Limit Theorems for Network Data without Metric Structure

TL;DR

This paper establishes new limit theorems for network-dependent data without relying on metric space assumptions, broadening applicability to various social and financial networks.

Contribution

It introduces a generalized framework for limit theorems in network data that does not depend on underlying metric structures, unlike previous approaches.

Findings

Derived inequalities, laws of large numbers, and central limit theorems for network data.

Provided primitive conditions for spatial autoregressive models.

Applicable to a wider range of networks beyond metric-based assumptions.

Abstract

This paper develops limit theorems for random variables with network dependence, without requiring the individuals in the network to be located in a Euclidean or metric space. This distinguishes our approach from most existing limit theorems in network statistics and econometrics, which are based on weak dependence concepts such as strong mixing, near-epoch dependence, or $\psi$-dependence. All these weak dependence concepts presuppose an underlying metric. By relaxing the assumption of an underlying metric space, our theorems can be applied to a broader range of network data, including financial and social networks. To derive the limit theorems, we generalize the concept of functional dependence (also known as physical dependence) from time series to random variables with network dependence. Using this framework, we establish several inequalities, a law of large numbers, and central limit theorems. Furthermore, we demonstrate the verifiability of our high-level conditions by deriving primitive sufficient conditions for spatial autoregressive models, which are widely used in network data analysis.