The Leiden algorithm guarantees all communities to be connected, but it may yield badly connected communities. We typically reduce the dimensionality of the data first by running PCA, then construct a neighbor graph in the reduced space. Leiden keeps finding better partitions for empirical networks also after the first 10 iterations of the algorithm. Leiden is both faster than Louvain and finds better partitions. 2(a). E 81, 046106, https://doi.org/10.1103/PhysRevE.81.046106 (2010). In particular, in an attempt to find better partitions, multiple consecutive iterations of the algorithm can be performed, using the partition identified in one iteration as starting point for the next iteration. The corresponding results are presented in the Supplementary Fig. Rev. Reichardt, J. V. A. Traag. 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. Communities may even be disconnected. This function takes a cell_data_set as input, clusters the cells using . Second, to study the scaling of the Louvain and the Leiden algorithm, we use benchmark networks, allowing us to compare the algorithms in terms of both computational time and quality of the partitions. Leiden is faster than Louvain especially for larger networks. This is because Louvain only moves individual nodes at a time. Modularity scores of +1 mean that all the edges in a community are connecting nodes within the community. Such algorithms are rather slow, making them ineffective for large networks. Speed and quality for the first 10 iterations of the Louvain and the Leiden algorithm for benchmark networks (n=106 and n=107). 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. 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. 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. A partition of clusters as a vector of integers Examples Therefore, clustering algorithms look for similarities or dissimilarities among data points. The algorithm then moves individual nodes in the aggregate network (d). 69 (2 Pt 2): 026113. http://dx.doi.org/10.1103/PhysRevE.69.026113. Excluding node mergers that decrease the quality function makes the refinement phase more efficient. The random component also makes the algorithm more explorative, which might help to find better community structures. The aggregate network is created based on the partition \({{\mathscr{P}}}_{{\rm{refined}}}\). Theory Exp. 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. b, The elephant graph (in a) is clustered using the Leiden clustering algorithm 51 (resolution r = 0.5). Rev. J. Comput. MathSciNet Fortunato, Santo, and Marc Barthlemy. https://doi.org/10.1038/s41598-019-41695-z, DOI: https://doi.org/10.1038/s41598-019-41695-z. 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. It maximizes a modularity score for each community, where the modularity quantifies the quality of an assignment of nodes to communities. Here we can see partitions in the plotted results. leidenalg. Furthermore, by relying on a fast local move approach, the Leiden algorithm runs faster than the Louvain algorithm. This represents the following graph structure. Narrow scope for resolution-limit-free community detection. Later iterations of the Louvain algorithm are very fast, but this is only because the partition remains the same. 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). The quality of such an asymptotically stable partition provides an upper bound on the quality of an optimal partition. V.A.T. For higher values of , Leiden finds better partitions than Louvain. In many complex networks, nodes cluster and form relatively dense groupsoften called communities1,2. Using the fast local move procedure, the first visit to all nodes in a network in the Leiden algorithm is the same as in the Louvain algorithm. The Leiden algorithm starts from a singleton partition (a). . We consider these ideas to represent the most promising directions in which the Louvain algorithm can be improved, even though we recognise that other improvements have been suggested as well22. These steps are repeated until no further improvements can be made. Some of these nodes may very well act as bridges, similarly to node 0 in the above example. The nodes that are more interconnected have been partitioned into separate clusters. 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. J. Assoc. The Leiden algorithm provides several guarantees. 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. However, nodes 16 are still locally optimally assigned, and therefore these nodes will stay in the red community. In our experimental analysis, we observe that up to 25% of the communities are badly connected and up to 16% are disconnected. Correspondence to Communities in Networks. Soft Matter Phys. We show that this algorithm has a major defect that largely went unnoticed until now: the Louvain algorithm may yield arbitrarily badly connected communities. For each set of parameters, we repeated the experiment 10 times. http://dx.doi.org/10.1073/pnas.0605965104. We conclude that the Leiden algorithm is strongly preferable to the Louvain algorithm. For each community in a partition that was uncovered by the Louvain algorithm, we determined whether it is internally connected or not. 