Predicting Adoption Probabilities in Social Networks

Xiao Fang; Paul J. Hu; Zhepeng Li; Weiyu Tsai

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Predicting Adoption Probabilities in Social Networks

Xiao Fang, Paul J. Hu, Zhepeng Li, Weiyu Tsai

arXiv:1309.6369v1Sep 24, 2013cs.SIphysics.soc-ph

TL;DR

The paper studies how to predict future adoption probabilities, an important but underexplored problem for social-network analysis and applications. It develops a locally-weighted expectation-maximization method for Naïve Bayesian learning that incorporates social influence, structural equivalence, entity similarity, and confounding factors. Results from two large-scale social networks indicate that social-influence-only cascade methods have limited predictive power and that confounding factors are important.

Problem
Predicting adoption probabilities is important for social-network analysis and applications but has received insufficient research attention, especially when confounding factors are unobserved.
Method
The paper develops a locally-weighted expectation-maximization method for Naïve Bayesian learning to predict adoption probabilities from multiple social-network factors.
Results
Across two social networks, cascade methods relying primarily on social influence offered limited predictive power, while confounding factors were critical to adoption-probability predictions.
Takeaways & Limitations
Adoption probabilities should be analyzed and predicted using several related but distinct forces and comprehensive social-network data beyond social influence.
Takeaways & Limitations
The method may not scale to social networks with millions of nodes because of its computational requirements.

Abstract

from arXiv · show

In a social network, adoption probability refers to the probability that a social entity will adopt a product, service, or opinion in the foreseeable future. Such probabilities are central to fundamental issues in social network analysis, including the influence maximization problem. In practice, adoption probabilities have significant implications for applications ranging from social network-based target marketing to political campaigns; yet, predicting adoption probabilities has not received sufficient research attention. Building on relevant social network theories, we identify and operationalize key factors that affect adoption decisions: social influence, structural equivalence, entity similarity, and confounding factors. We then develop the locally-weighted expectation-maximization method for Naïve Bayesian learning to predict adoption probabilities on the basis of these factors. The principal challenge addressed in this study is how to predict adoption probabilities in the presence of confounding factors that are generally unobserved. Using data from two large-scale social networks, we demonstrate the effectiveness of the proposed method. The empirical results also suggest that cascade methods primarily using social influence to predict adoption probabilities offer limited predictive power, and that confounding factors are critical to adoption probability predictions.

1. Introduction

The paper addresses the underexplored problem of predicting future adoption probabilities for nonadopters, which supports social-network applications and influence maximization. It argues that reliable prediction requires factors beyond social influence, including structural equivalence, entity characteristics, and confounding factors.

1. Introduction: Adoption-probability predictions support applications such as viral marketing, target marketing, demand prediction, public health, and politics.Viral marketing in particular depends on predicting which consumers will adopt after initially targeted consumers adopt.
1. Introduction: Predicting future adoption probabilities for individuals who have not yet adopted is an underexplored problem with implications for influence maximization and social-network applications.Existing influence-maximization methods often treat adoption probabilities as arbitrary or given, although they are not generally available.
1. Introduction: Cascade methods model adoption through neighbors’ social influence and commonly assume that adopter neighbors act independently.Influence probabilities may be fixed, set to 1/k, or learned while retaining these assumptions.
1. Introduction: Existing methods may provide unreliable predictions because they focus mainly on social influence and neighboring adopters while omitting structural equivalence, entity characteristics, interactions, and confounding factors.Adoption decisions may also involve adopters who are not direct neighbors, as well as intrinsic characteristics and interaction relations.
1. Introduction: The paper develops a Bayesian learning method that predicts adoption probabilities using multiple factors and addresses the challenge of generally unobserved confounding factors.The method is evaluated with data from two large-scale social networks.
1. Introduction: Empirical results suggest that cascade methods relying exclusively on social influence have limited effectiveness, whereas incorporating confounding factors substantially improves adoption-probability prediction.The findings support analyzing adoption decisions through several related but distinct forces and more comprehensive social-network data.

2. Key Factors Underlying Adoption Decision: Theoretical Foundations and Operationalization

The paper identifies social influence, structural equivalence, entity similarity, and unobserved confounding factors as forces underlying adoption decisions, and operationalizes the observable factors with social-network data.

Social influence: Stronger social ties generally produce greater influence because communication, trust, and reciprocal investment connect individuals’ opinions and behaviors.
Structural equivalence: Structural equivalence links similar opinions and behaviors to occupying comparable network positions, even without direct communication.
Operationalization: The paper therefore proposes weighting entities by both structural equivalence and communications rather than relying only on direct communications.
Entity similarity: Entity similarity captures demographic and behavioral resemblance, which is associated with similar needs, preferences, prior responses, and future adoption behavior.
Confounding factors: Unobserved confounding factors may account for significant variance in observed adoption behavior and can bias social-influence analyses when omitted.
2. Key Factors Underlying Adoption Decision: Theoretical Foundations and Operationalization: Theoretical foundations identify social influence, structural equivalence, entity similarity, and unobserved confounding factors as key forces underlying adoption decisions.

