It maximizes a modularity score for each community, where the modularity quantifies the quality of an assignment of nodes to 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. Data 11, 130, https://doi.org/10.1145/2992785 (2017). Speed of the first iteration of the Louvain and the Leiden algorithm for benchmark networks with increasingly difficult partitions (n=107). where >0 is a resolution parameter4. The Leiden algorithm is typically iterated: the output of one iteration is used as the input for the next iteration. 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 \({\mathscr{P}}\). Rev. leiden_clsutering is distributed under a BSD 3-Clause License (see LICENSE). In this case we know the answer is exactly 10. This way of defining the expected number of edges is based on the so-called configuration model. In the local move procedure in the Leiden algorithm, only nodes whose neighborhood . 9 shows that more than 10 iterations of the Leiden algorithm can be performed before the Louvain algorithm has finished its first iteration. We show that this algorithm has a major defect that largely went unnoticed until now: the Louvain algorithm may yield arbitrarily badly connected communities. 10008, 6, https://doi.org/10.1088/1742-5468/2008/10/P10008 (2008). volume9, Articlenumber:5233 (2019) Source Code (2018). & Clauset, A. Cluster cells using Louvain/Leiden community detection Description. 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. Sci Rep 9, 5233 (2019). It is a directed graph if the adjacency matrix is not symmetric. Soft Matter Phys. & Bornholdt, S. Statistical mechanics of community detection. 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. Article Int. A Simple Acceleration Method for the Louvain Algorithm. Int. 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. Fortunato, Santo, and Marc Barthlemy. Similarly, in citation networks, such as the Web of Science network, nodes in a community are usually considered to share a common topic26,27. CAS Leiden algorithm. We use six empirical networks in our analysis. The algorithm moves individual nodes from one community to another to find a partition (b), which is then refined (c). As discussed earlier, the Louvain algorithm does not guarantee connectivity. Clearly, it would be better to split up the community. Indeed, the percentage of disconnected communities becomes more comparable to the percentage of badly connected communities in later iterations. However, as shown in this paper, the Louvain algorithm has a major shortcoming: the algorithm yields communities that may be arbitrarily badly connected. Phys. All communities are subpartition -dense. 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). We also suggested that the Leiden algorithm is faster than the Louvain algorithm, because of the fast local move approach. Article Unsupervised clustering of cells is a common step in many single-cell expression workflows. E 74, 016110, https://doi.org/10.1103/PhysRevE.74.016110 (2006). Phys. Guimer, R. & Nunes Amaral, L. A. Functional cartography of complex metabolic networks. & Girvan, M. Finding and evaluating community structure in networks. where nc is the number of nodes in community c. The interpretation of the resolution parameter is quite straightforward. 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. E 76, 036106, https://doi.org/10.1103/PhysRevE.76.036106 (2007). J. From Louvain to Leiden: Guaranteeing Well-Connected Communities, October. 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. By submitting a comment you agree to abide by our Terms and Community Guidelines. While smart local moving and multilevel refinement can improve the communities found, the next two improvements on Louvain that Ill discuss focus on the speed/efficiency of the algorithm. As we will demonstrate in our experimental analysis, the problem occurs frequently in practice when using the Louvain algorithm. In this new situation, nodes 2, 3, 5 and 6 have only internal connections. CAS In the fast local move procedure in the Leiden algorithm, only nodes whose neighbourhood has changed are visited. However, for higher values of , Leiden becomes orders of magnitude faster than Louvain, reaching 10100 times faster runtimes for the largest networks. Acad. 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. https://leidenalg.readthedocs.io/en/latest/reference.html. Later iterations of the Louvain algorithm are very fast, but this is only because the partition remains the same. This will compute the Leiden clusters and add them to the Seurat Object Class. Phys. N.J.v.E. Slider with three articles shown per slide. (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. 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. Louvain quickly converges to a partition and is then unable to make further improvements. In many complex networks, nodes cluster and form relatively dense groupsoften called communities1,2. import leidenalg as la import igraph as ig Example output. 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. A structure that is more informative than the unstructured set of clusters returned by flat clustering. At each iteration all clusters are guaranteed to be connected and well-separated. For this network, Leiden requires over 750 iterations on average to reach a stable iteration. (2) and m is the number of edges. To ensure readability of the paper to the broadest possible audience, we have chosen to relegate all technical details to the Supplementary Information. 