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MCL is an algorithm for data clustering based on simulations of stochastic flows (random walks) in graphs. It takes as input an unweighted or weighted network, where the weights are interpreted as similarities, and it works with an iterative process by applying in alternation two operators called expansion and inflation, which update a stochastic matrix representing the probabilities of random walks. The expansion operator corresponds to the computation of random walks of higher length (many steps), associating new probabilities between each pair of nodes. Since it is more frequent to have higher length paths within clusters rather than between different clusters, the probabilities related with node pairs from the same cluster will in general be larger. The inflation operator will then have the effect of increasing intra-cluster probabilities and lowering inter-cluster walks (van Dongen 2000). The iteration of expansion and inflation eventually leads to the separation of the graph into segments without paths between them, which is interpreted as the clustering result. The inflation operator has a parameter, which serves to detect clustering patterns on different scales of granularity. In our simulations, given the correct number of communities to detect, we implemented a binary search in the range [1.1, 20] in order to choose the inflation parameter value that produces a number of communities as close as possible to the correct one.
The proposed rationale states that, in order to favour the simulation of random walks, the graph similarities (or dissimilarities) should approximate the closeness (or distances) on the hidden nonlinear manifold that characterizes the graph geometry (Papadopoulos et al. 2012b; Bogu et al. 2008). Indeed, in many networks the information can efficiently flow according to a greedy routing procedure because their topology is emerging from this hidden geometry (Bogu et al. 2008), whose hyperbolic and tree-like structure facilitates the greedy propagation (Papadopoulos et al. 2012b; Bogu et al. 2008; Cannistraci et al. 2010, 2013; Muscoloni and Cannistraci 2019). Recently, Muscoloni et al. (2017) and Muscoloni and Cannistraci (2018a) proposed two latent geometry based pre-weighting techniques (one local and one global) as valuable strategies for approximating the pairwise geometrical distances between connected nodes of an unweighted network. In a later study of the same authors, the clustering algorithm affinity propagation was applied to the community detection task adopting two related dissimilarity matrices, containing dissimilarity values both for connected and disconnected nodes, which proved to simulate a more navigable geometry than other kernels previously designed for this purpose (Cannistraci and Muscoloni 2018). Here, in accordance with the MCL algorithm requirements, we converted the previous pre-weighting techniques in similarity measures. They contain and merge two fundamental properties that characterize the hidden geometry of many real complex networks and thus might serve to improve stochastic flow simulations: node similarity (proximity or homophily), related with the network clustering and the concept of local attraction between common neighbours, and node popularity (centrality), related with the node degree (Papadopoulos et al. 2012b).
Although inspired by the same rationale, the second similarity is global (exploits the entire network topology to compute each similarity value between pairs of nodes), in fact as a first step it makes a global-information-based pre-weighting of the links, using the edge-betweenness-centrality (EBC) to approximate distances between nodes and regions of the network (Muscoloni et al. 2017). EBC is indeed a global topological network measure that assigns to each link a value of centrality related to its importance in propagating information across different regions of the network. The assumption is that central edges are bridges that tend to connect geometrically distant regions of the network, while peripheral edges tend to connect nodes in the same neighbourhood. The higher the EBC value of a network link, the more information will pass through that link. The algorithm to compute the EBC similarity for each link (i, j) in the network is the following:
The community detection algorithms Infomap (Rosvall and Bergstrom 2011) and Louvain (Blondel et al. 2008) are two state of the art approaches that have been shown to provide high performances on synthetic benchmarks (Lancichinetti and Fortunato 2009; Yang et al. 2016; Orman and Labatut 2009). Recently, they have been tested also on small-size and large-size real networks, resulting overall among the best performing on recovering ground-truth communities associated to metadata (Hric et al. 2014).
