It is good at identifying small clusters. 8, the Leiden algorithm is significantly faster than the Louvain algorithm also in empirical networks. Natl. Removing such a node from its old community disconnects the old community. The new algorithm integrates several earlier improvements, incorporating a combination of smart local move15, fast local move16,17 and random neighbour move18. Cluster cells using Louvain/Leiden community detection Description. To address this problem, we introduce the Leiden algorithm. Disconnected community. In the initial stage of Louvain (when all nodes belong to their own community), nearly any move will result in a modularity gain, and it doesnt matter too much which move is chosen. 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). Similarly, in citation networks, such as the Web of Science network, nodes in a community are usually considered to share a common topic26,27. https://doi.org/10.1038/s41598-019-41695-z, DOI: https://doi.org/10.1038/s41598-019-41695-z. Newman, M. E. J. To obtain You are using a browser version with limited support for CSS. First calculate k-nearest neighbors and construct the SNN graph. Note that this code is . leiden_clustering Description Class wrapper based on scanpy to use the Leiden algorithm to directly cluster your data matrix with a scikit-learn flavor. A Smart Local Moving Algorithm for Large-Scale Modularity-Based Community Detection. Eur. 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). The quality of such an asymptotically stable partition provides an upper bound on the quality of an optimal partition. Brandes, U. et al. Complex brain networks: graph theoretical analysis of structural and functional systems. Rev. See the documentation on the leidenalg Python module for more information: https://leidenalg.readthedocs.io/en/latest/reference.html. Google Scholar. Neurosci. 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. Rev. As we will demonstrate in our experimental analysis, the problem occurs frequently in practice when using the Louvain algorithm. For example, after four iterations, the Web UK network has 8% disconnected communities, but twice as many badly connected communities. The above results shows that the problem of disconnected and badly connected communities is quite pervasive in practice. Due to the resolution limit, modularity may cause smaller communities to be clustered into larger communities. 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. 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. Other networks show an almost tenfold increase in the percentage of disconnected communities. Elect. Traag, V A. One may expect that other nodes in the old community will then also be moved to other communities. Rev. For the Amazon, DBLP and Web UK networks, Louvain yields on average respectively 23%, 16% and 14% badly connected communities. That is, no subset can be moved to a different community. Google Scholar. The horizontal axis indicates the cumulative time taken to obtain the quality indicated on the vertical axis. This commit does not belong to any branch on this repository, and may belong to a fork outside of the repository. Rev. 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. 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). E 92, 032801, https://doi.org/10.1103/PhysRevE.92.032801 (2015). 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 Traag, V.A., Waltman, L. & van Eck, N.J. From Louvain to Leiden: guaranteeing well-connected communities. The horizontal axis indicates the cumulative time taken to obtain the quality indicated on the vertical axis. In this case we know the answer is exactly 10. Porter, M. A., Onnela, J.-P. & Mucha, P. J. Knowl. conda install -c conda-forge leidenalg pip install leiden-clustering Used via. We find that the Leiden algorithm commonly finds partitions of higher quality in less time. The resolution limit describes a limitation where there is a minimum community size able to be resolved by optimizing modularity (or other related functions). This will compute the Leiden clusters and add them to the Seurat Object Class. This will compute the Leiden clusters and add them to the Seurat Object Class. As the problem of modularity optimization is NP-hard, we need heuristic methods to optimize modularity (or CPM). 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. 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. An iteration of the Leiden algorithm in which the partition does not change is called a stable iteration. Waltman, L. & van Eck, N. J. 4, in the first iteration of the Louvain algorithm, the percentage of badly connected communities can be quite high. Such algorithms are rather slow, making them ineffective for large networks. Any sub-networks that are found are treated as different communities in the next aggregation step. For each set of parameters, we repeated the experiment 10 times. We also suggested that the Leiden algorithm is faster than the Louvain algorithm, because of the fast local move approach. For each community, modularity measures the number of edges within the community and the number of edges going outside the community, and gives a value between -1 and +1. In this post, I will cover one of the common approaches which is hierarchical clustering. 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 Louvain algorithm is illustrated in Fig. After the first iteration of the Louvain algorithm, some partition has been obtained. In this case, refinement does not change the partition (f). However, for higher values of , Leiden becomes orders of magnitude faster than Louvain, reaching 10100 times faster runtimes for the largest networks. This is similar to ideas proposed recently as pruning16 and in a slightly different form as prioritisation17. Technol. 