Phases of Information Spreading

• Phases of Information Spreading are multi-stage diffusion processes characterized by stochastic branching, heavy-tailed delays, and human decision-making.
• The two-step Bellman–Harris model distinguishes between Seeds and Viral nodes to predict cascade sizes, critical thresholds, and non-Markovian dynamics.
• Empirical insights reveal that super-spreader events and irregular waiting times drive prolonged cascade growth and impulsive viral expansion.

The phases of information spreading describe the dynamic evolution of how content, messages, innovations, or ideas propagate through populations, particularly in social and communications networks. Theoretical and empirical studies have revealed that this process is typically not monolithic but proceeds through distinct stages, governed by stochastic branching, heterogeneous temporal delays, and the conscious decisions of human agents. Advanced models such as the non-Markovian two-step Bellman–Harris (BH) branching process generalize beyond epidemic or static percolation analogues by explicitly incorporating irregular message fanout and heavy-tailed activity times, providing a refined understanding of the critical thresholds and dynamical regimes that govern viral diffusion.

1. Structural Foundations and Heterogeneous Dynamics

Information spreading in peer-to-peer settings fundamentally differs from classical epidemic models. Rather than automatic transmission per contact, individuals make deliberate decisions about whether and how broadly to forward a message. This results in diffusion processes that form tree-like cascades, typically initiated by a small number of Seed nodes activated by exogenous exposure. Subsequent propagation occurs through Viral nodes, each deciding probabilistically to forward the message and, if so, selecting a random number of recipients from individualized waiting time and offspring distributions.

The stochastic process is characterized by:

• Highly variable number of recommendations per node, drawn from potentially heavy-tailed distributions.
• Heterogeneous waiting (response) times, often modeled as log-normal or gamma distributions rather than exponential.
• Branching structures with a strong bias toward finite or vanishing clustering, in contrast to the underlying social network.

This leads to non-Markovian temporal correlations that are essential for accurately describing real-world information spread, particularly because long-tailed response delays frequently dominate system-level dynamics.

2. The Two-Step Bellman–Harris Branching Process

The BH branching process provides the mathematical foundation for analyzing information cascades. This approach generalizes classical Galton–Watson and static percolation models by introducing both discrete fanout and continuous, non-exponential delays. The two-step variant distinguishes between Seeds and Viral nodes with separate generation mechanisms and waiting time distributions.

Key mathematical constructs:

• Offspring (fanout) generating functions:

$$f_0(x) = \sum_{r=0}^\infty \tilde{p}_0(r) x^r, \quad f_1(x) = \sum_{r=0}^\infty \tilde{p}_1(r) x^r$$

• Waiting time cumulative distributions $G_0(t)$ and $G_1(t)$ for Seeds and Viral nodes, respectively.
• Self-consistent evolution of the active node process via generating functions:

$$F_1(s, t) = s[1 - G_1(t)] + \int_0^t dG_1(\tau) f_1[F_1(s, t-\tau)]$$

• Basic reproductive number for the Viral node process:

$R_1 = \lambda_1 \langle r_1 \rangle$

where $\lambda_1$ is the probability a node forwards the message, and $\langle r_1 \rangle$ is the mean number of recommendations sent by a Viral node.

This framework captures both the non-Markovian structure and branching heterogeneity essential for realistic modeling.

3. Temporal Regimes and the Tipping-Point

A central feature of the dynamics is the existence of a critical reproductive number ("tipping-point") that separates regions with dying-out versus explosive cascading. The extinction probability $q_1$ is determined by the generating function fixed-point:

$q_1 = f_1(q_1)$

For convex $f_1$, the extinction probability is unity for $R_1 \leq 1$, and there is a non-zero chance of extensive spreading only when $R_1 > 1$. The average cascade size initiated by a Seed (with always-active Seeds, $\lambda_0=1$) is:

$$\langle s^* \rangle = 1 + \frac{\langle r_0 \rangle}{1 - R_1}$$

which diverges as $R_1 \to 1$ from below, marking the threshold for explosive information propagation.

