Scream Effect Propagation Mechanism and Simulation

Pulse-driven information diffusion and intervention simulation on directed social networks with SEIR-style state transitions

Jiang Zhengxu¹ · Yao Jiaqi¹ · Meng Lingjun² · Wang Jian² · Cao Yongchang²

¹ Haide College, Ocean University of China
² School of Mathematical Sciences, Ocean University of China

Mathematical Modeling and Its Applications, 2023, 12(4)

English | 简体中文

Background · Roadmap · Model · Experiment · Results · Interventions · Quick start · Citation

Scream Effect Propagation Simulation studies how repeated attention pulses interact with network influence, heterogeneous interest thresholds, and SEIR-style states to trigger bursty information diffusion. On a SNAP-derived directed network with 324 nodes and 5,028 edges, the study compares threshold, pulse-intensity, and structural interventions; removing network cores produces the strongest suppression in the reported experiments.

From formation to control: repeated attention pulses create successive propagation peaks, while core-aware intervention sharply reduces diffusion reach.

324 Nodes

5,028 Directed edges

7 Attention pulses

3 Interventions

_{Published experiment at a glance}

Background and contributions

Online information does not always diffuse at a steady rate. A striking post can be amplified by influential users, repeatedly resurfaced through recommendation and discussion, and pushed through a network in a succession of attention bursts. The paper calls this sudden, high-frequency amplification the scream effect: an information item temporarily dominates the surrounding information environment, attracts large-scale participation, and may reinforce information cocoons and homogeneous exposure.

A static transmission rate cannot explain the whole process. The model must capture who can amplify a message, when that influence surges, which users decide to participate, and how the diffusion can be interrupted. This project addresses those questions through three connected contributions:

1. Pulse-driven temporal mechanism. A family of attention pulses represents repeated outbreaks rather than smooth, constant influence.
2. Influence–threshold state dynamics. LeaderRank, normalized incoming-edge weights, heterogeneous interest thresholds, and stochastic latent periods drive the S → E → I → R process.
3. Interpretable intervention design. Raising thresholds, reducing pulse intensity, and removing selected k-cores target individual response, temporal amplification, and network structure, respectively.

Technical roadmap

AI-assisted technical roadmap created by the authors using Gemini 3.5 Flash. This figure provides a conceptual overview; the definitions and equations below are the authoritative project description.

Model overview

Method at a glance

flowchart LR
    A["SNAP ego-Facebook data"] --> B["Directed weighted network"]
    B --> C["LeaderRank node influence"]
    C --> D["Pulse-driven proactive influence"]
    D --> E["Weighted passive influence"]
    E --> F{"Above interest threshold?"}
    F -- No --> S["Remain S"]
    F -- Yes --> X["Enter E"]
    X --> L["Stochastic latent period"]
    L --> I["Enter I and propagate"]
    I --> R["Enter R after propagation"]
    R --> O["State curves and intervention comparisons"]

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Propagation process and state dynamics

Figure 2. The outbreak source enters I. Downstream users remain in S while influence stays below threshold; after crossing it, they enter E, I, and finally R.

State	Interpretation in information diffusion	Transition
S — Susceptible	Has not joined the diffusion process	Remains S until passive influence exceeds the threshold
E — Exposed	Has been activated but is not yet propagating	Enters I after a latent time $T\sim\mathrm{Geo}(p)$
I — Propagating	Actively passes the information to downstream users	Enters R after its propagation paths are completed
R — Removed	Has completed propagation and no longer participates	Terminal state for the current information item

Influence and threshold equations

Let $G=(V,E)$ be a directed social network. For an edge from upstream user $v_k$ to user $v_i$, the normalized incoming-edge bias is

$$ r_{ki} = \frac{w_{ki}} {\sum_{v_j\in N^-(v_i)} w_{ji}}. $$

LeaderRank determines the baseline ability of a user to affect others, while a pulse family determines when that potential proactive influence is amplified:

$$ w_i^+(t) = LR(v_i) \left(1 + \sum_{\ell=1}^{m}\delta_\ell(t-t_0)\right). $$

The outbreak source $v_{i_0}$ can immediately express this influence. A non-source user must first cross its threshold, after which the propagation factor $\mu_i(t)$ scales the influence by the accumulated response:

$$ \widehat{w_i^+}(t)= \begin{cases} w_i^+(t), & i=i_0,\cr \chi_i(t)\mu_i(t)w_i^+(t), & i\ne i_0. \end{cases} $$

$$ \mu_i(t)=\frac{w_i^-(t)}{\theta_i}. $$

Each user accumulates passive influence from upstream neighbors, with a decayed contribution from the previous time step:

$$ w_i^-(t) = \sum_{v_k\in N^-(v_i)} \widehat{w_k^+}(t-1)r_{ki} \quad + e^{-\lambda}w_i^-(t-1). $$

The user joins the propagation process only after this accumulated influence crosses the interest threshold $\theta_i$:

$$ \chi_i(t)=\mathbf{1}\left[w_i^-(t)>\theta_i\right]. $$

In short, node importance determines who can amplify, the pulse family determines when amplification occurs, and edge weights plus interest thresholds determine whether that influence triggers a new propagator.

