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Modeling Diffusion of Competing Products and Conventions in Social Media

Isabel Valera, Manuel Gomez-Rodriguez, Krishna Gummadi

arXiv:1406.0516v3cs.SIphysics.soc-ph

TL;DR

The paper addresses how competing products and social conventions are adopted and repeatedly used by individuals, a question enabled by increasingly fine-grained social-media activity data. It develops a scalable continuous-time Hawkes model with cooperation and competition, fits it through decoupled convex optimization, and validates it on synthetic and Twitter data. The model provides predictive performance and interpretable parameters that reveal factors associated with adoption and usage success.

  • Problem

    Fine-grained evidence for modeling individual adoption and recurrent use of competing products and social conventions has been limited, despite longstanding interest in product adoption and competition.

    Source: §1 · p. 1

  • Method

    The paper uses multidimensional continuous-time Hawkes processes with cooperation and competition, fitting their parameters through a decoupled convex maximum-likelihood program.

    Source: §1 · p. 2

  • Results

    The model provides better fit than a Poisson baseline, typically outperforms Poisson and linear regression for per-user usage prediction, and predicts true product rankings for more than 70% of users.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Interpretable model parameters identify factors associated with the success and usage dynamics of competing products and conventions.

    Source: §1 · p. 2, §4 · p. 9

    External global events can substantially degrade predicted product rankings, and the model uses fixed exponential kernels and constant baseline rates.

    Source: §3.2 · p. 8, §4 · p. 9

Abstract

from arXiv · show

The emergence and wide-spread use of social networks and microblogging sites has led to a dramatic increase on the availability of users' activity data. Importantly, this data can be exploited to solve some of the problems that have captured the attention of economists and marketers for decades as, e.g., product adoption, product competition and product life cycle. In this paper, we leverage on users' activity data from a popular microblogging site to model and predict the competing dynamics of products and social conventions adoptions. To this aim, we propose a data-driven model, based on continuous-time Hawkes processes, for the adoption and frequency of use of competing products and conventions. 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 method over synthetic and real diffusion data gathered from Twitter, and show that the proposed model does not only present a good predictive power but also provides interpretable model parameters, which allow us to gain insights into the fundamental principles that drive product and convention adoptions.

1. INTRODUCTION

The paper uses fine-grained social-media activity data to model competing product and convention diffusion at both individual and aggregate levels. It proposes a scalable continuous-time Hawkes framework that captures recurrent use, cooperation, and competition, and reports predictive and interpretive benefits on synthetic and Twitter data.

Motivation

  • Fine-grained social-media activity data makes individual adoptions and recurrent product uses observable, enabling models of both microscopic and macroscopic diffusion dynamics.Earlier economic and marketing models generally captured macroscopic evolution rather than specific individuals’ adoption and usage.Source: §1 · p. 1

Modeling approach

  • Multidimensional Hawkes processes model self-excitation and are generalized to represent both cooperative and competitive effects among products or conventions.The parameters are fitted by maximum likelihood through a convex program that decouples into smaller, parallelizable problems for networks with hundreds of thousands of nodes.Source: §1 · pp. 1–2

Empirical validation

  • On synthetic and Twitter data, the Hawkes model improves fit over a Poisson baseline, typically predicts per-user usage better than Poisson and linear regression, and predicts product rankings for more than 70% of users.The Twitter corpus contains 1.7 billion tweets from 52 million users collected between March 2006 and September 2009.Source: §1 · p. 2

Interpretation

  • The model indicates that popular products and conventions are associated with self-excitation or cooperation, while less popular ones are used more spontaneously and inhibit popular alternatives more strongly.These interpretable effects are presented as factors associated with adoption and usage success across Twitter users.Source: §1 · p. 2

Modeling approach

  • The model represents competing products and conventions as asynchronous continuous-time events, avoiding arbitrary time bins and restrictive exponential inter-event assumptions.Prior diffusion work often modeled contagions independently, whereas this framework incorporates competition and cooperation.Source: §1 · p. 2
  • Users may repeatedly use products, stop using them, and use multiple competing products simultaneously with different time-varying frequencies.This extends prior adoption models that treated use as a one-time, progressive process.Source: §1 · p. 2

2. MODEL AND PROPOSED METHOD

The paper models repeated, competing product uses as a continuous-time, network-aware counting process and fits its parameters through a convex maximum-likelihood problem. The formulation captures self-excitation, cross-product inhibition, and neighbor influence while decomposing into parallel subproblems that scale to very large networks.