The Infomap algorithm (Rosvall and Bergstrom 2011) finds the community structure by minimizing the expected description length of a random walker trajectory using the Huffman coding process. It uses the hierarchical map equation, a further development of the map equation, to detect community structures on more than one level. The hierarchical map equation indicates the theoretical limit of how concisely a network path can be specified using a given partition structure. In order to calculate the optimal partition (community) structure, this limit can be computed for different partitions and the community annotation that gives the shortest path length is chosen. We used the C implementation released by the authors at
The Louvain algorithm (Blondel et al. 2008) is separated into two phases, which are repeated iteratively. At first every node in the (weighted) network represents a community in itself. In the first phase, for each node i, it considers its neighbours j and evaluates the gain in modularity that would take place by removing i from its community and placing it in the community of j. The node i is then placed in the community j for which this gain is maximum, but only if the gain is positive. If no gain is possible node i stays in its original community. This process is applied until no further improvement can be achieved. In the second phase the algorithm builds a new network whose nodes are the communities found in the first phase, whereas the weights of the links between the new nodes are given by the sum of the weight of the links between nodes in the corresponding two communities. Links between nodes of the same community lead to self-loops for this community in the new network. Once the new network has been built, the two phase process is iterated until there are no more changes and a maximum of modularity has been obtained. The number of iterations determines the height of the hierarchy of communities detected by the algorithm. We used the R function multilevel.community, an implementation of the method available in the igraph package (Csrdi and Nepusz 2006). For each hierarchical level there is a possible partition to compare to the ground-truth annotation. In this case, the hierarchical level considered is the one that guarantees the best match, therefore the detected partition that gives the highest NMI value. We let notice that most of this Methods section is equivalent to an analogous Methods section present in other studies of the authors (Cannistraci and Muscoloni 2018; Muscoloni et al. 2017).
Different similarity measures have been developed for evaluating the matching between two partitions (the communities detected by the method and the ground-truth). They are mainly based on three categories: pair counting, cluster matching and information theory (Fortunato and Hric 2016). Although there is not yet one measure without any drawback, the most adopted in community detection studies is the Normalized Mutual Information (NMI) (Danon et al. 2005).
If we consider a partition of the nodes in communities as a distribution (probability of one node falling into one community), the previous equations allow us to compute the matching between the annotations obtained by the community detection algorithm and the ground-truth communities of a network. We used the MATLAB implementation available at As suggested in the code, when \(\fracNC \le 100\), where N represents the number of nodes and C the number of communities, the NMI should be adjusted in order to correct for chance (Vinh et al. 2010). We let notice that most of this Methods section is equivalent to an analogous Methods section present in other studies of the authors (Cannistraci and Muscoloni 2018; Muscoloni et al. 2017).
The Polblogs (Adamic and Glance 2005) network consists of links between blogs about the politics in the 2004 US presidential election. The ground-truth communities represent the political opinions of the blogs (right/conservative and left/liberal). We let notice that most of this Methods section is equivalent to an analogous Methods section present in other studies of the authors (Cannistraci and Muscoloni 2018; Muscoloni et al. 2017).
The Popularity-Similarity-Optimization (PSO) model (Papadopoulos et al. 2012b) is a generative network model recently introduced in order to describe how random geometric graphs grow in the hyperbolic space. In this model the networks evolve optimizing a trade-off between node popularity, abstracted by the radial coordinate, and similarity, represented by the angular distance. The PSO model can reproduce many structural properties of real networks: clustering, small-worldness (concurrent low characteristic path length and high clustering), node degree heterogeneity with power-law degree distribution and rich-clubness. However, being the nodes uniformly distributed over the angular coordinate, the model lacks a non-trivial community structure.
The first investigation of this study has been carried out on real datasets. In Table 2 we report the comparison of MCL in its original form, the three LGI-MCL variants (EBC, RA and ER) and the state of the art methods for community detection Infomap and Louvain. In addition, we made two in-silico experiments to test the robustness of the techniques in case of noise injection in the real topologies. In the first case we perturbed the network structure by random deletion of 10% of the links. We repeated this procedure for 100 realizations, and the average results are reported in Table 3. This experiment simulates the behaviour of our algorithms in case of partial (10%) missing topological information. In the second case we perturbed the network structure by random addition of 10% of the links. We repeated this procedure for 100 realizations, and the average results are reported in Table 4. This experiment simulates the behaviour of our algorithms in case of partial (10%) addition of wrong topological information.
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