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Learning and Forecasting Opinion Dynamics in Social Networks

Abir De, Isabel Valera, Niloy Ganguly, Sourangshu Bhattacharya, Manuel Gomez Rodriguez

arXiv:1506.05474v3cs.SIphysics.soc-ph

TL;DR

Accurately forecasting users’ opinions from social media remains a scarce capability despite extensive theoretical work on opinion dynamics. SLANT models latent opinions and asynchronously shared sentiment messages, and its predictive formulas outperform alternatives in the reported experiments.

  • Problem

    Methods for accurately forecasting opinions remain scarce despite extensive theoretical work on opinion dynamics and growing interest in sensing and forecasting social-media opinions.

    Source: §1 · p. 1

  • Method

    SLANT represents latent user opinions as continuous-time stochastic processes influenced by neighbors’ asynchronously expressed sentiment messages, with efficient estimation, simulation, and predictive formulas.

    Source: §1 · pp. 1–2

  • Results

    SLANT’s predictive formulas achieve more accurate opinion forecasting than alternatives, and the model provides a good fit to the data.

    Source: §1 · p. 2, §5 · p. 8

  • Takeaways & Limitations

    The formulas identify conditions under which opinions converge to a steady state of consensus or polarization.

    Source: §5 · p. 8

    The model assumes linear dependence between users’ opinions, which the authors note may be a coarse approximation in some scenarios.

    Source: §5 · p. 8

Abstract

from arXiv · show

Social media and social networking sites have become a global pinboard for exposition and discussion of news, topics, and ideas, where social media users often update their opinions about a particular topic by learning from the opinions shared by their friends. In this context, can we learn a data-driven model of opinion dynamics that is able to accurately forecast opinions from users? In this paper, we introduce SLANT, a probabilistic modeling framework of opinion dynamics, which represents users opinions over time by means of marked jump diffusion stochastic differential equations, and allows for efficient model simulation and parameter estimation from historical fine grained event data. We then leverage our framework to derive a set of efficient predictive formulas for opinion forecasting and identify conditions under which opinions converge to a steady state. Experiments on data gathered from Twitter show that our model provides a good fit to the data and our formulas achieve more accurate forecasting than alternatives.

1 Introduction

Forecasting opinions from social media remains scarce despite interest in sensing opinion and extensive theoretical work on opinion dynamics. SLANT models hidden opinions and shared sentiment messages, with experiments reporting better forecasting than alternatives.

1 Introduction

  • The paper addresses the scarcity of accurate opinion-forecasting methods despite extensive theoretical work on opinion dynamics.The introduction also notes applications in politics, investment, and marketing.Source: §1 · p. 1
  • SLANT models latent opinions that evolve with asynchronously shared neighbor opinions, treating each sentiment message as a noisy observation of the user’s current opinion.The framework addresses the distinction between latent opinion and expressed sentiment that prior models often omit.Source: §1 · pp. 1–2
  • The framework uses the Markov property for efficient parameter estimation by convex programming and scalable simulation of millions of sentiment messages.The paper reports that simulation can sample millions of messages in minutes.Source: §1 · pp. 1–2
  • Its predictive formulas support efficient opinion forecasting and identify conditions for convergence to consensus or polarization.Experiments use synthetic and Twitter data.Source: §1 · p. 2
  • SLANT’s predictive formulas achieve more accurate opinion forecasting than alternatives.The framework also provides a good fit to the data.Source: §1 · p. 2

2 Proposed model

The model represents users’ opinions as latent continuous-time processes influenced by neighbors’ asynchronously posted sentiment messages. Markov structure supports efficient estimation and simulation.

2 Proposed model

  • Message intensities can be modeled as constant Poisson rates or as Hawkes processes in which prior events influence later posting rates.In the Hawkes formulation, users’ own initiative and messages from followed users contribute to intensity.Source: §2 · pp. 2–3
  • Each user’s latent opinion is modulated by neighbors’ previously posted opinions, while a posted sentiment is a noisy estimate of the user’s current latent opinion.Messages are recorded as user, sentiment, and time, and sentiment may follow Gaussian or logistic distributions.Source: §2 · pp. 2–3

2.1 Model parameter estimation

  • For log-concave sentiment distributions, maximum-likelihood estimation is concave and decomposes into parallelizable per-user subproblems.The method uses spectral projected gradient descent for intensity parameters and solves the Gaussian sentiment component as least squares.Source: §2.1 · pp. 3–4

2.2 Model simulation

  • The simulation algorithm updates opinions and intensities in O(1) between consecutive events and exploits sparse networks to reuse most samples.Only a small local neighborhood changes when a node expresses an opinion.Source: §2.2 · p. 4

3 Opinion forecasting

The section treats forecasting as the conditional expected opinion given message history and develops analytical and simulation-based methods. The analytical results also characterize conditions for steady-state opinions.

