leiden clustering explained

Later iterations of the Louvain algorithm are very fast, but this is only because the partition remains the same. contrastive-sc works best on datasets with fewer clusters when using the KMeans clustering and conversely for Leiden. First calculate k-nearest neighbors and construct the SNN graph. In fact, if we keep iterating the Leiden algorithm, it will converge to a partition without any badly connected communities, as discussed earlier. Conversely, if Leiden does not find subcommunities, there is no guarantee that modularity cannot be increased by splitting up the community. We used the CPM quality function. 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. Figure4 shows how well it does compared to the Louvain algorithm. Please Introduction The Louvain method is an algorithm to detect communities in large networks. MathSciNet MathSciNet Fast Unfolding of Communities in Large Networks. Journal of Statistical , January. Below, the quality of a partition is reported as \(\frac{ {\mathcal H} }{2m}\), where H is defined in Eq. The leidenalg package facilitates community detection of networks and builds on the package igraph. Proc. The algorithm continues to move nodes in the rest of the network. Importantly, the problem of disconnected communities is not just a theoretical curiosity. J. Comput. For larger networks and higher values of , Louvain is much slower than Leiden. 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. In practice, this means that small clusters can hide inside larger clusters, making their identification difficult. 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. Optimising modularity is NP-hard5, and consequentially many heuristic algorithms have been proposed, such as hierarchical agglomeration6, extremal optimisation7, simulated annealing4,8 and spectral9 algorithms. Higher resolutions lead to more communities, while lower resolutions lead to fewer communities. Google Scholar. Source Code (2018). The main ideas of our algorithm are explained in an intuitive way in the main text of the paper. E 92, 032801, https://doi.org/10.1103/PhysRevE.92.032801 (2015). First, we show that the Louvain algorithm finds disconnected communities, and more generally, badly connected communities in the empirical networks. Community detection in complex networks using extremal optimization. partition_type : Optional [ Type [ MutableVertexPartition ]] (default: None) Type of partition to use. 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. For all networks, Leiden identifies substantially better partitions than Louvain. In an experiment containing a mixture of cell types, each cluster might correspond to a different cell type. To address this problem, we introduce the Leiden algorithm. We now show that the Louvain algorithm may find arbitrarily badly connected communities. ACM Trans. We provide the full definitions of the properties as well as the mathematical proofs in SectionD of the Supplementary Information. 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. Phys. In the case of modularity, communities may have significant substructure both because of the resolution limit and because of the shortcomings of Louvain. It is a directed graph if the adjacency matrix is not symmetric. This amounts to a clustering problem, where we aim to learn an optimal set of groups (communities) from the observed data. Eng. However, as shown in this paper, the Louvain algorithm has a major shortcoming: the algorithm yields communities that may be arbitrarily badly connected. For the Amazon, DBLP and Web UK networks, Louvain yields on average respectively 23%, 16% and 14% badly connected communities. Badly connected communities. The resulting clusters are shown as colors on the 3D model (top) and t -SNE embedding . Rev. However, the initial partition for the aggregate network is based on P, just like in the Louvain algorithm. 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. 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. Communities in \({\mathscr{P}}\) may be split into multiple subcommunities in \({{\mathscr{P}}}_{{\rm{refined}}}\). Sci. Speed of the first iteration of the Louvain and the Leiden algorithm for six empirical networks. The random component also makes the algorithm more explorative, which might help to find better community structures. http://iopscience.iop.org/article/10.1088/1742-5468/2008/10/P10008/meta. Later iterations of the Louvain algorithm only aggravate the problem of disconnected communities, even though the quality function (i.e. 