idea of moving nodes to random neighbours18. Rev. Google Scholar. Louvain has two phases: local moving and aggregation. PubMed One may expect that other nodes in the old community will then also be moved to other communities. 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. In the aggregation phase, an aggregate network is created based on the partition obtained in the local moving phase. In the first step of the next iteration, Louvain will again move individual nodes in the network. Agglomerative clustering is a bottom-up approach. Run the code above in your browser using DataCamp Workspace. 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. Higher resolutions lead to more communities, while lower resolutions lead to fewer communities. However, the initial partition for the aggregate network is based on P, just like in the Louvain algorithm. This package implements the Leiden algorithm in C++ and exposes it to python.It relies on (python-)igraph for it to function. Table2 provides an overview of the six networks. We generated benchmark networks in the following way. How many iterations of the Leiden clustering algorithm to perform. Runtime versus quality for benchmark networks. Use the Previous and Next buttons to navigate the slides or the slide controller buttons at the end to navigate through each slide. This enables us to find cases where its beneficial to split a community. This contrasts with optimisation algorithms such as simulated annealing, which do allow the quality function to decrease4,8. Theory Exp. The aggregate network is created based on the partition \({{\mathscr{P}}}_{{\rm{refined}}}\). While smart local moving and multilevel refinement can improve the communities found, the next two improvements on Louvain that Ill discuss focus on the speed/efficiency of the algorithm. 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. The Leiden community detection algorithm outperforms other clustering methods. For example, after four iterations, the Web UK network has 8% disconnected communities, but twice as many badly connected communities. Fast Unfolding of Communities in Large Networks. Journal of Statistical , January. CAS Newman, M E J, and M Girvan. It states that there are no communities that can be merged. Provided by the Springer Nature SharedIt content-sharing initiative. 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. It implies uniform -density and all the other above-mentioned properties. However, focussing only on disconnected communities masks the more fundamental issue: Louvain finds arbitrarily badly connected communities. Once no further increase in modularity is possible by moving any node to its neighboring community, we move to the second phase of the algorithm: aggregation. Louvain community detection algorithm was originally proposed in 2008 as a fast community unfolding method for large networks. ADS We conclude that the Leiden algorithm is strongly preferable to the Louvain algorithm. Although originally defined for modularity, the Louvain algorithm can also be used to optimise other quality functions. This is not the case when nodes are greedily merged with the community that yields the largest increase in the quality function. Rev. Narrow scope for resolution-limit-free community detection. Conversely, if Leiden does not find subcommunities, there is no guarantee that modularity cannot be increased by splitting up 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 ). & Clauset, A. Eur. An iteration of the Leiden algorithm in which the partition does not change is called a stable iteration. We gratefully acknowledge computational facilities provided by the LIACS Data Science Lab Computing Facilities through Frank Takes. Later iterations of the Louvain algorithm only aggravate the problem of disconnected communities, even though the quality function (i.e. Article The solution provided by Leiden is based on the smart local moving algorithm. IEEE Trans. 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 Cite this article. Google Scholar. The images or other third party material in this article are included in the articles Creative Commons license, unless indicated otherwise in a credit line to the material. Class wrapper based on scanpy to use the Leiden algorithm to directly cluster your data matrix with a scikit-learn flavor. ADS The quality of such an asymptotically stable partition provides an upper bound on the quality of an optimal partition. Leiden is faster than Louvain especially for larger networks. Sci. 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. As can be seen in Fig. Random moving is a very simple adjustment to Louvain local moving proposed in 2015 (Traag 2015). Modularity is given by. 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. For both algorithms, 10 iterations were performed. These steps are repeated until no further improvements can be made. http://dx.doi.org/10.1073/pnas.0605965104. Article Importantly, the output of the local moving stage will depend on the order that the nodes are considered in. Sci. Phys. Usually, the Louvain algorithm starts from a singleton partition, in which each node is in its own community. Good, B. H., De Montjoye, Y. In this case we know the answer is exactly 10. J. Nodes 06 are in the same community. The second iteration of Louvain shows a large increase in the percentage of disconnected communities. 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. Such a modular structure is usually not known beforehand. 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). A score of -1 means that there are no edges connecting nodes within the community, and they instead all connect nodes outside the community. Ronhovde, Peter, and Zohar Nussinov. 