3 Opinion forecasting

  • Forecasting targets the conditional expectation of a user's opinion at a future time given the message history up to an earlier time.The section also uses analytical forecasts to identify when average opinion converges to a steady state and what that steady-state opinion is.Source: §3 · p. 4

3.1 Analytical forecasting

  • Analytical forecasts cover Poisson intensities and a restricted multivariate Hawkes case; the Hawkes derivation assumes zero off-diagonal influence parameters and computes conditional means through a differential equation.The differential equation can be solved numerically, with matrix-exponential and matrix-inversion methods used to compute its terms.Source: §3.1 · pp. 4–5
  • For Poisson and Hawkes settings, the section derives stability conditions and steady-state conditional average opinions.For the Poisson case, the text also identifies matrix-exponential and matrix-inversion methods for efficiently computing the predictive formula.Source: §3.1 · p. 5

3.2 Simulation based forecasting

  • Simulation-based forecasting estimates conditional average opinions by averaging simulated trajectories, with a theoretical bound on estimation error.The stated guarantee bounds the error by ε with probability at least 1 − δ.Source: §3.2 · p. 6

4 Experiments

Synthetic experiments show that the model produces consensus or polarization and scales to large networks. On Twitter datasets, its forecasts outperform six methods, with performance depending on the forecast horizon and message-intensity model.

4.1 Experiments on synthetic data

  • Synthetic experiments produce consensus and polarization, show close agreement between forecasting formulas and simulation estimates, and find that dynamics depend on message-intensity form.The experiments also report examples where Poisson intensities lead to consensus and Hawkes intensities to polarization, as well as examples with the reverse outcomes.Source: §4.1 · pp. 6–7
  • 20 minutes suffices to estimate parameters from 10 million events generated by one million nodes on a single 24-core, 64 GB machine.The scalability experiments cover estimation, simulation, and analytical predictive formulas.Source: Figure 2 · p. 7, §4.1 · p. 7

4.2 Experiments on real data

  • SLANT consistently outperforms six state-of-the-art methods on Twitter sentiment prediction in MSE, often by an order of magnitude, and polarity failure rate; its forecasting degrades gracefully with horizon.Failure rates are close to zero on datasets where most users post messages with the same polarity.Source: §4.2 · p. 7, Figure 3 · p. 8
  • For network-level forecasts, accuracy decreases as the horizon grows, and multivariate Hawkes intensities yield more accurate predictions than Poisson intensities.The reported qualitative pattern also holds for the remaining datasets beyond the two shown.Source: Figure 4 · p. 8, §4.2 · p. 8

5 Conclusions

SLANT models latent opinions as continuous-time stochastic processes influenced by neighbors’ asynchronous sentiment messages, and uses the model for scalable estimation, simulation, forecasting, and steady-state analysis. The authors report more accurate forecasting than state-of-the-art methods on Twitter data, while identifying linear opinion dependence as a limitation.

5 Conclusions

  • SLANT models latent opinions as continuous-time stochastic processes driven by neighbors’ asynchronously expressed sentiment messages.The framework uses the Markov property for estimation and simulation algorithms that scale to networks with millions of nodes.Source: §5 · p. 8
  • The framework derives predictive formulas for opinion forecasting and identifies conditions for convergence to consensus or polarization.Experiments on Twitter data show more accurate forecasting than state-of-the-art methods.Source: §5 · p. 8
  • Linear dependence between users’ opinions may be a coarse approximation in some scenarios; nonlinear dependence is proposed as a forecasting-accuracy follow-up.Source: §5 · p. 8
  • The authors propose extending the model to correlate topics and generalizing the noisy-discrete-sample idea to other event data, with recovery guarantees as a research direction.Source: §5 · p. 9

A Proof of Proposition 1

These appendix proofs recast the model for all users, state an auxiliary differential equation for expected opinions, and use Laplace transforms in deriving a time-domain expression.

A Proof of Proposition 1

B Auxiliary theoretical results

  • For exponential triggering kernels with parameters ω and ν, the auxiliary lemma states a differential equation for expected opinions conditioned on prior history.Source: B Auxiliary theoretical results · p. 11

C Proof of Theorem 2

  • The Theorem 2 proof uses Laplace and inverse Laplace transforms to obtain the conditional average opinion in the time domain.It conditions on prior history, under which the opinion is non-random.Source: C Proof of Theorem 2 · p. 12

D Proof of Theorem 3

The appendix states that Theorem 2 gives the time-domain average opinion and develops Theorem 4 under an assumption that excludes cross-user triggering.