92 (3): 032801. http://dx.doi.org/10.1103/PhysRevE.92.032801. This algorithm provides a number of explicit guarantees. Any sub-networks that are found are treated as different communities in the next aggregation step. In doing so, Louvain keeps visiting nodes that cannot be moved to a different community. Electr. 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. In short, the problem of badly connected communities has important practical consequences. Rep. 486, 75174, https://doi.org/10.1016/j.physrep.2009.11.002 (2010). 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. The Louvain algorithm guarantees that modularity cannot be increased by merging communities (it finds a locally optimal solution). 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 Louvain method for community detection is a popular way to discover communities from single-cell data. 1 and summarised in pseudo-code in AlgorithmA.1 in SectionA of the Supplementary Information. the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Four popular community detection algorithms are explained . Performance of modularity maximization in practical contexts. We conclude that the Leiden algorithm is strongly preferable to the Louvain algorithm. 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. 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. Modularity optimization. While current approaches are successful in reducing the number of sequence alignments performed, the generated clusters are . Communities in Networks. In this paper, we show that the Louvain algorithm has a major problem, for both modularity and CPM. The corresponding results are presented in the Supplementary Fig. The nodes are added to the queue in a random order. Is modularity with a resolution parameter equivalent to leidenalg.RBConfigurationVertexPartition? 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. Weights for edges an also be passed to the leiden algorithm either as a separate vector or weights or a weighted adjacency matrix. Networks with high modularity have dense connections between the nodes within modules but sparse connections between nodes in different modules. Perhaps surprisingly, iterating the algorithm aggravates the problem, even though it does increase the quality function. As shown in Fig. You are using a browser version with limited support for CSS. The Web of Science network is the most difficult one. 4. Phys. The algorithm moves individual nodes from one community to another to find a partition (b). 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. We now compare how the Leiden and the Louvain algorithm perform for the six empirical networks listed in Table2. Note that the object for Seurat version 3 has changed. Technol. This contrasts with the Leiden algorithm. Each point corresponds to a certain iteration of an algorithm, with results averaged over 10 experiments. The Leiden algorithm is considerably more complex than the Louvain algorithm. If we move the node to a different community, we add to the rear of the queue all neighbours of the node that do not belong to the nodes new community and that are not yet in the queue. The Louvain algorithm10 is very simple and elegant. There was a problem preparing your codespace, please try again. Figure3 provides an illustration of the algorithm. We typically reduce the dimensionality of the data first by running PCA, then construct a neighbor graph in the reduced space. Newman, M. E. J. The idea of the refinement phase in the Leiden algorithm is to identify a partition \({{\mathscr{P}}}_{{\rm{refined}}}\) that is a refinement of \({\mathscr{P}}\). Duch, J. ACM Trans. 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 . Below, the quality of a partition is reported as \(\frac{ {\mathcal H} }{2m}\), where H is defined in Eq. 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 ). This should be the first preference when choosing an algorithm. Due to the resolution limit, modularity may cause smaller communities to be clustered into larger communities. Hence, no further improvements can be made after a stable iteration of the Louvain algorithm. A community is subpartition -dense if it can be partitioned into two parts such that: (1) the two parts are well connected to each other; (2) neither part can be separated from its community; and (3) each part is also subpartition -dense itself. The resolution limit describes a limitation where there is a minimum community size able to be resolved by optimizing modularity (or other related functions). 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. 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"). Louvain keeps visiting all nodes in a network until there are no more node movements that increase the quality function. 