## Electronic Journal of Statistics

- Electron. J. Statist.
- Volume 12, Number 2 (2018), 4697-4740.

### Estimating a network from multiple noisy realizations

Can M. Le, Keith Levin, and Elizaveta Levina

#### Abstract

Complex interactions between entities are often represented as edges in a network. In practice, the network is often constructed from noisy measurements and inevitably contains some errors. In this paper we consider the problem of estimating a network from multiple noisy observations where edges of the original network are recorded with both false positives and false negatives. This problem is motivated by neuroimaging applications where brain networks of a group of patients with a particular brain condition could be viewed as noisy versions of an unobserved true network corresponding to the disease. The key to optimally leveraging these multiple observations is to take advantage of network structure, and here we focus on the case where the true network contains communities. Communities are common in real networks in general and in particular are believed to be presented in brain networks. Under a community structure assumption on the truth, we derive an efficient method to estimate the noise levels and the original network, with theoretical guarantees on the convergence of our estimates. We show on synthetic networks that the performance of our method is close to an oracle method using the true parameter values, and apply our method to fMRI brain data, demonstrating that it constructs stable and plausible estimates of the population network.

#### Article information

**Source**

Electron. J. Statist., Volume 12, Number 2 (2018), 4697-4740.

**Dates**

Received: October 2017

First available in Project Euclid: 22 December 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.ejs/1545448230

**Digital Object Identifier**

doi:10.1214/18-EJS1521

**Mathematical Reviews number (MathSciNet)**

MR3894068

**Zentralblatt MATH identifier**

07003255

**Subjects**

Primary: 62H12: Estimation

Secondary: 62H30: Classification and discrimination; cluster analysis [See also 68T10, 91C20] 62F12: Asymptotic properties of estimators

**Keywords**

Noisy networks stochastic block model brain networks EM algorithm

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Le, Can M.; Levin, Keith; Levina, Elizaveta. Estimating a network from multiple noisy realizations. Electron. J. Statist. 12 (2018), no. 2, 4697--4740. doi:10.1214/18-EJS1521. https://projecteuclid.org/euclid.ejs/1545448230