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. wrote the manuscript. A score of 0 would mean that the community has half its edges connecting nodes within the same community, and half connecting nodes outside the community. 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. The high percentage of badly connected communities attests to this. Source Code (2018). Detecting communities in a network is therefore an important problem. On Modularity Clustering. We now show that the Louvain algorithm may find arbitrarily badly connected communities. By submitting a comment you agree to abide by our Terms and Community Guidelines. Importantly, the problem of disconnected communities is not just a theoretical curiosity. A score of -1 means that there are no edges connecting nodes within the community, and they instead all connect nodes outside the community. Such a modular structure is usually not known beforehand. All experiments were run on a computer with 64 Intel Xeon E5-4667v3 2GHz CPUs and 1TB internal memory. The algorithm moves individual nodes from one community to another to find a partition (b), which is then refined (c). Publishers note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. There are many different approaches and algorithms to perform clustering tasks. E 81, 046106, https://doi.org/10.1103/PhysRevE.81.046106 (2010). However, nodes 16 are still locally optimally assigned, and therefore these nodes will stay in the red community. Leiden consists of the following steps: The refinement step allows badly connected communities to be split before creating the aggregate network. Rev. 10008, 6, https://doi.org/10.1088/1742-5468/2008/10/P10008 (2008). 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. http://iopscience.iop.org/article/10.1088/1742-5468/2008/10/P10008/meta. Nevertheless, depending on the relative strengths of the different connections, these nodes may still be optimally assigned to their current community. MathSciNet 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. They show that the original Louvain algorithm that can result in badly connected communities (even communities that are completely disconnected internally) and propose an alternative method, Leiden, that guarantees that communities are well connected. At each iteration all clusters are guaranteed to be connected and well-separated. Furthermore, by relying on a fast local move approach, the Leiden algorithm runs faster than the Louvain algorithm. Nodes 13 should form a community and nodes 46 should form another community. Electr. We show that this algorithm has a major defect that largely went unnoticed until now: the Louvain algorithm may yield arbitrarily badly connected communities. Phys. Consider the partition shown in (a). 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. J. Fast Unfolding of Communities in Large Networks. Journal of Statistical , January. We keep removing nodes from the front of the queue, possibly moving these nodes to a different community. Rotta, R. & Noack, A. Multilevel local search algorithms for modularity clustering. As can be seen in the figure, Louvain quickly reaches a state in which it is unable to find better partitions. Please For empirical networks, it may take quite some time before the Leiden algorithm reaches its first stable iteration. Sci Rep 9, 5233 (2019). First, we created a specified number of nodes and we assigned each node to a community. Centre for Science and Technology Studies, Leiden University, Leiden, The Netherlands, You can also search for this author in Importantly, the first iteration of the Leiden algorithm is the most computationally intensive one, and subsequent iterations are faster. Cluster Determination Source: R/generics.R, R/clustering.R Identify clusters of cells by a shared nearest neighbor (SNN) modularity optimization based clustering algorithm. & Bornholdt, S. Statistical mechanics of community detection. 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"). Rev. Rev. We conclude that the Leiden algorithm is strongly preferable to the Louvain algorithm. A Simple Acceleration Method for the Louvain Algorithm. Int. This enables us to find cases where its beneficial to split a community. 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. Some of these nodes may very well act as bridges, similarly to node 0 in the above example. In particular, we show that Louvain may identify communities that are internally disconnected. modularity) increases. Finally, we demonstrate the excellent performance of the algorithm for several benchmark and real-world networks. The solution proposed in smart local moving is to alter how the local moving step in Louvain works. Louvain algorithm. (We implemented both algorithms in Java, available from https://github.com/CWTSLeiden/networkanalysis and deposited at Zenodo23. Run the code above in your browser using DataCamp Workspace. Rev. Reichardt, J. Source Code (2018). Soft Matter Phys. Analyses based on benchmark networks have only a limited value because these networks are not representative of empirical real-world networks. Speed of the first iteration of the Louvain and the Leiden algorithm for six empirical networks. Article Article J. Traag, V. A. leidenalg 0.7.0. 2010. To find an optimal grouping of cells into communities, we need some way of evaluating different partitions in the graph. 