PurposePurchase conversion prediction aims to improve user experience and convert visitors into real buyers to drive sales of firms; however, the total conversion rate is low, especially for e-retailers. To date, little is known about how e-retailers can scientifically detect users' intents within a purchase conversion funnel during their ongoing sessions and strategically optimize real-time marketing tactics corresponding to dynamic intent states. This study mainly aims to detect a real-time state of the conversion funnel based on graph theory, which refers to a five-class classification problem in the overt real-time choice decisions (RTCDs)—click, tag-to-wishlist, add-to-cart, remove-from-cart and purchase—during an ongoing session.Design/methodology/approachThe authors propose a novel graph-theoretic framework to detect different states of the conversion funnel by identifying a user's unobserved mindset revealed from their navigation process graph, namely clickstream graph. First, the raw clickstream data are identified into individual sessions based on a 30-min time-out heuristic approach. Then, the authors convert each session into a sequence of temporal item-level clickstream graphs and conduct a temporal graph feature engineering according to the basic, single-, dyadic- and triadic-node and global characteristics. Furthermore, the synthetic minority oversampling technique is adopted to address with the problem of classifying imbalanced data. Finally, the authors train and test the proposed approach with several popular artificial intelligence algorithms.FindingsThe graph-theoretic approach validates that users' latent intent states within the conversion funnel can be interpreted as time-varying natures of their online graph footprints. In particular, the experimental results indicate that the graph-theoretic feature-oriented models achieve a substantial improvement of over 27% in line with the macro-average and micro-average area under the precision-recall curve, as compared to the conventional ones. In addition, the top five informative graph features for RTCDs are found to be Transitivity, Edge, Node, Degree and Reciprocity. In view of interpretability, the basic, single-, dyadic- and triadic-node and global characteristics of clickstream graphs have their specific advantages.Practical implicationsThe findings suggest that the temporal graph-theoretic approach can form an efficient and powerful AI-based real-time intent detecting decision-support system. Different levels of graph features have their specific interpretability on RTCDs from the perspectives of consumer behavior and psychology, which provides a theoretical basis for the design of computer information systems and the optimization of the ongoing session intervention or recommendation in e-commerce.Originality/valueTo the best of the authors' knowledge, this is the first study to apply clickstream graphs and real-time decision choices in conversion prediction and detection. Most studies have only meditated on a binary classification problem, while this study applies a graph-theoretic approach in a five-class classification problem. In addition, this study constructs temporal item-level graphs to represent the original structure of clickstream session data based on graph theory. The time-varying characteristics of the proposed approach enhance the performance of purchase conversion detection during an ongoing session.