Hence, the community remains disconnected, unless it is merged with another community that happens to act as a bridge. The two phases are repeated until the quality function cannot be increased further. 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. 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. Runtime versus quality for benchmark networks. Louvain community detection algorithm was originally proposed in 2008 as a fast community unfolding method for large networks. Crucially, however, the percentage of badly connected communities decreases with each iteration of the Leiden algorithm. 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. Local Resolution-Limit-Free Potts Model for Community Detection. Phys. This problem is different from the well-known issue of the resolution limit of modularity14. This step will involve reducing the dimensionality of our data into two dimensions using uniform manifold approximation (UMAP), allowing us to visualize our cell populations as they are binned into discrete populations using Leiden clustering. Knowl. 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. Acad. 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. This package implements the Leiden algorithm in C++ and exposes it to python.It relies on (python-)igraph for it to function. The aggregate network is created based on the partition \({{\mathscr{P}}}_{{\rm{refined}}}\). The value of the resolution parameter was determined based on the so-called mixing parameter 13. The Leiden algorithm provides several guarantees. E 74, 036104, https://doi.org/10.1103/PhysRevE.74.036104 (2006). 92 (3): 032801. http://dx.doi.org/10.1103/PhysRevE.92.032801. A smart local moving algorithm for large-scale modularity-based community detection. Figure3 provides an illustration of the algorithm. Klavans, R. & Boyack, K. W. Which Type of Citation Analysis Generates the Most Accurate Taxonomy of Scientific and Technical Knowledge? Leiden keeps finding better partitions for empirical networks also after the first 10 iterations of the algorithm. In the Louvain algorithm, an aggregate network is created based on the partition \({\mathscr{P}}\) resulting from the local moving phase. These are the same networks that were also studied in an earlier paper introducing the smart local move algorithm15. 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. Hence, no further improvements can be made after a stable iteration of the Louvain algorithm. I tracked the number of clusters post-clustering at each step. The problem of disconnected communities has been observed before19,20, also in the context of the label propagation algorithm21. 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. Due to the resolution limit, modularity may cause smaller communities to be clustered into larger communities. As can be seen in the figure, Louvain quickly reaches a state in which it is unable to find better partitions. Theory Exp. The R implementation of Leiden can be run directly on the snn igraph object in Seurat. Sci. J. Stat. First, we show that the Louvain algorithm finds disconnected communities, and more generally, badly connected communities in the empirical networks. 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. In fact, for the Web of Science and Web UK networks, Fig. Phys. Agglomerative clustering is a bottom-up approach. Modularity is given by. Rev. 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. Nodes 06 are in the same community. Leiden algorithm. Work fast with our official CLI. The algorithm moves individual nodes from one community to another to find a partition (b), which is then refined (c). We used the CPM quality function. https://doi.org/10.1038/s41598-019-41695-z, DOI: https://doi.org/10.1038/s41598-019-41695-z. Hence, by counting the number of communities that have been split up, we obtained a lower bound on the number of communities that are badly connected. However, the Louvain algorithm does not consider this possibility, since it considers only individual node movements. We will use sklearns K-Means implementation looking for 10 clusters in the original 784 dimensional data. It implies uniform -density and all the other above-mentioned properties. An overview of the various guarantees is presented in Table1. The algorithm then moves individual nodes in the aggregate network (e). Initially, \({{\mathscr{P}}}_{{\rm{refined}}}\) is set to a singleton partition, in which each node is in its own community. The algorithm is run iteratively, using the partition identified in one iteration as starting point for the next iteration. A. Basically, there are two types of hierarchical cluster analysis strategies - 1. 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). An aggregate. The minimum resolvable community size depends on the total size of the network and the degree of interconnectedness of the modules. E 84, 016114, https://doi.org/10.1103/PhysRevE.84.016114 (2011). MathSciNet CAS Louvain keeps visiting all nodes in a network until there are no more node movements that increase the quality function. Community detection in complex networks using extremal optimization. Runtime versus quality for empirical networks. Modularity is used most commonly, but is subject to the resolution limit. How many iterations of the Leiden clustering algorithm to perform. It states that there are no communities that can be merged. In addition, we prove that, when the Leiden algorithm is applied iteratively, it converges to a partition in which all subsets of all communities are locally optimally assigned. These steps are repeated until no further improvements can be made. 