  Centrality is a fundamental network property which ranks nodes by their
structural importance. However, structural importance may not suffice to
predict successful diffusions in a wide range of applications, such as
word-of-mouth marketing and political campaigns. In particular, nodes with high
structural importance may contribute negatively to the objective of the
diffusion. To address this problem, we propose contextual centrality, which
integrates structural positions, the diffusion process, and, most importantly,
nodal contributions to the objective of the diffusion. We perform an empirical
analysis of the adoption of microfinance in Indian villages and weather
insurance in Chinese villages. Results show that contextual centrality of the
first-informed individuals has higher predictive power towards the eventual
adoption outcomes than other standard centrality measures. Interestingly, when
the product of diffusion rate $p$ and the largest eigenvalue $\lambda_1$ is
larger than one and diffusion period is long, contextual centrality linearly
scales with eigenvector centrality. This approximation reveals that contextual
centrality identifies scenarios where a higher diffusion rate of individuals
may negatively influence the cascade payoff. Further simulations on the
synthetic and real-world networks show that contextual centrality has the
advantage of selecting an individual whose local neighborhood generates a high
cascade payoff when $p \lambda_1 < 1$. Under this condition, stronger homophily
leads to higher cascade payoff. Our results suggest that contextual centrality
captures more complicated dynamics on networks and has significant implications
for applications, such as information diffusion, viral marketing, and political
campaigns.
