leiden clustering explained

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. J. Electr. 2018. As can be seen in Fig. Unlike the Louvain algorithm, the Leiden algorithm uses a fast local move procedure in this phase. As such, we scored leiden-clustering popularity level to be Limited. Blondel, V D, J L Guillaume, and R Lambiotte. 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. modularity) increases. 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 . We now consider the guarantees provided by the Leiden algorithm. Louvain community detection algorithm was originally proposed in 2008 as a fast community unfolding method for large networks. 69 (2 Pt 2): 026113. http://dx.doi.org/10.1103/PhysRevE.69.026113. 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. In a stable iteration, the partition is guaranteed to be node optimal and subpartition -dense. 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 Beginner's Guide to Dimensionality Reduction. Acad. ADS This commit does not belong to any branch on this repository, and may belong to a fork outside of the repository. The speed difference is especially large for larger networks. Disconnected community. In subsequent iterations, the percentage of disconnected communities remains fairly stable. This is because Louvain only moves individual nodes at a time. Nevertheless, depending on the relative strengths of the different connections, these nodes may still be optimally assigned to their current community. Modularity is given by. Inf. We used the CPM quality function. 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. 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. Starting from the second iteration, Leiden outperformed Louvain in terms of the percentage of badly connected communities. A community size of 50 nodes was used for the results presented below, but larger community sizes yielded qualitatively similar results. Knowl. Note that communities found by the Leiden algorithm are guaranteed to be connected. In the meantime, to ensure continued support, we are displaying the site without styles Each community in this partition becomes a node in the aggregate network. Phys. Natl. Each point corresponds to a certain iteration of an algorithm, with results averaged over 10 experiments. In this post, I will cover one of the common approaches which is hierarchical clustering. This package allows calling the Leiden algorithm for clustering on an igraph object from R. See the Python and Java implementations for more details: https://github.com/CWTSLeiden/networkanalysis. It is good at identifying small clusters. Modularity is a scale value between 0.5 (non-modular clustering) and 1 (fully modular clustering) that measures the relative density of edges inside communities with respect to edges outside communities. Luecken, M. D. Application of multi-resolution partitioning of interaction networks to the study of complex disease. It does not guarantee that modularity cant be increased by moving nodes between communities. 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. 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. Similarly, in citation networks, such as the Web of Science network, nodes in a community are usually considered to share a common topic26,27. 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. 68, 984998, https://doi.org/10.1002/asi.23734 (2017). CPM does not suffer from this issue13. Moreover, Louvain has no mechanism for fixing these communities. An overview of the various guarantees is presented in Table1. Performance of modularity maximization in practical contexts. Presumably, many of the badly connected communities in the first iteration of Louvain become disconnected in the second iteration. Fortunato, Santo, and Marc Barthlemy. Hence, no further improvements can be made after a stable iteration of the Louvain algorithm. B 86, 471, https://doi.org/10.1140/epjb/e2013-40829-0 (2013). 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 \({\mathscr{P}}\). 63, 23782392, https://doi.org/10.1002/asi.22748 (2012). The quality of such an asymptotically stable partition provides an upper bound on the quality of an optimal partition. All experiments were run on a computer with 64 Intel Xeon E5-4667v3 2GHz CPUs and 1TB internal memory. The two phases are repeated until the quality function cannot be increased further. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate. The property of -separation is also guaranteed by the Louvain algorithm. This method tries to maximise the difference between the actual number of edges in a community and the expected number of such edges. Subpartition -density is not guaranteed by the Louvain algorithm. Nonlin. In our experimental analysis, we observe that up to 25% of the communities are badly connected and up to 16% are disconnected. 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. This can be a shared nearest neighbours matrix derived from a graph object. It states that there are no communities that can be merged. Positive values above 2 define the total number of iterations to perform, -1 has the algorithm run until it reaches its optimal clustering. It implies uniform -density and all the other above-mentioned properties. This problem is different from the well-known issue of the resolution limit of modularity14. However, if communities are badly connected, this may lead to incorrect attributions of shared functionality. This algorithm provides a number of explicit guarantees. For empirical networks, it may take quite some time before the Leiden algorithm reaches its first stable iteration. The corresponding results are presented in the Supplementary Fig. 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. Each of these can be used as an objective function for graph-based community detection methods, with our goal being to maximize this value. In fact, if we keep iterating the Leiden algorithm, it will converge to a partition without any badly connected communities, as discussed earlier. The leidenalg package facilitates community detection of networks and builds on the package igraph. It partitions the data space and identifies the sub-spaces using the Apriori principle. Hence, the complex structure of empirical networks creates an even stronger need for the use of the Leiden algorithm. The algorithm moves individual nodes from one community to another to find a partition (b), which is then refined (c). Number of iterations until stability. 2(b). In the case of modularity, communities may have significant substructure both because of the resolution limit and because of the shortcomings of Louvain. Int. Hence, the Leiden algorithm effectively addresses the problem of badly connected communities. 