Resolution limit in community detection

Abstract

Detecting community structure is fundamental for uncovering the links between structure and function in complex networks and for practical applications in many disciplines such as biology and sociology. A popular method now widely used relies on the optimization of a quantity called modularity, which is a quality index for a partition of a network into communities. We find that modularity optimization may fail to identify modules smaller than a scale which depends on the total size of the network and on the degree of interconnectedness of the modules, even in cases where modules are unambiguously defined. This finding is confirmed through several examples, both in artificial and in real social, biological, and technological networks, where we show that modularity optimization indeed does not resolve a large number of modules. A check of the modules obtained through modularity optimization is thus necessary, and we provide here key elements for the assessment of the reliability of this community detection method.complex networks ͉ modular structure ͉ metabolic networks ͉ social networks C ommunity detection in complex networks has attracted a lot of attention in recent years (for a review, see refs. 1 and 2). The main reason is that complex networks (3-7) are made of a large number of nodes and most previous quantitative investigations focused on statistical properties disregarding the roles played by specific subgraphs. Detecting communities (or modules) can be a way to identify substructures which could correspond to important functions. This is, for example, confirmed in the case of the World Wide Web, where communities are sets of Web pages dealing with the same topic (8). In biological networks, it is widely believed that the modular structure results from evolutionary constraints and plays a crucial role in biological functions (9-11), which makes community detection very relevant (12)(13)(14). Relevant community structures were also found in social networks (15-17), the Internet (18), food webs (19,20), and in networks of sexual contacts (21,22).Loosely speaking, a community is a subgraph of a network whose nodes are more tightly connected with each other than with nodes outside the subgraph. A decisive advance in community detection was made by Newman and Girvan (23), who introduced a quantitative measure for the quality of a partition of a network into communities, the modularity. This measure essentially compares the number of links inside a given module with the expected value for a randomized graph of the same size and same degree sequence. If one chooses modularity as the relevant quality function, the problem of community detection becomes equivalent to modularity optimization. The latter is not trivial, as the number of possible partitions of a network into clusters increases at least exponentially with the size of the network, making exhaustive optimization computationally unfeasible even for relatively small graphs. Therefore, a number of algorithms have been devised to find a good optimization t...

Modularity-Driven Clustering of Dynamic Graphs

“…In this work we set our focus on modularity [3], a measure for the goodness of a clustering. Just like any other quality index for clusterings (see, e.g., [20,2]), modularity does have certain drawbacks such as non-locality and scaling behavior [1] or resolution limit [21]. However, being aware of these peculiarities, modularity can very well be considered a robust and useful measure that closely agrees with intuition on a wide range of real-world graphs, as observed by myriad studies.…”

“…These graphs consist solely of glued cliques of authors (papers), established within single timesteps where potentially many new nodes and edges are introduced. Together with modularity's resolution limit [21] and its fondness of balanced clusters and a non-arbitrary number thereof in large graphs [29], these degenerate dynamics are adequate for fooling local algorithms that cannot regroup cliques all over as to modularity's liking: Static algorithms constantly reassess a growing component (Figs. 3a-3c), while dynamics using N or BN will sometimes have no choice but to further enlarge some growing cluster.…”

k-Clique Percolation and Clustering

“…The majority of these algorithms classify the nodes into disjoint communities, and in most cases a global quantity called modularity [56,55] is used to evaluate the quality of the partitioning. However, as pointed out in [29,49], the modularity optimisation introduces a resolution limit in the clustering, and communities containing a smaller number of edges than √ M (where M is the total number of edges) cannot be resolved. One of the big advantages of the clique percolation method (CPM) is that it identifies communities as k-clique percolation clusters, and therefore, the algorithm is local, and does not suffer from resolution problems of this type [64,21].…”

Parallel Community Detection for Massive Graphs

“…Modularity has known limitations. Fortunato and Barthélemy [10] demonstrate that global modularity optimization cannot distinguish between a single community and a group of smaller communities. Berry et al [4] provide a weighting mechanism that overcomes this resolution limit.…”