Finding community structure in networks using the eigenvectors of matrices

We consider the problem of detecting communities or modules in networks, groups of vertices with a higher-than-average density of edges connecting them. Previous work indicates that a robust approach to this problem is the maximization of the benefit function known as “modularity” over possible divisions of a network. Here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix, which plays a role in community detection similar to that played by the graph Laplacian in graph partitioning calculations. This result leads us to a number of possible algorithms for detecting community structure, as well as several other results, including a spectral measure of bipartite structure in networks and a centrality measure that identifies vertices that occupy central positions within the communities to which they belong. The algorithms and measures proposed are illustrated with applications to a variety of real-world complex networks.

Figure 1

(Color online) (a) The mesh network of Bern et al. [50]. (b) The best division into equal-sized parts found by the spectral partitioning algorithm based on the Laplacian matrix.

Figure 2

(Color online) The dolphin social network of Lusseau et al. [70]. The dashed curve represents the division into two equally sized parts found by a standard spectral partitioning calculation (Sec. 2). The solid curve represents the division found by the modularity-based method of this section. And the squares and circles represent the actual division of the network observed when the dolphin community split into two as a result of the departure of a keystone individual. (The individual who departed is represented by the triangle.)

Figure 3

(Color online) The network of political books described in the text. Vertex shades represent the values of the corresponding elements of the leading eigenvector of the modularity matrix.

Figure 4

(Color online) A plot of the vertex vectors ${\mathbf{r}}_i$ for a small network with $p=2$. The dotted line represents one of the $n$ possible topologically distinct cut planes.

Figure 5

Division by the method of optimal modularity of a simple network consisting of eight vertices in a line. (a) The optimal division into just two parts separates the network symmetrically into two groups of four vertices each. (b) The optimal division into any number of parts divides the network into three groups as shown here.

Figure 6

(Color online) A small example of an approximately bipartite network. The network is composed of two groups of vertices, and most edges run between vertices in different groups.

Figure 7

(Color online) (a) The network of commonly occurring English adjectives (circles) and nouns (squares) described in the text. (b) The same network redrawn with the nodes grouped so as to minimize the modularity of the grouping. The network is now revealed to be approximately bipartite, with one group consisting almost entirely of adjectives and the other of nouns.

Figure 8

(Color online) A network of coauthorships between 379 scientists whose research centers on the properties of networks of one kind or another. Vertex diameters indicate the community centrality, and the ten vertices with highest centralities are highlighted. For those readers curious about the identities of the vertices, an annotated version of this figure, names and all, can be found in Ref. 84 Inset: a scatter plot of community centrality against vertex degrees. Like most centrality measures, this one is correlated with degree, though only moderately strongly.