leiden clustering explained leiden clustering explained

Rev. Note that this code is . The Louvain algorithm is illustrated in Fig. 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. Presumably, many of the badly connected communities in the first iteration of Louvain become disconnected in the second iteration. It partitions the data space and identifies the sub-spaces using the Apriori principle. Later iterations of the Louvain algorithm only aggravate the problem of disconnected communities, even though the quality function (i.e. Such algorithms are rather slow, making them ineffective for large networks. Am. Although originally defined for modularity, the Louvain algorithm can also be used to optimise other quality functions. Moreover, when no more nodes can be moved, the algorithm will aggregate the network. Soft Matter Phys. 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. Traag, V. A. E 92, 032801, https://doi.org/10.1103/PhysRevE.92.032801 (2015). Number of iterations until stability. Due to the resolution limit, modularity may cause smaller communities to be clustered into larger communities. In the first iteration, Leiden is roughly 220 times faster than Louvain. Nature 433, 895900, https://doi.org/10.1038/nature03288 (2005). Newman, M. E. J. Even worse, the Amazon network has 5% disconnected communities, but 25% badly connected communities. We study the problem of badly connected communities when using the Louvain algorithm for several empirical networks. Number of iterations before the Leiden algorithm has reached a stable iteration for six empirical networks. & Clauset, A. 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. This function takes a cell_data_set as input, clusters the cells using . The nodes are added to the queue in a random order. E 80, 056117, https://doi.org/10.1103/PhysRevE.80.056117 (2009). Fast Unfolding of Communities in Large Networks. Journal of Statistical , January. Unsupervised clustering of cells is a common step in many single-cell expression workflows. S3. Node mergers that cause the quality function to decrease are not considered. The Leiden algorithm has been specifically designed to address the problem of badly connected communities. Hence, the complex structure of empirical networks creates an even stronger need for the use of the Leiden algorithm. Sci. 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 . Finding community structure in networks using the eigenvectors of matrices. After a stable iteration of the Leiden algorithm, it is guaranteed that: All nodes are locally optimally assigned. In this paper, we show that the Louvain algorithm has a major problem, for both modularity and CPM. The difference in computational time is especially pronounced for larger networks, with Leiden being up to 20 times faster than Louvain in empirical networks. However, the initial partition for the aggregate network is based on P, just like in the Louvain algorithm. A community size of 50 nodes was used for the results presented below, but larger community sizes yielded qualitatively similar results. An aggregate. 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. These nodes can be approximately identified based on whether neighbouring nodes have changed communities. It states that there are no communities that can be merged. In this post Ive mainly focused on the optimisation methods for community detection, rather than the different objective functions that can be used. 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. At some point, node 0 is considered for moving. All authors conceived the algorithm and contributed to the source code. Later iterations of the Louvain algorithm are very fast, but this is only because the partition remains the same. MATH Modularity optimization. Luecken, M. D. Application of multi-resolution partitioning of interaction networks to the study of complex disease. All communities are subpartition -dense. modularity) increases. 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. Excluding node mergers that decrease the quality function makes the refinement phase more efficient. However, as increases, the Leiden algorithm starts to outperform the Louvain algorithm. For each network, Table2 reports the maximal modularity obtained using the Louvain and the Leiden algorithm. In addition, we prove that the algorithm converges to an asymptotically stable partition in which all subsets of all communities are locally optimally assigned. In the first step of the next iteration, Louvain will again move individual nodes in the network. Random moving is a very simple adjustment to Louvain local moving proposed in 2015 (Traag 2015). Runtime versus quality for benchmark networks. Zenodo, https://doi.org/10.5281/zenodo.1466831 https://github.com/CWTSLeiden/networkanalysis. 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. 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 algorithm then moves individual nodes in the aggregate network (e). For both algorithms, 10 iterations were performed. Neurosci. Duch, J. The Louvain local moving phase consists of the following steps: This process is repeated for every node in the network until no further improvement in modularity is possible. Here we can see partitions in the plotted results. The percentage of disconnected communities is more limited, usually around 1%. 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. Subset optimality is the strongest guarantee that is provided by the Leiden algorithm. 