We prove that the Leiden algorithm yields communities that are guaranteed to be connected. 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. Each point corresponds to a certain iteration of an algorithm, with results averaged over 10 experiments. J. This contrasts with optimisation algorithms such as simulated annealing, which do allow the quality function to decrease4,8. Rev. The degree of randomness in the selection of a community is determined by a parameter >0. 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. We name our algorithm the Leiden algorithm, after the location of its authors. Speed and quality for the first 10 iterations of the Louvain and the Leiden algorithm for benchmark networks (n=106 and n=107). 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. However, if communities are badly connected, this may lead to incorrect attributions of shared functionality. The aggregate network is created based on the partition \({{\mathscr{P}}}_{{\rm{refined}}}\). The algorithm is described in pseudo-code in AlgorithmA.2 in SectionA of the Supplementary Information. 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. Communities in \({\mathscr{P}}\) may be split into multiple subcommunities in \({{\mathscr{P}}}_{{\rm{refined}}}\). Blondel, V D, J L Guillaume, and R Lambiotte. Blondel, V. D., Guillaume, J.-L., Lambiotte, R. & Lefebvre, E. Fast unfolding of communities in large networks. MATH ADS Community detection is often used to understand the structure of large and complex networks. Higher resolutions lead to more communities, while lower resolutions lead to fewer communities. The thick edges in Fig. 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. 6 show that Leiden outperforms Louvain in terms of both computational time and quality of the partitions. E 74, 036104, https://doi.org/10.1103/PhysRevE.74.036104 (2006). 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. Data Eng. Powered by DataCamp DataCamp However, modularity suffers from a difficult problem known as the resolution limit (Fortunato and Barthlemy 2007). The reasoning behind this is that the best community to join will usually be the one that most of the nodes neighbors already belong to. Speed of the first iteration of the Louvain and the Leiden algorithm for benchmark networks with increasingly difficult partitions (n=107). All authors conceived the algorithm and contributed to the source code. 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. Networks with high modularity have dense connections between the nodes within modules but sparse connections between nodes in different modules. The Leiden algorithm provides several guarantees. 8, 207218, https://doi.org/10.17706/IJCEE.2016.8.3.207-218 (2016). However, as shown in this paper, the Louvain algorithm has a major shortcoming: the algorithm yields communities that may be arbitrarily badly connected. J. Stat. Default behaviour is calling cluster_leiden in igraph with Modularity (for undirected graphs) and CPM cost functions. Google Scholar. Discov. V.A.T. Waltman, L. & van Eck, N. J. Later iterations of the Louvain algorithm only aggravate the problem of disconnected communities, even though the quality function (i.e. Proc. Sci. Hence, the complex structure of empirical networks creates an even stronger need for the use of the Leiden algorithm. 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. The Leiden algorithm is considerably more complex than the Louvain algorithm. 9, the Leiden algorithm also performs better than the Louvain algorithm in terms of the quality of the partitions that are obtained. 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 disconnected communities is more limited, usually around 1%. The Leiden algorithm is typically iterated: the output of one iteration is used as the input for the next iteration. However, after all nodes have been visited once, Leiden visits only nodes whose neighbourhood has changed, whereas Louvain keeps visiting all nodes in the network. 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 . However, in the case of the Web of Science network, more than 5% of the communities are disconnected in the first iteration. Zenodo, https://doi.org/10.5281/zenodo.1466831 https://github.com/CWTSLeiden/networkanalysis. Starting from the second iteration, Leiden outperformed Louvain in terms of the percentage of badly connected communities. The nodes that are more interconnected have been partitioned into separate clusters. As can be seen in Fig. For example, for the Web of Science network, the first iteration takes about 110120 seconds, while subsequent iterations require about 40 seconds. 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. The second iteration of Louvain shows a large increase in the percentage of disconnected communities. 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. Yang, Z., Algesheimer, R. & Tessone, C. J. Phys. Natl. E 78, 046110, https://doi.org/10.1103/PhysRevE.78.046110 (2008). Although originally defined for modularity, the Louvain algorithm can also be used to optimise other quality functions. In this case, refinement does not change the partition (f). Am. We also suggested that the Leiden algorithm is faster than the Louvain algorithm, because of the fast local move approach. It states that there are no communities that can be merged. Natl. E 81, 046106, https://doi.org/10.1103/PhysRevE.81.046106 (2010). A structure that is more informative than the unstructured set of clusters returned by flat clustering. For larger networks and higher values of , Louvain is much slower than Leiden. Speed of the first iteration of the Louvain and the Leiden algorithm for six empirical networks. We demonstrate the performance of the Leiden algorithm for several benchmark