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Dynamics of social contagions with memory of non-redundant information

Wei Wang, Ming Tang, Hai-Feng Zhang, Ying-Cheng Lai

arXiv:1507.04062v1physics.soc-phcs.SI

TL;DR

Social contagion theory has not adequately analyzed reinforcement from memory of non-redundant information. The paper proposes a general model and edge-based theory, finding that memory affects final adoption size and can change its dependence on parameters from discontinuous to continuous.

  • Problem

    The effects of memory for non-redundant information on social contagion have been neglected, while existing approximations can deviate from simulations, especially on structurally heterogeneous networks.

    Source: §I · p. 2

  • Method

    The paper formulates reinforcement through memory of distinct information received from neighbors and develops a unified edge-based compartmental theory, studying the spreading threshold model and alternative models.

    Source: §II · p. 2, §I · p. 2, §V · p. 10

  • Results

    The memory characteristic markedly affects final adoption size, and its dependence on parameters can cross from discontinuous to continuous as thresholds, seed size, or network heterogeneity change.

    Source: §I · p. 2, §VI · p. 12

  • Takeaways & Limitations

    In the spreading threshold model, lower adoption thresholds and larger initial seeds can facilitate adoption, while heterogeneity can promote spreading but impede it at relatively large transmission probabilities.

    Source: §VI · p. 12

Abstract

from arXiv · show

A key ingredient in social contagion dynamics is reinforcement, as adopting a certain social behavior requires verification of its credibility and legitimacy. Memory of non-redundant information plays an important role in reinforcement, which so far has eluded theoretical analysis. We first propose a general social contagion model with reinforcement derived from non-redundant information memory. Then, we develop a unified edge-based compartmental theory to analyze this model, and a remarkable agreement with numerics is obtained on some specific models. Using a spreading threshold model as a specific example to understand the memory effect, in which each individual adopts a social behavior only when the cumulative pieces of information that the individual received from his/her neighbors exceeds an adoption threshold. Through analysis and numerical simulations, we find that the memory characteristic markedly affects the dynamics as quantified by the final adoption size. Strikingly, we uncover a transition phenomenon in which the dependence of the final adoption size on some key parameters, such as the transmission probability, can change from being discontinuous to being continuous. The transition can be triggered by proper parameters and structural perturbations to the system, such as decreasing individuals' adoption threshold, increasing initial seed size, or enhancing the network heterogeneity.

I. INTRODUCTION

Social contagion involves reinforcement because people weigh prior contacts when deciding whether to adopt a behavior. The paper addresses the neglected role of memory for non-redundant information and develops a theory to analyze its effects.

I. INTRODUCTION

  • Social contagion differs from biological contagion because adoption probability depends on previous contacts, reflecting the need for repeated confirmation of credibility and legitimacy.The introduction illustrates this with an individual considering three friends who have adopted the behavior.Source: §I · p. 1
  • Prior models largely neglected reinforcement from non-redundant information, where each pair can transmit information successfully only once.Such transmission is relevant when repeated information between the same individuals is unnecessary.Source: §I · p. 2
  • Existing approximations can deviate from simulations because non-Markovian memory induces strong correlations between connected individuals, especially on structurally heterogeneous networks.The paper identifies a need for a general theory that covers different forms of adoption dependence on non-redundant information.Source: §I · p. 2
  • The final adoption size can shift from discontinuous to continuous dependence, with the crossover triggered by lower thresholds, larger initial seeds, or greater network heterogeneity.The paper reports that its theoretical predictions agree well with numerical simulations.Source: §I · p. 2

II. A GENERAL SOCIAL CONTAGION MODEL

The model represents social reinforcement through memory of distinct information received from neighbors. Individuals can adopt based on cumulative information, with transmission and abandonment shaping the dynamics.

