leiden clustering explained leiden clustering explained

Cluster your data matrix with the Leiden algorithm. A structure that is more informative than the unstructured set of clusters returned by flat clustering. J. Exp. 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). The inspiration for this method of community detection is the optimization of modularity as the algorithm progresses. To address this problem, we introduce the Leiden algorithm. Here we can see partitions in the plotted results. With one exception (=0.2 and n=107), all results in Fig. 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. In fact, for the Web of Science and Web UK networks, Fig. 2. 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. 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 \({\mathscr{P}}\). Later iterations of the Louvain algorithm are very fast, but this is only because the partition remains the same. We now consider the guarantees provided by the Leiden algorithm. All experiments were run on a computer with 64 Intel Xeon E5-4667v3 2GHz CPUs and 1TB internal memory. The corresponding results are presented in the Supplementary Fig. Eur. Communities in \({\mathscr{P}}\) may be split into multiple subcommunities in \({{\mathscr{P}}}_{{\rm{refined}}}\). Luecken, M. D. Application of multi-resolution partitioning of interaction networks to the study of complex disease. Ozaki, Naoto, Hiroshi Tezuka, and Mary Inaba. Nonlin. As discussed earlier, the Louvain algorithm does not guarantee connectivity. We use six empirical networks in our analysis. Therefore, clustering algorithms look for similarities or dissimilarities among data points. Knowl. Scaling of benchmark results for network size. ADS 2(a). Positive values above 2 define the total number of iterations to perform, -1 has the algorithm run until it reaches its optimal clustering. This algorithm provides a number of explicit guarantees. Such algorithms are rather slow, making them ineffective for large networks. Such a modular structure is usually not known beforehand. A community size of 50 nodes was used for the results presented below, but larger community sizes yielded qualitatively similar results. The algorithm may yield arbitrarily badly connected communities, over and above the well-known issue of the resolution limit14. In doing so, Louvain keeps visiting nodes that cannot be moved to a different community. We can guarantee a number of properties of the partitions found by the Leiden algorithm at various stages of the iterative process. We provide the full definitions of the properties as well as the mathematical proofs in SectionD of the Supplementary Information. The main ideas of our algorithm are explained in an intuitive way in the main text of the paper. Eng. Technol. Basically, there are two types of hierarchical cluster analysis strategies - 1. Disconnected community. We typically reduce the dimensionality of the data first by running PCA, then construct a neighbor graph in the reduced space. MathSciNet Google Scholar. The minimum resolvable community size depends on the total size of the network and the degree of interconnectedness of the modules. It was found to be one of the fastest and best performing algorithms in comparative analyses11,12, and it is one of the most-cited works in the community detection literature. Eng. 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. Speed and quality for the first 10 iterations of the Louvain and the Leiden algorithm for benchmark networks (n=106 and n=107). In particular, we show that Louvain may identify communities that are internally disconnected. Directed Undirected Homogeneous Heterogeneous Weighted 1. Moreover, Louvain has no mechanism for fixing these communities. However, the initial partition for the aggregate network is based on P, just like in the Louvain algorithm. Computer Syst. By moving these nodes, Louvain creates badly connected communities. Communities were all of equal size. Sci. It was originally developed for modularity optimization, although the same method can be applied to optimize CPM. 10, 186198, https://doi.org/10.1038/nrn2575 (2009). 8, 207218, https://doi.org/10.17706/IJCEE.2016.8.3.207-218 (2016). The docs are here. 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. We used modularity with a resolution parameter of =1 for the experiments. & Bornholdt, S. Statistical mechanics of community detection. The Leiden algorithm has been specifically designed to address the problem of badly connected communities. Similarly, in citation networks, such as the Web of Science network, nodes in a community are usually considered to share a common topic26,27. Speed and quality for the first 10 iterations of the Louvain and the Leiden algorithm for six empirical networks. We therefore require a more principled solution, which we will introduce in the next section. Sci. We applied the Louvain and the Leiden algorithm to exactly the same networks, using the same seed for the random number generator. An aggregate network (d) is created based on the refined partition, using the non-refined partition to create an initial partition for the aggregate network. In the refinement phase, nodes are not necessarily greedily merged with the community that yields the largest increase in the quality function. In this case we can solve one of the hard problems for K-Means clustering - choosing the right k value, giving the number of clusters we are looking for. 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. We used the CPM quality function. In addition, we prove that the algorithm converges to an asymptotically stable partition in which all subsets of all communities are locally optimally assigned. 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. Slider with three articles shown per slide. Learn more. reviewed the manuscript. A number of iterations of the Leiden algorithm can be performed before the Louvain algorithm has finished its first iteration. In short, the problem of badly connected communities has important practical consequences. Sci. When the Leiden algorithm found that a community could be split into multiple subcommunities, we counted the community as badly connected. It only implies that individual nodes are well connected to their community. Waltman, L. & van Eck, N. J. In other words, communities are guaranteed to be well separated. Unsupervised clustering of cells is a common step in many single-cell expression workflows. Louvain has two phases: local moving and aggregation. 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). 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. PubMedGoogle Scholar. To ensure readability of the paper to the broadest possible audience, we have chosen to relegate all technical details to the Supplementary Information. 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. Powered by DataCamp DataCamp CPM is defined as. In this way, the constant acts as a resolution parameter, and setting the constant higher will result in fewer communities. Not. Unlike the Louvain algorithm, the Leiden algorithm uses a fast local move procedure in this phase. Waltman, Ludo, and Nees Jan van Eck. B 86 (11): 471. https://doi.org/10.1140/epjb/e2013-40829-0. The R implementation of Leiden can be run directly on the snn igraph object in Seurat. o CLIQUE (Clustering in Quest): - CLIQUE is a combination of density-based and grid-based clustering algorithm. J. In this stage we essentially collapse communities down into a single representative node, creating a new simplified graph. We will use sklearns K-Means implementation looking for 10 clusters in the original 784 dimensional data. This way of defining the expected number of edges is based on the so-called configuration model. Once aggregation is complete we restart the local moving phase, and continue to iterate until everything converges down to one node. This is very similar to what the smart local moving algorithm does. 