[Paper Review] Maximum-Likelihood Network Reconstruction for SIS Processes is NP-Hard

Bastian Prasse, Piet Van Mieghem|arXiv (Cornell University)|Jul 23, 2018

Complex Network Analysis Techniques8 references4 citations

TL;DR

This paper proves that maximum-likelihood network reconstruction for continuous-time SIS epidemic processes is NP-hard, even when the infection and curing rates are known. By constructing a polynomial-time reduction from the NP-complete maximum cut problem to the SIS network reconstruction problem, the authors show that inferring the true network topology from viral state observations is computationally intractable for any connected graph.

ABSTRACT

The knowledge of the network topology is imperative to precisely describing the viral dynamics of an SIS epidemic process. In scenarios for which the network topology is unknown, one resorts to reconstructing the network from observing the viral state trace. This work focusses on the impact of the viral state observations on the computational complexity of the resulting network reconstruction problem. We propose a novel method of constructing a specific class of viral state traces from which the inference of the presence or absence of links is either easy or difficult. In particular, we use this construction to prove that the maximum-likelihood SIS network reconstruction is NP-hard. The NP-hardness holds for any adjacency matrix of a graph which is connected.

Motivation & Objective

• To determine the computational complexity of maximum-likelihood (ML) network reconstruction in SIS epidemic processes given observed viral state traces.
• To investigate whether the ML estimation problem for SIS processes is tractable or computationally hard.
• To establish a formal connection between SIS network reconstruction and known NP-hard problems.
• To prove that the ML network reconstruction problem remains NP-hard for all connected graphs, regardless of structure.

Proposed method

• Constructs a specific class of viral state traces from which the presence or absence of links in the network can be inferred.
• Uses a polynomial-time reduction from the maximum cut problem—a known NP-complete problem—to the ML SIS network reconstruction problem.
• Demonstrates that solving the ML estimation problem is equivalent to solving a zero-one unconstrained quadratic program (UQP), which is NP-hard.
• Employs a formal transformation of the log-likelihood objective function into a form resembling the UQP, showing equivalence under specific parameter settings.
• Applies asymptotic analysis and parameter tuning (e.g., transition counts and probabilities) to ensure that the ML solution aligns with the solution of the maximum cut problem.
• Uses lemmas to show that the ML estimate must preserve known link states in the constructed trace, reducing the problem to a core optimization over a subset of edges.

Experimental results

Research questions

• RQ1 Is the maximum-likelihood network reconstruction problem for SIS processes computationally tractable?
• RQ2 Can the SIS network reconstruction problem be reduced to a known NP-hard problem?
• RQ3 Does the NP-hardness of SIS reconstruction hold for all connected graphs?
• RQ4 What is the relationship between the log-likelihood objective in SIS reconstruction and the zero-one unconstrained quadratic program (UQP) formulation?
• RQ5 Can a viral state trace be engineered such that solving the ML reconstruction is equivalent to solving the maximum cut problem?

Key findings

• The maximum-likelihood SIS network reconstruction problem is NP-hard for any connected adjacency matrix.
• The NP-hardness is established via a polynomial-time reduction from the maximum cut problem to the ML estimation problem.
• The log-likelihood objective function in the SIS reconstruction problem can be made equivalent to a zero-one unconstrained quadratic program (UQP), which is NP-hard.
• The reduction holds regardless of the specific structure of the underlying graph, as long as it is connected.
• The solution to the ML estimation problem must satisfy the same constraints as the maximum cut problem when the viral state trace is constructed appropriately.
• The computational intractability persists even when infection and curing rates are known, and the network is undirected and unweighted.