II. A GENERAL SOCIAL CONTAGION MODEL

  • Each edge can transmit information successfully only once, while individuals remember the cumulative information received from their neighbors, making the process non-Markovian.The adoption probability π(k, m) generally increases with the number m of received pieces of information for a given degree k.Source: §II · pp. 2–3
  • The SAR model places individuals in susceptible, adopted, or recovered states on a configuration network, with recovered individuals no longer transmitting information.The model assumes large sparse networks without degree-degree correlations.Source: §II · p. 2
  • Adopted individuals transmit independently to neighbors with probability λ, and a susceptible person evaluates adoption after each new piece of information using π(k, m).People who have already received m−1 pieces of information update the cumulative count when another distinct neighbor transmits; adopted individuals recover with probability γ.Source: §II · pp. 2–3
  • By choosing parameters, the model recovers the Watts threshold model and several existing non-Markovian contagion models as special cases.With constant adoption probability and the adopted state treated as infected, the process reduces to the standard SIR model.Source: §II · p. 3
  • In the spreading threshold model, adoption occurs only when the number of distinct pieces of information exceeds an absolute adoption threshold.Unlike the Watts model, individuals learn neighbor states through transmission, and adopted individuals may lose interest and stop spreading information.Source: §II · pp. 3–4

III. THEORY

The paper develops a unified edge-based compartmental theory for the general contagion model with reinforcement from memory of non-redundant information.

III. THEORY

  • The unified edge-based compartmental theory analyzes the general model’s reinforcement mechanism based on non-redundant information memory.The theory assumes large, sparse networks without degree-degree correlations and continuously evolving contagion dynamics.Source: §III · p. 4
  • The framework tracks susceptible, adopted, and recovered population densities over time and defines R(∞) as the final fraction that adopted the behavior.The states remain unchanged as time tends to infinity.Source: §III · p. 4
  • The theory is used both to analyze the general model and to examine memory effects in the spreading threshold model.The final adoption size and its dependence on dynamical and topological parameters are the focus.Source: §III · p. 4

A. General theoretical framework

The framework uses edge-level probabilities and a cavity construction to account for correlations between neighboring node states. It derives time-dependent and steady-state quantities, including the final adoption size and outbreak threshold.

A. General theoretical framework

  • Edge-based compartmental theory addresses correlations between adjacent node states that can make existing approximations inaccurate, particularly on structurally heterogeneous networks.The framework draws on cavity-theory insights to analyze contagion with strong nodal-state correlations.Source: A. General theoretical framework · p. 4
  • The cavity construction tracks θ(t), the probability that a randomly chosen edge has not transmitted information, while preventing the focal individual from transmitting along its edges.The analysis assumes identical non-transmission probabilities across edges and no dynamical correlations among an individual’s neighbors.Source: A. General theoretical framework · p. 4
  • The theory computes susceptible-state probabilities by summing over the numbers of distinct information pieces received, accounting for adoption after each piece.It then averages over degrees to obtain population-level state fractions.Source: A. General theoretical framework · p. 4
  • The edge probability θ(t) is decomposed by whether a neighbor is susceptible, adopted, or recovered and has not yet transmitted to the focal individual.The susceptible, adopted, and recovered edge probabilities are solved as part of the framework.Source: A. General theoretical framework · p. 4
  • The model’s equations provide a complete description of state densities over time, and the steady-state solution yields the final adoption size.The largest stable fixed point is physically meaningful when multiple stable solutions exist; it determines S(∞) and R(∞).Source: A. General theoretical framework · p. 5
  • The critical condition is found when a nontrivial steady-state solution appears, allowing the critical transmission probability λc to be calculated.λc depends on adoption probability, initial seed size, abandon probability, and network topology.Source: A. General theoretical framework · p. 5

B. Spreading threshold model

The spreading-threshold model makes adoption depend on information received from distinct neighbors, and its final adoption size can rise continuously or discontinuously with transmission probability. The paper analyzes this behavior through fixed points of Eq. (15), including the effects of threshold and seed size.

