1 I am using the leiden algorithm implementation in iGraph, and noticed that when I repeat clustering using the same resolution parameter, I get different results. Raghavan, U., Albert, R. & Kumara, S. Near linear time algorithm to detect community structures in large-scale networks. Int. In the first iteration, Leiden is roughly 220 times faster than Louvain. When the Leiden algorithm found that a community could be split into multiple subcommunities, we counted the community as badly connected. As we will demonstrate in our experimental analysis, the problem occurs frequently in practice when using the Louvain algorithm. Hence, the community remains disconnected, unless it is merged with another community that happens to act as a bridge. Many Git commands accept both tag and branch names, so creating this branch may cause unexpected behavior. In the fast local move procedure in the Leiden algorithm, only nodes whose neighbourhood has changed are visited. The problem of disconnected communities has been observed before19,20, also in the context of the label propagation algorithm21. . Note that the object for Seurat version 3 has changed. E 76, 036106, https://doi.org/10.1103/PhysRevE.76.036106 (2007). The community with which a node is merged is selected randomly18. The 'devtools' package will be used to install 'leiden' and the dependancies (igraph and reticulate). Initially, \({{\mathscr{P}}}_{{\rm{refined}}}\) is set to a singleton partition, in which each node is in its own community. First calculate k-nearest neighbors and construct the SNN graph. We can guarantee a number of properties of the partitions found by the Leiden algorithm at various stages of the iterative process. The Leiden algorithm provides several guarantees. N.J.v.E. Algorithmics 16, 2.1, https://doi.org/10.1145/1963190.1970376 (2011). For example, after four iterations, the Web UK network has 8% disconnected communities, but twice as many badly connected communities. 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. Elect. Phys. Figure3 provides an illustration of the algorithm. & Moore, C. Finding community structure in very large networks. leiden_clustering Description Class wrapper based on scanpy to use the Leiden algorithm to directly cluster your data matrix with a scikit-learn flavor. Modularity is a popular objective function used with the Louvain method for community detection. The resolution limit describes a limitation where there is a minimum community size able to be resolved by optimizing modularity (or other related functions). It identifies the clusters by calculating the densities of the cells. In terms of the percentage of badly connected communities in the first iteration, Leiden performs even worse than Louvain, as can be seen in Fig. Faster unfolding of communities: Speeding up the Louvain algorithm. However, the initial partition for the aggregate network is based on P, just like in the Louvain algorithm. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/. Run the code above in your browser using DataCamp Workspace. In subsequent iterations, the percentage of disconnected communities remains fairly stable. The algorithm then moves individual nodes in the aggregate network (e). Introduction The Louvain method is an algorithm to detect communities in large networks. Bullmore, E. & Sporns, O. Blondel, V D, J L Guillaume, and R Lambiotte. The property of -connectivity is a slightly stronger variant of ordinary connectivity. Random moving is a very simple adjustment to Louvain local moving proposed in 2015 (Traag 2015). modularity) increases. The Leiden algorithm has been specifically designed to address the problem of badly connected communities. These nodes can be approximately identified based on whether neighbouring nodes have changed communities. 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). Crucially, however, the percentage of badly connected communities decreases with each iteration of the Leiden algorithm. V. A. Traag. Nat. Theory Exp. 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. Zenodo, https://doi.org/10.5281/zenodo.1466831 https://github.com/CWTSLeiden/networkanalysis. Hence, the Leiden algorithm effectively addresses the problem of badly connected communities. The numerical details of the example can be found in SectionB of the Supplementary Information. The Leiden algorithm is typically iterated: the output of one iteration is used as the input for the next iteration. Somewhat stronger guarantees can be obtained by iterating the algorithm, using the partition obtained in one iteration of the algorithm as starting point for the next iteration. Lancichinetti, A. 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. 8, 207218, https://doi.org/10.17706/IJCEE.2016.8.3.207-218 (2016). 10, 186198, https://doi.org/10.1038/nrn2575 (2009). We generated networks with n=103 to n=107 nodes. 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. J. Comput. The corresponding results are presented in the Supplementary Fig. 8 (3): 207. https://pdfs.semanticscholar.org/4ea9/74f0fadb57a0b1ec35cbc5b3eb28e9b966d8.pdf. The algorithm may yield arbitrarily badly connected communities, over and above the well-known issue of the resolution limit14. The algorithm then moves individual nodes in the aggregate network (d). 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. & Clauset, A. 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. In the local moving phase, individual nodes are moved to the community that yields the largest increase in the quality function. The new algorithm integrates several earlier improvements, incorporating a combination of smart local move15, fast local move16,17 and random neighbour move18. Louvain community detection algorithm was originally proposed in 2008 as a fast community unfolding method for large networks. Modules smaller than the minimum size may not be resolved through modularity optimization, even in the extreme case where they are only connected to the rest of the network through a single edge. We prove that the Leiden algorithm yields communities that are guaranteed to be connected. Node mergers that cause the quality function to decrease are not considered. The Louvain algorithm is a simple and popular method for community detection (Blondel, Guillaume, and Lambiotte 2008). The constant Potts model might give better communities in some cases, as it is not subject to the resolution limit. 