3. Predicting Adoption Probabilities: Problem Formulation and Proposed Method

The paper formulates next-step adoption-probability prediction from observed adoption histories and develops LEMNB to learn with incomplete data and hidden confounding power.

Problem formulation: Given adoption decisions and adoption times observed through time J, the task is to predict each nonadopter’s probability of adopting at time J + 1.
3.1 Analysis of Powers Underlying Adoption Decision: Observed adoption information is converted into training data by comparing adopters with nonadopters and calculating influence, equivalence, and similarity powers over time.
3.2 Predicting Adoption Probabilities: The target probability conditions on influence, equivalence, similarity, and hidden confounding power, using Bayes rule and Naïve Bayes factorization.
Problem formulation: Training records contain influence, equivalence, similarity, and adoption decision but omit hidden confounding power, making both training and prediction incomplete.
3.2.1 Learning z: LEMNB combines expectation-maximization with local learning to estimate parameters from incomplete training data while weighting records near the test case.

4. Empirical Evaluations

The proposed method was evaluated on mobile and avatar social networks using weekly adoption snapshots, then compared with eight benchmark methods. It substantially outperformed the benchmarks, while results showed that influence-only cascade methods have limited predictive power and hidden confounding factors are important.

4.1 Data and Evaluation Procedure: Evaluations used weekly mobile-network snapshots to predict each nonadopter’s adoption probability in the following week.The mobile dataset covered 34,797 users and 52 weekly snapshots.
4.2 Evaluation Results and Analyses: Across 50 evaluations, the proposed method achieved an average AUC of 0.8029 versus 0.6910 for k-NN.The proposed method significantly outperformed each benchmark method, with p < 0.001.
4.2 Evaluation Results and Analyses: Cascade and influence-probability methods had AUCs barely above 0.5 because they often assigned zero probability when users had no adopting neighbors.Across evaluations, 95.78% of adopters and 98.51% of nonadopters had zero neighbors adopting in the prior week.
4.2 Evaluation Results and Analyses: The proposed method outperformed NB, LWNB, SVM, and k-NN, consistent with its explicit consideration of hidden confounding power.The results also indicate that NB significantly outperformed LWNB, with p < 0.01.
4.2 Evaluation Results and Analyses: Similar comparative results emerged in the avatar social network, supporting the method’s effectiveness across both evaluated networks.The study links this performance to considering a comprehensive set of adoption-related factors, including unobserved confounding factors.

5. Discussion and Conclusion

The study develops a broader approach to predicting adoption probabilities and finds that social-influence-only cascade methods have limited predictive power. It also highlights confounding factors as important and discusses practical uses and scope boundaries of the method.

5. Discussion and Conclusion: The study operationalizes a locally weighted EM method for Naïve Bayes learning using a more comprehensive set of adoption factors, including unobserved confounding factors.The factors are intended to support prediction from observed adoption data despite generally unobserved confounding factors.
5. Discussion and Conclusion: The proposed method outperformed all benchmark methods across the two studied social networks.The evaluation used real-world data from two large-scale social networks.
5. Discussion and Conclusion: Cascade methods relying exclusively on social influence showed limited predictive power, without diminishing the value of social influence itself.The findings motivate incorporating additional forces into adoption-probability prediction.
5. Discussion and Conclusion: The method can support target selection, personalized incentives, aggregate-demand estimation, viral-marketing seed selection, and dynamic resource allocation.These applications include ranking potential adopters, summing probabilities to estimate expected adopters, and selecting seeds over time.
5. Discussion and Conclusion: Future extensions include adding omitted factors, examining interaction effects, testing alternative operationalizations, and conducting systematic analyses of LWNB versus NB.The paper specifically mentions viral product features, weak ties, adopter-neighbor connectedness, and hidden confounding information.
5. Discussion and Conclusion: The current implementation may not scale to networks with millions of nodes because pairwise distance calculation requires O(n^2) time.The paper suggests tree-based distance calculation and parallel computing as possible ways to reduce computational requirements.

A2. Construction of training data from observed adoption information

The algorithm constructs training data by processing adopters in adoption-time order and computing power measures for adopters and nonadopters. These computed powers are then added as labeled training instances, with innovators attributed to hidden confounding power.

A2. Construction of training data from observed adoption information: Training data are constructed by applying the Figure 1 algorithm step by step, as illustrated in Table A1.The example begins with empty training data and processes adoption information sequentially.
A2. Construction of training data from observed adoption information: The first loop computes powers from the earliest adopter through the latest adopter in adoption-time order.The earliest adopter is processed first, followed by later adopters.
A2. Construction of training data from observed adoption information: An innovator receives no influence, equivalence, or similarity power, so its adoption is attributed to hidden confounding power.This assumption follows the cited Bass-style innovator treatment.
A2. Construction of training data from observed adoption information: Computed power tuples for adopters and nonadopters are added to the training data according to their respective algorithmic formulas.The second loop computes powers for each nonadopter, while adopter examples are added during the first loop.

