Closed benchmarks for network community structure characterization

Rodrigo Aldecoa and Ignacio Marín

Phys. Rev. E 85, 026109 – Published 15 February 2012

Abstract

Characterizing the community structure of complex networks is a key challenge in many scientific fields. Very diverse algorithms and methods have been proposed to this end, many working reasonably well in specific situations. However, no consensus has emerged on which of these methods is the best to use in practice. In part, this is due to the fact that testing their performance requires the generation of a comprehensive, standard set of synthetic benchmarks, a goal not yet fully achieved. Here, we present a type of benchmark that we call “closed,” in which an initial network of known community structure is progressively converted into a second network whose communities are also known. This approach differs from all previously published ones, in which networks evolve toward randomness. The use of this type of benchmark allows us to monitor the transformation of the community structure of a network. Moreover, we can predict the optimal behavior of the variation of information, a measure of the quality of the partitions obtained, at any moment of the process. This enables us in many cases to determine the best partition among those suggested by different algorithms. Also, since any network can be used as a starting point, extensive studies and comparisons can be performed using a heterogeneous set of structures, including random ones. These properties make our benchmarks a general standard for comparing community detection algorithms.

Received 14 September 2011

DOI:https://doi.org/10.1103/PhysRevE.85.026109

Authors & Affiliations

Rodrigo Aldecoa^* and Ignacio Marín^†

Instituto de Biomedicina de Valencia. Consejo Superior de Investigaciones Científicas (IBV-CSIC). Calle Jaime Roig 11, E-46010, Valencia, Spain

^*raldecoa@ibv.csic.es
^†imarin@ibv.csic.es

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Vol. 85, Iss. 2 — February 2012

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Images

Figure 1
Transformation process in a closed benchmark. In this case, the starting network is the GN benchmark. Links are progressively rewired from the initial ( $C = 0$ $%$ ) to the final network ( $C$ $=$ 100 $%$ ). Nodes color is defined by the initial community to which they belong, whereas their shape corresponds to the final community in which they are contained.Reuse & Permissions
Figure 2
Graphical view of the adjacency matrices of the four initial networks used in the tests. Nodes are ordered according to the communities to which they belong. Black indicates that two nodes are connected. Differences in relative community sizes are evident. In the GN network, the nodes of the four equal-sized communities are sparsely connected. The groups in the LFR $_{S}$ are also sparse. However, there are so many of them (195) that visualization is difficult at this resolution level. The RC initial networks (RC75 and RC50) are formed by 16 cliques and the distribution of community sizes is highly skewed, especially in RC50, where a single community dominates the network.Reuse & Permissions
Figure 3
Variation of information behavior in the four closed benchmarks used in this study. Black circles depict the $V$ between the initial and the estimated partition ( $V_{I E}$ ). Red (gray) squares show the $V$ between the estimated and the final partition ( $V_{E F}$ ). $\bar{V}$ appears as a dashed line, which should follow a straight line if the performance of the algorithm is optimal during the whole process of conversion (i.e., $V_{I E}$ $+$ $V_{E F}$ $=$ $V_{I F}$ ).Reuse & Permissions
Figure 4
Open benchmarks with starting structures identical to the initial structures of the closed benchmarks shown in Fig. 3. These structures are progressively degraded by randomly shuffling links. The percentage of rewired links is indicated on the $x$ axis. The dashed line indicates the $V_{I F} / 2$ value of the corresponding closed benchmark. Stars indicate the partitions with the highest surprise values.Reuse & Permissions
Figure 5
Random networks with the same number of nodes as the corresponding closed benchmarks indicated on top. As in Fig. 3, the dashed line corresponds to the $\bar{V}$ value, while red (gray) dots correspond to $V_{E F}$ values and black squares to $V_{I E}$ values. The values of $V_{I E}$ , $V_{E F}$ , and $\bar{V}$ largely/fully coincide in Infomap analyses, appearing as a single line or close parallel lines. Notice that as soon as conversion starts, $V_{I E}$ $+$ V $_{E F}$ $≫$ $V_{I F}$ . Differences between Infomap and SCluster are due to the different way they interpret the random structures present, i.e., as a single cluster (Infomap) or as many individual clusters (SCluster).Reuse & Permissions

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