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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. Each point corresponds to a certain iteration of an algorithm, with results averaged over 10 experiments. These nodes are therefore optimally assigned to their current community. The Leiden algorithm starts from a singleton partition (a). This phenomenon can be explained by the documented tendency KMeans has to identify equal-sized , combined with the significant class imbalance associated with the datasets having more than 8 clusters (Table 1). ADS Value. S3. As the problem of modularity optimization is NP-hard, we need heuristic methods to optimize modularity (or CPM). However, if communities are badly connected, this may lead to incorrect attributions of shared functionality. Phys. 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. The horizontal axis indicates the cumulative time taken to obtain the quality indicated on the vertical axis. Weights for edges an also be passed to the leiden algorithm either as a separate vector or weights or a weighted adjacency matrix. When a sufficient number of neighbours of node 0 have formed a community in the rest of the network, it may be optimal to move node 0 to this community, thus creating the situation depicted in Fig. In the first step of the next iteration, Louvain will again move individual nodes in the network. The Louvain algorithm is illustrated in Fig. The count of badly connected communities also included disconnected communities. Google Scholar. 2. The parameter functions as a sort of threshold: communities should have a density of at least , while the density between communities should be lower than . Speed and quality for the first 10 iterations of the Louvain and the Leiden algorithm for benchmark networks (n=106 and n=107). Besides the Louvain algorithm and the Leiden algorithm (see the "Methods" section), there are several widely-used network clustering algorithms, such as the Markov clustering algorithm [], Infomap algorithm [], and label propagation algorithm [].Markov clustering and Infomap algorithm are both based on flow . 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 an experiment containing a mixture of cell types, each cluster might correspond to a different cell type. When iterating Louvain, the quality of the partitions will keep increasing until the algorithm is unable to make any further improvements. It means that there are no individual nodes that can be moved to a different community. Discov. I tracked the number of clusters post-clustering at each step. Louvain algorithm. 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). By moving these nodes, Louvain creates badly connected communities. The aggregate network is created based on the partition \({{\mathscr{P}}}_{{\rm{refined}}}\). Communities in Networks. Iterating the Louvain algorithm can therefore be seen as a double-edged sword: it improves the partition in some way, but degrades it in another way. o CLIQUE (Clustering in Quest): - CLIQUE is a combination of density-based and grid-based clustering algorithm. From Louvain to Leiden: Guaranteeing Well-Connected Communities, October. However, for higher values of , Leiden becomes orders of magnitude faster than Louvain, reaching 10100 times faster runtimes for the largest networks. 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. Four popular community detection algorithms are explained . We start by initialising a queue with all nodes in the network. Runtime versus quality for empirical networks. Communities may even be internally disconnected. 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). Phys. 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 Louvain method for community detection is a popular way to discover communities from single-cell data. 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. We then remove the first node from the front of the queue and we determine whether the quality function can be increased by moving this node from its current community to a different one. However, in the case of the Web of Science network, more than 5% of the communities are disconnected in the first iteration. PubMedGoogle Scholar. 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 refined partition \({{\mathscr{P}}}_{{\rm{refined}}}\) is obtained as follows. This way of defining the expected number of edges is based on the so-called configuration model. 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. Subset optimality is the strongest guarantee that is provided by the Leiden algorithm. We now show that the Louvain algorithm may find arbitrarily badly connected communities. reviewed the manuscript. The constant Potts model tries to maximize the number of internal edges in a community, while simultaneously trying to keep community sizes small, and the constant parameter balances these two characteristics. http://dx.doi.org/10.1073/pnas.0605965104. To obtain First, we created a specified number of nodes and we assigned each node to a community. In that case, some optimal partitions cannot be found, as we show in SectionC2 of the Supplementary Information. In this stage we essentially collapse communities down into a single representative node, creating a new simplified graph. 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 For lower values of , the correct partition is easy to find and Leiden is only about twice as fast as Louvain. This is very similar to what the smart local moving algorithm does. Rotta, R. & Noack, A. Multilevel local search algorithms for modularity clustering. We name our algorithm the Leiden algorithm, after the location of its authors. MATH Obviously, this is a worst case example, showing that disconnected communities may be identified by the Louvain algorithm. Article As can be seen in Fig. Rev. Hence, in general, Louvain may find arbitrarily badly connected communities. This is not the case when nodes are greedily merged with the community that yields the largest increase in the quality function. Article For higher values of , Leiden finds better partitions than Louvain. Once aggregation is complete we restart the local moving phase, and continue to iterate until everything converges down to one node. Ozaki, N., Tezuka, H. & Inaba, M. A Simple Acceleration Method for the Louvain Algorithm. Even worse, the Amazon network has 5% disconnected communities, but 25% badly connected communities. In addition, to analyse whether a community is badly connected, we ran the Leiden algorithm on the subnetwork consisting of all nodes belonging to the community. As shown in Fig. Rev. 