Community Detection in Networks via Nonlinear Modularity Eigenvectors

Abstract

Revealing a community structure in a network or dataset is a central problem arising in many scientific areas. The modularity function Q is an established measure quantifying the quality of a community, being identified as a set of nodes having high modularity. In our terminology, a set of nodes with positive modularity is called a module and a set that maximizes Q is thus called leading module. Finding a leading module in a network is an important task, however the dimension of real-world problems makes the maximization of Q unfeasible. This poses the need of approximation techniques which are typically based on a linear relaxation of Q, induced by the spectrum of the modularity matrix M . In this work we propose a nonlinear relaxation which is instead based on the spectrum of a nonlinear modularity operator M. We show that extremal eigenvalues of M provide an exact relaxation of the modularity measure Q, in the sense that the maximum eigenvalue of M is equal to the maximum value of Q, however at the price of being more challenging to be computed than those of M . Thus we extend the work made on nonlinear Laplacians, by proposing a computational scheme, named generalized RatioDCA, to address such extremal eigenvalues. We show monotonic ascent and convergence of the method. We finally apply the new method to several synthetic and real-world data sets, showing both effectiveness of the model and performance of the method.

A Nonlinear Spectral Method for Core--Periphery Detection in Networks

“…This philosophy is in line with classical and widely used reordering and clustering methods that make use of the Fiedler or the Perron-Frobenius eigenvectors [14]. However, in the core-periphery setting considered here, the resulting relaxed problem is equivalent to an eigenvalue problem that is inherently nonlinear and is reminiscent of more recent clustering and reordering techniques that exploit nonlinear eigenvectors [7,32,33,34]. Hence, we developed new results in nonlinear Perron-Frobenius theory in order to derive and analyze the algorithm.…”

The self-consistent field iteration for p-spectral clustering

“…While being widely explored in various areas, it is known that clustering methods based on linear graph operators may perform poorly on real-world data. To overcome this issue, nonlinear graph operators have been introduced in recent years as a generalization of more standard linear graph mappings [4,6,37]. Among them, the p-Laplacian received considerable attention due to its simple definition, its connection with Cheeger (isoperimetric) constants and nodal domains on graphs and manifolds, as well as its remarkable clustering performance [6,19,35,36].…”

“…While being theoretically better suited for clustering purposes, computing the spectrum of the p-Laplacian for p = 2 is, in general, severely more challenging than the linear case p = 2. This has motivated several authors to focus on algorithms to compute eigenpairs of the p-Laplacian for general p ≥ 1 An inverse power method for the 1-Laplacian was proposed in [14], later extended into the RatioDCA algorithm and its generalizations [15,18,37]. A modified gradient descent method is used to find eigenvectors that satisfy an orthogonality condition in [22].…”

Generalized K-Core Percolation in Networks with Community Structure

“…Transportation networks have more connections within cities than between cities. Due to numerous applications across disciplines such as technology, biology, and social sciences, detecting and characterizing community structure has been a very prolific research area for many years; see, e.g., [11,28,29,35,27] and references therein.…”