B. Spreading threshold model

  • For T > 1, a vanishingly small seed fraction cannot cause a finite fraction of individuals to adopt.For T = 1, the continuous critical transmission probability has the same form as the epidemic outbreak threshold.Source: B. Spreading threshold model · p. 6
  • The graphical solutions illustrate continuous R(∞) growth for T = 1 and a discontinuous change for T = 3 on ER networks.The figure marks tangent points in the graphical solutions of Eq. (15).Source: Figure 2 · p. 6
  • With non-vanishing initial seeds, changing transmission probability can produce either continuous growth or a discontinuous jump in R(∞), associated with a saddle-node bifurcation of Eq. (15).The critical transmission probability for the jump is obtained by solving Eqs. (15) and (17)–(18).Source: B. Spreading threshold model · p. 6
  • For fixed T and degree distribution, the critical values of other system parameters, including λ_s^c and ρ_s^0, can be determined from θ_s(∞).The passage describes combining Eqs. (15), (16), and (25) to obtain θ_s(∞).Source: B. Spreading threshold model · p. 7

IV. NUMERICAL VERIFICATION

The numerical verification uses ER and configuration networks with power-law degree distributions, averaging repeated dynamical runs across network realizations.

IV. NUMERICAL VERIFICATION

  • Simulations use Erdős–Rényi and configuration networks with power-law degree distributions.The network size is N = 10^4 and mean degree is ⟨k⟩ = 10.Source: §IV · p. 7
  • At least 2 × 10^3 dynamical realizations on a fixed network are averaged to calculate each pertinent quantity.The averages are further averaged over 100 network realizations.Source: §IV · p. 7
  • The numerical analysis separately examines dynamical and topological parameter effects.Source: §IV · p. 7

A. Effects of dynamical parameters

On ER networks, the threshold and initial seed size shape both the final adoption size and whether it grows continuously or discontinuously with transmission probability. The analysis relates abrupt growth to the subcritical population and reports agreement between theory and simulations.

A. Effects of dynamical parameters

  • At λ = 0.8, ρ0 = 0.1, T = 3, and γ = 0.5, susceptible density falls and recovered density rises over time, while adopted density first falls and then peaks near t ≈ 5.The time-course results agree with predictions from the edge-based compartmental theory.Source: A. Effects of dynamical parameters · p. 7, Figure 3 · p. 7
  • For T ≤ 4, R(∞) grows continuously with λ at T = 1 but discontinuously for slightly larger thresholds; larger T impedes adoption, and theory agrees well with simulations.For relatively large thresholds such as T ≥ 5, individuals hardly adopt the behavior.Source: Figure 3 · p. 7, A. Effects of dynamical parameters · p. 8
  • For 2 ≤ T < 5, the subcritical fraction peaks at λI_c, after which a finite fraction adopts together and R(∞) jumps; for T = 1, the subcritical state is essentially absent and growth is continuous.This identifies the subcritical population as the reservoir associated with discontinuous growth.Source: Figure 4 · p. 8, A. Effects of dynamical parameters · p. 9
  • Increasing the initial seed size raises R(∞); at T = 3, growth versus λ is discontinuous for 0.04 ≤ ρ0 ≤ 0.15 and continuous for 0.15 < ρ0 ≤ 1.The crossover is associated with a decreasing number of individuals in the subcritical state as ρ0 rises.Source: Figure 5 · p. 8, A. Effects of dynamical parameters · p. 9

B. Effects of topological parameters

Network topology changes both the final adoption size and how it responds to transmission probability. Increasing mean degree shifts growth from continuous to discontinuous, while greater heterogeneity can shift it back to continuous.