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). Blondel, V D, J L Guillaume, and R Lambiotte. CPM is defined as. Learn more. 2007. Nodes 06 are in the same community. We gratefully acknowledge computational facilities provided by the LIACS Data Science Lab Computing Facilities through Frank Takes. Scaling of benchmark results for network size. The Leiden algorithm has been specifically designed to address the problem of badly connected communities. Louvain can also be quite slow, as it spends a lot of time revisiting nodes that may not have changed neighborhoods. Subpartition -density does not imply that individual nodes are locally optimally assigned. This function takes a cell_data_set as input, clusters the cells using . Rev. Communities may even be disconnected. Internet Explorer). These nodes can be approximately identified based on whether neighbouring nodes have changed communities. The R implementation of Leiden can be run directly on the snn igraph object in Seurat. conda install -c conda-forge leidenalg pip install leiden-clustering Used via. 8 (3): 207. https://pdfs.semanticscholar.org/4ea9/74f0fadb57a0b1ec35cbc5b3eb28e9b966d8.pdf. The new algorithm integrates several earlier improvements, incorporating a combination of smart local move15, fast local move16,17 and random neighbour move18. Work fast with our official CLI. It does not guarantee that modularity cant be increased by moving nodes between communities. J. Rev. Rev. You will not need much Python to use it. 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. One of the most popular algorithms to optimise modularity is the so-called Louvain algorithm10, named after the location of its authors. Article Leiden is faster than Louvain especially for larger networks. We find that the Leiden algorithm is faster than the Louvain algorithm and uncovers better partitions, in addition to providing explicit guarantees. Importantly, the output of the local moving stage will depend on the order that the nodes are considered in. The refined partition \({{\mathscr{P}}}_{{\rm{refined}}}\) is obtained as follows. AMS 56, 10821097 (2009). Electr. Modularity optimization. Louvain quickly converges to a partition and is then unable to make further improvements. 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. The count of badly connected communities also included disconnected communities. * (2018). Both conda and PyPI have leiden clustering in Python which operates via iGraph. http://arxiv.org/abs/1810.08473. 2013. At some point, node 0 is considered for moving. 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"). In other words, communities are guaranteed to be well separated. The 'devtools' package will be used to install 'leiden' and the dependancies (igraph and reticulate). Weights for edges an also be passed to the leiden algorithm either as a separate vector or weights or a weighted adjacency matrix. Even worse, the Amazon network has 5% disconnected communities, but 25% badly connected communities. There is an entire Leiden package in R-cran here 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. 10, for the IMDB and Amazon networks, Leiden reaches a stable iteration relatively quickly, presumably because these networks have a fairly simple community structure. E 80, 056117, https://doi.org/10.1103/PhysRevE.80.056117 (2009). Are you sure you want to create this branch? In particular, we show that Louvain may identify communities that are internally disconnected. Traag, V.A., Waltman, L. & van Eck, N.J. From Louvain to Leiden: guaranteeing well-connected communities. The aggregate network is created based on the partition \({{\mathscr{P}}}_{{\rm{refined}}}\). Correspondence to A structure that is more informative than the unstructured set of clusters returned by flat clustering. Local Resolution-Limit-Free Potts Model for Community Detection. Phys. Nonlin. Perhaps surprisingly, iterating the algorithm aggravates the problem, even though it does increase the quality function. Hence, for lower values of , the difference in quality is negligible. At this point, it is guaranteed that each individual node is optimally assigned. Phys. A. We can guarantee a number of properties of the partitions found by the Leiden algorithm at various stages of the iterative process. Scaling of benchmark results for difficulty of the partition. Wolf, F. A. et al. The percentage of disconnected communities is more limited, usually around 1%. 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. Any sub-networks that are found are treated as different communities in the next aggregation step. B 86 (11): 471. https://doi.org/10.1140/epjb/e2013-40829-0. Sign up for the Nature Briefing newsletter what matters in science, free to your inbox daily. As the problem of modularity optimization is NP-hard, we need heuristic methods to optimize modularity (or CPM). The two phases are repeated until the quality function cannot be increased further. Random moving can result in some huge speedups, since Louvain spends about 95% of its time computing the modularity gain from moving nodes. ADS 1 and summarised in pseudo-code in AlgorithmA.1 in SectionA of the Supplementary Information. In this post, I will cover one of the common approaches which is hierarchical clustering. As far as I can tell, Leiden seems to essentially be smart local moving with the additional improvements of random moving and Louvain pruning added. For lower values of , the correct partition is easy to find and Leiden is only about twice as fast as Louvain. Rev. 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. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/. The degree of randomness in the selection of a community is determined by a parameter >0. The R implementation of Leiden can be run directly on the snn igraph object in Seurat. Phys. A number of iterations of the Leiden algorithm can be performed before the Louvain algorithm has finished its first iteration. 