2 represent stronger connections, while the other edges represent weaker connections. Here is some small debugging code I wrote to find this. In the aggregation phase, an aggregate network is created based on the partition obtained in the local moving phase. The R implementation of Leiden can be run directly on the snn igraph object in Seurat. 2(b). Traag, V A. Google Scholar. Presumably, many of the badly connected communities in the first iteration of Louvain become disconnected in the second iteration. 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. As can be seen in Fig. 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 \({\mathscr{P}}\). http://iopscience.iop.org/article/10.1088/1742-5468/2008/10/P10008/meta. There is an entire Leiden package in R-cran here At this point, it is guaranteed that each individual node is optimally assigned. Rev. Narrow scope for resolution-limit-free community detection. The PyPI package leiden-clustering receives a total of 15 downloads a week. Publishers note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. It implies uniform -density and all the other above-mentioned properties. running Leiden clustering finished: found 16 clusters and added 'leiden_1.0', the cluster labels (adata.obs, categorical) (0:00:00) running Leiden clustering finished: found 12 clusters and added 'leiden_0.6', the cluster labels (adata.obs, categorical) (0:00:00) running Leiden clustering finished: found 9 clusters and added 'leiden_0.4', the 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. Guimer, R. & Nunes Amaral, L. A. Functional cartography of complex metabolic networks. E 81, 046106, https://doi.org/10.1103/PhysRevE.81.046106 (2010). Scaling of benchmark results for network size. 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. Article Article Waltman, Ludo, and Nees Jan van Eck. In an experiment containing a mixture of cell types, each cluster might correspond to a different cell type. The numerical details of the example can be found in SectionB of the Supplementary Information. 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. The Leiden algorithm guarantees all communities to be connected, but it may yield badly connected communities. This may have serious consequences for analyses based on the resulting partitions. We here introduce the Leiden algorithm, which guarantees that communities are well connected. However, as shown in this paper, the Louvain algorithm has a major shortcoming: the algorithm yields communities that may be arbitrarily badly connected. Note that this code is designed for Seurat version 2 releases. & Fortunato, S. Community detection algorithms: A comparative analysis. We also suggested that the Leiden algorithm is faster than the Louvain algorithm, because of the fast local move approach. First iteration runtime for empirical networks. The property of -separation is also guaranteed by the Louvain algorithm. In the first step of the next iteration, Louvain will again move individual nodes in the network. Perhaps surprisingly, iterating the algorithm aggravates the problem, even though it does increase the quality function. Luecken, M. D. Application of multi-resolution partitioning of interaction networks to the study of complex disease. In the fast local move procedure in the Leiden algorithm, only nodes whose neighbourhood has changed are visited. The Leiden algorithm also takes advantage of the idea of speeding up the local moving of nodes16,17 and the idea of moving nodes to random neighbours18. In other words, communities are guaranteed to be well separated. Clustering the neighborhood graph As with Seurat and many other frameworks, we recommend the Leiden graph-clustering method (community detection based on optimizing modularity) by Traag *et al. Default behaviour is calling cluster_leiden in igraph with Modularity (for undirected graphs) and CPM cost functions. See the documentation for these functions. Later iterations of the Louvain algorithm only aggravate the problem of disconnected communities, even though the quality function (i.e. In particular, we show that Louvain may identify communities that are internally disconnected. GitHub on Feb 15, 2020 Do you think the performance improvements will also be implemented in leidenalg? The percentage of badly connected communities is less affected by the number of iterations of the Louvain algorithm. Good, B. H., De Montjoye, Y. Rev. Rev. PubMed First calculate k-nearest neighbors and construct the SNN graph. In many complex networks, nodes cluster and form relatively dense groupsoften called communities1,2. Soft Matter Phys. Phys. Each of these can be used as an objective function for graph-based community detection methods, with our goal being to maximize this value. Scaling of benchmark results for difficulty of the partition. The Leiden algorithm is typically iterated: the output of one iteration is used as the input for the next iteration. In subsequent iterations, the percentage of disconnected communities remains fairly stable. The algorithm may yield arbitrarily badly connected communities, over and above the well-known issue of the resolution limit14. This package requires the 'leidenalg' and 'igraph' modules for python (2) to be installed on your system. We first applied the Scanpy pipeline, including its clustering method (Leiden clustering), on the PBMC dataset. CAS For larger networks and higher values of , Louvain is much slower than Leiden. The algorithm then moves individual nodes in the aggregate network (e). Sci Rep 9, 5233 (2019). After a stable iteration of the Leiden algorithm, it is guaranteed that: All nodes