Modeling Adoption and Usage of Competing Products

The emergence and wide-spread use of online social networks has led to a dramatic increase on the availability of social activity data. Importantly, this data can be exploited to investigate, at a microscopic level, some of the problems that have captured the attention of economists, marketers and sociologists for decades, such as, e.g., product adoption, usage and competition. In this paper, we propose a continuous-time probabilistic model, based on temporal point processes, for the adoption and frequency of use of competing products, where the frequency of use of one product can be modulated by those of others. This model allows us to efficiently simulate the adoption and recurrent usages of competing products, and generate traces in which we can easily recognize the effect of social influence, recency and competition. We then develop an inference method to efficiently fit the model parameters by solving a convex program. The problem decouples into a collection of smaller subproblems, thus scaling easily to networks with hundred of thousands of nodes. We validate our model over synthetic and real diffusion data gathered from Twitter, and show that the proposed model does not only provides a good fit to the data and more accurate predictions than alternatives but also provides interpretable model parameters, which allow us to gain insights into some of the factors driving product adoption and frequency of use.

💡 Research Summary

The paper introduces a continuous‑time probabilistic framework for modeling how users adopt and repeatedly use competing products in online social networks. By representing each product‑use event as a timestamped point in time, the authors employ multivariate Hawkes processes—extended to allow both excitation and inhibition—to capture three fundamental mechanisms identified in the literature: (1) social influence, where exposure to neighbors’ product usage raises adoption likelihood; (2) recency, where a user’s own recent usage of a product increases the chance of future use; and (3) competition, where usage of a competing product can suppress the propensity to use another.

The intensity function for a user‑product pair consists of three additive components: a baseline spontaneous adoption rate μₚ, a recency term weighted by parameters a_{lp} that quantify how past use of product l by the same user affects the current intensity for product p, and a social influence term weighted by b_{lp} that captures the effect of neighbors’ past uses of product l on the user’s intensity for product p. Positive a_{pp} and b_{pp} encourage repeat usage of the same product, while negative a_{lp} or b_{lp} for l ≠ p model competitive inhibition. An exponential decay kernel g(t)=exp(−ωt) is used for computational convenience, though the inference method is kernel‑agnostic.

Parameter estimation is performed by maximizing the log‑likelihood of observed event times, which yields a convex optimization problem. Crucially, the problem decomposes across users and products, enabling parallel solution and scaling to networks with hundreds of thousands of nodes. The authors exploit the sparsity of social graphs and the memoryless property of the exponential kernel to implement an O(nd|V|) simulation algorithm based on Ogata’s thinning method, where d is the maximum node degree.

Empirical evaluation proceeds in two stages. First, synthetic data generated from known parameters demonstrate that the inference procedure reliably recovers those parameters, with accuracy improving as more events are observed. Second, real‑world data from Twitter are used to study two competitive settings: (i) the adoption of URL‑shortening services (e.g., bit.ly vs. tinyurl) and (ii) the usage of two conventions for indicating retweets in 2009. The proposed model outperforms several baselines—including homogeneous Poisson processes, Weibull renewal processes, and standard Hawkes models—both in terms of log‑likelihood fit and predictive performance on held‑out data. Moreover, the learned parameters provide interpretable insights: popular products tend to be driven primarily by recency effects, whereas exposure to less popular products exerts a strong inhibitory influence on the usage of the popular ones.

In summary, the work makes three key contributions: (1) a novel continuous‑time point‑process model that jointly captures social influence, recency, and competition among multiple products; (2) scalable algorithms for both simulation and maximum‑likelihood estimation that leverage network sparsity and exponential kernels; and (3) a demonstration that the model not only fits real diffusion data better than existing approaches but also yields actionable, interpretable parameters for understanding and potentially steering product competition dynamics. Future directions suggested include non‑parametric kernel learning, hierarchical modeling of product families, and integration of the framework into prescriptive marketing decision tools.