Communities may even be internally disconnected. Rev. where >0 is a resolution parameter4. You signed in with another tab or window. We show that this algorithm has a major defect that largely went unnoticed until now: the Louvain algorithm may yield arbitrarily badly connected communities. For this network, Leiden requires over 750 iterations on average to reach a stable iteration. 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. Therefore, clustering algorithms look for similarities or dissimilarities among data points. The leidenalg package facilitates community detection of networks and builds on the package igraph. Nature 433, 895900, https://doi.org/10.1038/nature03288 (2005). 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. Phys. Resolution Limit in Community Detection. Proc. These steps are repeated until no further improvements can be made. Each of these can be used as an objective function for graph-based community detection methods, with our goal being to maximize this value. Phys. From Louvain to Leiden: Guaranteeing Well-Connected Communities, October. J. Comput. import leidenalg as la import igraph as ig Example output. In this way, Leiden implements the local moving phase more efficiently than Louvain. The Louvain algorithm is illustrated in Fig. 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. The algorithm then moves individual nodes in the aggregate network (e). Communities may even be disconnected. The Leiden algorithm is considerably more complex than the Louvain algorithm. In subsequent iterations, the percentage of disconnected communities remains fairly stable. Nonlin. Google Scholar. Faster Unfolding of Communities: Speeding up the Louvain Algorithm. Phys. Brandes, U. et al. 2007. As we will demonstrate in our experimental analysis, the problem occurs frequently in practice when using the Louvain algorithm. In all experiments reported here, we used a value of 0.01 for the parameter that determines the degree of randomness in the refinement phase of the Leiden algorithm. We provide the full definitions of the properties as well as the mathematical proofs in SectionD of the Supplementary Information. ADS The increase in the percentage of disconnected communities is relatively limited for the Live Journal and Web of Science networks. Data 11, 130, https://doi.org/10.1145/2992785 (2017). An aggregate network (d) is created based on the refined partition, using the non-refined partition to create an initial partition for the aggregate network. Randomness in the selection of a community allows the partition space to be explored more broadly. 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. By submitting a comment you agree to abide by our Terms and Community Guidelines. This package implements the Leiden algorithm in C++ and exposes it to python.It relies on (python-)igraph for it to function. & Fortunato, S. Community detection algorithms: A comparative analysis. The triumphs and limitations of computational methods for - Nature Are you sure you want to create this branch? Ozaki, Naoto, Hiroshi Tezuka, and Mary Inaba. For each set of parameters, we repeated the experiment 10 times. 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 Fortunato, S. Community detection in graphs. 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. We gratefully acknowledge computational facilities provided by the LIACS Data Science Lab Computing Facilities through Frank Takes. 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. Leiden consists of the following steps: The refinement step allows badly connected communities to be split before creating the aggregate network. 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 ). 2(a). Clustering with the Leiden Algorithm in R These nodes can be approximately identified based on whether neighbouring nodes have changed communities. The above results shows that the problem of disconnected and badly connected communities is quite pervasive in practice. 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. A partition of clusters as a vector of integers Examples Rev. More subtle problems may occur as well, causing Louvain to find communities that are connected, but only in a very weak sense. Am. the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in First calculate k-nearest neighbors and construct the SNN graph. Speed and quality for the first 10 iterations of the Louvain and the Leiden algorithm for benchmark networks (n=106 and n=107). Discovering cell types using manifold learning and enhanced We will use sklearns K-Means implementation looking for 10 clusters in the original 784 dimensional data. Blondel, V D, J L Guillaume, and R Lambiotte. MathSciNet Finding communities in large networks is far from trivial: algorithms need to be fast, but they also need to provide high-quality results. Contrastive self-supervised clustering of scRNA-seq data Again, if communities are badly connected, this may lead to incorrect inferences of topics, which will affect bibliometric analyses relying on the inferred topics. Preprocessing and clustering 3k PBMCs Scanpy documentation As discussed earlier, the Louvain algorithm does not guarantee connectivity. Nevertheless, depending on the relative strengths of the different connections, these nodes may still be optimally assigned to their current community. E 76, 036106, https://doi.org/10.1103/PhysRevE.76.036106 (2007). Soft Matter Phys. Louvain keeps visiting all nodes in a network until there are no more node movements that increase the quality function. Inf. Class wrapper based on scanpy to use the Leiden algorithm to directly cluster your data matrix with a scikit-learn flavor. 2010. leiden_clustering Description Class wrapper based on scanpy to use the Leiden algorithm to directly cluster your data matrix with a scikit-learn flavor. CAS 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. The value of the resolution parameter was determined based on the so-called mixing parameter 13. In fact, for the Web of Science and Web UK networks, Fig. Waltman, L. & van Eck, N. J. CAS In many complex networks, nodes cluster and form relatively dense groupsoften called communities1,2. Then the Leiden algorithm can be run on the adjacency matrix. Importantly, the output of the local moving stage will depend on the order that the nodes are considered in. See the documentation for these functions. Speed and quality for the first 10 iterations of the Louvain and the Leiden algorithm for six empirical networks. As can be seen in Fig. The algorithm is described in pseudo-code in AlgorithmA.2 in SectionA of the Supplementary Information. The idea of the refinement phase in the Leiden algorithm is to identify a partition \({{\mathscr{P}}}_{{\rm{refined}}}\) that is a refinement of \({\mathscr{P}}\). Rotta, R. & Noack, A. Multilevel local search algorithms for modularity clustering. This contrasts with the Leiden algorithm. Subpartition -density is not guaranteed by the Louvain algorithm. Each point corresponds to a certain iteration of an algorithm, with results averaged over 10 experiments. Each community in this partition becomes a node in the aggregate network. This can be a shared nearest neighbours matrix derived from a graph object. 