B. Effects of topological parameters

  • Increasing mean degree generally raises R(∞), lowers the critical transmission probability, and shifts its growth from continuous to discontinuous.At ⟨k⟩=5 growth is continuous; for ⟨k⟩>5 it becomes discontinuous, associated with a sizable subcritical population.Source: B. Effects of topological parameters · p. 9
  • In scale-free networks, increasing heterogeneity promotes R(∞) at small λ but suppresses it at large λ.With smaller degree exponent ν, hubs aid spreading at small λ, while many low-degree individuals hinder adoption at large λ.Source: B. Effects of topological parameters · p. 9
  • Below the critical degree exponent νs ≈ 4.0, R(∞) varies continuously with λ; above it, the variation is discontinuous.Greater heterogeneity also lowers the critical transmission probability; theory agrees well with numerical results.Source: B. Effects of topological parameters · p. 9

V. ALTERNATIVE MODELS OF SOCIAL CONTAGION DYNAMICS

The edge-based compartmental theory is applied to alternative contagion models with degree-correlated thresholds and more general adoption probabilities. The supplied figures specify how final adoption size is plotted for network degree and heterogeneity.

V. ALTERNATIVE MODELS OF SOCIAL CONTAGION DYNAMICS

  • For ER networks, Fig. 6 plots R(∞) against λ across mean degrees ⟨k⟩=5, 8, 10, 15, and 20, with theoretical curves.The other parameters are ρ0=0.1, γ=1.0, and T=3.Source: Figure 6 · p. 10
  • For scale-free networks, Fig. 7 plots R(∞) against λ across degree exponents ν=2.2, 3.0, 4.0, and ν→∞, with theoretical curves.The ν→∞ case is a random regular network with identical degree.Source: Figure 7 · p. 10
  • The edge-based compartmental theory is applied to two alternative models: degree-correlated adoption thresholds and adoption probability π(k, m) increasing with m.The latter model generalizes how adoption probability depends on received information.Source: §V · pp. 9–10

A. Correlated spreading threshold model

The model correlates individuals’ adoption thresholds with network degree through a tunable exponent. This correlation changes the final adoption pattern: negative α can remove discontinuous growth, while positive α produces a non-monotonic shift in the critical point.

A. Correlated spreading threshold model

  • For α = 0, adoption threshold is uncorrelated with degree and all individuals share the same threshold; for α > 0, larger-degree individuals have higher thresholds, while α < 0 reverses this relation.The mean adoption threshold is set somewhat arbitrarily to ⟨T⟩ = 3.Source: A. Correlated spreading threshold model · p. 10
  • For α>0, increasing α first raises and then lowers the critical point λcI.Moderate α raises thresholds near the mean degree; larger α raises hub thresholds and lowers those near the mean degree, where most ER-network individuals lie.Source: A. Correlated spreading threshold model · p. 10
  • For α<0, decreasing α changes R(∞)’s dependence on λ from discontinuous to continuous.Lower-degree individuals then have higher thresholds and face more difficulty adopting, reducing the subcritical population until discontinuous growth disappears.Source: A. Correlated spreading threshold model · pp. 10–11
  • For α<0, final adoption sizes are smaller than for α=0 in the large-λ regime.Higher thresholds among low-degree individuals make adoption more difficult.Source: A. Correlated spreading threshold model · p. 10

B. A generalized social contagion model

The generalized model makes adoption probability depend on exposures from distinct neighbors, and its edge-based analysis agrees well with simulations. In the reported results, R(∞) varies continuously with λ.

B. A generalized social contagion model

VI. CONCLUSIONS

The paper models reinforcement from non-redundant information memory and develops a unified edge-based theory tested across contagion models and networks. In the spreading threshold model, the dependence of final adoption size on λ can change between continuous and discontinuous patterns.

VI. CONCLUSIONS

  • The unified edge-based compartmental theory is tested on contagion models of varying complexity and different model networks.Source: §VI · p. 12
  • Non-redundant information memory forbids repeated transmission on each edge while retaining cumulative pieces of distinct information.Source: §VI · p. 12
  • R(∞) can shift between continuous and discontinuous dependence on λ when adoption threshold, initial seed size, or network heterogeneity is adjusted.Source: §VI · p. 12
  • Lower adoption thresholds, larger initial seeds, and higher mean degree facilitate individual adoption, making the system less resilient to social contagions.Source: §VI · p. 12
  • Greater structural heterogeneity can promote spreading but impede it for relatively large λ.Source: §VI · p. 12

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