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Complex contagion process in spreading of online innovation

Márton Karsai, Gerardo Iñiguez, Kimmo Kaski, János Kertész

arXiv:1405.6879v2physics.soc-phcs.SInlin.AOphysics.comp-phphysics.data-an

TL;DR

The paper addresses long-standing questions about how innovations spread. It analyzes adoption and termination dynamics and models the identified mechanisms, finding constant rates after a transition and linearly increasing neighbor influence on adoption.

  • Problem

    The mechanisms driving innovation spreading remain a long-standing question.

    Source: Discussion · p. 9

  • Method

    The paper analyzes service adoption and termination as competing processes, including influences from individual factors or external media independent of the social network.

    Source: The adoption dynamics · p. 3, The model process · p. 6

  • Results

    After a transition time, innovation spreading has constant rates and linearly increasing neighbor influence on adoption; the minimal model provides accurate medium-term predictions.

    Source: Discussion · p. 10

  • Takeaways & Limitations

    The study links average account lifetime with general civil-liberty measures.

    Source: Validation and socio-economic… · p. 9

    Online interactions alone cannot map society's complete structure.

    Source: Discussion · p. 10

Abstract

from arXiv · show

Diffusion of innovation can be interpreted as a social spreading phenomena governed by the impact of media and social interactions. Although these mechanisms have been identified by quantitative theories, their role and relative importance are not entirely understood, since empirical verification has so far been hindered by the lack of appropriate data. Here we analyse a dataset recording the spreading dynamics of the world's largest Voice over Internet Protocol service to empirically support the assumptions behind models of social contagion. We show that the rate of spontaneous service adoption is constant, the probability of adoption via social influence is linearly proportional to the fraction of adopting neighbours, and the rate of service termination is time-invariant and independent of the behaviour of peers. By implementing the detected diffusion mechanisms into a dynamical agent-based model, we are able to emulate the adoption dynamics of the service in several countries worldwide. This approach enables us to make medium-term predictions of service adoption and disclose dependencies between the dynamics of innovation spreading and the socioeconomic development of a country.

Results

The results characterize spontaneous adoption, peer influence, and termination rates, then incorporate these mechanisms into a three-state model. Simulations assess network correlations and show how model rates capture cross-country differences in adoption speed and account use.

Mechanisms of adoption

  • After an initial transient, spontaneous adoption and termination rates become constant, while the social-influence contribution increases approximately linearly with the fraction of adopting neighbors.The authors describe this as empirical validation of a fundamental assumption commonly used in social-spreading models.Source: Figure 2 · p. 4, Mechanisms of adoption · pp. 5–6

The adoption dynamics

  • Spontaneous adoption is typical early in diffusion, peer-pressure adoption is more common later, and termination appears largely independent of social influence.Termination-network users were mostly isolated even when surrounded by adopters.Source: The adoption dynamics · p. 4

The model process

  • The model represents people as susceptible, adopter, or removed agents, with spontaneous and peer-pressure adoption plus temporary and permanent termination.The model maps these processes to S→A, A→S, and A→R transitions.Source: The model process · pp. 6–7
  • Empirical and matched uncorrelated networks show similar rate behavior, suggesting structural correlations only slightly slow spreading and negligibly affect overall adoption rates.Source: The model process · p. 7, Figure 3 · p. 8

Validation and socio-economic correlations

Discussion

The study identifies online innovation diffusion as competition between adoption and termination, with constant rates after a transition and influence from adopting neighbours that increases linearly. It incorporates these mechanisms into a minimal model that provides accurate medium-term predictions, while noting limits in what online interaction data can reveal.

Discussion

  • After a transition, diffusion has constant rates and neighbour influence on adoption increases linearly.The process is framed as competition between service adoption and termination.Source: Discussion · p. 10
  • The identified mechanisms support a minimal modelling framework that provides accurate medium-term predictions for online innovation spread.The study also reports strong evidence for key mechanisms driving complex contagion at a level of detail and scale not previously possible.Source: Discussion · p. 10
  • Online interactions underestimate an individual's real contact peers, so the network data do not map society's complete structure.This incompleteness constrains what can be inferred from the observed social network.Source: Discussion · p. 10
  • Correlated adoption does not necessarily establish actual social influence, and the dataset cannot capture other mechanisms such as homophily.The paper presents these as limitations on interpreting the observed adoption patterns.Source: Discussion · p. 10
  • The results may help bridge theoretical understanding and empirical observation of social contagion, supporting earlier studies on constant adoption rates and social influence.The paper also relates novel-technology adoption to a country's societal and economic development.Source: Discussion · p. 10
  • The observations may also provide insight into differences in modern online societies, beyond their stated value for marketing and business planning.The discussion links adoption dynamics with a country's development.Source: Discussion · p. 10

A Model description

The model represents adoption and termination as transitions among degree-based agent states, with adoption driven by spontaneous and peer effects. It derives rate equations under degree-class assumptions and uses a first-order moment closure that ignores higher moments.