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. 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. Newman, M. E. J. Are you sure you want to create this branch? Nonlin. In the case of modularity, communities may have significant substructure both because of the resolution limit and because of the shortcomings of Louvain. Although originally defined for modularity, the Louvain algorithm can also be used to optimise other quality functions. Rev. This contrasts with the Leiden algorithm. Besides being pervasive, the problem is also sizeable. Phys. 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 ). In this way, the constant acts as a resolution parameter, and setting the constant higher will result in fewer communities. We also suggested that the Leiden algorithm is faster than the Louvain algorithm, because of the fast local move approach. E 72, 027104, https://doi.org/10.1103/PhysRevE.72.027104 (2005). In fact, by implementing the refinement phase in the right way, several attractive guarantees can be given for partitions produced by the Leiden algorithm. Newman, M. E. J. Traag, V. A., Van Dooren, P. & Nesterov, Y. IEEE Trans. Sci. Complex brain networks: graph theoretical analysis of structural and functional systems. Speed and quality of the Louvain and the Leiden algorithm for benchmark networks of increasing size (two iterations). Internet Explorer). 63, 23782392, https://doi.org/10.1002/asi.22748 (2012). In the fast local move procedure in the Leiden algorithm, only nodes whose neighbourhood has changed are visited. In particular, it yields communities that are guaranteed to be connected. Somewhat stronger guarantees can be obtained by iterating the algorithm, using the partition obtained in one iteration of the algorithm as starting point for the next iteration. In the local moving phase, individual nodes are moved to the community that yields the largest increase in the quality function. USA 104, 36, https://doi.org/10.1073/pnas.0605965104 (2007). As such, we scored leiden-clustering popularity level to be Limited. Importantly, the output of the local moving stage will depend on the order that the nodes are considered in. 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. We therefore require a more principled solution, which we will introduce in the next section. http://dx.doi.org/10.1073/pnas.0605965104. For example, nodes in a community in biological or neurological networks are often assumed to share similar functions or behaviour25. Google Scholar. 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. Rev. There was a problem preparing your codespace, please try again. The quality improvement realised by the Leiden algorithm relative to the Louvain algorithm is larger for empirical networks than for benchmark networks. Such a modular structure is usually not known beforehand. 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. Both conda and PyPI have leiden clustering in Python which operates via iGraph. * (2018). 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. Yang, Z., Algesheimer, R. & Tessone, C. J. CPM does not suffer from this issue13. The algorithm moves individual nodes from one community to another to find a partition (b). Conversely, if Leiden does not find subcommunities, there is no guarantee that modularity cannot be increased by splitting up the community. In a stable iteration, the partition is guaranteed to be node optimal and subpartition -dense. However, in the case of the Web of Science network, more than 5% of the communities are disconnected in the first iteration. 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. 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. In the most difficult case (=0.9), Louvain requires almost 2.5 days, while Leiden needs fewer than 10 minutes. This is because Louvain only moves individual nodes at a time. The docs are here. Nonlin. In subsequent iterations, the percentage of disconnected communities remains fairly stable. 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. PubMed Central 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. 2(a). We generated networks with n=103 to n=107 nodes. Rev. b, The elephant graph (in a) is clustered using the Leiden clustering algorithm 51 (resolution r = 0.5). & Arenas, A. 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. https://leidenalg.readthedocs.io/en/latest/reference.html. Introduction The Louvain method is an algorithm to detect communities in large networks. Sci. The authors act as bibliometric consultants to CWTS B.V., which makes use of community detection algorithms in commercial products and services. Finally, we demonstrate the excellent performance of the algorithm for several benchmark and real-world networks. Lancichinetti, A., Fortunato, S. & Radicchi, F. Benchmark graphs for testing community detection algorithms. E 78, 046110, https://doi.org/10.1103/PhysRevE.78.046110 (2008). https://doi.org/10.1038/s41598-019-41695-z. After a stable iteration of the Leiden algorithm, it is guaranteed that: All nodes are locally optimally assigned. Sign up for the Nature Briefing newsletter what matters in science, free to your inbox daily. Please Higher resolutions lead to more communities and lower resolutions lead to fewer communities, similarly to the resolution parameter for modularity. Good, B. H., De Montjoye, Y. 