See the documentation for these functions. Google Scholar. Then the Leiden algorithm can be run on the adjacency matrix. Based on project statistics from the GitHub repository for the PyPI package leiden-clustering, we found that it has been starred 1 times. If nothing happens, download Xcode and try again. 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. It is a directed graph if the adjacency matrix is not symmetric. Leiden is the most recent major development in this space, and highlighted a flaw in the original Louvain algorithm (Traag, Waltman, and Eck 2018). Similarly, in citation networks, such as the Web of Science network, nodes in a community are usually considered to share a common topic26,27. This makes sense, because after phase one the total size of the graph should be significantly reduced. J. Comput. Clauset, A., Newman, M. E. J. Sci. where nc is the number of nodes in community c. The interpretation of the resolution parameter is quite straightforward. E 70, 066111, https://doi.org/10.1103/PhysRevE.70.066111 (2004). Rev. We keep removing nodes from the front of the queue, possibly moving these nodes to a different community. 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. First, we created a specified number of nodes and we assigned each node to a community. The idea of the refinement phase in the Leiden algorithm is to identify a partition \({{\mathscr{P}}}_{{\rm{refined}}}\) that is a refinement of \({\mathscr{P}}\). By moving these nodes, Louvain creates badly connected communities. In the fast local move procedure in the Leiden algorithm, only nodes whose neighbourhood has changed are visited. Below, the quality of a partition is reported as \(\frac{ {\mathcal H} }{2m}\), where H is defined in Eq. Phys. 9, the Leiden algorithm also performs better than the Louvain algorithm in terms of the quality of the partitions that are obtained. Nonlin. Contrary to what might be expected, iterating the Louvain algorithm aggravates the problem of badly connected communities, as we will also see in our experimental analysis. b, The elephant graph (in a) is clustered using the Leiden clustering algorithm 51 (resolution r = 0.5). 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. 10008, 6, https://doi.org/10.1088/1742-5468/2008/10/P10008 (2008). Traag, V. A. leidenalg 0.7.0. In the local move procedure in the Leiden algorithm, only nodes whose neighborhood . The random component also makes the algorithm more explorative, which might help to find better community structures. Traag, V. A., Van Dooren, P. & Nesterov, Y. E 81, 046106, https://doi.org/10.1103/PhysRevE.81.046106 (2010). Random moving is a very simple adjustment to Louvain local moving proposed in 2015 (Traag 2015). The corresponding results are presented in the Supplementary Fig. Other networks show an almost tenfold increase in the percentage of disconnected communities. AMS 56, 10821097 (2009). Work fast with our official CLI. Randomness in the selection of a community allows the partition space to be explored more broadly. Louvain has two phases: local moving and aggregation. J. Stat. 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. Subset optimality is the strongest guarantee that is provided by the Leiden algorithm. 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. Leiden consists of the following steps: Local moving of nodes Partition refinement Network aggregation The refinement step allows badly connected communities to be split before creating the aggregate network. Disconnected 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. The thick edges in Fig. The percentage of disconnected communities is more limited, usually around 1%. 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. 2(b). Modularity is used most commonly, but is subject to the resolution limit. We name our algorithm the Leiden algorithm, after the location of its authors. When the Leiden algorithm found that a community could be split into multiple subcommunities, we counted the community as badly connected. Rev. Rev. Neurosci. Some of these nodes may very well act as bridges, similarly to node 0 in the above example. Then, in order . Table2 provides an overview of the six networks. We denote by ec the actual number of edges in community c. The expected number of edges can be expressed as \(\frac{{K}_{c}^{2}}{2m}\), where Kc is the sum of the degrees of the nodes in community c and m is the total number of edges in the network. https://doi.org/10.1038/s41598-019-41695-z. Run the code above in your browser using DataCamp Workspace. For each network, we repeated the experiment 10 times. These are the same networks that were also studied in an earlier paper introducing the smart local move algorithm15. Traag, V. A. There was a problem preparing your codespace, please try again. For each network, Table2 reports the maximal modularity obtained using the Louvain and the Leiden algorithm. Hence, in general, Louvain may find arbitrarily badly connected communities. If nothing happens, download GitHub Desktop and try again. Blondel, V D, J L Guillaume, and R Lambiotte. We find that the Leiden algorithm commonly finds partitions of higher quality in less time. First, we show that the Louvain algorithm finds disconnected communities, and more generally, badly connected communities in the empirical networks. Phys. Electr. Scientific Reports (Sci Rep) 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 . 2018. Excluding node mergers that decrease the quality function makes the refinement phase more efficient. Subpartition -density is not guaranteed by the Louvain algorithm. A tag already exists with the provided branch name. Community detection is an important task in the analysis of complex networks. Thank you for visiting nature.com. Article For higher values of , Leiden finds better partitions than Louvain. Raghavan, U., Albert, R. & Kumara, S. Near linear time algorithm to detect community structures in large-scale networks. Scaling of benchmark results for network size. 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. Newman, M. E. J. Fortunato, Santo, and Marc Barthlemy. 1 and summarised in pseudo-code in AlgorithmA.1 in SectionA of the Supplementary Information. Centre for Science and Technology Studies, Leiden University, Leiden, The Netherlands, You can also search for this author in PubMed Central The Leiden algorithm is clearly faster than the Louvain algorithm. E 92, 032801, https://doi.org/10.1103/PhysRevE.92.032801 (2015). The phase one loop can be greatly accelerated by finding the nodes that have the potential to change community and only revisit those nodes. Phys. Nonetheless, some networks still show large differences. 6 show that Leiden outperforms Louvain in terms of both computational time and quality of the partitions. 