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  • Open Access

Hierarchical Block Structures and High-Resolution Model Selection in Large Networks

Tiago P. Peixoto*

  • Institut für Theoretische Physik, Universität Bremen, Hochschulring 18, D-28359 Bremen, Germany

  • *tiago@itp.uni-bremen.de
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Phys. Rev. X 4, 011047 – Published 24 March, 2014

DOI: https://doi.org/10.1103/PhysRevX.4.011047

Abstract

Discovering and characterizing the large-scale topological features in empirical networks are crucial steps in understanding how complex systems function. However, most existing methods used to obtain the modular structure of networks suffer from serious problems, such as being oblivious to the statistical evidence supporting the discovered patterns, which results in the inability to separate actual structure from noise. In addition to this, one also observes a resolution limit on the size of communities, where smaller but well-defined clusters are not detectable when the network becomes large. This phenomenon occurs for the very popular approach of modularity optimization, which lacks built-in statistical validation, but also for more principled methods based on statistical inference and model selection, which do incorporate statistical validation in a formally correct way. Here, we construct a nested generative model that, through a complete description of the entire network hierarchy at multiple scales, is capable of avoiding this limitation and enables the detection of modular structure at levels far beyond those possible with current approaches. Even with this increased resolution, the method is based on the principle of parsimony, and is capable of separating signal from noise, and thus will not lead to the identification of spurious modules even on sparse networks. Furthermore, it fully generalizes other approaches in that it is not restricted to purely assortative mixing patterns, directed or undirected graphs, and ad hoc hierarchical structures such as binary trees. Despite its general character, the approach is tractable and can be combined with advanced techniques of community detection to yield an efficient algorithm that scales well for very large networks.

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Popular Summary

Many social, technological, and biological networks are known to organize into modules or “communities,” groups of nodes with a similar connectivity pattern. At first sight, the task of identifying and characterizing these modules seems a simple one. This simplicity is, however, deceiving. Despite a large amount of effort aimed at implementing this task, several fundamental questions still remain open: How should one formally quantify the network’s modular structure? Given a formal characterization, what is the optimal method to identify modules from empirical or simulated data? How should one distinguish modules that arise from some underlying organizational principle from those that occur purely by chance? Why do some existing methods have a “resolution” limit, in other words, fail to detect modules smaller than a certain size? And, how does one overcome such limits? In this work, we show that the concepts of statistical inference and modular hierarchy may hold the key to all these questions, in particular, in terms of providing a principled means for distinguishing structure from noise and a detection method without a resolution limit.

We approach the module-detection problem in two steps. First, we view networks as the results of a nested generative model that is capable of incorporating arbitrary hierarchical modular structures present at different size scales. Indeed, this model serves as a general and formally satisfying definition of arbitrary modular structure and provides a more faithful description for many empirical networks. Based on this view, we then use principled statistical inference methods to detect modules of different sizes and show that this detection approach can reliably distinguish genuine modules from spurious ones arising from noise while at the same time not suffering from severe resolution limits present in other approaches.

Our modular-hierarchy approach does not impose any specific constraints on the types of modular structures, and the statistical inference methods are derived from basic principles rather than being ad hoc. Despite its general character, the approach can be effectively combined with some of the existing community-detection techniques for efficient implementations on very large empirical networks.

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  97. This specification generalizes other hierarchical constructions in a straightforward manner. For instance, the generative model of Refs. [43, 44] can be recovered as a special case by forcing a binary tree hierarchy, terminating at the individual nodes, and a strictly assortative modular structure. A similar argument holds for the variant of Ref. [51] as well.

  98. In Ref. [32] the degree sequence entropy was taken to be NH({pk}), with pk=∑rnrpkr/N, which implicitly assumed that the degrees are uncorrelated with the block partitions, and, hence, should be interpreted only as an upper bound to the actual description length given by Eq. (9).

  99. Note that MDL can still be used to select the simpler model in this case: Although the complete description length Σ will be asymptotically the same with both models for networks sampled from the traditional block model, we still have that Lt<Lc, since the degree-corrected version still needs to include the information on the degree sequence, as in Eq. (9).

  100. The normalized mutual information (NMI) is defined as 2I({bi},{ci})/[H({bi})+H({ci})], where I({bi},{ci})=∑rspbc(r,s)ln[pbc(r,s)/pb(r)pc(r)] and H({xi})=−∑rpx(r)lnpx(r), where {bi} and {ci} are two partitions of the network.

  101. The fact that the NMI between the true and inferred partitions remains slightly above zero in Fig. 2 for ⟨k⟩<1 with the incomplete BMS criterion is a finite size effect, as it tends increasingly to zero as N→∞. On the other hand, according to this criterion, the inferred value of B in this region increases as N becomes larger.

  102. This threshold corresponds simply to the point where it becomes impossible to fully encode the block partition in the network structure, i.e., for uniform blocks −ElnB+NlnB=0, which leads to E=N and, hence, ⟨k⟩=2.

  103. This limit cannot be significantly changed even if one introduces scale parameters to the definition of modularity [15, 62].

  104. In the model selection context, adding a single edge between the blocks is not a necessary condition for the observation of the resolution limit, and has a negligible effect, differently from the modularity approach, where it is a deciding factor.

  105. An efficient and fully documented C++ implementation of the algorithm described here is freely available as part of the graph-tool Python library at http://graph-tool.skewed.de.

  106. IPv4 Routed /24 AS Links Dataset, http://www.caida.org/data/active/ipv4-routed-topology-aslinks-dataset.xml.

  107. Note that this is slightly different than in Ref. [94], which parametrized the fraction of internal and external degrees via a local mixing parameter μ, which is the same for all communities. That choice corresponds to a different parametrization of the degree-corrected block model than the one used here. However, since the blocks have different sizes, and the degrees are approximately the same in all blocks, in general, there is no choice of μ that would allow one to recover the fully random configuration model, since the intrinsic mixing would be different for each block in this case. Because of this, we have opted for the parametrization used here; however, this should not alter the interpretation of the benchmark and the comparison with Ref. [94] in a significant way.

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Phys. Rev. X 4, 011047– Published 24 March, 2014
  • Received 5 November 2013
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