A Model description

  • Adoption probability combines spontaneous adoption with peer influence, weighted by the fraction of an agent’s neighbors who have already adopted.The model gives p_i = p_a + p_p(1 − p_a)n_i, where n_i = N_i/k_i, N_i is the number of adopting neighbors, and k_i is the agent’s degree.Source: A Model description · p. 10
  • Termination probability combines spontaneous termination with peer-pressure termination.The model gives p_i = p_r + p_s(1 − p_r).Source: A Model description · p. 10
  • Agents with the same degree are treated as statistically equivalent, and susceptible, adopter, and removed states are summarized by their average probabilities.The model denotes these probabilities by s_k, a_k, and r_k, respectively.Source: A Model description · p. 10
  • A first-order moment closure approximates the average adoption probability using the mean fraction of adopting neighbors, ignoring higher moments of the dynamical quantities.The approximation is ⟨p_i^+⟩s_k = [p_a + p_p(1 − p_a)n_a]s_k, with n_a = ⟨n_i⟩.Source: A Model description · p. 10
  • Degree-class rate equations describe adoption, with degree-degree correlations represented by conditional neighbor-degree probabilities when present.Similar rate equations for susceptible, adopter, and removed states form a nonlinear ODE system.Source: A Model description · p. 10

1 Data description

The study uses a temporally detailed Skype dataset to track registration, contact-network evolution, adoption, and termination. It retains confirmed connections and uses registered users’ countries to construct country-level adoption histories.

1 Data description

  • The dataset follows Skype users from September 2003 to March 2011, recording registration events and contact-network evolution for registered users.It also records termination history; the passages give 2738 days for the registration and network history and 2373 days for the termination history.Source: §1 · p. 14
  • For each user, the dataset provides registration country and date, plus time-stamped link additions and confirmations.A contact invitation is retained as a trusted social link only after confirmation.Source: §1 · p. 14
  • The analysis selects accounts with identified registration countries and retains confirmed connections, including international ties, to estimate users’ network degrees.Degrees are estimated by integrating the evolving network over the available period and counting each user’s confirmed relationships.Source: §1 · p. 14
  • Adoption is dated at registration, while termination is dated at last Skype activity and assigned only when that activity predates the observation-period end by more than one year.These dates are used to build the product’s adoption and termination history.Source: §1 · p. 14

2 Empirical measures of spreading parameters

This section describes how the study estimates spreading parameters and aligns model and empirical time scales, including rescaling data values as p′, while noting limitations from incomplete social-network observations and data coverage.

2.1 Average number of social ties

  • The estimated average number of social ties uses an aggregated Skype network as a proxy for society, but incomplete ego networks make the estimate a lower bound on true degree.The proxy is more representative where online-service use is high.Source: §2.1 · pp. 14–15

2.2 Probability of spontaneous adoption

  • Spontaneous adoption probability stabilizes after an initial transient, but only 11 of 34 countries provide enough observation time to estimate it empirically.The study therefore leaves pa as a free fitted parameter where stationarity is not observed; its direct data estimate is unbiased.Source: §2.2 · p. 15

2.3 Probability of peer-pressure adoption

  • Peer-pressure effects rise linearly with the fraction of adopting neighbors, robustly across degree, but weaken sharply beyond an effective fraction of about 0.8.Because the Skype graph reveals only effective degrees, the slope-based estimate of peer-pressure adoption is an upper limit on its true value.Source: §2.3 · pp. 16–17

2.4 Probability of termination

  • Overall termination probability reaches a constant after an initial transient, and decoupled termination probabilities also approach steady states, consistent with constant-rate termination.The dataset cannot distinguish temporary from permanent termination, so the two probabilities cannot be measured independently.Source: §2.4 · p. 17