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. Rev. AMS 56, 10821097 (2009). The quality improvement realised by the Leiden algorithm relative to the Louvain algorithm is larger for empirical networks than for benchmark networks. Such algorithms are rather slow, making them ineffective for large networks. Rev. We demonstrate the performance of the Leiden algorithm for several benchmark and real-world networks. Google Scholar. For larger networks and higher values of , Louvain is much slower than Leiden. The Leiden algorithm is considerably more complex than the Louvain algorithm. Both conda and PyPI have leiden clustering in Python which operates via iGraph. In that case, some optimal partitions cannot be found, as we show in SectionC2 of the Supplementary Information. Slider with three articles shown per slide. Random moving is a very simple adjustment to Louvain local moving proposed in 2015 (Traag 2015). The degree of randomness in the selection of a community is determined by a parameter >0. 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. By moving these nodes, Louvain creates badly connected communities. 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. Then, in order . Once aggregation is complete we restart the local moving phase, and continue to iterate until everything converges down to one node. While current approaches are successful in reducing the number of sequence alignments performed, the generated clusters are . After a stable iteration of the Leiden algorithm, the algorithm may still be able to make further improvements in later iterations. The nodes that are more interconnected have been partitioned into separate clusters. CPM is defined as. The main ideas of our algorithm are explained in an intuitive way in the main text of the paper. Source Code (2018). Even though clustering can be applied to networks, it is a broader field in unsupervised machine learning which deals with multiple attribute types. 2010. To ensure readability of the paper to the broadest possible audience, we have chosen to relegate all technical details to the Supplementary Information. Randomness in the selection of a community allows the partition space to be explored more broadly. Newman, M. E. J. We show that this algorithm has a major defect that largely went unnoticed until now: the Louvain algorithm may yield arbitrarily badly connected communities. The value of the resolution parameter was determined based on the so-called mixing parameter 13. Source Code (2018). Brandes, U. et al. One of the most popular algorithms for uncovering community structure is the so-called Louvain algorithm. Duch, J. Consider the partition shown in (a). ACM Trans. Article 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. To do this we just sum all the edge weights between nodes of the corresponding communities to get a single weighted edge between them, and collapse each community down to a single new node. The Leiden algorithm guarantees all communities to be connected, but it may yield badly connected communities. Fast Unfolding of Communities in Large Networks. Journal of Statistical , January. The R implementation of Leiden can be run directly on the snn igraph object in Seurat. For this network, Leiden requires over 750 iterations on average to reach a stable iteration. Nonetheless, some networks still show large differences. 2.3. Finding community structure in networks using the eigenvectors of matrices. 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. 2004. 8 (3): 207. https://pdfs.semanticscholar.org/4ea9/74f0fadb57a0b1ec35cbc5b3eb28e9b966d8.pdf. Figure6 presents total runtime versus quality for all iterations of the Louvain and the Leiden algorithm. Rev. Obviously, this is a worst case example, showing that disconnected communities may be identified by the Louvain algorithm. performed the experimental analysis. These are the same networks that were also studied in an earlier paper introducing the smart local move algorithm15. Rev. The minimum resolvable community size depends on the total size of the network and the degree of interconnectedness of the modules. 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. 81 (4 Pt 2): 046114. http://dx.doi.org/10.1103/PhysRevE.81.046114. This is very similar to what the smart local moving algorithm does. 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 V. A. Traag. https://doi.org/10.1038/s41598-019-41695-z. The Louvain algorithm is illustrated in Fig. GitHub on Feb 15, 2020 Do you think the performance improvements will also be implemented in leidenalg? As can be seen in Fig. 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. These nodes are therefore optimally assigned to their current community. The horizontal axis indicates the cumulative time taken to obtain the quality indicated on the vertical axis. Scaling of benchmark results for difficulty of the partition. The PyPI package leiden-clustering receives a total of 15 downloads a week. Clauset, A., Newman, M. E. J. MathSciNet Phys. However, after all nodes have been visited once, Leiden visits only nodes whose neighbourhood has changed, whereas Louvain keeps visiting all nodes in the network. 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. 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. In the refinement phase, nodes are not necessarily greedily merged with the community that yields the largest increase in the quality function. 9 shows that more than 10 iterations of the Leiden algorithm can be performed before the Louvain algorithm has finished its first iteration. In this iterative scheme, Louvain provides two guarantees: (1) no communities can be merged and (2) no nodes can be moved. 8, the Leiden algorithm is significantly faster than the Louvain algorithm also in empirical networks. Eur. The Louvain algorithm guarantees that modularity cannot be increased by merging communities (it finds a locally optimal solution). When a disconnected community has become a node in an aggregate network, there are no more possibilities to split up the community. In this way, Leiden implements the local moving phase more efficiently than Louvain. Google Scholar. Traag, V A. 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. In the Louvain algorithm, an aggregate network is created based on the partition \({\mathscr{P}}\) resulting from the local moving phase. Wolf, F. A. et al. This may have serious consequences for analyses based on the resulting partitions. J. Stat. 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. The algorithm then moves individual nodes in the aggregate network (d). (We implemented both algorithms in Java, available from https://github.com/CWTSLeiden/networkanalysis and deposited at Zenodo23. In other words, communities are guaranteed to be well separated. We keep removing nodes from the front of the queue, possibly moving these nodes to a different community. & Bornholdt, S. Statistical mechanics of community detection.

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