are locally optimally assigned. Rather than evaluating the modularity gain for moving a node to each neighboring communities, we choose a neighboring node at random and evaluate whether there is a gain in modularity if we were to move the node to that neighbors community. This continues until the queue is empty. Natl. Rev. For each network, Table2 reports the maximal modularity obtained using the Louvain and the Leiden algorithm. V.A.T. V. A. Traag. Later iterations of the Louvain algorithm are very fast, but this is only because the partition remains the same. The value of the resolution parameter was determined based on the so-called mixing parameter 13. I tracked the number of clusters post-clustering at each step. This can be a shared nearest neighbours matrix derived from a graph object. Moreover, Louvain has no mechanism for fixing these communities. For example, the red community in (b) is refined into two subcommunities in (c), which after aggregation become two separate nodes in (d), both belonging to the same community. This method tries to maximise the difference between the actual number of edges in a community and the expected number of such edges. Class wrapper based on scanpy to use the Leiden algorithm to directly cluster your data matrix with a scikit-learn flavor. The Leiden algorithm is partly based on the previously introduced smart local move algorithm15, which itself can be seen as an improvement of the Louvain algorithm. Clearly, it would be better to split up the community. However, for higher values of , Leiden becomes orders of magnitude faster than Louvain, reaching 10100 times faster runtimes for the largest networks. First, we show that the Louvain algorithm finds disconnected communities, and more generally, badly connected communities in the empirical networks. On Modularity Clustering. sign in 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. & Bornholdt, S. Statistical mechanics of community detection. Unlike the Louvain algorithm, the Leiden algorithm uses a fast local move procedure in this phase. It states that there are no communities that can be merged. Lancichinetti, A., Fortunato, S. & Radicchi, F. Benchmark graphs for testing community detection algorithms. and JavaScript. 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. wrote the manuscript. ADS The phase one loop can be greatly accelerated by finding the nodes that have the potential to change community and only revisit those nodes. These steps are repeated until the quality cannot be increased further. It was originally developed for modularity optimization, although the same method can be applied to optimize CPM. To overcome the problem of arbitrarily badly connected communities, we introduced a new algorithm, which we refer to as the Leiden algorithm. The resulting clusters are shown as colors on the 3D model (top) and t -SNE embedding . Empirical networks show a much richer and more complex structure. In the previous section, we showed that the Leiden algorithm guarantees a number of properties of the partitions uncovered at different stages of the algorithm. Bullmore, E. & Sporns, O. In fact, by implementing the refinement phase in the right way, several attractive guarantees can be given for partitions produced by the Leiden algorithm. The Louvain algorithm is illustrated in Fig. Correspondence to We use six empirical networks in our analysis. 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. Our analysis is based on modularity with resolution parameter =1. J. Comput. J. Assoc. A tag already exists with the provided branch name. Traag, V. A. As shown in Fig. 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. volume9, Articlenumber:5233 (2019) Hence, the problem of Louvain outlined above is independent from the issue of the resolution limit. 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). Initially, \({{\mathscr{P}}}_{{\rm{refined}}}\) is set to a singleton partition, in which each node is in its own community. Furthermore, by relying on a fast local move approach, the Leiden algorithm runs faster than the Louvain algorithm. Even worse, the Amazon network has 5% disconnected communities, but 25% badly connected communities. The resolution limit describes a limitation where there is a minimum community size able to be resolved by optimizing modularity (or other related functions). E 92, 032801, https://doi.org/10.1103/PhysRevE.92.032801 (2015). 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. 2(a). The horizontal axis indicates the cumulative time taken to obtain the quality indicated on the vertical axis. Complex brain networks: graph theoretical analysis of structural and functional systems. In fact, if we keep iterating the Leiden algorithm, it will converge to a partition without any badly connected communities, as discussed earlier. This is the crux of the Leiden paper, and the authors show that this exact problem happens frequently in practice. On the other hand, Leiden keeps finding better partitions, especially for higher values of , for which it is more difficult to identify good partitions. Analyses based on benchmark networks have only a limited value because these networks are not representative of empirical real-world networks. PubMed Central 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. For lower values of , the correct partition is easy to find and Leiden is only about twice as fast as Louvain. Instead, a node may be merged with any community for which the quality function increases. Wolf, F. A. et al. 