and real-world networks. Article performed the experimental analysis. J. Assoc. We generated benchmark networks in the following way. It partitions the data space and identifies the sub-spaces using the Apriori principle. Introduction The Louvain method is an algorithm to detect communities in large networks. Eng. Later iterations of the Louvain algorithm are very fast, but this is only because the partition remains the same. Runtime versus quality for benchmark networks. That is, no subset can be moved to a different community. In short, the problem of badly connected communities has important practical consequences. Faster unfolding of communities: Speeding up the Louvain algorithm. The count of badly connected communities also included disconnected communities. It starts clustering by treating the individual data points as a single cluster then it is merged continuously based on similarity until it forms one big cluster containing all objects. 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. 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. 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. ISSN 2045-2322 (online). 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. E 92, 032801, https://doi.org/10.1103/PhysRevE.92.032801 (2015). N.J.v.E. Rep. 486, 75174, https://doi.org/10.1016/j.physrep.2009.11.002 (2010). At some point, the Louvain algorithm may end up in the community structure shown in Fig. 68, 984998, https://doi.org/10.1002/asi.23734 (2017). ADS For example, nodes in a community in biological or neurological networks are often assumed to share similar functions or behaviour25. Note that this code is . It was originally developed for modularity optimization, although the same method can be applied to optimize CPM. leiden_clsutering is distributed under a BSD 3-Clause License (see LICENSE). SPATA2 currently offers the functions findSeuratClusters (), findMonocleClusters () and findNearestNeighbourClusters () which are wrapper around widely used clustering algorithms. modularity) increases. 2004. For empirical networks, it may take quite some time before the Leiden algorithm reaches its first stable iteration. Community detection in complex networks using extremal optimization. There are many different approaches and algorithms to perform clustering tasks. First iteration runtime for empirical networks. 8, the Leiden algorithm is significantly faster than the Louvain algorithm also in empirical networks. & Bornholdt, S. Statistical mechanics of community detection. We now show that the Louvain algorithm may find arbitrarily badly connected communities. For the results reported below, the average degree was set to \(\langle k\rangle =10\). Indeed, the percentage of disconnected communities becomes more comparable to the percentage of badly connected communities in later iterations. The algorithm may yield arbitrarily badly connected communities, over and above the well-known issue of the resolution limit14. Publishers note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 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). Clustering is a machine learning technique in which similar data points are grouped into the same cluster based on their attributes. Number of iterations until stability. 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. Even worse, the Amazon network has 5% disconnected communities, but 25% badly connected communities. Electr. Class wrapper based on scanpy to use the Leiden algorithm to directly cluster your data matrix with a scikit-learn flavor. However, so far this problem has never been studied for the Louvain algorithm. Then the Leiden algorithm can be run on the adjacency matrix. B 86, 471, https://doi.org/10.1140/epjb/e2013-40829-0 (2013). where >0 is a resolution parameter4. Run the code above in your browser using DataCamp Workspace. IEEE Trans. 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. There was a problem preparing your codespace, please try again. Badly connected communities. This enables us to find cases where its beneficial to split a community. The algorithm then locally merges nodes in \({{\mathscr{P}}}_{{\rm{refined}}}\): nodes that are on their own in a community in \({{\mathscr{P}}}_{{\rm{refined}}}\) can be merged with a different community. As shown in Fig. 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. We abbreviate the leidenalg package as la and the igraph package as ig in all Python code throughout this documentation. These steps are repeated until the quality cannot be increased further. A smart local moving algorithm for large-scale modularity-based community detection. On the other hand, Leiden keeps finding better partitions, especially for higher values of , for which it is more difficult to identify good partitions. The horizontal axis indicates the cumulative time taken to obtain the quality indicated on the vertical axis. https://doi.org/10.1038/s41598-019-41695-z, DOI: https://doi.org/10.1038/s41598-019-41695-z. Even though clustering can be applied to networks, it is a broader field in unsupervised machine learning which deals with multiple attribute types. CAS When iterating Louvain, the quality of the partitions will keep increasing until the algorithm is unable to make any further improvements. The algorithm continues to move nodes in the rest of the network. ADS The Leiden algorithm has been specifically designed to address the problem of badly connected communities. In addition, a node is merged with a community in \({{\mathscr{P}}}_{{\rm{refined}}}\) only if both are sufficiently well connected to their community in \({\mathscr{P}}\). Ozaki, N., Tezuka, H. & Inaba, M. A Simple Acceleration Method for the Louvain Algorithm. However, focussing only on disconnected communities masks the more fundamental issue: Louvain finds arbitrarily badly connected communities. This is the crux of the Leiden paper, and the authors show that this exact problem happens frequently in practice. 