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. The corresponding results are presented in the Supplementary Fig. 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. There are many different approaches and algorithms to perform clustering tasks. Article Natl. This is very similar to what the smart local moving algorithm does. The Louvain algorithm is illustrated in Fig. For higher values of , Leiden finds better partitions than Louvain. Agglomerative Clustering: Also known as bottom-up approach or hierarchical agglomerative clustering (HAC). Nonetheless, some networks still show large differences. This amounts to a clustering problem, where we aim to learn an optimal set of groups (communities) from the observed data. CAS Guimer, R. & Nunes Amaral, L. A. Functional cartography of complex metabolic networks. Sci. To overcome the problem of arbitrarily badly connected communities, we introduced a new algorithm, which we refer to as the Leiden algorithm. conda install -c conda-forge leidenalg pip install leiden-clustering Used via. We prove that the new algorithm is guaranteed to produce partitions in which all communities are internally connected. 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"). To obtain MATH 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. Article bioRxiv, https://doi.org/10.1101/208819 (2018). Provided by the Springer Nature SharedIt content-sharing initiative. Usually, the Louvain algorithm starts from a singleton partition, in which each node is in its own community. Rotta, R. & Noack, A. Multilevel local search algorithms for modularity clustering. Based on this partition, an aggregate network is created (c). Rev. Once aggregation is complete we restart the local moving phase, and continue to iterate until everything converges down to one node. Phys. PubMedGoogle Scholar. Communities were all of equal size. For lower values of , the correct partition is easy to find and Leiden is only about twice as fast as Louvain. Leiden is both faster than Louvain and finds better partitions. Acad. In that case, some optimal partitions cannot be found, as we show in SectionC2 of the Supplementary Information. Leiden is faster than Louvain especially for larger networks. In terms of the percentage of badly connected communities in the first iteration, Leiden performs even worse than Louvain, as can be seen in Fig. More subtle problems may occur as well, causing Louvain to find communities that are connected, but only in a very weak sense. The above results shows that the problem of disconnected and badly connected communities is quite pervasive in practice. The new algorithm integrates several earlier improvements, incorporating a combination of smart local move15, fast local move16,17 and random neighbour move18. 2 represent stronger connections, while the other edges represent weaker connections. Similarly, in citation networks, such as the Web of Science network, nodes in a community are usually considered to share a common topic26,27. In many complex networks, nodes cluster and form relatively dense groupsoften called communities1,2. Blondel, V. D., Guillaume, J.-L., Lambiotte, R. & Lefebvre, E. Fast unfolding of communities in large networks. Traag, V. A. Hence, the problem of Louvain outlined above is independent from the issue of the resolution limit. Cluster your data matrix with the Leiden algorithm. sign in The 'devtools' package will be used to install 'leiden' and the dependancies (igraph and reticulate). We demonstrate the performance of the Leiden algorithm for several benchmark and real-world networks. In this case, refinement does not change the partition (f). While current approaches are successful in reducing the number of sequence alignments performed, the generated clusters are . 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. Article MathSciNet http://iopscience.iop.org/article/10.1088/1742-5468/2008/10/P10008/meta. Importantly, mergers are performed only within each community of the partition \({\mathscr{P}}\). Inf. CPM has the advantage that it is not subject to the resolution limit. The high percentage of badly connected communities attests to this. As can be seen 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). 4, in the first iteration of the Louvain algorithm, the percentage of badly connected communities can be quite high. By submitting a comment you agree to abide by our Terms and Community Guidelines. 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. According to CPM, it is better to split into two communities when the link density between the communities is lower than the constant. We now consider the guarantees provided by the Leiden algorithm. Node mergers that cause the quality function to decrease are not considered. 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. Clustering algorithms look for similarities or dissimilarities among data points so that similar ones can be grouped together. http://arxiv.org/abs/1810.08473. The Leiden algorithm has been specifically designed to address the problem of badly connected communities. In this new situation, nodes 2, 3, 5 and 6 have only internal connections. Electr. The main ideas of our algorithm are explained in an intuitive way in the main text of the paper. Eng. Run the code above in your browser using DataCamp Workspace. (2) and m is the number of edges. Phys. 