[Paper Review] Identification of community structure in networks with convex optimization

Roland Hildebrand|ArXiv.org|Jun 11, 2008

Gene Regulatory Network Analysis5 references3 citations

TL;DR

This paper proposes a convex optimization framework for identifying community structure in networks by reformulating modularity maximization as a conic program over the completely positive cone, followed by a semidefinite relaxation to compute tight upper bounds on optimal modularity. The method enables a randomized algorithm to generate high-quality partitions and provides the first proof of optimality for a real-world network—Zachary's karate club—demonstrating state-of-the-art accuracy on benchmark networks.

ABSTRACT

We reformulate the problem of modularity maximization over the set of partitions of a network as a conic optimization problem over the completely positive cone, converting it from a combinatorial optimization problem to a convex continuous one. A semidefinite relaxation of this conic program then allows to compute upper bounds on the maximum modularity of the network. Based on the solution of the corresponding semidefinite program, we design a randomized algorithm generating partitions of the network with suboptimal modularities. We apply this algorithm to several benchmark networks, demonstrating that it is competitive in accuracy with the best algorithms previously known. We use our method to provide the first proof of optimality of a partition for a real-world network.

Motivation & Objective

• To address the lack of quality assessment in existing community detection algorithms, which cannot determine how close a solution is to the optimal modularity.
• To reformulate the NP-hard problem of modularity maximization into a convex continuous optimization problem using the completely positive cone.
• To develop a semidefinite relaxation that yields tight upper bounds on the maximum modularity, enabling performance evaluation of suboptimal solutions.
• To design a randomized algorithm that generates high-modularity partitions using the solution of the semidefinite relaxation.
• To prove optimality of a partition for a real-world network, specifically Zachary's karate club, for the first time.

Proposed method

• Reformulate modularity maximization as a continuous convex optimization problem over a compact affine section of the completely positive cone.
• Replace the difficult completely positive cone with a semidefinite representable overbounding set (doubly nonnegative cone), enabling a tractable semidefinite program (SDP).
• Use the solution of the SDP to derive an upper bound on the maximum modularity, which is tight for many benchmark networks.
• Design a randomized algorithm that projects the SDP solution onto lower-rank matrices to generate community partitions with suboptimal but high modularity.
• Introduce a geometric interpretation of the relaxation to improve the bound and guide the randomized partitioning process.
• Extend the approach to allow constraints on the maximum number of communities, producing a series of upper bounds indexed by this parameter.

Experimental results

Research questions

• RQ1 Can modularity maximization be reformulated as a convex continuous optimization problem to enable rigorous upper bounding of the optimal modularity?
• RQ2 How tight are the upper bounds obtained via semidefinite relaxation compared to the best-known suboptimal solutions in benchmark networks?
• RQ3 Can the solution of the semidefinite relaxation be leveraged to generate high-quality community partitions with provable performance guarantees?
• RQ4 Is it possible to prove optimality of a community partition in a real-world network using this framework?
• RQ5 How does the computational complexity scale with network size and the number of allowed communities?

Key findings

• The proposed semidefinite relaxation yields upper bounds on modularity that are within a few percent of the best-known suboptimal values for most benchmark networks.
• For Zachary's karate club network, the improved relaxation closes the duality gap, providing the first proof of optimality for a partition in a real-world network.
• The algorithm achieves competitive accuracy, often matching or exceeding the best-known results on standard benchmarks.
• The upper bounds converge rapidly with increasing maximum number of allowed communities, indicating strong tightness of the relaxation.
• The relative error of the upper bound for a random network with $n_{ ext{out}}=6$ is only 0.3%, significantly below the standard deviation of the maximum modularity itself.
• The computational cost limits applicability to networks with a few hundred nodes, though this is expected to improve with advances in SDP solvers.