leiden clustering explained

This is similar to what we have seen for benchmark networks. First iteration runtime for empirical networks. In particular, benchmark networks have a rather simple structure. For each community in a partition that was uncovered by the Louvain algorithm, we determined whether it is internally connected or not. Phys. E 74, 016110, https://doi.org/10.1103/PhysRevE.74.016110 (2006). 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. Number of iterations until stability. They identified an inefficiency in the Louvain algorithm: computes modularity gain for all neighbouring nodes per loop in local moving phase, even though many of these nodes will not have moved. From Louvain to Leiden: Guaranteeing Well-Connected Communities, October. Networks with high modularity have dense connections between the nodes within modules but sparse connections between nodes in different modules. Google Scholar. 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 . S3. Modularity optimization. This will compute the Leiden clusters and add them to the Seurat Object Class. 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. Discov. For example, after four iterations, the Web UK network has 8% disconnected communities, but twice as many badly connected communities. This amounts to a clustering problem, where we aim to learn an optimal set of groups (communities) from the observed data. 10, for the IMDB and Amazon networks, Leiden reaches a stable iteration relatively quickly, presumably because these networks have a fairly simple community structure. The minimum resolvable community size depends on the total size of the network and the degree of interconnectedness of the modules. The 'devtools' package will be used to install 'leiden' and the dependancies (igraph and reticulate). While current approaches are successful in reducing the number of sequence alignments performed, the generated clusters are . The authors act as bibliometric consultants to CWTS B.V., which makes use of community detection algorithms in commercial products and services. In this case, refinement does not change the partition (f). It implies uniform -density and all the other above-mentioned properties. 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. Technol. Finding community structure in networks using the eigenvectors of matrices. Zenodo, https://doi.org/10.5281/zenodo.1469357 https://github.com/vtraag/leidenalg. Thank you for visiting nature.com. 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. We now show that the Louvain algorithm may find arbitrarily badly connected communities. One of the best-known methods for community detection is called modularity3. Conversely, if Leiden does not find subcommunities, there is no guarantee that modularity cannot be increased by splitting up the community. Soft Matter Phys. Computer Syst. In this iterative scheme, Louvain provides two guarantees: (1) no communities can be merged and (2) no nodes can be moved. Work fast with our official CLI. At each iteration all clusters are guaranteed to be connected and well-separated. 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. As such, we scored leiden-clustering popularity level to be Limited. Runtime versus quality for empirical networks. At this point, it is guaranteed that each individual node is optimally assigned. Finding communities in large networks is far from trivial: algorithms need to be fast, but they also need to provide high-quality results. Good, B. H., De Montjoye, Y. E 74, 036104, https://doi.org/10.1103/PhysRevE.74.036104 (2006). We abbreviate the leidenalg package as la and the igraph package as ig in all Python code throughout this documentation. Community detection in complex networks using extremal optimization. A community size of 50 nodes was used for the results presented below, but larger community sizes yielded qualitatively similar results. 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. Use the Previous and Next buttons to navigate the slides or the slide controller buttons at the end to navigate through each slide. To overcome the problem of arbitrarily badly connected communities, we introduced a new algorithm, which we refer to as the Leiden algorithm. Phys. Google Scholar. ADS 7, whereas Louvain becomes much slower for more difficult partitions, Leiden is much less affected by the difficulty of the partition. Are you sure you want to create this branch? Nature 433, 895900, https://doi.org/10.1038/nature03288 (2005). Excluding node mergers that decrease the quality function makes the refinement phase more efficient. These steps are repeated until no further improvements can be made. The algorithm is run iteratively, using the partition identified in one iteration as starting point for the next iteration. 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. Presumably, many of the badly connected communities in the first iteration of Louvain become disconnected in the second iteration. 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. For all networks, Leiden identifies substantially better partitions than Louvain. 