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=== Independent cascades models ===
== Independent Cascade (IC) Model ==
The Independent Cascade (IC) model is a probabilistic framework for modeling the spread of influence through a social network. In this model, the diffusion process unfolds in discrete time steps. When a node becomes active (i.e., influenced), it has a single chance to activate each of its inactive neighbors. The activation attempt succeeds with a certain probability, specific to the edge connecting the nodes. If the attempt fails, the neighbor cannot be activated by that node in future steps. This process continues until no more activations are possible.
Mathematically, the probability that an active node ''u'' activates an inactive neighbor ''v'' is denoted as <math>p_{uv}</math>. The activation process can be described as follows:
# At time <math>t = 0</math>, a set of seed nodes <math>A_0</math> are activated.
# For each subsequent time step <math>t \geq 1</math>, any node <math>u</math> that became active at time <math>t - 1</math> has one chance to activate each inactive neighbor <math>v</math>, with probability <math>p_{uv}</math>.
# If <math>v</math> is activated, it becomes active at time <math>t</math>; otherwise, it remains inactive.
# The process continues until no new activations occur.
This model captures scenarios where individuals independently decide to adopt a behavior or product after being influenced by their peers. It is particularly useful for modeling viral marketing and information dissemination in networks.
<ref name="Kempe2003">Kempe, D., Kleinberg, J., & Tardos, É. (2003). Maximizing the Spread of Influence through a Social Network. In Proceedings of the Ninth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (pp. 137–146). https://doi.org/10.1145/956750.956769</ref>
=== Linear threshold models ===
The Linear Threshold (LT) model is another approach to modeling influence spread in social networks. In this model, each node is assigned a threshold value, representing the fraction of its neighbors that need to be active before it becomes active itself. Each edge from a neighbor to a node has an associated weight, indicating the strength of influence. The sum of incoming edge weights to a node is typically constrained to be at most 1.
Mathematically, for a node <math>v</math>, let <math>N(v)</math> be the set of its neighbors, and <math>w_{uv}</math> be the weight of the edge from neighbor <math>u</math> to <math>v</math>, such that <math>\sum_{u \in N(v)} w_{uv} \leq 1</math>. Each node <math>v</math> is assigned a threshold <math>\theta_v</math>, drawn uniformly at random from the interval [0,1].
The activation process proceeds as follows:
# At time <math>t = 0</math>, a set of seed nodes <math>A_0</math> are activated.
# For each subsequent time step <math>t \geq 1</math>, any inactive node <math>v</math> becomes active if the total weight of its active neighbors meets or exceeds its threshold:
<math>\sum_{u \in A_{t-1} \cap N(v)} w_{uv} \geq \theta_v</math>
# The process continues until no new activations occur.
The LT model reflects situations where individuals adopt behaviors based on the collective influence of their peers, such as adopting a new technology or participating in social movements.
<ref name="Kempe2003" />
=== Energy model ===
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== HISB Model ==
In recent years, the rampant spread of misinformation and rumors on online social networks (OSNs) has raised significant concerns within the scientific and technological communities. OSNs are powerful tools for disseminating information, but they can also accelerate the propagation of false or misleading content, which may have serious societal consequences. Consequently, the study of rumor dynamics and the development of effective mitigation strategies have become crucial areas of research.
Several models have been proposed to analyze and control rumor spread in OSNs. Among them, the HISB model (Human Individual and Social Behavior model) stands out as a comprehensive approach that incorporates both individual cognitive factors and social influence mechanisms. This model was first introduced in,<ref>A.I.E. Hosni, K. Li, S. Ahmed, HISBmodel : A Rumor Diffusion Model Based on Human Individual and Social Behaviors in Online Social Networks, in: Springer, 2018.</ref> where the authors proposed a rumor diffusion framework based on human behaviors in online social environments. It was later refined and applied in a more practical context for rumor mitigation in.<ref>A.I.E. Hosni, K. Li, S. Ahmed, Minimizing rumor influence in multiplex online social networks based on human individual and social behaviors, in: Elsivier, 2019 Information science .</ref>
Subsequent enhancements were made in,<ref>A.I.E. Hosni, K. Li, Minimizing the influence of rumors during breaking news events in online social networks, in: Elsivier, 2019 knowledge based systems .</ref> where the model was adapted for breaking news scenarios, addressing the challenges of real-time rumor control. Most recently, a hybrid approach incorporating the HISB model was proposed by.<ref>Hosni, A.I.E., Baira, I., Merini, H. et al. Efficient hybrid rumor mitigation in dynamic and multilayer online social networks. Soc. Netw. Anal. Min. 14, 224 (2024). https://doi.org/10.1007/s13278-024-01381-z </ref>
These successive contributions illustrate the robustness and adaptability of the HISB framework in tackling the evolving nature of misinformation in OSNs. This article provides a comprehensive overview of rumor propagation models, with a focus on those incorporating behavioral dynamics, and discusses state-of-the-art strategies for minimizing rumor influence.
This article adopts the HISBmodel for simulating rumor diffusion. Below, we provide an overview of its key components.
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