2.5 Time scales

  • To align model and empirical time scales, the study rescales model time as t′ = qt, where q = Nint/Npop, while preserving real-time fits and predictions independently of q.Nint and Npop are the numbers of internet users and the population of a country, respectively.Source: §2.5 · pp. 17–18
  • The data values need to be rescaled as p′.The supplied passage does not specify the rescaling rule.Source: §2.5 · p. 18

3 Model

This section develops an agent-based model of Skype adoption and termination on a social network, derives aggregate rate equations under simplifying network assumptions, and tests them in synthetic-network simulations. It examines long-run outcomes with and without a removed state and, for the no-removal case, characterizes the approach to equilibrium and the balance of adoption and termination.

3.1 Model description

  • Agents move among susceptible, adopting, and removed states: adoption is spontaneous or driven by the fraction of adopting neighbours, while stopping can be temporary or permanent.The model uses a static social network, and the S → A, A → R, and A → S transitions are regulated by pa and pp, pr, and ps, respectively.Source: §3.1 · p. 18, §3 · p. 18

3.2 Uncorrelated random networks

  • For pr ≠ 0, the only fixed point is (a∞, s∞) = (0, 0); when pa, pr, and ps are nonzero, it is a stable node and final state; for large ⟨k⟩, ppk ∼ pp.Source: §3.2 · pp. 20–21

3.3 Numerical simulations

  • At large average degree, simulated and theoretical rates fit very accurately, validating the theoretical solution; their discrepancy shrinks as average degree increases.The simulations average 100 realizations of the adoption process.Source: Figure 7 · p. 21, §3.3 · p. 22
  • The solution predicts no system-size dependence in normalized characteristic rates, and simulations tested this by varying network size at average degree 32.Figure 8 compares sizes N = 10^3, 10^4, and 10^5; the supplied passage does not state the outcome of that comparison.Source: §3.3 · p. 22, Figure 8 · p. 22
  • In numerical simulations, averages overlap the theoretical curve despite larger deviations.Source: §3.3 · p. 22

3.4 Spreading scenarios

  • With pr=0, adoption accelerates, slows as termination rises, then reaches equilibrium when adoption and termination balance.The active-user total remains constant at equilibrium; allowing removal instead leads to a trivial final state with no susceptible nodes and a shrinking adoption network once termination dominates.Source: §3.4 · p. 22
  • Allowing agents to enter a removed state leads to a trivial final state with no susceptible nodes, after termination becomes dominant and the adoption network shrinks.The third regime begins at the crossover where termination becomes dominant.Source: §3.4 · pp. 22–23

3.5 Non-trivial equilibrium states

  • For nonzero pr, the planar system has a single fixed point (0, 0), where all individuals are removed and never use the product; setting pr = 0 yields a non-trivial equilibrium with nonzero adoption and termination rates.Source: §3.5 · p. 23
  • Analytical rate equations link equilibrium and maximal rates to model parameters, enabling inverse estimation of parameters from measured rates.The model also derives expressions for adoption and net-change rates over time and for the times at which those rates peak.Source: §3.5 · p. 24

4 Empirical fits

Across most countries, both two- and three-free-parameter fits closely matched predicted rates and supported predictions of asymptotic adoption dynamics. The analysis also found shorter account lifetimes in countries with weaker press, political, or civil liberties.

4 Empirical fits

  • Empirical rates Ra, Rt, and Rn were fitted simultaneously by bounded nonlinear least squares, with parameters estimated from data or treated as free.For 11 of the 34 countries, the spontaneous-adoption parameter could also be estimated empirically, enabling fits with two free parameters.Source: §4 · p. 25

4.1 Two vs. three free parameter fits

  • In most countries, two- and three-free-parameter fits showed excellent agreement between predicted rates and supported predictions of asymptotic adoption dynamics.The predicted-rate symbols generally collapsed along a unit-slope line.Source: §4.1 · p. 25

4.2 Goodness of fits

  • A comparable combined RSS evaluates grouped empirical and model rates over the first five years, while the start date varies with each country's initial transient.Source: §4.2 · pp. 25–26

4.3 Correlations with liberty measures

  • Weaker press, political, or civil liberties were associated with shorter account lifetimes, indicating a dependence between innovation-spreading dynamics and country socioeconomic status.Source: §4.3 · pp. 26–27

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