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. For empirical networks, it may take quite some time before the Leiden algorithm reaches its first stable iteration. The algorithm then moves individual nodes in the aggregate network (d). We gratefully acknowledge computational facilities provided by the LIACS Data Science Lab Computing Facilities through Frank Takes. Louvain keeps visiting all nodes in a network until there are no more node movements that increase the quality function. reviewed the manuscript. (We ensured that modularity optimisation for the subnetwork was fully consistent with modularity optimisation for the whole network13) The Leiden algorithm was run until a stable iteration was obtained. Work fast with our official CLI. The count of badly connected communities also included disconnected communities. Article 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. 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. Article One of the best-known methods for community detection is called modularity3. PubMed Nodes 13 should form a community and nodes 46 should form another community. We abbreviate the leidenalg package as la and the igraph package as ig in all Python code throughout this documentation. 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. The Leiden algorithm guarantees all communities to be connected, but it may yield badly connected communities. Rev. Scaling of benchmark results for difficulty of the partition. 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. Are you sure you want to create this branch? Note that this code is designed for Seurat version 2 releases. Hence, the Leiden algorithm effectively addresses the problem of 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. In the fast local move procedure in the Leiden algorithm, only nodes whose neighbourhood has changed are visited. Nonlin. In the local moving phase, individual nodes are moved to the community that yields the largest increase in the quality function. For the Amazon, DBLP and Web UK networks, Louvain yields on average respectively 23%, 16% and 14% badly connected communities. Anyone you share the following link with will be able to read this content: Sorry, a shareable link is not currently available for this article. For larger networks and higher values of , Louvain is much slower than Leiden. Louvain algorithm. AMS 56, 10821097 (2009). The Leiden algorithm is clearly faster than the Louvain algorithm. CAS 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 second iteration of Louvain shows a large increase in the percentage of disconnected communities. Phys. In addition, we prove that, when the Leiden algorithm is applied iteratively, it converges to a partition in which all subsets of all communities are locally optimally assigned. At this point, it is guaranteed that each individual node is optimally assigned. 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. MATH We can guarantee a number of properties of the partitions found by the Leiden algorithm at various stages of the iterative process. Rev. Waltman, Ludo, and Nees Jan van Eck. Positive values above 2 define the total number of iterations to perform, -1 has the algorithm run until it reaches its optimal clustering. 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). For this network, Leiden requires over 750 iterations on average to reach a stable iteration. The authors act as bibliometric consultants to CWTS B.V., which makes use of community detection algorithms in commercial products and services. Instead, a node may be merged with any community for which the quality function increases. The smart local moving algorithm (Waltman and Eck 2013) identified another limitation in the original Louvain method: it isnt able to split communities once theyre merged, even when it may be very beneficial to do so. It therefore does not guarantee -connectivity either. * (2018). E 76, 036106, https://doi.org/10.1103/PhysRevE.76.036106 (2007). MathSciNet You will not need much Python to use it. However, it is also possible to start the algorithm from a different partition15. Default behaviour is calling cluster_leiden in igraph with Modularity (for undirected graphs) and CPM cost functions. 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. Please An alternative quality function is the Constant Potts Model (CPM)13, which overcomes some limitations of modularity. 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). 2004. Guimer, R. & Nunes Amaral, L. A. Functional cartography of complex metabolic networks. Cluster cells using Louvain/Leiden community detection Description. In particular, benchmark networks have a rather simple structure. Publishers note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 63, 23782392, https://doi.org/10.1002/asi.22748 (2012). The property of -connectivity is a slightly stronger variant of ordinary connectivity. 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. The Leiden algorithm is considerably more complex than the Louvain algorithm. Hence, the problem of Louvain outlined above is independent from the issue of the resolution limit. However, as shown in this paper, the Louvain algorithm has a major shortcoming: the algorithm yields communities that may be arbitrarily badly connected. These steps are repeated until no further improvements can be made. 2010. In a stable iteration, the partition is guaranteed to be node optimal and subpartition -dense. In the case of modularity, communities may have significant substructure both because of the resolution limit and because of the shortcomings of Louvain. The algorithm is described in pseudo-code in AlgorithmA.2 in SectionA of the Supplementary Information. We consider these ideas to represent the most promising directions in which the Louvain algorithm can be improved, even though we recognise that other improvements have been suggested as well22. 