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. Random moving is a very simple adjustment to Louvain local moving proposed in 2015 (Traag 2015). For each set of parameters, we repeated the experiment 10 times. We generated networks with n=103 to n=107 nodes. 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. That is, one part of such an internally disconnected community can reach another part only through a path going outside the community. Ph.D. thesis, (University of Oxford, 2016). 9, the Leiden algorithm also performs better than the Louvain algorithm in terms of the quality of the partitions that are obtained. For each network, Table2 reports the maximal modularity obtained using the Louvain and the Leiden algorithm. CAS The Leiden algorithm has been specifically designed to address the problem of badly connected communities. A new methodology for constructing a publication-level classification system of science. Based on project statistics from the GitHub repository for the PyPI package leiden-clustering, we found that it has been starred 1 times. In our experimental analysis, we observe that up to 25% of the communities are badly connected and up to 16% are disconnected. Article It was originally developed for modularity optimization, although the same method can be applied to optimize CPM. The larger the increase in the quality function, the more likely a community is to be selected. 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. 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. Because the percentage of disconnected communities in the first iteration of the Louvain algorithm usually seems to be relatively low, the problem may have escaped attention from users of the algorithm. Nonlin. Rev. 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). Value. However, so far this problem has never been studied for the Louvain algorithm. E 84, 016114, https://doi.org/10.1103/PhysRevE.84.016114 (2011). Weights for edges an also be passed to the leiden algorithm either as a separate vector or weights or a weighted adjacency matrix. Inf. We show that this algorithm has a major defect that largely went unnoticed until now: the Louvain algorithm may yield arbitrarily badly connected communities. In the most difficult case (=0.9), Louvain requires almost 2.5 days, while Leiden needs fewer than 10 minutes. In particular, it yields communities that are guaranteed to be connected. 8, 207218, https://doi.org/10.17706/IJCEE.2016.8.3.207-218 (2016). 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. Nevertheless, depending on the relative strengths of the different connections, these nodes may still be optimally assigned to their current community. Blondel, V. D., Guillaume, J.-L., Lambiotte, R. & Lefebvre, E. Fast unfolding of communities in large networks. B 86, 471, https://doi.org/10.1140/epjb/e2013-40829-0 (2013). Finally, we demonstrate the excellent performance of the algorithm for several benchmark and real-world networks. * (2018). The docs are here. Wolf, F. A. et al. 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. 8 (3): 207. https://pdfs.semanticscholar.org/4ea9/74f0fadb57a0b1ec35cbc5b3eb28e9b966d8.pdf. Rev. Article conda install -c conda-forge leidenalg pip install leiden-clustering Used via. This enables us to find cases where its beneficial to split a community. To obtain Figure4 shows how well it does compared to the Louvain algorithm. Rev. 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. Introduction The Louvain method is an algorithm to detect communities in large networks. 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. Louvain can also be quite slow, as it spends a lot of time revisiting nodes that may not have changed neighborhoods. In fact, for the Web of Science and Web UK networks, Fig. We start by initialising a queue with all nodes in the network. Use Git or checkout with SVN using the web URL. Instead, a node may be merged with any community for which the quality function increases. In the Louvain algorithm, a node may be moved to a different community while it may have acted as a bridge between different components of its old community. However, in the case of the Web of Science network, more than 5% of the communities are disconnected in the first iteration. The count of badly connected communities also included disconnected communities. Sci. 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. IEEE Trans. Importantly, mergers are performed only within each community of the partition \({\mathscr{P}}\). Even worse, the Amazon network has 5% disconnected communities, but 25% badly connected communities. However, nodes 16 are still locally optimally assigned, and therefore these nodes will stay in the red community. A. Raghavan, U., Albert, R. & Kumara, S. Near linear time algorithm to detect community structures in large-scale networks. We conclude that the Leiden algorithm is strongly preferable to the Louvain algorithm. Scaling of benchmark results for difficulty of the partition. However, modularity suffers from a difficult problem known as the resolution limit (Fortunato and Barthlemy 2007). The R implementation of Leiden can be run directly on the snn igraph object in Seurat. However, the initial partition for the aggregate network is based on P, just like in the Louvain algorithm. The thick edges in Fig. Modularity is a measure of the structure of networks or graphs which measures the strength of division of a network into modules (also called groups, clusters or communities). Article Leiden is both faster than Louvain and finds better partitions. E 92, 032801, https://doi.org/10.1103/PhysRevE.92.032801 (2015). 