A Perturbation-Theoretic Model for Fact-Checker Deployment in Dynamic Disinformation Networks

1 A Perturbation-Theoretic Model for Fact-Checker Deployment in Dynamic Disinformation Networks Spyridon Evangelatos‡, Mariza Konidi†,§, Eleni Veroni‡,§, Sophia Karagiorgou†and Christos Nikolopoulos§ ‡Research & Innovation Development, Netcompany-Intrasoft S.A., Luxembourg, Luxembourg †Data and AI/ML Systems (DAI) Research Group, Ubitech Ltd, Limassol, Cyprus §Department of Electronic Engineering, Hellenic Mediterranean University, Crete, Greece Emails: [email protected], mk[email protected], [email protected], [email protected], cnik[email protected] ✦ Abstract—Disinformation undermines public trust and weakens democratic institutions. It is without doubt that ffective interventions are necessary to mitigate these risks. Counter narratives based on empirical evidence assist in limiting the dissemination of false information and preserve the integrity of public discourse. In this paper, we present a perturbation-based framework that integrates message passing algorithms to model and optimize the deployment of fact-checkers in social media networks. Instead of removing key influential accounts that spread disinformation, we dynamically introduce fact-checking nodes at optimal times. We analyze how small perturbations in network structure affect global disinformation spread, incorporating a message passing mechanism to iteratively track belief evolution across users. The proposed framework leverages activation strategies based on belief thresholds and network acceleration metrics, ensuring that fact-checkers intervene only when it is necessary. We validate the effectiveness of our method through extensive simulations, demonstrating its capability to suppress disinformation while preserving network stability. Index Terms—Fake News, Disinformation Spread, Fact-checkers, Perturbation Theory, Message Passing 1 INTRODUCTION Disinformation spreads dynamically across social networks, often adapting to countermeasures. Combating disinformation often involves removing disinformation sources such as deplatforming users, banning bot accounts, shutting down disinformation websites, etc. While this approach may seem effective in the short term, it is often inefficient and raises significant ethical concerns. From an efficiency perspective, removing disinformation sources does not eliminate the underlying conditions that allow disinformation to spread uncontrollably [1]. Many disinformation sources are highly adaptable and can quickly regenerate through new accounts or alternative social media platforms. It has been frequently observed that banned social media influencers migrating to less-regulated platforms [2] where their reach may be diminished but their narratives become more extreme. Automated bot networks can be easily reactivated with minor modifications, turning takedown efforts into a continuous and costly cycle of suppression and reemergence. Moreover, when major disinformation sources are removed, the network may re-organize in ways that make it more difficult to track and contain disinformation. Studies on censorship effects show that when content is forcibly removed, disinformation communities frequently reinforce their beliefs even more strongly, migrating to encrypted spaces or adopting coded language to evade detection [3]. This can lead to a fragmented but more resilient disinformation landscape, where fact-checking and counter-narratives become even less effective. Additionally, suppressing certain voices - regardless of their intent - may foster a persecution narrative, where deplatformed users gain credibility among their followers, reinforcing the perception that mainstream institutions are suppressing the truth. Beyond efficiency concerns, the removal of disinformation sources also raises critical ethical issues. Deplatforming individuals or organizations often involves subjective decisions about what constitutes harmful content, which can be influenced by political or cultural biases [4]. This leads to concerns about freedom of speech, particularly when decisions are made by private corporations with limited transparency and accountability. If not applied consistently, content moderation practices may be perceived as unfair, leading to further distrust in media and institutions. Moreover, removing disinformation sources does not necessarily educate users or improve media literacy; instead, it may create an environment where people are more likely to seek out alternative sources that reinforce their existing biases. Given these challenges, in this paper we describe an alternative approach, introducing adaptive fact-checking interventions rather than removing disinformation sources. Instead of eliminating nodes, fact-checkers can be introduced dynamically into the network, countering disinformation at optimal times to minimize its spread. Our