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). In many complex networks, nodes cluster and form relatively dense groupsoften called communities1,2. https://doi.org/10.1038/s41598-019-41695-z. Furthermore, by relying on a fast local move approach, the Leiden algorithm runs faster than the Louvain algorithm. E Stat. 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. First iteration runtime for empirical networks. 2(b). sign in Scientific Reports (Sci Rep) ADS This contrasts with the Leiden algorithm. J. 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. Google Scholar. Rev. 4. Article The Leiden algorithm starts from a singleton partition (a). 7, whereas Louvain becomes much slower for more difficult partitions, Leiden is much less affected by the difficulty of the partition. See the documentation on the leidenalg Python module for more information: https://leidenalg.readthedocs.io/en/latest/reference.html. Clauset, A., Newman, M. E. J. First, we show that the Louvain algorithm finds disconnected communities, and more generally, badly connected communities in the empirical networks. Discov. 10, 186198, https://doi.org/10.1038/nrn2575 (2009). Acad. 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. This continues until the queue is empty. The leidenalg package facilitates community detection of networks and builds on the package igraph. Mech. (We implemented both algorithms in Java, available from https://github.com/CWTSLeiden/networkanalysis and deposited at Zenodo23. The constant Potts model might give better communities in some cases, as it is not subject to the resolution limit. When the Leiden algorithm found that a community could be split into multiple subcommunities, we counted the community as badly connected. For the results reported below, the average degree was set to \(\langle k\rangle =10\). Proc. Another important difference between the Leiden algorithm and the Louvain algorithm is the implementation of the local moving phase. and JavaScript. 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. Sci. Porter, M. A., Onnela, J.-P. & Mucha, P. J. PubMedGoogle Scholar. For each network, we repeated the experiment 10 times. J. Assoc. The above results shows that the problem of disconnected and badly connected communities is quite pervasive in practice. E 72, 027104, https://doi.org/10.1103/PhysRevE.72.027104 (2005). ADS The degree of randomness in the selection of a community is determined by a parameter >0. volume9, Articlenumber:5233 (2019) 2.3. In our experimental analysis, we observe that up to 25% of the communities are badly connected and up to 16% are disconnected. Rep. 486, 75174, https://doi.org/10.1016/j.physrep.2009.11.002 (2010). PubMed Phys. Even though clustering can be applied to networks, it is a broader field in unsupervised machine learning which deals with multiple attribute types. On Modularity Clustering. Louvain community detection algorithm was originally proposed in 2008 as a fast community unfolding method for large networks. where nc is the number of nodes in community c. The interpretation of the resolution parameter is quite straightforward. The Louvain algorithm is a simple and popular method for community detection (Blondel, Guillaume, and Lambiotte 2008). Traag, V. A., Waltman, L. & van Eck, N. J. networkanalysis. Powered by DataCamp DataCamp Nevertheless, depending on the relative strengths of the different connections, these nodes may still be optimally assigned to their current community. Louvain pruning keeps track of a list of nodes that have the potential to change communities, and only revisits nodes in this list, which is much smaller than the total number of nodes. https://doi.org/10.1038/s41598-019-41695-z, DOI: https://doi.org/10.1038/s41598-019-41695-z. Use Git or checkout with SVN using the web URL. To address this important shortcoming, we introduce a new algorithm that is faster, finds better partitions and provides explicit guarantees and bounds. We now show that the Louvain algorithm may find arbitrarily badly connected communities. Bullmore, E. & Sporns, O. 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. 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. In short, the problem of badly connected communities has important practical consequences. Scaling of benchmark results for network size. Louvain has two phases: local moving and aggregation. In this case, refinement does not change the partition (f). Finally, we compare the performance of the algorithms on the empirical networks. In this post, I will cover one of the common approaches which is hierarchical clustering. Moreover, Louvain has no mechanism for fixing these communities. Performance of modularity maximization in practical contexts. Ronhovde, Peter, and Zohar Nussinov. Perhaps surprisingly, iterating the algorithm aggravates the problem, even though it does increase the quality function. In other words, communities are guaranteed to be well separated.