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. Reichardt, J. 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. In the case of the Louvain algorithm, after a stable iteration, all subsequent iterations will be stable as well. Elect. An alternative quality function is the Constant Potts Model (CPM)13, which overcomes some limitations of modularity. The nodes are added to the queue in a random order. Yang, Z., Algesheimer, R. & Tessone, C. J. Algorithmics 16, 2.1, https://doi.org/10.1145/1963190.1970376 (2011). 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. Presumably, many of the badly connected communities in the first iteration of Louvain become disconnected in the second iteration. 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. E Stat. As discussed earlier, the Louvain algorithm does not guarantee connectivity. However, as increases, the Leiden algorithm starts to outperform the Louvain algorithm. E Stat. Rather than evaluating the modularity gain for moving a node to each neighboring communities, we choose a neighboring node at random and evaluate whether there is a gain in modularity if we were to move the node to that neighbors community. It maximizes a modularity score for each community, where the modularity quantifies the quality of an assignment of nodes to communities. However, nodes 16 are still locally optimally assigned, and therefore these nodes will stay in the red community. The quality improvement realised by the Leiden algorithm relative to the Louvain algorithm is larger for empirical networks than for benchmark networks. 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. The percentage of badly connected communities is less affected by the number of iterations of the Louvain algorithm. S3. After the refinement phase is concluded, communities in \({\mathscr{P}}\) often will have been split into multiple communities in \({{\mathscr{P}}}_{{\rm{refined}}}\), but not always. In doing so, Louvain keeps visiting nodes that cannot be moved to a different community. Importantly, the problem of disconnected communities is not just a theoretical curiosity. 1 and summarised in pseudo-code in AlgorithmA.1 in SectionA of the Supplementary Information. It therefore does not guarantee -connectivity either. This is similar to what we have seen for benchmark networks. One of the best-known methods for community detection is called modularity3. contrastive-sc works best on datasets with fewer clusters when using the KMeans clustering and conversely for Leiden. The Leiden algorithm is typically iterated: the output of one iteration is used as the input for the next iteration. 2010. USA 104, 36, https://doi.org/10.1073/pnas.0605965104 (2007). 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. Rev. Then, in order . Communities in \({\mathscr{P}}\) may be split into multiple subcommunities in \({{\mathscr{P}}}_{{\rm{refined}}}\). Electr. In our experimental analysis, we observe that up to 25% of the communities are badly connected and up to 16% are disconnected. 2004. Duch, J. Note that the object for Seurat version 3 has changed. where >0 is a resolution parameter4. Moreover, when the algorithm is applied iteratively, it converges to a partition in which all subsets of all communities are guaranteed to be locally optimally assigned. Louvain can also be quite slow, as it spends a lot of time revisiting nodes that may not have changed neighborhoods. SPATA2 currently offers the functions findSeuratClusters (), findMonocleClusters () and findNearestNeighbourClusters () which are wrapper around widely used clustering algorithms. 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 prove that the new algorithm is guaranteed to produce partitions in which all communities are internally connected. In addition, we prove that the algorithm converges to an asymptotically stable partition in which all subsets of all communities are locally optimally assigned. & Arenas, A. We first applied the Scanpy pipeline, including its clustering method (Leiden clustering), on the PBMC dataset. Please After the first iteration of the Louvain algorithm, some partition has been obtained. A. 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. CAS Similarly, in citation networks, such as the Web of Science network, nodes in a community are usually considered to share a common topic26,27. However, the Louvain algorithm does not consider this possibility, since it considers only individual node movements. These are the same networks that were also studied in an earlier paper introducing the smart local move algorithm15. In the local moving phase, individual nodes are moved to the community that yields the largest increase in the quality function. This is well illustrated by figure 2 in the Leiden paper: When a community becomes disconnected like this, there is no way for Louvain to easily split it into two separate communities. Each of these can be used as an objective function for graph-based community detection methods, with our goal being to maximize this value. Are you sure you want to create this branch? You will not need much Python to use it. ADS Discov. Get the most important science stories of the day, free in your inbox. Then the Leiden algorithm can be run on the adjacency matrix. To elucidate the problem, we consider the example illustrated in Fig. 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). Soft Matter Phys. Complex brain networks: graph theoretical analysis of structural and functional systems. As shown in Fig. MATH See the documentation for these functions. N.J.v.E. Second, to study the scaling of the Louvain and the Leiden algorithm, we use benchmark networks, allowing us to compare the algorithms in terms of both computational time and quality of the partitions.