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. We start by initialising a queue with all nodes in the network. Resolution Limit in Community Detection. Proc. CPM has the advantage that it is not subject to the resolution limit. Soft Matter Phys. Four popular community detection algorithms are explained . IEEE Trans. Use Git or checkout with SVN using the web URL. The classic Louvain algorithm should be avoided due to the known problem with disconnected communities. Louvain can also be quite slow, as it spends a lot of time revisiting nodes that may not have changed neighborhoods. This is very similar to what the smart local moving algorithm does. Obviously, this is a worst case example, showing that disconnected communities may be identified by the Louvain 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. Agglomerative Clustering: Also known as bottom-up approach or hierarchical agglomerative clustering (HAC). This aspect of the Louvain algorithm can be used to give information about the hierarchical relationships between communities by tracking at which stage the nodes in the communities were aggregated. To overcome the problem of arbitrarily badly connected communities, we introduced a new algorithm, which we refer to as the Leiden algorithm. Fortunato, Santo, and Marc Barthlemy. Subset optimality is the strongest guarantee that is provided by the Leiden algorithm. Clauset, A., Newman, M. E. J. Soft Matter Phys. For example, for the Web of Science network, the first iteration takes about 110120 seconds, while subsequent iterations require about 40 seconds. This is the crux of the Leiden paper, and the authors show that this exact problem happens frequently in practice. In practice, this means that small clusters can hide inside larger clusters, making their identification difficult. See the documentation for these functions. Nevertheless, when CPM is used as the quality function, the Louvain algorithm may still find arbitrarily badly connected communities. 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. 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. Sci. The algorithm then moves individual nodes in the aggregate network (e). After running local moving, we end up with a set of communities where we cant increase the objective function (eg, modularity) by moving any node to any neighboring community. B 86 (11): 471. https://doi.org/10.1140/epjb/e2013-40829-0. These steps are repeated until the quality cannot be increased further. As shown in Fig. Community detection is an important task in the analysis of complex networks. 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. However, in the case of the Web of Science network, more than 5% of the communities are disconnected in the first iteration. E 76, 036106, https://doi.org/10.1103/PhysRevE.76.036106 (2007). 2007. 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. Use the Previous and Next buttons to navigate the slides or the slide controller buttons at the end to navigate through each slide. 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). Communities were all of equal size. More subtle problems may occur as well, causing Louvain to find communities that are connected, but only in a very weak sense. Starting from the second iteration, Leiden outperformed Louvain in terms of the percentage of badly connected communities. In particular, it yields communities that are guaranteed to be connected. We will use sklearns K-Means implementation looking for 10 clusters in the original 784 dimensional data. For those wanting to read more, I highly recommend starting with the Leiden paper (Traag, Waltman, and Eck 2018) or the smart local moving paper (Waltman and Eck 2013). 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. This should be the first preference when choosing an algorithm. As can be seen in Fig. and L.W. In fact, if we keep iterating the Leiden algorithm, it will converge to a partition without any badly connected communities, as discussed earlier. 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. This contrasts to benchmark networks, for which Leiden often converges after a few iterations. Then the Leiden algorithm can be run on the adjacency matrix. However, focussing only on disconnected communities masks the more fundamental issue: Louvain finds arbitrarily badly connected communities. 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. The algorithm may yield arbitrarily badly connected communities, over and above the well-known issue of the resolution limit14. 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. volume9, Articlenumber:5233 (2019) Random moving is a very simple adjustment to Louvain local moving proposed in 2015 (Traag 2015). The random component also makes the algorithm more explorative, which might help to find better community structures. V. A. Traag. 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. Communities may even be disconnected. You signed in with another tab or window. This is not too difficult to explain. ADS 8 (3): 207. https://pdfs.semanticscholar.org/4ea9/74f0fadb57a0b1ec35cbc5b3eb28e9b966d8.pdf. At some point, node 0 is considered for moving. 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. Positive values above 2 define the total number of iterations to perform, -1 has the algorithm run until it reaches its optimal clustering. Crucially, however, the percentage of badly connected communities decreases with each iteration of the Leiden algorithm. This will compute the Leiden clusters and add them to the Seurat Object Class. CAS J. Comput. The main ideas of our algorithm are explained in an intuitive way in the main text of the paper. One of the best-known methods for community detection is called modularity3. The percentage of disconnected communities even jumps to 16% for the DBLP network. Ozaki, N., Tezuka, H. & Inaba, M. A Simple Acceleration Method for the Louvain Algorithm. This is not the case when nodes are greedily merged with the community that yields the largest increase in the quality function. In this way, the constant acts as a resolution parameter, and setting the constant higher will result in fewer communities. Scaling of benchmark results for network size. USA 104, 36, https://doi.org/10.1073/pnas.0605965104 (2007). Eng. The Leiden community detection algorithm outperforms other clustering methods. Default behaviour is calling cluster_leiden in igraph with Modularity (for undirected graphs) and CPM cost functions.
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