approach leverages network perturbation theory to disrupt disinformation without triggering unintended consequences like fragmentation or radicalization. Notably, disinformation not only distorts public understanding but can escalate into online and offline harms [5], including incitement to violence or even real-world criminal activity [6]. This is a pre-print version of the paper entitled “A Perturbation-Theoretic Model for Fact-Checker Deployment in Dynamic Disinformation Networks”. The published version is available at the following link:https://dl.acm.org/doi/10.1145/3746275.3762200 2 The strategically deployment of fact-checking mechanisms at the right moment, i.e., when disinformation gains traction but before becoming deeply entrenched, allows for a more efficient and ethically sound way to manage information ecosystems. The rest of the paper is organized as follows: Section 2 provides an overview of prior studies on disinformation propagation, mitigation strategies and network intervention techniques. Section 3 introduces the network representation model, detailing the formalism used to describe disinformation spread, fact-checker influence and network perturbations. Section 4 presents the perturbation-theoretic framework, explaining how small structural modifications impact network stability and belief evolution. Section 5 formulates the perturbed Laplacian matrix and analyzes its spectral properties to determine the effectiveness of different intervention strategies. Section 6 describes the adaptive fact-checker deployment mechanism based on the message passing equations that govern belief updates across the network. Section 7 presents the performance evaluation of the proposed framework. Finally, Section 8 concludes the paper by summarizing key insights and outlining future research directions. 2 RELATED WORK The effectiveness of various intervention strategies, including the removal of disinformation sources, has been a focal point in recent research studies [7]–[9], with findings indicating that while removal can suppress harmful content in the short term, it often leads to unintended consequences such as migration to alternative platforms and increased distrust in authorities [10]. In contrast, corrective interventions such as fact-checking, inoculation strategies and algorithmic content moderation have been shown to be more effective in reducing disinformation’s long-term influence by improving audience resilience and fostering critical engagement [11]. In their landmark study [12], the authors compared the diffusion of true versus false news on social media, highlighting how false information can spread more rapidly and broadly than verified information, and provided an empirical basis for considering various interventions to stem the flow of disinformation. Nevertheless, their study also revealed the persistent challenges in countering disinformation, as corrections often fail to reach the same audience penetration as the initial false narratives. In another seminal work [13], the authors provided a thorough review of the challenges posed by fake news and disinformation, discussing both the underlying dynamics of online information diffusion and a range of potential interventions. They analysed various intervention strategies, from content flagging and corrections to the more proactive removal or limiting of sources that repeatedly disseminate falsehoods. Despite their extensive analysis, their study acknowledged several challenges in implementing these interventions effectively. They highlighted issues such as the trade-offs between content moderation and free speech, the potential unintended consequences of selective factchecking (e.g., the implied truth effect) and the limitations of algorithmic interventions in distinguishing satire, opinion and deliberate disinformation. Furthermore, the authors emphasized the role of cognitive biases in shaping users’ receptiveness to corrections, arguing that interventions must account for psychological resilience to disinformation. Last, the authors in [14] conducted a bibliometric analysis of intervention strategies aimed at mitigating disinformation sharing on social media, identifying key research trends, influential works and thematic clusters in the field. Their findings highlighted the increasing academic interest in disinformation interventions, emphasizing the role of fact-checking, debunking, algorithmic moderation and user awareness campaigns. Their study also underscored the interdisciplinary nature of disinformation research, drawing from computer science and psychology. However, the paper relied heavily on bibliometric data, overlooking qualitative insights and the real-world effectiveness of intervention strategies. Additionally, while their analysis identified research gaps, it did not propose concrete solutions or assessed the ethical and regulatory challenges associated with disinformation interventions. 