Appendix B: Derivation of (19)

The derivation chooses K to satisfy a positivity constraint for all indexed cases. It uses known conditional densities and the binary nature of the relevant variable to obtain the required form of K.

Appendix B: Derivation of (19): K is chosen so that the required density-related expression remains positive for every indexed case.The constraint is stated for all values of the index.
Appendix B: Derivation of (19): Given z, the conditional densities are treated as known, including an exponential example for the density of p when the binary variable equals 1.The example writes the density using parameters associated with the conditional distribution.
Appendix B: Derivation of (19): The derivation decomposes the marginal density over the two possible values of the binary variable before solving for K.It explicitly uses the cases where the variable equals 1 and 0.
Appendix B: Derivation of (19): The stated constraint is satisfied by setting K according to the derived expression.The appendix concludes that the proposed choice of K meets the required condition.

Appendix C: Proofs of Theorems

The appendix derives parameter equations for the objective and establishes curvature properties of its Hessian. It then concludes the proof by showing that the Hessian is negative definite.

Appendix C: Proofs of Theorems: Naïve Bayes conditional independence decomposes the log-likelihood into terms for the component variables and their conditional distributions.The decomposition separates terms involving the relevant modeled quantities.
Appendix C: Proofs of Theorems: Mixed second-order derivatives vanish for distinct parameters, making all off-diagonal Hessian entries zero.Each first-order derivative depends only on its corresponding parameter under the displayed derivation.
Appendix C: Proofs of Theorems: The proof establishes that the Hessian matrix is negative definite, completing the theorem’s concavity argument.Its principal minors alternate in sign because the diagonal second derivatives are negative and off-diagonal terms vanish.
Appendix C: Proofs of Theorems: First-order derivatives are taken with respect to each parameter in z, and the resulting equations are rearranged into the forms used in Theorem 2.The parameter vector includes separate components for the two modeled cases.

Appendix D: Evaluation Results with the Avatar Social Network

The avatar-network evaluation uses one year of adoption and interaction data for 25,500 avatars, with profile and behavioral attributes as predictors. The proposed method significantly outperforms every benchmark, while cascade and influence-probability methods are only marginally above random guessing.

Appendix D: Evaluation Results with the Avatar Social Network: The dataset contains 25,500 avatars observed over one year, including 263.8 million dialog records and weekly virtual-item adoption data.Avatar profiles include time-invariant attributes and time-variant behavioral attributes.
Appendix D: Evaluation Results with the Avatar Social Network: The evaluation uses 52 weekly network snapshots with dialog-based tie strength and avatar profile attributes as predictors.Additional virtual-item adoption data support implementation of the influence probability method.
Appendix D: Evaluation Results with the Avatar Social Network: p < 0.001: the proposed method significantly outperforms each benchmark method under the Wilcoxon test.The result is reported for the avatar social network evaluation and is consistent with the mobile social network findings.
Appendix D: Evaluation Results with the Avatar Social Network: AUCs for cascade and influence probability methods are only marginally larger than random guess at 0.5.This comparison indicates limited predictive performance for methods centered on those signals.

Appendix E: AUC Diagrams: Proposed Method versus Benchmark Methods

The appendix presents AUC trajectories across 50 evaluation weeks for LEMNB and benchmark methods. Learning-based methods vary over time, whereas cascade methods remain relatively stable.

AUCs of learning-based methods vary across evaluation weeks, while cascade methods remain relatively stable.The learning-based methods are LEMNB, NB, LWNB, SVM, and k-NN; cascade methods are CM1, CM2, CM3, and IP.
Different learning-based methods can show different AUC trajectories because they learn different patterns.

Appendix F: Extending Proposed Method with Additional Factor: A Preliminary Analysis

This appendix extends LEMNB with a connectedness factor describing ties among adopter neighbors. The resulting LEMNB+ significantly outperforms the original method on both studied social networks.

Connectedness measures the social ties among an entity’s adopter neighbors and requires calculating the factor for each training and test record.The method then adjusts adoption-probability prediction to include connectedness alongside influence, equivalence, similarity, and hidden confounding powers.
The extension initializes and updates connectedness parameters analogously to those for influence power.Theorem validity is retained because of Naïve Bayes conditional independence among factors.
LEMNB+ significantly outperforms LEMNB on each studied social network, with Wilcoxon tests yielding p < 0.01.LEMNB+ is the adjusted method that incorporates connectedness of adopter neighbors.
The analysis suggests connectedness of adopter neighbors is an important predictor of adoption behavior and demonstrates how additional factors can be incorporated.

Similarity

The appendix tests whether an alternative entity-similarity distance function changes LEMNB’s performance. Across the studied networks, the performance difference is not significant, suggesting robustness across the evaluated distance functions.

The alternative distance function supports mixed nominal, real-valued, and integer attributes while integrating feature-level distances differently.It computes feature-level entity distances using the paper’s existing equations before combining them into an overall distance.
The performance difference between LEMNB and LEMNBA is not significant on either studied social network, with p > 0.1.LEMNBA uses the alternative distance function of Aha et al. (1991).
The results suggest that LEMNB is robust across the commonly used distance functions evaluated in the appendix.