Data and experiment setup

The simulation network is constructed from the Stanford SNAP ego-Facebook dataset. After structural preprocessing, the directed graph contains 324 user nodes and 5,028 directed edges. Dense cores, local communities, and cross-community links provide a heterogeneous topology in which a small number of central users can strongly amplify diffusion.

Paper Figure 1 · Social-network structure Small-world, clustered, and heterogeneous structure motivating the directed-network model.

Project network visualization A Gephi view of communities, dense regions, and structurally central users.

Published simulation configuration

Setting	Value reported in the paper
Network	324 nodes, 5,028 directed edges
Pulse family	$m=7$ attention pulses
Maximum pulse intensity	$3$
Pulse interval radius	$\varepsilon=3$
Initial mean interest threshold	$0.6$
Outbreak source	Randomly selected from the top 10% of nodes by degree
Edge influence weight	Independently sampled from $U[0,1]$
Intervention outcome	Proportion of nodes in state R; the terminal value summarizes completed diffusion reach

Baseline propagation

The baseline experiment asks whether repeated attention shocks can reproduce the two defining signatures of the scream effect: a rapid, large-scale transition into propagation and a sequence of short-lived participation peaks.

Figure 5 · State evolution S falls rapidly; E and I rise in sequence; R eventually dominates.

Figure 6 · Pulse-aligned participation Each new pulse creates a visible peak in the per-iteration propagation population.

Observation. The susceptible population contracts quickly, while exposed and propagating users rise and reach successive peaks. At the main outbreak peak, more than 40% of users are simultaneously in the propagating state. The per-iteration propagation curve also exhibits a sequence of local peaks aligned with repeated information pulses.

Mechanism. Early pulses activate densely connected and influential users, opening many downstream paths at once. As more users complete propagation and enter R, fewer unvisited paths remain, so later pulses generate progressively weaker peaks.

Takeaway. The model reproduces both the suddenness and the repeated high-frequency bursts that define the scream effect.

Controlling the scream effect

The intervention experiments modify one mechanism at a time and compare the resulting diffusion coverage.

1. Raising interest thresholds

Raising the threshold means that a user must accumulate stronger passive influence before entering E. The comparison below shows how this keeps more users in S and interrupts paths that would otherwise become active.

Figure 3. Baseline final state (left) and final state after raising the mean interest threshold (right).

Figure 7 experiment, re-plotted with the threshold settings 0.6, 0.7, 0.9, 1.0, and 1.1.

Observation. As the mean threshold increases from $0.6$ to $1.1$, diffusion begins later and the final participating proportion declines substantially.

Mechanism. More users fail to accumulate enough influence to cross their thresholds, so they remain in S and do not create new downstream propagation paths.

Takeaway. Increasing users' resistance to low-quality or misleading information can delay a burst and reduce its final reach.

2. Reducing pulse intensity

Pulse intensity controls the temporary amplification of proactive influence. Lowering the peak weakens both the outbreak source and subsequent propagators.

Figure 4. Baseline final state (left) and final state after reducing the pulse intensity (right).

Figure 8. Propagation outcomes under progressively weaker pulse peaks.

Observation. Weaker pulse peaks delay growth and reduce the fraction of users that ever participate in diffusion.

Mechanism. Downstream users accumulate interest more slowly; some never cross their thresholds before the attention burst disappears.

Takeaway. Attenuating repeated amplification reduces both diffusion speed and final coverage.

3. Blocking structurally central users

A k-core captures a densely interconnected group of central users. Removing a selected core blocks both within-core diffusion and paths that connect the core to surrounding communities.

Figure 9. Propagation outcomes after removing different network `k`-cores.

Observation. Removing the tested k-cores produces a much larger reduction than the two parameter-only interventions. Lower $k$ values remove a broader core set and produce stronger suppression; in the strongest setting shown, the final participating proportion remains below $0.2$.

Mechanism. Central users are not merely high-degree individuals: they form mutually reinforcing structures and bridge propagation between communities. Blocking these structures eliminates many paths simultaneously.