2.1 Data

  • The observed data are temporally ordered product-use events for users, where each user may use a product once or repeatedly during an observation window.The model represents adoption and recurrent usage rather than only first adoption.Source: §2.1 · p. 3

2.2 Adoption and frequency of use as a multidimensional counting process

  • A user’s product-specific intensity is the instantaneous frequency of use conditioned on the usage history of the user and her neighbors.This history supports predicting both which products a user will use and how frequently.Source: §2.2 · p. 3

2.3 Nodes’ intensities and multidimensional Hawkes processes

  • The multidimensional Hawkes model combines spontaneous adoption, self-excitation, cross-product inhibition, neighbor influence, and time-decaying event effects.The base intensity represents spontaneous use, while signed cross-product parameters model competition and a positive kernel models temporal decay.Source: §2.3 · p. 3

2.4 Likelihood of product uses

  • The likelihood factorizes over observed uses and survival periods up to the observation-window cutoff, enabling maximum-likelihood fitting from users’ event histories.The survival term accounts for the interval from the last use of each product to the cutoff T.Source: §2.4 · p. 4

2.5 Competing adoption inference problem

  • The method is implemented by solving a regularized maximum-likelihood optimization problem for each user’s base, within-product, and neighbor-related parameters.The objective combines negative log-likelihood with a regularization term controlled by β; CVX is used in the experiments.Source: §2.5 · pp. 4–5
  • The competing adoption inference problem is jointly convex, decomposes into |V| × |P| independent subproblems, and can be solved in parallel with globally optimal local solutions.This decomposition is reported to support scaling to networks on the order of hundreds of thousands of nodes.Source: §2.5 · p. 5

3. EXPERIMENTAL EVALUATION

Experiments on synthetic and Twitter data evaluate parameter recovery, predictive performance, ranking accuracy, and interpretability of the competing-product and convention model.

3.1 Experiments on synthetic data

  • With β = 10 and w = 1, MSE decreases as average events per user increases, reaching below 0.18 for all three 512-user networks, while inference performance remains remarkably stable despite their different global structures.Source: §3.1 · p. 5

3.2 Experiments on real data

  • Hawkes-based models achieve the minimum AIC for more than 90% of URL-shortening-service users and 80% of retweet-convention users, with WCP followed by Full performing best.WNET is rarely best by AIC because of its larger parameter count and lower log-likelihood for popular products and conventions.Source: Table 3 · p. 6, §3.2 · p. 7
  • All three Hawkes models outperform Poisson in test log-likelihood for every evaluated service and convention except ‘RT’, where WNET still beats Poisson.Among Hawkes variants, the best model depends on the product or convention; WCP best fits ‘tinyURL’, while Full best fits ‘via’ and ‘retweeting’ and WNET best fits ‘RT’.Source: Table 4 · p. 6, Figure 2 · p. 6, Figure 3 · p. 6
  • More than 72% of users have correctly recovered product or convention rankings, with WNET and Poisson achieving indistinguishable accuracy.For cumulative test events, Figure 4 compares true and estimated counts over time for WNET.Source: Table 5 · p. 6, Figure 4 · p. 6, §3.2 · p. 8
  • At least one Hawkes model outperforms Poisson and linear regression on Spearman correlation for all but ‘tinyURL’ and ‘Retweeting’, where Poisson is best.WNET leads for three URL services and three conventions, while Full leads for one service and one convention.Source: Figure 5 · p. 7, Figure 6 · p. 7, §3.2 · p. 8
  • A global default-service change reduces ranking accuracy by up to 20%, indicating that model parameters may require retraining after external interventions.Twitter’s switch from ‘tinyurl.com’ to ‘bit.ly’ triggered substantial adoption changes shortly afterward.Source: §3.2 · p. 8, Figure 7 · p. 8, Table 6 · p. 8
  • Popular products and conventions are mainly driven by self-excitement or neighbor influence, whereas less popular ones are mainly adopted spontaneously.Less popular products and conventions also exert strong inhibitory effects on more popular alternatives; the inferred Full model supports these interaction patterns.Source: §3.2 · p. 8

4. CONCLUSIONS

The paper presents a Hawkes-process model and efficient inference algorithm for competing product and convention diffusion, with interpretable parameters for analyzing adoption drivers. Its validation and conclusions also identify important scope boundaries for the model and data.

  • The model combines interpretable cooperation and competition dynamics with an efficient inference algorithm for predicting individual adoption and usage patterns.The inference problem is solved through smaller decoupled subproblems, supporting scalability to large networks.Source: §4 · p. 9
  • The validation relies exclusively on Twitter data, leaving evaluation on other microblogging and social networking services as future work.The paper specifically mentions Weibo, Tumblr, Facebook, G+, and Tuenti as possible additional settings.Source: §4 · p. 9
  • Table 10 reports retweet conventions using the percentage of users with µu p > 0, au pp > 0, and au pp > 0.Source: Table 10 · p. 9
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