Communications in Mathematical Sciences

Volume 19 (2021)

Number 1

An MBO scheme for clustering and semi-supervised clustering of signed networks

Pages: 73 – 109



Mihai Cucuringu (Department of Statistics and Mathematical Institute, University of Oxford, United Kingdom)

Andrea Pizzoferrato (Department of Mathematical Sciences, University of Bath, United Kingdom; and Alan Turing Institute, London, United Kingdom)

Yves van Gennip (Delft Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands)


We introduce a principled method for the signed clustering problem, where the goal is to partition a weighted undirected graph whose edge weights take both positive and negative values, such that edges within the same cluster are mostly positive, while edges spanning across clusters are mostly negative. Our method relies on a graph-based diffuse interface model formulation utilizing the Ginzburg–Landau functional, based on an adaptation of the classic numerical Merriman–Bence–Osher (MBO) scheme for minimizing such graph-based functionals. The proposed objective function aims to minimize the total weight of inter-cluster positively-weighted edges, while maximizing the total weight of the inter-cluster negatively-weighted edges. Our method scales to large sparse networks, and can be easily adjusted to incorporate labelled data information, as is often the case in the context of semisupervised learning. We tested our method on a number of both synthetic stochastic block models and real-world data sets (including financial correlation matrices), and obtained promising results that compare favourably against a number of state-of-the-art approaches from the recent literature.


Merriman–Bence–Osher scheme, threshold dynamics, clustering, signed networks, graph Laplacians, spectral methods, time series

2010 Mathematics Subject Classification

05C22, 05C85, 15A99, 35Q56, 35R02, 49K15, 62H30, 68R10

Received 5 February 2020

Accepted 30 July 2020

Published 24 March 2021