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. Is modularity with a resolution parameter equivalent to leidenalg.RBConfigurationVertexPartition? Randomness in the selection of a community allows the partition space to be explored more broadly. Due to the resolution limit, modularity may cause smaller communities to be clustered into larger communities. The Leiden algorithm is considerably more complex than the Louvain algorithm. Phys. DBSCAN Clustering Explained Detailed theorotical explanation and scikit-learn implementation Clustering is a way to group a set of data points in a way that similar data points are grouped together. In the fast local move procedure in the Leiden algorithm, only nodes whose neighbourhood has changed are visited. Note that if Leiden finds subcommunities, splitting up the community is guaranteed to increase modularity. To address this important shortcoming, we introduce a new algorithm that is faster, finds better partitions and provides explicit guarantees and bounds. Both conda and PyPI have leiden clustering in Python which operates via iGraph. E 72, 027104, https://doi.org/10.1103/PhysRevE.72.027104 (2005). We therefore require a more principled solution, which we will introduce in the next section. Ph.D. thesis, (University of Oxford, 2016). This is similar to what we have seen for benchmark networks. 2007. 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. A Comparative Analysis of Community Detection Algorithms on Artificial Networks. Waltman, L. & van Eck, N. J. Node optimality is also guaranteed after a stable iteration of the Louvain algorithm. Source Code (2018). 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. E 70, 066111, https://doi.org/10.1103/PhysRevE.70.066111 (2004). Nat. Clearly, it would be better to split up the community. 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. This package requires the 'leidenalg' and 'igraph' modules for python (2) to be installed on your system. Clustering algorithms look for similarities or dissimilarities among data points so that similar ones can be grouped together. Rev. For the Amazon, DBLP and Web UK networks, Louvain yields on average respectively 23%, 16% and 14% badly connected communities. 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. While current approaches are successful in reducing the number of sequence alignments performed, the generated clusters are . In the first step of the next iteration, Louvain will again move individual nodes in the network. Louvain keeps visiting all nodes in a network until there are no more node movements that increase the quality function. 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. It maximizes a modularity score for each community, where the modularity quantifies the quality of an assignment of nodes to communities. The problem of disconnected communities has been observed before19,20, also in the context of the label propagation algorithm21. This may have serious consequences for analyses based on the resulting partitions. The algorithm is run iteratively, using the partition identified in one iteration as starting point for the next iteration. 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. Instead, a node may be merged with any community for which the quality function increases. volume9, Articlenumber:5233 (2019) In the first iteration, Leiden is roughly 220 times faster than Louvain. Knowl. The phase one loop can be greatly accelerated by finding the nodes that have the potential to change community and only revisit those nodes. In particular, it yields communities that are guaranteed to be connected. J. Comput. We used modularity with a resolution parameter of =1 for the experiments. At each iteration all clusters are guaranteed to be connected and well-separated. Hence, the Leiden algorithm effectively addresses the problem of badly connected communities. 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. 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 Web of Science network is the most difficult one. For lower values of , the correct partition is easy to find and Leiden is only about twice as fast as Louvain. Based on project statistics from the GitHub repository for the PyPI package leiden-clustering, we found that it has been starred 1 times. A. J. Note that Leiden clustering directly clusters the neighborhood graph of cells, which we already computed in the previous section. Nature 433, 895900, https://doi.org/10.1038/nature03288 (2005). Note that the object for Seurat version 3 has changed. o CLIQUE (Clustering in Quest): - CLIQUE is a combination of density-based and grid-based clustering algorithm. 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. Excluding node mergers that decrease the quality function makes the refinement phase more efficient. Traag, V. A. One may expect that other nodes in the old community will then also be moved to other communities. This package implements the Leiden algorithm in C++ and exposes it to python.It relies on (python-)igraph for it to function. Each point corresponds to a certain iteration of an algorithm, with results averaged over 10 experiments. Package 'leiden' October 13, 2022 Type Package Title R Implementation of Leiden Clustering Algorithm Version 0.4.3 Date 2022-09-10 Description Implements the 'Python leidenalg' module to be called in R. Enables clustering using the leiden algorithm for partition a graph into communities. Once aggregation is complete we restart the local moving phase, and continue to iterate until everything converges down to one node. 