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. Louvain quickly converges to a partition and is then unable to make further improvements. In fact, by implementing the refinement phase in the right way, several attractive guarantees can be given for partitions produced by the Leiden algorithm. The property of -separation is also guaranteed by the Louvain algorithm. Modularity optimization. Nodes 13 should form a community and nodes 46 should form another community. Subpartition -density does not imply that individual nodes are locally optimally assigned. Rotta, R. & Noack, A. Multilevel local search algorithms for modularity clustering. Analyses based on benchmark networks have only a limited value because these networks are not representative of empirical real-world networks. The authors act as bibliometric consultants to CWTS B.V., which makes use of community detection algorithms in commercial products and services. Although originally defined for modularity, the Louvain algorithm can also be used to optimise other quality functions. Obviously, this is a worst case example, showing that disconnected communities may be identified by the Louvain algorithm. Google Scholar. Louvain keeps visiting all nodes in a network until there are no more node movements that increase the quality function. 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. The inspiration for this method of community detection is the optimization of modularity as the algorithm progresses. An overview of the various guarantees is presented in Table1. Nodes 16 have connections only within this community, whereas node 0 also has many external connections. Blondel, V D, J L Guillaume, and R Lambiotte. CAS For the results reported below, the average degree was set to \(\langle k\rangle =10\). Phys. Scaling of benchmark results for network size. E 92, 032801, https://doi.org/10.1103/PhysRevE.92.032801 (2015). Randomness in the selection of a community allows the partition space to be explored more broadly. Blondel, V. D., Guillaume, J.-L., Lambiotte, R. & Lefebvre, E. Fast unfolding of communities in large networks. One of the best-known methods for community detection is called modularity3. However, as increases, the Leiden algorithm starts to outperform the Louvain algorithm. First calculate k-nearest neighbors and construct the SNN graph. Rep. 486, 75174, https://doi.org/10.1016/j.physrep.2009.11.002 (2010). * (2018). 81 (4 Pt 2): 046114. http://dx.doi.org/10.1103/PhysRevE.81.046114. Each point corresponds to a certain iteration of an algorithm, with results averaged over 10 experiments. Initially, \({{\mathscr{P}}}_{{\rm{refined}}}\) is set to a singleton partition, in which each node is in its own community. A Smart Local Moving Algorithm for Large-Scale Modularity-Based Community Detection. Eur. Hence, in general, Louvain may find arbitrarily badly connected communities. We gratefully acknowledge computational facilities provided by the LIACS Data Science Lab Computing Facilities through Frank Takes. It was originally developed for modularity optimization, although the same method can be applied to optimize CPM. The algorithm then moves individual nodes in the aggregate network (e). Rep. 6, 30750, https://doi.org/10.1038/srep30750 (2016). 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. The 'devtools' package will be used to install 'leiden' and the dependancies (igraph and reticulate). In the initial stage of Louvain (when all nodes belong to their own community), nearly any move will result in a modularity gain, and it doesnt matter too much which move is chosen. 10, 186198, https://doi.org/10.1038/nrn2575 (2009). The constant Potts model tries to maximize the number of internal edges in a community, while simultaneously trying to keep community sizes small, and the constant parameter balances these two characteristics. Higher resolutions lead to more communities, while lower resolutions lead to fewer communities. See the documentation on the leidenalg Python module for more information: https://leidenalg.readthedocs.io/en/latest/reference.html. 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. 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. partition_type : Optional [ Type [ MutableVertexPartition ]] (default: None) Type of partition to use. Finding communities in large networks is far from trivial: algorithms need to be fast, but they also need to provide high-quality results. The high percentage of badly connected communities attests to this. 2013. S3. Int. Google Scholar. Finally, we compare the performance of the algorithms on the empirical networks. Electr. Proc. 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). Phys. Introduction The Louvain method is an algorithm to detect communities in large networks. The difference in computational time is especially pronounced for larger networks, with Leiden being up to 20 times faster than Louvain in empirical networks. E 76, 036106, https://doi.org/10.1103/PhysRevE.76.036106 (2007). Phys. 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. In short, the problem of badly connected communities has important practical consequences. 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. Then, in order . 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. 