A Robust Method for Inferring Network Structures

Abstract

Inferring the network structure from limited observable data is significant in molecular biology, communication and many other areas. It is challenging, primarily because the observable data are sparse, finite and noisy. The development of machine learning and network structure study provides a great chance to solve the problem. In this paper, we propose an iterative smoothing algorithm with structure sparsity (ISSS) method. The elastic penalty in the model is introduced for the sparse solution, identifying group features and avoiding over-fitting, and the total variation (TV) penalty in the model can effectively utilize the structure information to identify the neighborhood of the vertices. Due to the non-smoothness of the elastic and structural TV penalties, an efficient algorithm with the Nesterov's smoothing optimization technique is proposed to solve the non-smooth problem. The experimental results on both synthetic and real-world networks show that the proposed model is robust against insufficient data and high noise. In addition, we investigate many factors that play important roles in identifying the performance of ISSS.Almost everything in our daily life can be modeled as complex networks, such as the social network 1, 2 , transportation network 3 , protein-to-protein network 4 and knowledge graph 5 . The interactions in these networks play an important role in identifying the networks' structure, functionality and dynamics. Hence, many researchers utilize the detailed interaction information to do a lot of interesting research works, such as the centrality measures 6, 7 , community detection 8 , robustness analysis 9-11 , controllability 12 and network game [13][14][15] . However, the link information of networks is often invisible in many cases. Hence, it's very important to propose a robust method in inferring the network structure from very few observed values (measurements) with noise, which are indirectly generated from the network.Many researchers have made great efforts in solving the network inference problem. For instance, Han and Di et al. 16 proposed the state-of-art method i.e., Lasso (Least Absolute Shrinkage and Selection Operator) 17 , which provides an estimation of a network with limited connectivity and low model prediction error. Hempe 18 proposed an inference algorithm based on inner composition alignment to identify the network structure 19, 20 on the time series data 21 . Timme 22 and Domenico Napoletani et al. 23 inferred the complete connectivity of a network from its stable response dynamics. Daniel Marbach et al. 24 summarized many robust gene network inference methods and compared their performances. Soheil Feizi et al. 25 proposed network deconvolution as a general method to distinguish direct dependencies in gene expression regulatory networks.Though many researchers have made great efforts in solving the network inference problem, it is still very challenging mainly on the following reasons. Firstly, the interactions in the network are very sparse. For most of the vertices...

Local Tomography of Large Networks Under the Low-Observability Regime

“…Regarding F 1 , most works focus on linear systems. Nonlinear dynamics are often dealt with by linearizing via considering variational characterizations of the dynamics (under small-noise regimes) [33]- [35] or by appropriately increasing the dimension of the observable space [36], [37]. In the context of May 24, 2018 DRAFT 9 linear (or linearized) systems, particular attention is paid to autoregressive diffusion models [28], [30], [38]- [40].…”

“…Step 1: Relating the error to the distance between nodes belonging to S . It is shown in [31] that the error matrix in (36) can be represented as: 5…”

“…First, we use the fact (proved in [31]), that the entries of the error matrix in (36) are nonnegative, implying, for i, j ∈ S:…”

“…Let us now examine part ii). Using one result in [31], we rewrite the entries of the error matrix in (36) as:…”

“…We first prove part i). It is shown in [31] that the entries of the error matrix defined in (36) are nonnegative, i.e., E S ≥ 0 S×S , and, hence, we can write, for i, j ∈ S:…”