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. Runtime versus quality for empirical networks. Louvain keeps visiting all nodes in a network until there are no more node movements that increase the quality function. Slider with three articles shown per slide. We therefore require a more principled solution, which we will introduce in the next section. However, nodes 16 are still locally optimally assigned, and therefore these nodes will stay in the red community. However, for higher values of , Leiden becomes orders of magnitude faster than Louvain, reaching 10100 times faster runtimes for the largest networks. A smart local moving algorithm for large-scale modularity-based community detection. We then created a certain number of edges such that a specified average degree \(\langle k\rangle \) was obtained. 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 CAS Eng. Newman, M. E. J. 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. This is similar to ideas proposed recently as pruning16 and in a slightly different form as prioritisation17. Faster unfolding of communities: Speeding up the Louvain algorithm. E Stat. A new methodology for constructing a publication-level classification system of science. In doing so, Louvain keeps visiting nodes that cannot be moved to a different 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. Importantly, the first iteration of the Leiden algorithm is the most computationally intensive one, and subsequent iterations are faster. Directed Undirected Homogeneous Heterogeneous Weighted 1. Luecken, M. D. Application of multi-resolution partitioning of interaction networks to the study of complex disease. For example, after four iterations, the Web UK network has 8% disconnected communities, but twice as many badly connected communities. We thank Lovro Subelj for his comments on an earlier version of this paper. This contrasts with the Leiden algorithm. Each point corresponds to a certain iteration of an algorithm, with results averaged over 10 experiments. 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 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. Due to the resolution limit, modularity may cause smaller communities to be clustered into larger communities. 2(b). 2018. Fortunato, Santo, and Marc Barthlemy. We will use sklearns K-Means implementation looking for 10 clusters in the original 784 dimensional data. Furthermore, by relying on a fast local move approach, the Leiden algorithm runs faster than the Louvain algorithm. leiden_clustering Description Class wrapper based on scanpy to use the Leiden algorithm to directly cluster your data matrix with a scikit-learn flavor. Publishers note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Ronhovde, Peter, and Zohar Nussinov. Besides the relative flexibility of the implementation, it also scales well, and can be run on graphs of millions of nodes (as long as they can fit in memory). CAS 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. To obtain Moreover, when no more nodes can be moved, the algorithm will aggregate the network. In this paper, we show that the Louvain algorithm has a major problem, for both modularity and CPM. 2 represent stronger connections, while the other edges represent weaker connections. The Louvain algorithm guarantees that modularity cannot be increased by merging communities (it finds a locally optimal solution). An aggregate. Google Scholar. Discov. Indeed, the percentage of disconnected communities becomes more comparable to the percentage of badly connected communities in later iterations. The Louvain algorithm is illustrated in Fig. Crucially, however, the percentage of badly connected communities decreases with each iteration of the Leiden algorithm. 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. A community size of 50 nodes was used for the results presented below, but larger community sizes yielded qualitatively similar results. Acad. The DBLP network is somewhat more challenging, requiring almost 80 iterations on average to reach a stable iteration. Blondel, V. D., Guillaume, J.-L., Lambiotte, R. & Lefebvre, E. Fast unfolding of communities in large networks. Google Scholar. Article 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. Node optimality is also guaranteed after a stable iteration of the Louvain algorithm. However, focussing only on disconnected communities masks the more fundamental issue: Louvain finds arbitrarily badly connected communities. Google Scholar. V.A.T. 