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. This algorithm provides a number of explicit guarantees. A Simple Acceleration Method for the Louvain Algorithm. Int. There is an entire Leiden package in R-cran here 5, for lower values of the partition is well defined, and neither the Louvain nor the Leiden algorithm has a problem in determining the correct partition in only two iterations. In this paper, we show that the Louvain algorithm has a major problem, for both modularity and CPM. At some point, the Louvain algorithm may end up in the community structure shown in Fig. J. Stat. Empirical networks show a much richer and more complex structure. Leiden consists of the following steps: The refinement step allows badly connected communities to be split before creating the aggregate network. PubMedGoogle Scholar. Cluster cells using Louvain/Leiden community detection Description. E 74, 036104, https://doi.org/10.1103/PhysRevE.74.036104 (2006). However, as shown in this paper, the Louvain algorithm has a major shortcoming: the algorithm yields communities that may be arbitrarily badly connected. J. & Girvan, M. Finding and evaluating community structure in networks. The algorithm is run iteratively, using the partition identified in one iteration as starting point for the next iteration. Traag, V. A. leidenalg 0.7.0. We name our algorithm the Leiden algorithm, after the location of its authors. However, the initial partition for the aggregate network is based on P, just like in the Louvain algorithm. 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. 10008, 6, https://doi.org/10.1088/1742-5468/2008/10/P10008 (2008). These nodes can be approximately identified based on whether neighbouring nodes have changed communities. In general, Leiden is both faster than Louvain and finds better partitions. Finding community structure in networks using the eigenvectors of matrices. The degree of randomness in the selection of a community is determined by a parameter >0. The Louvain method for community detection is a popular way to discover communities from single-cell data. Nat. In particular, we show that Louvain may identify communities that are internally disconnected. CAS We therefore require a more principled solution, which we will introduce in the next section. The algorithm may yield arbitrarily badly connected communities, over and above the well-known issue of the resolution limit14. In this post Ive mainly focused on the optimisation methods for community detection, rather than the different objective functions that can be used. MATH Any sub-networks that are found are treated as different communities in the next aggregation step. A community size of 50 nodes was used for the results presented below, but larger community sizes yielded qualitatively similar results. Note that this code is designed for Seurat version 2 releases. We keep removing nodes from the front of the queue, possibly moving these nodes to a different community. Raghavan, U., Albert, R. & Kumara, S. Near linear time algorithm to detect community structures in large-scale networks. Sci. To elucidate the problem, we consider the example illustrated in Fig. On the other hand, Leiden keeps finding better partitions, especially for higher values of , for which it is more difficult to identify good partitions. Unlike the Louvain algorithm, the Leiden algorithm uses a fast local move procedure in this phase. These steps are repeated until the quality cannot be increased further. One may expect that other nodes in the old community will then also be moved to other communities. AMS 56, 10821097 (2009). Technol. ADS One of the most popular algorithms to optimise modularity is the so-called Louvain algorithm10, named after the location of its authors. 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. At some point, node 0 is considered for moving. Rev. 8, the Leiden algorithm is significantly faster than the Louvain algorithm also in empirical networks. http://iopscience.iop.org/article/10.1088/1742-5468/2008/10/P10008/meta. 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. Soft Matter Phys. 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. 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. Performance of modularity maximization in practical contexts. Inf. In other words, modularity may hide smaller communities and may yield communities containing significant substructure. However, it is also possible to start the algorithm from a different partition15. The PyPI package leiden-clustering receives a total of 15 downloads a week. Acad. 2. PubMed Central Waltman, Ludo, and Nees Jan van Eck. Badly connected communities. To obtain Data 11, 130, https://doi.org/10.1145/2992785 (2017). A. On Modularity Clustering. ADS Google Scholar. First, we created a specified number of nodes and we assigned each node to a community. Louvain pruning keeps track of a list of nodes that have the potential to change communities, and only revisits nodes in this list, which is much smaller than the total number of nodes. 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. As the problem of modularity optimization is NP-hard, we need heuristic methods to optimize modularity (or CPM).
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