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). This contrasts to benchmark networks, for which Leiden often converges after a few iterations. 9 shows that more than 10 iterations of the Leiden algorithm can be performed before the Louvain algorithm has finished its first iteration. However, the Louvain algorithm does not consider this possibility, since it considers only individual node movements. We find that the Leiden algorithm commonly finds partitions of higher quality in less time. In doing so, Louvain keeps visiting nodes that cannot be moved to a different community. A major goal of single-cell analysis is to study the cell-state heterogeneity within a sample by discovering groups within the population of cells. With one exception (=0.2 and n=107), all results in Fig. Provided by the Springer Nature SharedIt content-sharing initiative. 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 numerical details of the example can be found in SectionB of the Supplementary Information. We now compare how the Leiden and the Louvain algorithm perform for the six empirical networks listed in Table2. 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. Unlike the Louvain algorithm, the Leiden algorithm uses a fast local move procedure in this phase. and JavaScript. Nonetheless, some networks still show large differences. The value of the resolution parameter was determined based on the so-called mixing parameter 13. Cite this article. Crucially, however, the percentage of badly connected communities decreases with each iteration of the Leiden algorithm. 4. 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. As discussed earlier, the Louvain algorithm does not guarantee connectivity. To find an optimal grouping of cells into communities, we need some way of evaluating different partitions in the graph. performed the experimental analysis. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate. Basically, there are two types of hierarchical cluster analysis strategies - 1. In a stable iteration, the partition is guaranteed to be node optimal and subpartition -dense. Traag, Vincent, Ludo Waltman, and Nees Jan van Eck. 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. 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). Community detection is often used to understand the structure of large and complex networks. The algorithm moves individual nodes from one community to another to find a partition (b), which is then refined (c). Louvain algorithm. To elucidate the problem, we consider the example illustrated in Fig. Later iterations of the Louvain algorithm only aggravate the problem of disconnected communities, even though the quality function (i.e. Finally, we compare the performance of the algorithms on the empirical networks. ADS The Leiden algorithm is clearly faster than the Louvain algorithm. 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. The aggregate network is created based on the partition \({{\mathscr{P}}}_{{\rm{refined}}}\). Graph abstraction reconciles clustering with trajectory inference through a topology preserving map of single cells. The Louvain algorithm is illustrated in Fig. MathSciNet 20, 172188, https://doi.org/10.1109/TKDE.2007.190689 (2008). This is the crux of the Leiden paper, and the authors show that this exact problem happens frequently in practice. Furthermore, by relying on a fast local move approach, the Leiden algorithm runs faster than the Louvain algorithm. Eur. E 70, 066111, https://doi.org/10.1103/PhysRevE.70.066111 (2004). In this new situation, nodes 2, 3, 5 and 6 have only internal connections. Mech. It means that there are no individual nodes that can be moved to a different community. where >0 is a resolution parameter4. How many iterations of the Leiden clustering algorithm to perform. 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. We applied the Louvain and the Leiden algorithm to exactly the same networks, using the same seed for the random number generator. In short, the problem of badly connected communities has important practical consequences. Our analysis is based on modularity with resolution parameter =1. We also suggested that the Leiden algorithm is faster than the Louvain algorithm, because of the fast local move approach. 2(a). N.J.v.E. and L.W. In the refinement phase, nodes are not necessarily greedily merged with the community that yields the largest increase in the quality function. 