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. It only implies that individual nodes are well connected to their community. https://leidenalg.readthedocs.io/en/latest/reference.html. If nothing happens, download Xcode and try again. 81 (4 Pt 2): 046114. http://dx.doi.org/10.1103/PhysRevE.81.046114. It means that there are no individual nodes that can be moved to a different community. Below we offer an intuitive explanation of these properties. the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in 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. Article As discussed earlier, the Louvain algorithm does not guarantee connectivity. 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. In the aggregation phase, an aggregate network is created based on the partition obtained in the local moving phase. CAS 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. Eng. J. 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). It does not guarantee that modularity cant be increased by moving nodes between communities. Not. Communities in Networks. This is very similar to what the smart local moving algorithm does. Basically, there are two types of hierarchical cluster analysis strategies - 1. Moreover, when no more nodes can be moved, the algorithm will aggregate the network. https://doi.org/10.1038/s41598-019-41695-z. In fact, if we keep iterating the Leiden algorithm, it will converge to a partition without any badly connected communities, as discussed earlier. 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. Inf. leidenalg. Faster Unfolding of Communities: Speeding up the Louvain Algorithm. Phys. Cluster your data matrix with the Leiden algorithm. The new algorithm integrates several earlier improvements, incorporating a combination of smart local move15, fast local move16,17 and random neighbour move18. In the Louvain algorithm, an aggregate network is created based on the partition \({\mathscr{P}}\) resulting from the local moving phase. Modularity is a popular objective function used with the Louvain method for community detection. When the Leiden algorithm found that a community could be split into multiple subcommunities, we counted the community as badly connected. The nodes are added to the queue in a random order. Acad. In the case of modularity, communities may have significant substructure both because of the resolution limit and because of the shortcomings of Louvain. The docs are here. The Leiden algorithm starts from a singleton partition (a). Guimer, R. & Nunes Amaral, L. A. Functional cartography of complex metabolic networks. The property of -separation is also guaranteed by the Louvain algorithm. A partition of clusters as a vector of integers Examples import leidenalg as la import igraph as ig Example output. where nc is the number of nodes in community c. The interpretation of the resolution parameter is quite straightforward. Traag, Vincent, Ludo Waltman, and Nees Jan van Eck. Scientific Reports (Sci Rep) The property of -connectivity is a slightly stronger variant of ordinary connectivity. Rotta, R. & Noack, A. Multilevel local search algorithms for modularity clustering. 2(a). Porter, M. A., Onnela, J.-P. & Mucha, P. J. Nonlin. At some point, node 0 is considered for moving. As can be seen in the figure, Louvain quickly reaches a state in which it is unable to find better partitions. 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). http://dx.doi.org/10.1073/pnas.0605965104. Nodes 16 have connections only within this community, whereas node 0 also has many external connections. These steps are repeated until no further improvements can be made. After the first iteration of the Louvain algorithm, some partition has been obtained. We provide the full definitions of the properties as well as the mathematical proofs in SectionD of the Supplementary Information. 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 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. Internet Explorer). For each network, Table2 reports the maximal modularity obtained using the Louvain and the Leiden algorithm. J. Exp. (We implemented both algorithms in Java, available from https://github.com/CWTSLeiden/networkanalysis and deposited at Zenodo23. 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. However, for higher values of , Leiden becomes orders of magnitude faster than Louvain, reaching 10100 times faster runtimes for the largest networks. 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. We use six empirical networks in our analysis. You are using a browser version with limited support for CSS. Fortunato, S. & Barthlemy, M. Resolution Limit in Community Detection. We now consider the guarantees provided by the Leiden algorithm. Data 11, 130, https://doi.org/10.1145/2992785 (2017). The solution provided by Leiden is based on the smart local moving algorithm. For each network, we repeated the experiment 10 times. We then created a certain number of edges such that a specified average degree \(\langle k\rangle \) was obtained. The algorithm moves individual nodes from one community to another to find a partition (b), which is then refined (c). In this way, Leiden implements the local moving phase more efficiently than Louvain. Rev. In this iterative scheme, Louvain provides two guarantees: (1) no communities can be merged and (2) no nodes can be moved. Graph abstraction reconciles clustering with trajectory inference through a topology preserving map of single cells. In fact, when we keep iterating the Leiden algorithm, it will converge to a partition for which it is guaranteed that: A community is uniformly -dense if there are no subsets of the community that can be separated from the community.
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