D Proof of Theorem 3

  • Theorem 2 states that the average users’ opinion in the time domain is given by a formula, with a stability condition involving the eigenvalues of AΛ1 and ω.The stated condition is Re[ξ(AΛ1)] < ω.Source: D Proof of Theorem 3 · p. 12

E Proof of Theorem 4

  • Theorem 4’s proof assumes bvu = 0 for v ≠ u and derives expected message intensities using users’ histories and neighbors.It then uses these expectations with the opinion dynamics to derive an average-opinion expression.Source: E Proof of Theorem 4 · pp. 12–13

F Proof of Theorem 5

The section states an average-opinion expression and gives a stability condition under which a steady-state solution exists.

F Proof of Theorem 5

  • Theorem 4 states that the average users’ opinion in the time domain is given by a formula.Source: F Proof of Theorem 5 · p. 13
  • The proof invokes linear-system solutions and defines the transition matrix Φ(t) through a matrix differential equation.Source: F Proof of Theorem 5 · p. 13
  • A steady-state solution follows when the transition matrix satisfies ||Φ(t)|| ≤ γe^-ct for all t > 0, with γ, c > 0.Source: F Proof of Theorem 5 · p. 13

G Proof of Theorem 6

The appendix proves a concentration bound, analyzes variance stability for Poisson-driven opinion dynamics, and describes estimation and simulation procedures. It also gives a stability condition for a steady-state covariance and a complexity bound for simulation.

G Proof of Theorem 6

  • Bernstein’s inequality bounds the deviation of simulated opinions from their conditional mean, yielding a sample-size condition for probability at least 1 − δ.The proof uses independent, zero-mean terms with bounded magnitude.Source: G Proof of Theorem 6 · p. 13

G.1 Dynamics of variance

  • For Poisson-driven opinion dynamics, the covariance has a steady-state solution when the stated stability condition holds.The analysis assumes constant baseline intensities and characterizes stability through an eigenvalue condition.Source: §G.1 · pp. 14–15

H Parameter estimation algorithm

  • The opinion-dynamics parameter estimation algorithm reduces estimation of α and A to a least-squares problem.It merges users’ and neighbors’ message times before forming the regression inputs and outputs.Source: H Parameter estimation algorithm · pp. 16–17

H.2 Estimation of (µ, B)

  • The procedure estimates µ and B using spectral projected gradient descent with a backtracking line search.The algorithm updates its step length and uses a finite-memory criterion.Source: §H.2 · p. 17

I Model simulation algorithm

  • The simulation algorithm exploits network sparsity and the model’s Markov property, with worst-case time complexity O(dmax|H(T)||V|).It updates rates and opinions locally, using O(log |V|) priority-queue operations and O(1) operations for rate and opinion updates.Source: I Model simulation algorithm · p. 17

J Experiments on Synthetic Data

Synthetic-network experiments assess parameter-estimation accuracy using mean squared error. Estimation becomes more accurate as more messages are provided.

J Experiments on Synthetic Data

  • The evaluation covers assortative, dissortative, and core-periphery Kronecker networks.The three network types use distinct parameter matrices.Source: J Experiments on Synthetic Data · p. 19
  • The experiments compare estimated and true parameters using mean squared error, with synthetic parameters drawn from the stated uniform and Gaussian distributions.The setup uses exponential kernels with ω = 100 for opinions and ν = 1 for intensities.Source: J Experiments on Synthetic Data · p. 19
  • Figure 5 shows parameter-estimation mean squared error decreasing as more messages are used across several 512-node Kronecker networks.The measured parameters are α and A for Hawkes intensities, and µ and B for opinion updates.Source: Figure 5 · p. 19, J Experiments on Synthetic Data · p. 19

K Twitter dataset description

The Twitter datasets comprise event-related tweets collected over short periods, with follower-followee networks built for participating users. The dataset statistics are presented in Table 1.

K Twitter dataset description

  • Tweets were collected through hashtags related to five events: Politics, Movie, Fight, Bollywood, and US.Collection covered roughly two to three weeks around each event date.Source: K Twitter dataset description · p. 20
  • The follower-followee network was built for users who posted collected tweets, then users below activity and follower thresholds were filtered out.Excluded users had fewer than 200 lifetime tweets, followed fewer than 100 users, or had fewer than 50 followers.Source: K Twitter dataset description · p. 20
  • Table 1 reports statistics for the real Twitter datasets.Source: Table 1 · p. 20
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