Inward and Outward Network Influence Analysis

Abstract

Measuring heterogeneous influence across nodes in a network is critical in network analysis. This article proposes an inward and outward network influence (IONI) model to assess nodal heterogeneity. Specifically, we allow for two types of influence parameters; one measures the magnitude of influence that each node exerts on others (outward influence), while we introduce a new parameter to quantify the receptivity of each node to being influenced by others (inward influence). Accordingly, these two types of influence measures naturally classify all nodes into four quadrants (high inward and high outward, low inward and high outward, low inward and low outward, and high inward and low outward). To demonstrate our four-quadrant clustering method in practice, we apply the quasi-maximum likelihood approach to estimate the influence parameters, and we show the asymptotic properties of the resulting estimators. In addition, score tests are proposed to examine the homogeneity of the two types of influence parameters. To improve the accuracy of inferences about nodal influences, we introduce a Bayesian information criterion that selects the optimal influence model. The usefulness of the IONI model and the four-quadrant clustering method is illustrated via simulation studies and an empirical example involving customer segmentation.

Keywords:

• Customer segmentation
• Four-quadrant clustering
• Inward and outward network influence model
• Model selection
• Quasi-maximum likelihood estimation

Supplementary Materials

The online supplementary material includes six sections. Section 1 presents detailed expressions of the Fisher information matrix ${\mathcal{I}}_n(\theta )$ and the quantity ${\mathcal{J}}_n(\theta ,{\mu }_3,{\mu }_4)$ used in estimating the asymptotic covariance matrix of parameter estimators. Additional notations used for proving Theorem 3 and Propositions 1-2 are given in Section 2. Section 3 provides five technical lemmas. Section 4 presents the proofs of theorems and propositions. Simulation studies for mixture normal errors and for an overlapping design are provided in Sections 5 and 6, respectively. Both additional studies are used to demonstrate the robustness of our proposed estimators.

Acknowledgments

The authors are grateful to the editor, associate editor, and anonymous referees for their insightful comments and constructive suggestions.

Additional information

Funding

Wei Lan’s research was supported by the National Natural Science Foundation of China (NSFC,71991472, 12171395, 11931014, 71532001), the Joint Lab of Data Science and Business Intelligence at Southwestern University of Finance and Economics, and the Fundamental Research Funds for the Central Universities (JBK1806002). Yujia Wu’s research was supported by the Fundamental Research Funds for the Central Universities (JBK2107186). Tao Zou’s research was supported by ANU College of Business and Economics Early Career Researcher Grant, the RSFAS Cross Disciplinary Grant.