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. 104 (1): 3641. This is similar to ideas proposed recently as pruning16 and in a slightly different form as prioritisation17. MathSciNet ISSN 2045-2322 (online). In subsequent iterations, the percentage of disconnected communities remains fairly stable. Are you sure you want to create this branch? The resulting clusters are shown as colors on the 3D model (top) and t -SNE embedding . The percentage of badly connected communities is less affected by the number of iterations of the Louvain algorithm. E 76, 036106, https://doi.org/10.1103/PhysRevE.76.036106 (2007). We used the CPM quality function. We applied the Louvain and the Leiden algorithm to exactly the same networks, using the same seed for the random number generator. However, the initial partition for the aggregate network is based on P, just like in the Louvain algorithm. 8, the Leiden algorithm is significantly faster than the Louvain algorithm also in empirical networks. In this way, the constant acts as a resolution parameter, and setting the constant higher will result in fewer communities. However, the Louvain algorithm does not consider this possibility, since it considers only individual node movements. In this post Ive mainly focused on the optimisation methods for community detection, rather than the different objective functions that can be used. Importantly, the number of communities discovered is related only to the difference in edge density, and not the total number of nodes in the community. 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Rev. This problem is different from the well-known issue of the resolution limit of modularity14. The aggregate network is created based on the partition \({{\mathscr{P}}}_{{\rm{refined}}}\). Indeed, the percentage of disconnected communities becomes more comparable to the percentage of badly connected communities in later iterations. In this new situation, nodes 2, 3, 5 and 6 have only internal connections. For empirical networks, it may take quite some time before the Leiden algorithm reaches its first stable iteration. Newman, M. E. J. However, nodes 16 are still locally optimally assigned, and therefore these nodes will stay in the red community. We used modularity with a resolution parameter of =1 for the experiments. You are using a browser version with limited support for CSS. GitHub on Feb 15, 2020 Do you think the performance improvements will also be implemented in leidenalg? For the results reported below, the average degree was set to \(\langle k\rangle =10\). contrastive-sc works best on datasets with fewer clusters when using the KMeans clustering and conversely for Leiden. Nonlin. Phys. When node 0 is moved to a different community, the red community becomes internally disconnected, as shown in (b). Performance of modularity maximization in practical contexts. Even worse, the Amazon network has 5% disconnected communities, but 25% badly connected communities. The solution proposed in smart local moving is to alter how the local moving step in Louvain works. This commit does not belong to any branch on this repository, and may belong to a fork outside of the repository. The new algorithm integrates several earlier improvements, incorporating a combination of smart local move15, fast local move16,17 and random neighbour move18. We study the problem of badly connected communities when using the Louvain algorithm for several empirical networks. The resolution limit describes a limitation where there is a minimum community size able to be resolved by optimizing modularity (or other related functions). Fortunato, S. Community detection in graphs. Finding communities in large networks is far from trivial: algorithms need to be fast, but they also need to provide high-quality results. This enables us to find cases where its beneficial to split a community. Nodes 06 are in the same community. However, for higher values of , Leiden becomes orders of magnitude faster than Louvain, reaching 10100 times faster runtimes for the largest networks. Duch, J. The constant Potts model might give better communities in some cases, as it is not subject to the resolution limit. Data 11, 130, https://doi.org/10.1145/2992785 (2017). We demonstrate the performance of the Leiden algorithm for several benchmark and real-world networks. We prove that the new algorithm is guaranteed to produce partitions in which all communities are internally connected. The increase in the percentage of disconnected communities is relatively limited for the Live Journal and Web of Science networks. The steps for agglomerative clustering are as follows: Elect. 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). Google Scholar. Clustering is the task of grouping a set of objects with similar characteristics into one bucket and differentiating them from the rest of the group. 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. V. A. Traag. It does not guarantee that modularity cant be increased by moving nodes between communities. 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. An alternative quality function is the Constant Potts Model (CPM)13, which overcomes some limitations of modularity. The Leiden algorithm starts from a singleton partition (a). Scaling of benchmark results for difficulty of the partition. Rev. A Simple Acceleration Method for the Louvain Algorithm. Int. Mech. and L.W. Modularity optimization. For all networks, Leiden identifies substantially better partitions than Louvain. 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. Data Eng. 2016. Due to the resolution limit, modularity may cause smaller communities to be clustered into larger communities. Positive values above 2 define the total number of iterations to perform, -1 has the algorithm run until it reaches its optimal clustering. An iteration of the Leiden algorithm in which the partition does not change is called a stable iteration. The speed difference is especially large for larger networks. 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. Another important difference between the Leiden algorithm and the Louvain algorithm is the implementation of the local moving phase. Therefore, clustering algorithms look for similarities or dissimilarities among data points. A community is subset optimal if all subsets of the community are locally optimally assigned. It only implies that individual nodes are well connected to their community. Modularity is given by. However, as increases, the Leiden algorithm starts to outperform the Louvain algorithm. & Bornholdt, S. Statistical mechanics of community detection. Communities may even be internally disconnected. By submitting a comment you agree to abide by our Terms and Community Guidelines.