  Measuring heterogeneous influence across nodes in a network is critical in
network analysis. This paper 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, 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.