Takeaway. Targeting structurally central user groups is the strongest of the three tested strategies, and it can be combined with threshold or pulse-based controls.

Results at a glance

Finding	Evidence	Implication
Repeated pulses create repeated participation peaks	Per-iteration propagation curve	Bursty attention must be modeled as a time-varying process
Network heterogeneity concentrates amplification	Dense communities and high-influence nodes	Structural position matters alongside individual response
Higher interest thresholds increase delay and reduce reach	Threshold sensitivity experiment	User-side discernment can interrupt propagation paths
Lower pulse intensity weakens final coverage	Pulse sensitivity experiment	Limiting repeated amplification can slow diffusion
k-core removal gives the strongest suppression	Structural intervention experiment	Core-aware intervention can block many paths at once

Together, the experiments show that the scream effect emerges from the interaction of temporal shocks, heterogeneous influence, individual activation thresholds, and network topology.

Installation and quick start

Install

git clone https://github.com/Zysishuiyears/scream-effect-propagation-simulation.git
cd scream-effect-propagation-simulation

python -m venv .venv
.\.venv\Scripts\Activate.ps1
# macOS/Linux: source .venv/bin/activate

python -m pip install --upgrade pip
python -m pip install -r requirements.txt

Run the baseline

python main.py

Experiment entrypoints

Script	Controlled experiment	Displayed output	Updated CSV
main.py	Baseline propagation	S/E/I/R trajectories and a per-iteration propagation curve	df1.csv
main2.py	Interest threshold	Cumulative participation $E+I+R=1-S$	df2.csv
main3.py	Pulse intensity	Cumulative participation $E+I+R=1-S$	df3.csv
main4.py	k-core removal	$E+I+R$, normalized by the original 324-node network	df4.csv

The inputs nodes.csv and edges.csv are included in the repository. Each script opens interactive Matplotlib figures and updates its corresponding CSV table. For the structural experiment, removed core nodes are outside the four dynamic states, while the denominator remains the original network size.

Code and repository map

Research layer	Main files	Role
Network construction	read_data.py, random_function.py, leadrank.py	Build the directed graph, assign edge influence, and rank users
Dynamic propagation	Pulse_function.py, parameter_calculate.py, spread.py	Generate pulses, accumulate influence, apply thresholds, and update states
Experiments	main.py–main4.py	Run the baseline and three intervention families

Repository structure

.
├── figures/
│   ├── paper/                    # Figures 1–9 from the published article
│   ├── readme/                   # README-specific visual summary
│   └── source/                   # Full-resolution plots and visualization exports
├── gephi/                        # Gephi project files
├── facebook/                     # Stanford SNAP ego-Facebook data
├── 示例/                          # Small illustrative network inputs
├── main.py ... main4.py          # Baseline and intervention experiments
├── parameter_calculate.py        # Influence and threshold calculations
├── spread.py                     # SEIR-style state transitions
├── Pulse_function.py             # Pulse-family construction
├── nodes.csv / edges.csv         # Processed 324-node directed network
├── df1.csv ... df4.csv           # Experiment result tables
├── requirements.txt              # Python dependencies
└── CITATION.cff                  # Machine-readable citation metadata

The complete visual inventory is documented in figures/README.md.

Publication and citation

Jiang Zhengxu, Yao Jiaqi, Meng Lingjun, Wang Jian, and Cao Yongchang. “Research of the Propagation Mechanism of Scream Effect and its Simulation.” Mathematical Modeling and Its Applications, 12(4), 2023, pp. 64–72+106. https://doi.org/10.19943/j.2095-3070.jmmia.2023.04.09

@article{jiang2023scream,
  author  = {Jiang, Zhengxu and Yao, Jiaqi and Meng, Lingjun and Wang, Jian and Cao, Yongchang},
  title   = {Research of the Propagation Mechanism of Scream Effect and its Simulation},
  journal = {Mathematical Modeling and Its Applications},
  year    = {2023},
  volume  = {12},
  number  = {4},
  pages   = {64--72+106},
  doi     = {10.19943/j.2095-3070.jmmia.2023.04.09}
}

Machine-readable metadata is available in CITATION.cff.

Data citation and acknowledgements

The network data is derived from the Stanford SNAP ego-Facebook dataset, introduced by J. McAuley and J. Leskovec in Learning to Discover Social Circles in Ego Networks (NeurIPS 2012). We thank the SNAP project for making the anonymized ego-network data available.

License

No open-source license is currently declared. Please contact the authors before copying, modifying, or redistributing the code or figures; the SNAP data remains subject to its original terms.