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. Each community in this partition becomes a node in the aggregate network. The corresponding results are presented in the Supplementary Fig. 1 and summarised in pseudo-code in AlgorithmA.1 in SectionA of the Supplementary Information. The algorithm moves individual nodes from one community to another to find a partition (b), which is then refined (c). We demonstrate the performance of the Leiden algorithm for several benchmark and real-world networks. This is because Louvain only moves individual nodes at a time. import leidenalg as la import igraph as ig Example output. Please J. Exp. The nodes that are more interconnected have been partitioned into separate clusters. For example an SNN can be generated: For Seurat version 3 objects, the Leiden algorithm has been implemented in the Seurat version 3 package with Seurat::FindClusters and algorithm = "leiden"). 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. Not. Modularity scores of +1 mean that all the edges in a community are connecting nodes within the community. Article Ayan Sinha, David F. Gleich & Karthik Ramani, Marinka Zitnik, Rok Sosi & Jure Leskovec, Zhenqi Lu, Johan Wahlstrm & Arye Nehorai, Natalie Stanley, Roland Kwitt, Peter J. Mucha, Scientific Reports Is modularity with a resolution parameter equivalent to leidenalg.RBConfigurationVertexPartition? Runtime versus quality for benchmark networks. In practice, this means that small clusters can hide inside larger clusters, making their identification difficult. Leiden is both faster than Louvain and finds better partitions. Cluster your data matrix with the Leiden algorithm. This is the crux of the Leiden paper, and the authors show that this exact problem happens frequently in practice. In the refinement phase, nodes are not necessarily greedily merged with the community that yields the largest increase in the quality function. Networks with high modularity have dense connections between the nodes within modules but sparse connections between nodes in different modules. Below, the quality of a partition is reported as \(\frac{ {\mathcal H} }{2m}\), where H is defined in Eq. One may expect that other nodes in the old community will then also be moved to other communities. CPM does not suffer from this issue13. 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. 10008, 6, https://doi.org/10.1088/1742-5468/2008/10/P10008 (2008). Hence, no further improvements can be made after a stable iteration of the Louvain algorithm. Conversely, if Leiden does not find subcommunities, there is no guarantee that modularity cannot be increased by splitting up the community. Sci. A score of -1 means that there are no edges connecting nodes within the community, and they instead all connect nodes outside the community. Clauset, A., Newman, M. E. J. In the case of modularity, communities may have significant substructure both because of the resolution limit and because of the shortcomings of Louvain. 2(a). Here we can see partitions in the plotted results. The quality improvement realised by the Leiden algorithm relative to the Louvain algorithm is larger for empirical networks than for benchmark networks. Google Scholar. From Louvain to Leiden: guaranteeing well-connected communities, $$ {\mathcal H} =\frac{1}{2m}\,{\sum }_{c}({e}_{c}-{\rm{\gamma }}\frac{{K}_{c}^{2}}{2m}),$$, $$ {\mathcal H} ={\sum }_{c}[{e}_{c}-\gamma (\begin{array}{c}{n}_{c}\\ 2\end{array})],$$, https://doi.org/10.1038/s41598-019-41695-z. This continues until the queue is empty. sign in Agglomerative clustering is a bottom-up approach. Note that Leiden clustering directly clusters the neighborhood graph of cells, which we already computed in the previous section. 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. Article Google Scholar. In fact, by implementing the refinement phase in the right way, several attractive guarantees can be given for partitions produced by the Leiden algorithm. Complex brain networks: graph theoretical analysis of structural and functional systems. Community detection can then be performed using this graph. Am. Note that communities found by the Leiden algorithm are guaranteed to be 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. For example, nodes in a community in biological or neurological networks are often assumed to share similar functions or behaviour25. Rev. E 70, 066111, https://doi.org/10.1103/PhysRevE.70.066111 (2004). J. U. S. A. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate. E Stat. 92 (3): 032801. http://dx.doi.org/10.1103/PhysRevE.92.032801. We used the CPM quality function. Soft Matter Phys. Bae, S., Halperin, D., West, J. D., Rosvall, M. & Howe, B. Scalable and Efficient Flow-Based Community Detection for Large-Scale Graph Analysis. Moreover, when no more nodes can be moved, the algorithm will aggregate the network. Figure4 shows how well it does compared to the Louvain algorithm. See the documentation for these functions. When node 0 is moved to a different community, the red community becomes internally disconnected, as shown in (b). We study the problem of badly connected communities when using the Louvain algorithm for several empirical networks. The idea of the refinement phase in the Leiden algorithm is to identify a partition \({{\mathscr{P}}}_{{\rm{refined}}}\) that is a refinement of \({\mathscr{P}}\). Trying to fix the problem by simply considering the connected components of communities19,20,21 is unsatisfactory because it addresses only the most extreme case and does not resolve the more fundamental problem.
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