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. First, we created a specified number of nodes and we assigned each node to a community. The Leiden algorithm is clearly faster than the Louvain algorithm. One of the most popular algorithms to optimise modularity is the so-called Louvain algorithm10, named after the location of its authors. Empirical networks show a much richer and more complex structure. The Leiden algorithm guarantees all communities to be connected, but it may yield badly connected communities. For empirical networks, it may take quite some time before the Leiden algorithm reaches its first stable iteration. Source Code (2018). This is the crux of the Leiden paper, and the authors show that this exact problem happens frequently in practice. This continues until the queue is empty. In the Louvain algorithm, an aggregate network is created based on the partition \({\mathscr{P}}\) resulting from the local moving phase. Soc. Based on this partition, an aggregate network is created (c). Newman, M E J, and M Girvan. USA 104, 36, https://doi.org/10.1073/pnas.0605965104 (2007). Excluding node mergers that decrease the quality function makes the refinement phase more efficient. Phys. and L.W. Inf. Cluster your data matrix with the Leiden algorithm. Fortunato, S. & Barthlemy, M. Resolution Limit in Community Detection. http://dx.doi.org/10.1073/pnas.0605965104. Fast Unfolding of Communities in Large Networks. Journal of Statistical , January. Cite this article. Louvain community detection algorithm was originally proposed in 2008 as a fast community unfolding method for large networks. 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. In addition, we prove that the algorithm converges to an asymptotically stable partition in which all subsets of all communities are locally optimally assigned. The percentage of badly connected communities is less affected by the number of iterations of the Louvain algorithm. (We implemented both algorithms in Java, available from https://github.com/CWTSLeiden/networkanalysis and deposited at Zenodo23. S3. This commit does not belong to any branch on this repository, and may belong to a fork outside of the repository. CPM is defined as. Besides being pervasive, the problem is also sizeable. With one exception (=0.2 and n=107), all results in Fig. J. Exp. PubMed Subset optimality is the strongest guarantee that is provided by the Leiden algorithm. However, so far this problem has never been studied for the Louvain algorithm. The aggregate network is created based on the partition \({{\mathscr{P}}}_{{\rm{refined}}}\). ISSN 2045-2322 (online). MATH After a stable iteration of the Leiden algorithm, the algorithm may still be able to make further improvements in later iterations. For the results reported below, the average degree was set to \(\langle k\rangle =10\). Bullmore, E. & Sporns, O. 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. In the first step of the next iteration, Louvain will again move individual nodes in the network. Clauset, A., Newman, M. E. J. We applied the Louvain and the Leiden algorithm to exactly the same networks, using the same seed for the random number generator. Random moving can result in some huge speedups, since Louvain spends about 95% of its time computing the modularity gain from moving nodes. Lancichinetti, A., Fortunato, S. & Radicchi, F. Benchmark graphs for testing community detection algorithms. Hence, the problem of Louvain outlined above is independent from the issue of the resolution limit. However, the initial partition for the aggregate network is based on P, just like in the Louvain algorithm. Hence, the complex structure of empirical networks creates an even stronger need for the use of the Leiden algorithm. In our experimental analysis, we observe that up to 25% of the communities are badly connected and up to 16% are disconnected. 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. 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. The algorithm is run iteratively, using the partition identified in one iteration as starting point for the next iteration. Value. Initially, \({{\mathscr{P}}}_{{\rm{refined}}}\) is set to a singleton partition, in which each node is in its own community. Louvain algorithm. Both conda and PyPI have leiden clustering in Python which operates via iGraph. Scaling of benchmark results for network size. Nodes 06 are in the same community. This package requires the 'leidenalg' and 'igraph' modules for python (2) to be installed on your system. For each network, Table2 reports the maximal modularity obtained using the Louvain and the Leiden algorithm. It therefore does not guarantee -connectivity either. In this post Ive mainly focused on the optimisation methods for community detection, rather than the different objective functions that can be used. Rotta, R. & Noack, A. Multilevel local search algorithms for modularity clustering. Nonlin. It states that there are no communities that can be merged. However, focussing only on disconnected communities masks the more fundamental issue: Louvain finds arbitrarily badly connected communities. A community size of 50 nodes was used for the results presented below, but larger community sizes yielded qualitatively similar results. There are many different approaches and algorithms to perform clustering tasks. It only implies that individual nodes are well connected to their community. 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. In this way, the constant acts as a resolution parameter, and setting the constant higher will result in fewer communities.