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. Later iterations of the Louvain algorithm are very fast, but this is only because the partition remains the same. Additionally, we implemented a Python package, available from https://github.com/vtraag/leidenalg and deposited at Zenodo24). http://arxiv.org/abs/1810.08473. Proc. Traag, V. A., Van Dooren, P. & Nesterov, Y. Ozaki, N., Tezuka, H. & Inaba, M. A Simple Acceleration Method for the Louvain Algorithm. An iteration of the Leiden algorithm in which the partition does not change is called a stable iteration. The algorithm then moves individual nodes in the aggregate network (e). Clustering algorithms look for similarities or dissimilarities among data points so that similar ones can be grouped together. Leiden keeps finding better partitions for empirical networks also after the first 10 iterations of the algorithm. Phys. & Girvan, M. Finding and evaluating community structure in networks. 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). This package requires the 'leidenalg' and 'igraph' modules for python (2) to be installed on your system. J. In the Louvain algorithm, an aggregate network is created based on the partition \({\mathscr{P}}\) resulting from the local moving phase. Google Scholar. 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. ADS CPM is defined as. Porter, M. A., Onnela, J.-P. & Mucha, P. J. The algorithm continues to move nodes in the rest of the network. 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 . 2008. Phys. reviewed the manuscript. In the first iteration, Leiden is roughly 220 times faster than Louvain. Phys. At some point, node 0 is considered for moving. AMS 56, 10821097 (2009). Nonlin. Modularity is given by. . All communities are subpartition -dense. MathSciNet This will compute the Leiden clusters and add them to the Seurat Object Class. A Simple Acceleration Method for the Louvain Algorithm. Int. For this network, Leiden requires over 750 iterations on average to reach a stable iteration. A tag already exists with the provided branch name. The Louvain algorithm guarantees that modularity cannot be increased by merging communities (it finds a locally optimal solution). CAS First, we created a specified number of nodes and we assigned each node to a community. Each point corresponds to a certain iteration of an algorithm, with results averaged over 10 experiments. the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in A number of iterations of the Leiden algorithm can be performed before the Louvain algorithm has finished its first iteration. In the aggregation phase, an aggregate network is created based on the partition obtained in the local moving phase. E Stat. The solution provided by Leiden is based on the smart local moving algorithm. V.A.T. This commit does not belong to any branch on this repository, and may belong to a fork outside of the repository. Many Git commands accept both tag and branch names, so creating this branch may cause unexpected behavior. Randomness in the selection of a community allows the partition space to be explored more broadly. Each of these can be used as an objective function for graph-based community detection methods, with our goal being to maximize this value. In the fast local move procedure in the Leiden algorithm, only nodes whose neighbourhood has changed are visited. PubMed PubMed Communities may even be internally disconnected. According to CPM, it is better to split into two communities when the link density between the communities is lower than the constant. These steps are repeated until the quality cannot be increased further. It states that there are no communities that can be merged. 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. Biological sequence clustering is a complicated data clustering problem owing to the high computation costs incurred for pairwise sequence distance calculations through sequence alignments, as well as difficulties in determining parameters for deriving robust clusters. We use six empirical networks in our analysis. To ensure readability of the paper to the broadest possible audience, we have chosen to relegate all technical details to the Supplementary Information. 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. U. S. A. In contrast, Leiden keeps finding better partitions in each iteration. Louvain community detection algorithm was originally proposed in 2008 as a fast community unfolding method for large networks. Get the most important science stories of the day, free in your inbox. 10008, 6, https://doi.org/10.1088/1742-5468/2008/10/P10008 (2008). 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. When a sufficient number of neighbours of node 0 have formed a community in the rest of the network, it may be optimal to move node 0 to this community, thus creating the situation depicted in Fig. The phase one loop can be greatly accelerated by finding the nodes that have the potential to change community and only revisit those nodes. 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. Two ways of doing this are graph modularity (Newman and Girvan 2004) and the constant Potts model (Ronhovde and Nussinov 2010). 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.