Time evolution reveals two regimes:

• For exponential (memoryless) waiting times (Poissonian), growth or decay is exponential, defined by a Malthusian parameter $\alpha$. Specifically:

$$i_1(t) \sim C e^{\alpha t}, \quad \alpha = \rho_1 (R_1-1), \quad \rho_1 = 1/\langle \tau_1 \rangle$$

• For heavy-tailed waiting times (log-normal, gamma), the long-lived slow decay regime emerges:

$i_1(t) \sim \frac{1 - G(t)}{1 - R_1}$

For example, with log-normal delay density,

$$i_1(t) \sim \mathrm{Erfc}\left(\frac{\ln t - \mu}{\sigma \sqrt{2}}\right)$$

emphasizing that even rare, long response times can dominate the late-stage dynamics, yielding sub-exponential decay and extended persistence below the tipping-point.

4. Human Behavioral Variability and Model Generalization

Unlike homogeneous or Markovian models, the BH process explicitly encodes variability characteristic of human activity:

• Power-law or Harris offspring distributions account for super-spreader events, where a minority of highly active individuals contribute disproportionately to overall diffusion.
• Separation between Seed and Viral node functional roles allows for modeling trust asymmetry, source credibility, and behavioral biases.
• Non-exponential waiting times are calibrated to empirical measurements of response time distributions in realistic campaigns.

This structure enables accurate prediction of not only cascade sizes and reach, but also dynamical features such as low network clustering in observed cascades compared to the substrate social graph.

5. Mathematical Formulations and Predictive Implications

Key formulas central to the process include:

• Generating function characterization of the evolving process:

$$F_1(s, t) = s[1 - G_1(t)] + \int_0^t dG_1(\tau) f_1[F_1(s, t - \tau)]$$

• Reproductive number:

$R_1 = \lambda_1 \langle r_1 \rangle$

• Extinction fixed-point:

$q_1 = f_1(q_1)$

• Final cascade size (for $\lambda_0=1$):

$$\langle s^* \rangle = 1 + \frac{\langle r_0 \rangle}{1 - R_1}$$

• Malthusian parameter for gamma-distributed waiting times:

$\alpha_1 = \frac{R_1^{1/k} - 1}{\theta}$

where $k$ and $\theta$ are the gamma parameters.

By estimating relevant parameters early in a campaign (e.g., transmissibility and mean fanout), the model can forecast final or in-progress campaign reach, advise on optimal targeting of initial Seeds (favoring super-spreaders to accelerate threshold crossing), and forecast the time to saturation.

6. Practical Consequences and Management Strategies

The precision of the two-step BH model enables several practical uses:

• Real-time prediction of cascade size distributions, which typically exhibit overdispersion and rare, large super-spreader events—contrasting with the regularity of classical processes.
• Accurate forecasting of the expected campaign reach, supporting resource allocation and monitoring for marketing or information operations.
• Quantification of expected clustering in cascade trees, providing operators with diagnostic tools for anomaly detection or counter-influence efforts (as observed clustering is minimal).
• Management recommendations, such as prioritizing high-activity Seeds or identifying critical temporal windows where interventions may be most effective.

7. Synthesis: Beyond Markovian Frameworks in Information Diffusion

The integration of branching process theory with empirical behavioral variability marks a significant development in understanding the phases of information spreading. The existence of a well-defined tipping-point ($R_1 = 1$), the dominance of heavy-tailed delays, and the nuanced role of node heterogeneity collectively delineate a multi-phase picture:

• Initial seeding and activation,
• Growth and possible explosive expansion (above threshold),
• Protracted, anomalously slow decay (below threshold), especially in non-Poissonian systems.

Recognition of these nuanced regimes, and their divergence from the expectations of static, homogeneous, or Markovian models, has profound implications for any application seeking to predict, manage, or suppress viral information campaigns in complex social systems.