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. 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. 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. 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. As shown in Fig. Here is some small debugging code I wrote to find this. We study the problem of badly connected communities when using the Louvain algorithm for several empirical networks. Traag, V. A. leidenalg 0.7.0. 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. For all networks, Leiden identifies substantially better partitions than Louvain. Article For this network, Leiden requires over 750 iterations on average to reach a stable iteration. 104 (1): 3641. Fortunato, S. & Barthlemy, M. Resolution Limit in Community Detection. The random component also makes the algorithm more explorative, which might help to find better community structures. 2010. Brandes, U. et al. The property of -connectivity is a slightly stronger variant of ordinary connectivity. Based on this partition, an aggregate network is created (c). We thank Lovro Subelj for his comments on an earlier version of this paper. The corresponding results are presented in the Supplementary Fig. Note that if Leiden finds subcommunities, splitting up the community is guaranteed to increase modularity. J. Comput. The high percentage of badly connected communities attests to this. 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. A Simple Acceleration Method for the Louvain Algorithm. Int. 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. For the Amazon, DBLP and Web UK networks, Louvain yields on average respectively 23%, 16% and 14% badly connected communities. We prove that the new algorithm is guaranteed to produce partitions in which all communities are internally connected. Due to the resolution limit, modularity may cause smaller communities to be clustered into larger communities. However, in the case of the Web of Science network, more than 5% of the communities are disconnected in the first iteration. While smart local moving and multilevel refinement can improve the communities found, the next two improvements on Louvain that Ill discuss focus on the speed/efficiency of the algorithm. E 84, 016114, https://doi.org/10.1103/PhysRevE.84.016114 (2011). In the fast local move procedure in the Leiden algorithm, only nodes whose neighbourhood has changed are visited. 8 (3): 207. https://pdfs.semanticscholar.org/4ea9/74f0fadb57a0b1ec35cbc5b3eb28e9b966d8.pdf. Below we offer an intuitive explanation of these properties. This is not the case when nodes are greedily merged with the community that yields the largest increase in the quality function. Zenodo, https://doi.org/10.5281/zenodo.1466831 https://github.com/CWTSLeiden/networkanalysis. For each community, modularity measures the number of edges within the community and the number of edges going outside the community, and gives a value between -1 and +1. We also suggested that the Leiden algorithm is faster than the Louvain algorithm, because of the fast local move approach. 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. Phys. Cite this article. Ph.D. thesis, (University of Oxford, 2016). In many complex networks, nodes cluster and form relatively dense groupsoften called communities1,2. Modules smaller than the minimum size may not be resolved through modularity optimization, even in the extreme case where they are only connected to the rest of the network through a single edge. Natl. 9 shows that more than 10 iterations of the Leiden algorithm can be performed before the Louvain algorithm has finished its first iteration. The algorithm then moves individual nodes in the aggregate network (e). Discov. However, modularity suffers from a difficult problem known as the resolution limit (Fortunato and Barthlemy 2007). Algorithmics 16, 2.1, https://doi.org/10.1145/1963190.1970376 (2011). http://arxiv.org/abs/1810.08473. Data Eng. That is, one part of such an internally disconnected community can reach another part only through a path going outside the community. Nonetheless, some networks still show large differences. Louvain can also be quite slow, as it spends a lot of time revisiting nodes that may not have changed neighborhoods. As the use of clustering is highly depending on the biological question it makes sense to use several approaches and algorithms. Traag, V. A., Van Dooren, P. & Nesterov, Y. Rev. J. Assoc. Each point corresponds to a certain iteration of an algorithm, with results averaged over 10 experiments. Rev. How many iterations of the Leiden clustering algorithm to perform. Communities in Networks. Publishers note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. The constant Potts model might give better communities in some cases, as it is not subject to the resolution limit. This is very similar to what the smart local moving algorithm does. 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). Node optimality is also guaranteed after a stable iteration of the Louvain algorithm. This makes sense, because after phase one the total size of the graph should be significantly reduced. The Leiden algorithm guarantees all communities to be connected, but it may yield badly connected communities. 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. Rather than progress straight to the aggregation stage (as we would for the original Louvain), we next consider each community as a new sub-network and re-apply the local moving step within each community. J. http://iopscience.iop.org/article/10.1088/1742-5468/2008/10/P10008/meta. 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. Figure6 presents total runtime versus quality for all iterations of the Louvain and the Leiden algorithm. That is, no subset can be moved to a different community. 2 represent stronger connections, while the other edges represent weaker connections. Guimer, R. & Nunes Amaral, L. A. Functional cartography of complex metabolic networks. This continues until the queue is empty. 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. Rev. Community detection can then be performed using this graph. Complex brain networks: graph theoretical analysis of structural and functional systems. & Arenas, A. After the first iteration of the Louvain algorithm, some partition has been obtained. Both conda and PyPI have leiden clustering in Python which operates via iGraph. 2018. ADS The PyPI package leiden-clustering receives a total of 15 downloads a week. This should be the first preference when choosing an algorithm. Scaling of benchmark results for difficulty of the partition. In a stable iteration, the partition is guaranteed to be node optimal and subpartition -dense. Raghavan, U., Albert, R. & Kumara, S. Near linear time algorithm to detect community structures in large-scale networks. Phys. 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.