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. 2008. Phys. The Leiden algorithm consists of three phases: (1) local moving of nodes, (2) refinement of the partition and (3) aggregation of the network based on the refined partition, using the non-refined partition to create an initial partition for the aggregate network. The Leiden algorithm has been specifically designed to address the problem of badly connected communities. The resulting clusters are shown as colors on the 3D model (top) and t -SNE embedding . Traag, V. A., Waltman, L. & van Eck, N. J. networkanalysis. We also suggested that the Leiden algorithm is faster than the Louvain algorithm, because of the fast local move approach. Four popular community detection algorithms are explained . Data Eng. Conversely, if Leiden does not find subcommunities, there is no guarantee that modularity cannot be increased by splitting up the community. First, we show that the Louvain algorithm finds disconnected communities, and more generally, badly connected communities in the empirical networks. 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. Technol. Source Code (2018). For each network, we repeated the experiment 10 times. 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. This amounts to a clustering problem, where we aim to learn an optimal set of groups (communities) from the observed data. 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). Phys. Article The count of badly connected communities also included disconnected communities. A Simple Acceleration Method for the Louvain Algorithm. Int. However, modularity suffers from a difficult problem known as the resolution limit (Fortunato and Barthlemy 2007). However, it is also possible to start the algorithm from a different partition15. It therefore does not guarantee -connectivity either. Badly connected communities. Rep. 6, 30750, https://doi.org/10.1038/srep30750 (2016). Hence, the problem of Louvain outlined above is independent from the issue of the resolution limit. To ensure readability of the paper to the broadest possible audience, we have chosen to relegate all technical details to the Supplementary Information. Additionally, we implemented a Python package, available from https://github.com/vtraag/leidenalg and deposited at Zenodo24). 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. Segmentation & Clustering SPATA2 - GitHub Pages 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. http://arxiv.org/abs/1810.08473. Note that Leiden clustering directly clusters the neighborhood graph of cells, which we already computed in the previous section. 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. CAS Finally, we demonstrate the excellent performance of the algorithm for several benchmark and real-world networks. For larger networks and higher values of , Louvain is much slower than Leiden. 2.3. Rev. running Leiden clustering finished: found 16 clusters and added 'leiden_1.0', the cluster labels (adata.obs, categorical) (0:00:00) running Leiden clustering finished: found 12 clusters and added 'leiden_0.6', the cluster labels (adata.obs, categorical) (0:00:00) running Leiden clustering finished: found 9 clusters and added 'leiden_0.4', the The larger the increase in the quality function, the more likely a community is to be selected. 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. 68, 984998, https://doi.org/10.1002/asi.23734 (2017). These nodes are therefore optimally assigned to their current community. The algorithm moves individual nodes from one community to another to find a partition (b), which is then refined (c). E 81, 046106, https://doi.org/10.1103/PhysRevE.81.046106 (2010). Int. U. S. A. Directed Undirected Homogeneous Heterogeneous Weighted 1. The Leiden algorithm is clearly faster than the Louvain algorithm. The quality of such an asymptotically stable partition provides an upper bound on the quality of an optimal partition. The random component also makes the algorithm more explorative, which might help to find better community structures. 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. We thank Lovro Subelj for his comments on an earlier version of this paper. 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. Nodes 06 are in the same community. In this stage we essentially collapse communities down into a single representative node, creating a new simplified graph. Elect. Clauset, A., Newman, M. E. J. Note that this code is . 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. However, if communities are badly connected, this may lead to incorrect attributions of shared functionality. To address this important shortcoming, we introduce a new algorithm that is faster, finds better partitions and provides explicit guarantees and bounds. python - Leiden Clustering results are not always the same given the Indeed, the percentage of disconnected communities becomes more comparable to the percentage of badly connected communities in later iterations. The property of -connectivity is a slightly stronger variant of ordinary connectivity. . One may expect that other nodes in the old community will then also be moved to other communities. Centre for Science and Technology Studies, Leiden University, Leiden, The Netherlands, You can also search for this author in Introduction The Louvain method is an algorithm to detect communities in large networks. CAS Traag, V A. Publishers note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Value. This function takes a cell_data_set as input, clusters the cells using . To study the scaling of the Louvain and the Leiden algorithm, we rely on a variant of a well-known approach for constructing benchmark networks28. V.A.T. Higher resolutions lead to more communities, while lower resolutions lead to fewer communities. Rev. 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. Nonetheless, some networks still show large differences. To elucidate the problem, we consider the example illustrated in Fig. Leiden algorithm. To address this problem, we introduce the Leiden algorithm.
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