3 NETWORK REPRESENTATION Social networks are inherently dynamic, since both their structure and their interaction strength (how much influence each node has) evolve over time. Thus, in order to accurately capture and model the real-world information propagation, adaptive influence mechanisms and evolving relationships between users and content sources, we define a dynamic weighted network Gt= (Vt, Et, Wt),(1) where Vt=Ut∪St∪Ftrepresents the set of users, sources of disinformation and fact-checkers, respectively, Et⊆Vt×Vt is the set of time-dependent edges and Wt= [wij(t)] is the the time-dependent weighted adjacency matrix of a dynamic network, where each entry wij (t)quantifies the strength of influence from node ito node jat time t. Each node i∈Vtholds a continuous belief state xi(t)∈[−1,1], where xi(t) = +1 indicates full belief in disinformation, xi(t) = −1full belief in verified information and xi(t) = 0 represents a neutral or undecided stance. While user nodes dynamically update their belief states over time based on interactions and external influences, disinformation sources remain fixed at xi(t) = +1,∀i∈St, continuously propagating false narratives, and fact-checkers remain fixed at xi(t) = −1,∀i∈Ft, persistently promoting verified information. 4 PERTURBATION THEORY FOR ADAPTIVE FACTCHECKER DEPLOYMENT Removing sources of disinformation can often lead to unintended consequences. For this purpose, we introduce small perturbations in the network structure through the adaptive deployment of fact-checking nodes. Fact-checking interventions are introduced only when disinformation reaches critical levels, ensuring that corrective measures are both targeted and minimally disruptive to organic user interactions. The introduction of fact-checkers into the network modifies the network’s adjacency matrix, functioning as a controlled perturbation in the influence landscape. Rather than 3 abruptly eliminating high-influence disinformation nodes, the network structure is dynamically adjusted by adding counter-narrative nodes that selectively interact with users most exposed to disinformation. 4.1 Perturbed Adjacency Matrix Introducing a fact-checker at time tperturbs the network’s influence dynamics by modifying the influence weight matrix Wtas follows ∼ Wt=Wt+ϵ∆WF t,(2) where ϵis a perturbation parameter that controls the magnitude of intervention, ensuring that fact-checkers do not completely override organic interactions but instead introduce corrective influence gradually and ∆WF trepresents the change in influence weights due to fact-checker intervention, modifying how users process information over time. The elements of ∆WF tare defined as ∆WF ij (t) = −γ·wij(t)if xj(t)> θ. (3) This fact-checker influence term introduces a controlled suppression of disinformation influence through the parameter γwhich determines the strength of the fact-checker intervention. Since the proposed perturbation modifies the adjacency matrix smoothly, it prevents sudden disruptions that could cause instability in network influence dynamics. A sudden large-scale intervention, such as the removal of disinformation sources, might create structural holes in the network [15], leading to decreased trust in platform moderations and causing users to perceive fact-checking biased. The modification of the adjacency matrix inevitable leads to a new Laplacian matrix: ∼ L=L+ϵ∆LF(t).(4) with ∆LF(t)being the fact-checker influence modification, which adjusts the influence weights of nodes exposed to disinformation. Since ∆LF(t)arises due to fact-checker intervention, its elements are defined as: ∆LF(t) = DF−∆WF t,(5) where DFis the corresponding degree matrix, ensuring that row sums remain zero. To analyze the stability of the intervention, we examine the eigenvalues of ∼ L. The perturbed eigenvalues are given by ∼ λk=λk+ϵψ⊤ k∆LFψk,(6) where λkand ψkare the corresponding eigenvectors of L. The effectiveness of fact-checker deployment in suppressing disinformation can be analyzed through the spectral properties of the perturbed Laplacian matrix. The intervention is considered effective in mitigating the disinformation spread if ρ(∼ L) = max |∼ λk|<1.(7) This condition ensures that the influence of disinformation dissipates over time rather than persisting or amplifying within the network. A spectral radius greater than or equal to 1would indicate that disinformation can still propagate or even amplify, making the intervention ineffective. To effectively reduce the spectral radius, the perturbation ∆LFintroduced by fact-checkers must be designed in such a way that it targets the key influential disinforamtion pathways while minimizing disruptions to the rest of the network. A well-controlled intervention satisfies max |∼ λk| ≈ 1− O(ϵ),(8) where O(ϵ)represents the rate of spectral suppression induced by the intervention and ensures that disinformation loses its dominant eigenmodes while network stability is preserved and a natural information flow of information is maintained. In our approach, we included an adaptive perturbation scaling, ∆LF=κL ∥L∥,(9) where κis a tunable parameter ensuring that fact-checking influence scales proportionally to the existing disinformation diffusion patterns. 