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. 10, for the IMDB and Amazon networks, Leiden reaches a stable iteration relatively quickly, presumably because these networks have a fairly simple community structure. Note that nodes can be revisited several times within a single iteration of the local moving stage, as the possible increase in modularity will change as other nodes are moved to different 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. That is, no subset can be moved to a different community. One may expect that other nodes in the old community will then also be moved to other communities. Rev. The algorithm continues to move nodes in the rest of the network. To elucidate the problem, we consider the example illustrated in Fig. J. Assoc. At some point, the Louvain algorithm may end up in the community structure shown in Fig. CAS 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. The algorithm moves individual nodes from one community to another to find a partition (b), which is then refined (c). the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in o CLIQUE (Clustering in Quest): - CLIQUE is a combination of density-based and grid-based clustering algorithm. Note that Leiden clustering directly clusters the neighborhood graph of cells, which we already computed in the previous section. Article CAS wrote the manuscript. & Clauset, A. 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. With one exception (=0.2 and n=107), all results in Fig. Google Scholar. Nonlin. 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. Another important difference between the Leiden algorithm and the Louvain algorithm is the implementation of the local moving phase. Rev. Ph.D. thesis, (University of Oxford, 2016). Fast Unfolding of Communities in Large Networks. Journal of Statistical , January. A number of iterations of the Leiden algorithm can be performed before the Louvain algorithm has finished its first iteration. Waltman, L. & van Eck, N. J. Fortunato, S. & Barthlemy, M. Resolution Limit in Community Detection. We typically reduce the dimensionality of the data first by running PCA, then construct a neighbor graph in the reduced space. In the worst case, almost a quarter of the communities are badly connected. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/. In short, the problem of badly connected communities has important practical consequences. 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. The horizontal axis indicates the cumulative time taken to obtain the quality indicated on the vertical axis. 2013. For empirical networks, it may take quite some time before the Leiden algorithm reaches its first stable iteration. 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. Computer Syst. leidenalg. 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). That is, one part of such an internally disconnected community can reach another part only through a path going outside the community. In other words, modularity may hide smaller communities and may yield communities containing significant substructure. MathSciNet 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. Phys. Traag, V. A. leidenalg 0.7.0. 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. Any sub-networks that are found are treated as different communities in the next aggregation step. 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. In the refinement phase, nodes are not necessarily greedily merged with the community that yields the largest increase in the quality function. Data 11, 130, https://doi.org/10.1145/2992785 (2017). The constant Potts model might give better communities in some cases, as it is not subject to the resolution limit. Louvain has two phases: local moving and aggregation. In the case of the Louvain algorithm, after a stable iteration, all subsequent iterations will be stable as well. See the documentation on the leidenalg Python module for more information: https://leidenalg.readthedocs.io/en/latest/reference.html. 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. However, for higher values of , Leiden becomes orders of magnitude faster than Louvain, reaching 10100 times faster runtimes for the largest networks. 2(b). E 76, 036106, https://doi.org/10.1103/PhysRevE.76.036106 (2007). As we will demonstrate in our experimental analysis, the problem occurs frequently in practice when using the Louvain algorithm. Therefore, clustering algorithms look for similarities or dissimilarities among data points. 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 \({\mathscr{P}}\). Fortunato, S. Community detection in graphs. Importantly, the problem of disconnected communities is not just a theoretical curiosity. The Leiden algorithm starts from a singleton partition (a). In doing so, Louvain keeps visiting nodes that cannot be moved to a different community. When a disconnected community has become a node in an aggregate network, there are no more possibilities to split up the community. 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. In the first iteration, Leiden is roughly 220 times faster than Louvain. 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.