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. It is good at identifying small clusters. Phys. Traag, V. A., Waltman, L. & van Eck, N. J. networkanalysis. Rev. volume9, Articlenumber:5233 (2019) Newman, M. E. J. This method tries to maximise the difference between the actual number of edges in a community and the expected number of such edges. The Web of Science network is the most difficult one. Narrow scope for resolution-limit-free community detection. Louvain community detection algorithm was originally proposed in 2008 as a fast community unfolding method for large networks. Traag, V A. In this new situation, nodes 2, 3, 5 and 6 have only internal connections. Rep. 486, 75174, https://doi.org/10.1016/j.physrep.2009.11.002 (2010). This is similar to what we have seen for benchmark networks. E Stat. The Leiden algorithm is typically iterated: the output of one iteration is used as the input for the next iteration. E 78, 046110, https://doi.org/10.1103/PhysRevE.78.046110 (2008). The minimum resolvable community size depends on the total size of the network and the degree of interconnectedness of the modules. We generated benchmark networks in the following way. Here is some small debugging code I wrote to find this. Modularity is used most commonly, but is subject to the resolution limit. Powered by DataCamp DataCamp 10, 186198, https://doi.org/10.1038/nrn2575 (2009). & Fortunato, S. Community detection algorithms: A comparative analysis. On Modularity Clustering. Algorithmics 16, 2.1, https://doi.org/10.1145/1963190.1970376 (2011). Leiden algorithm. The above results shows that the problem of disconnected and badly connected communities is quite pervasive in practice. The property of -separation is also guaranteed by the Louvain algorithm. We here introduce the Leiden algorithm, which guarantees that communities are well connected. The algorithm is described in pseudo-code in AlgorithmA.2 in SectionA of the Supplementary Information. To study the scaling of the Louvain and the Leiden algorithm, we rely on a variant of a well-known approach for constructing benchmark networks28. Furthermore, if all communities in a partition are uniformly -dense, the quality of the partition is not too far from optimal, as shown in SectionE of the Supplementary Information. Google Scholar. Natl. MathSciNet Phys. Use the Previous and Next buttons to navigate the slides or the slide controller buttons at the end to navigate through each slide. Usually, the Louvain algorithm starts from a singleton partition, in which each node is in its own community. Two ways of doing this are graph modularity (Newman and Girvan 2004) and the constant Potts model (Ronhovde and Nussinov 2010). Similarly, in citation networks, such as the Web of Science network, nodes in a community are usually considered to share a common topic26,27. Obviously, this is a worst case example, showing that disconnected communities may be identified by the Louvain algorithm. Knowl. For both algorithms, 10 iterations were performed. We start by initialising a queue with all nodes in the network. Waltman, L. & van Eck, N. J. Nevertheless, when CPM is used as the quality function, the Louvain algorithm may still find arbitrarily badly connected communities. 104 (1): 3641. Rev. First iteration runtime for empirical networks. Graph abstraction reconciles clustering with trajectory inference through a topology preserving map of single cells. Additionally, we implemented a Python package, available from https://github.com/vtraag/leidenalg and deposited at Zenodo24). J. This aspect of the Louvain algorithm can be used to give information about the hierarchical relationships between communities by tracking at which stage the nodes in the communities were aggregated. Therefore, clustering algorithms look for similarities or dissimilarities among data points. 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. Some of these nodes may very well act as bridges, similarly to node 0 in the above example. Each point corresponds to a certain iteration of an algorithm, with results averaged over 10 experiments. These steps are repeated until no further improvements can be made. However, the Louvain algorithm does not consider this possibility, since it considers only individual node movements. This is very similar to what the smart local moving algorithm does. These nodes are therefore optimally assigned to their current community. The Leiden algorithm provides several guarantees. Rev. 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. Int. V. A. Traag. Phys. Moreover, Louvain has no mechanism for fixing these communities. Use Git or checkout with SVN using the web URL. 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 difference in computational time is especially pronounced for larger networks, with Leiden being up to 20 times faster than Louvain in empirical networks. This makes sense, because after phase one the total size of the graph should be significantly reduced. In the local move procedure in the Leiden algorithm, only nodes whose neighborhood . E Stat. It means that there are no individual nodes that can be moved to a different community. 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. IEEE Trans. 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. It states that there are no communities that can be merged. The algorithm moves individual nodes from one community to another to find a partition (b), which is then refined (c). The percentage of badly connected communities is less affected by the number of iterations of the Louvain algorithm. For higher values of , Leiden finds better partitions than Louvain. To find an optimal grouping of cells into communities, we need some way of evaluating different partitions in the graph. Each community in this partition becomes a node in the aggregate network. It only implies that individual nodes are well connected to their community. 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.