Theory of Rumour Spreading in Complex Social Networks (0807.1458v1)

Abstract: We introduce a general stochastic model for the spread of rumours, and derive mean-field equations that describe the dynamics of the model on complex social networks (in particular those mediated by the Internet). We use analytical and numerical solutions of these equations to examine the threshold behavior and dynamics of the model on several models of such networks: random graphs, uncorrelated scale-free networks and scale-free networks with assortative degree correlations. We show that in both homogeneous networks and random graphs the model exhibits a critical threshold in the rumour spreading rate below which a rumour cannot propagate in the system. In the case of scale-free networks, on the other hand, this threshold becomes vanishingly small in the limit of infinite system size. We find that the initial rate at which a rumour spreads is much higher in scale-free networks than in random graphs, and that the rate at which the spreading proceeds on scale-free networks is further increased when assortative degree correlations are introduced. The impact of degree correlations on the final fraction of nodes that ever hears a rumour, however, depends on the interplay between network topology and the rumour spreading rate. Our results show that scale-free social networks are prone to the spreading of rumours, just as they are to the spreading of infections. They are relevant to the spreading dynamics of chain emails, viral advertising and large-scale information dissemination algorithms on the Internet.

• The paper introduces a stochastic model that extends classic DK and MK frameworks by incorporating realistic social network topologies.
• The paper reveals that homogeneous networks display a critical spreading rate, while scale-free networks exhibit a vanishing threshold as size increases.
• The paper demonstrates that assortative degree correlations accelerate the initial spread, impacting final rumor size and informing control strategies.

Theory of Rumour Spreading in Complex Social Networks

The paper "Theory of rumour spreading in complex social networks" by M. Nekovee, Y. Moreno, G. Bianconi, and M. Marsili introduces a comprehensive stochastic model to describe the spread of rumors within complex social networks, with a special focus on those mediated by the internet.

Introduction to Rumor Spreading

Rumors are a fundamental aspect of social communication, significantly impacting areas ranging from public opinion to financial markets. The motivation behind studying rumor dynamics aligns with understanding viral marketing strategies and the spread of urgent information, including panic-inducing rumors during crises.

Model Description

The presented model builds upon the Daley-Kendall (DK) and Maki-Thompson (MK) models but incorporates a more realistic description of social networks. It characterizes individuals as ignorants, spreaders, and stiflers, with the rumor propagating through pairwise contacts dictated by the network topology.

• Ignorants: Individuals unaware of the rumor.
• Spreaders: Individuals actively spreading the rumor.
• Stiflers: Individuals who know the rumor but have ceased spreading it.

Key dynamics:

• Spreader contacts an ignorant → ignorant becomes a spreader with rate $\lambda$.
• Spreader contacts another spreader/stifler → initiating spreader becomes a stifler with rate $\alpha$.
• Spreader can also spontaneously forget the rumor with rate $\delta$.

Analytical and Numerical Methodology

The authors use Interacting Markov Chains (IMC) to derive mean-field equations describing the dynamics on various network topologies: Erdős-Rényi (ER) random graphs, uncorrelated scale-free (SF) networks, and assortatively correlated SF networks.

Key Findings

Homogeneous Networks

When applied to homogeneous networks, the model reveals a critical threshold for the spreading rate $\lambda$, below which the rumor cannot propagate. The presence of a forgetting mechanism leads to a critical behavior reminiscent of the SIR model of epidemic spreading. The threshold condition is $\frac{\lambda}{\delta}\bar{k} >1$.

Inhomogeneous Networks

For uncorrelated SF networks, the rumor spreading dynamics exhibit a vanishing threshold as the network size grows, showing higher initial rates of rumor spread compared to ER networks. The behavior aligns with the known properties of SF networks which promote faster diffusion due to the presence of "hubs".

In terms of final rumor size, SF networks displayed different regimes depending on the spreading rate $\lambda$. Scale-free networks show stretched exponential behavior, $R \sim \exp(-C/\lambda)$, indicating the absence of a critical threshold in infinite networks.

Assortative Degree Correlations

To account for realistic social network properties, the authors simulated assortatively correlated SF networks and found that such correlations further accelerate the initial spread. However, the overall size dependent on $\lambda$, with higher final sizes for larger $\lambda$ values in assortatively correlated networks compared to uncorrelated networks. This suggests that assortative correlations can either facilitate or impede spreading, conditional on $\lambda$.

Implications

The model's realistic incorporation of network topologies and individual behaviors provides insights into various spreading processes like viral advertising and email chain letters within modern social networks. Practically, understanding these dynamics aids in designing more effective information dissemination strategies while also controlling the unwanted spread of misinformation.

Future Directions

Future work aims to extend this static model to dynamic networks, accounting for time-varying interactions among nodes, which is particularly relevant for platforms like instant messaging and chat rooms.

Concluding Remarks

This paper contributes a rigorous analytical and numerical approach to understanding rumor spread in complex networks, highlighting significant differences due to network topology and dynamic interactions. It paves the way for further exploration into the effects of dynamic network topologies and other real-world social network characteristics on spreading processes.