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. Phys. Below, the quality of a partition is reported as \(\frac{ {\mathcal H} }{2m}\), where H is defined in Eq. In this paper, we show that the Louvain algorithm has a major problem, for both modularity and CPM. 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. 2007. For the Amazon and IMDB networks, the first iteration of the Leiden algorithm is only about 1.6 times faster than the first iteration of the Louvain 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. Soft Matter Phys. In this post Ive mainly focused on the optimisation methods for community detection, rather than the different objective functions that can be used. Source Code (2018). Traag, V. A., Van Dooren, P. & Nesterov, Y. Rev. Elect. Node mergers that cause the quality function to decrease are not considered. 7, whereas Louvain becomes much slower for more difficult partitions, Leiden is much less affected by the difficulty of the partition. Nodes 06 are in the same community. 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. The DBLP network is somewhat more challenging, requiring almost 80 iterations on average to reach a stable iteration. 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. CPM does not suffer from this issue13. 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. Some of these nodes may very well act as bridges, similarly to node 0 in the above example. Rep. 6, 30750, https://doi.org/10.1038/srep30750 (2016). Four popular community detection algorithms are explained . The corresponding results are presented in the Supplementary Fig. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate. Fortunato, Santo, and Marc Barthlemy. Louvain algorithm. Google Scholar. Moreover, when no more nodes can be moved, the algorithm will aggregate the network. Newman, M E J, and M Girvan. Google Scholar. However, if communities are badly connected, this may lead to incorrect attributions of shared functionality. 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. The minimum resolvable community size depends on the total size of the network and the degree of interconnectedness of the modules. PubMed Phys. We name our algorithm the Leiden algorithm, after the location of its authors. Article V.A.T. Indeed, the percentage of disconnected communities becomes more comparable to the percentage of badly connected communities in later iterations. The Leiden algorithm is clearly faster than the Louvain algorithm. In this way, Leiden implements the local moving phase more efficiently than Louvain. ADS Once aggregation is complete we restart the local moving phase, and continue to iterate until everything converges down to one node. However, the Louvain algorithm does not consider this possibility, since it considers only individual node movements. This makes sense, because after phase one the total size of the graph should be significantly reduced. After the first iteration of the Louvain algorithm, some partition has been obtained. Phys. Phys. http://dx.doi.org/10.1073/pnas.0605965104. 104 (1): 3641. For both algorithms, 10 iterations were performed. The random component also makes the algorithm more explorative, which might help to find better community structures. Detecting communities in a network is therefore an important problem. 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. We used modularity with a resolution parameter of =1 for the experiments. Waltman, L. & van Eck, N. J. E 78, 046110, https://doi.org/10.1103/PhysRevE.78.046110 (2008). Leiden keeps finding better partitions for empirical networks also after the first 10 iterations of the algorithm. After a stable iteration of the Leiden algorithm, the algorithm may still be able to make further improvements in later iterations. Positive values above 2 define the total number of iterations to perform, -1 has the algorithm run until it reaches its optimal clustering. 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. Use the Previous and Next buttons to navigate the slides or the slide controller buttons at the end to navigate through each slide. A structure that is more informative than the unstructured set of clusters returned by flat clustering. The Louvain algorithm is a simple and popular method for community detection (Blondel, Guillaume, and Lambiotte 2008). 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. In the meantime, to ensure continued support, we are displaying the site without styles The fast local move procedure can be summarised as follows. Blondel, V D, J L Guillaume, and R Lambiotte. where >0 is a resolution parameter4. B 86 (11): 471. https://doi.org/10.1140/epjb/e2013-40829-0. E 70, 066111, https://doi.org/10.1103/PhysRevE.70.066111 (2004). Rev. Unsupervised clustering of cells is a common step in many single-cell expression workflows. The property of -connectivity is a slightly stronger variant of ordinary connectivity. Rev. 9 shows that more than 10 iterations of the Leiden algorithm can be performed before the Louvain algorithm has finished its first iteration. In the most difficult case (=0.9), Louvain requires almost 2.5 days, while Leiden needs fewer than 10 minutes. Introduction The Louvain method is an algorithm to detect communities in large networks. https://doi.org/10.1038/s41598-019-41695-z. J. 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. Rev. We generated benchmark networks in the following way. The 'devtools' package will be used to install 'leiden' and the dependancies (igraph and reticulate). Communities in \({\mathscr{P}}\) may be split into multiple subcommunities in \({{\mathscr{P}}}_{{\rm{refined}}}\). All experiments were run on a computer with 64 Intel Xeon E5-4667v3 2GHz CPUs and 1TB internal memory.