5 ADAPTIVE FACT-CHECKER DEPLOYMENT STRATEGY Fact-checkers should be introduced dynamically at optimal times based on network conditions rather than being statically present throughout the network. Deploying factcheckers adaptively ensures that resources are efficiently allocated, minimizing intervention fatigue among network users and preventing unnecessary disruptions to organic discussions. 5.1 Fact-Checker Activation Function To determine the optimal time for introducing fact-checkers, we define an activation function that dynamically detects when disinformation reaches a critical threshold. This function uses real-time monitoring of user belief states and disinformation trends. For this purpose, we use a sigmoidbased activation function to determine the probability of deploying fact-checkers at time t, i.e., Pf(t) = 1 1 + exp [−β(E[x(t)] −θ)],(10) where E[x(t)] = 1 NPj∈Utxj(t)is the average disinformation belief level in the network at time t,θis the disinformation activation threshold which once exceeded, fact-checkers are deployed and βis the activation sensitivity parameter, determining how sharply fact-checkers respond to changes in disinformation prevalence. A fact-checker node is dynamically introduced into the network at time twhen the probability function Pf(t) which governs the likelihood of deployment, surpasses a predefined threshold τ, a tunable parameter that controls the level of disinformation required to trigger fact-checking interventions, i.e., Pf(t)> τ, (11) where τis the deployment probability threshold. This condition introduces a probabilistic decision-making mechanism that prevents fact-checkers from being deployed too frequently or too sparsely. Instead of a deterministic rule 4 where fact-checkers are always introduced at a fixed disinformation level, this approach allows for stochastic control, making interventions more natural and less predictable. Moreover, the choice of τdetermines how aggressively factcheckers are deployed, acting as a sensitivity parameter that balances intervention efficiency with user autonomy. 5.2 Stochastic Deployment Timing In order to prevent predictability, we introduce stochastic deployment timing by defining the actual fact-checker introduction time as: tf=t0+ξ, with ξ∼ N 0, σ2,(12) where t0is the expected activation time based on Pf(t)and ξis a random noise term drawn from a normal distribution N(0, σ2)that models timing uncertainty. The proposed unified deployment strategy integrates multiple decisionmaking criteria to ensure robust, adaptive and effective disinformation suppression since it combines threshold-based activation, ensuring that fact-checkers intervene only when disinformation reaches critical levels, and stochastic deployment timing, which prevents interventions from becoming too predictable. 6 MESSAGE PASSING EQUATIONS WITH ADAPTIVE FACT-CHECKING TIMING In a dynamic social network where disinformation spreads through user interactions, message passing algorithms [16], [17] provide a structured way to model how beliefs evolve over time. Each node in the network - whether a user, a disinformation source, or a fact-checker - exchanges messages with its neighbors, influencing their belief states. Even though fact-checkers and disinformation sources do not update their belief states, they still participate in message exchanges because they influence the users connected to them. Their participation in message exchange counteracts the persuasive strength of disinformation sources, ensuring that common users receive corrective information, which prevents disinformation from becoming the dominant narrative. Additionally, these exchanges assist in adjusting the influence weights over time, reducing the impact of disinformation sources on susceptible users and maintaining network stability by preventing disinformation from forming self-reinforcing loops. Last, fact-checker messages disrupt disinformation pathways, ensuring that disinformation does not spread unchecked and that users remain exposed to factual corrections, reinforcing the credibility of verified information across the network. The Message Passing Algorithm we propose, models in an accurate manner how beliefs evolve in a dynamic network where disinformation spreads and fact-checkers intervene adaptively. As it can been see in Figure 1 each s1 u3u4 u2 u1 u6 f2 u5 f1 ⋆ mu3→s1(t) ms1→u3(t) ⋆ mu4→u3(t) ⋆ mu3→u4(t) ⋆ mu2→u3(t) ⋆ mu3→u2(t) ⋆ mu5→s1(t) ⋆ mu1→u2(t) ⋆ mu2→u1(t) ms1→u1(t) ⋆ mu6→s1(t) ms1→u6(t) bmf2→s1(t) ms1→f2(t) ⋆ mu5→f2(t) bmf2→u5(t) ⋆ mu6→u5(t) ⋆ mu5→u6(t) bmf2→u4(t) ⋆ mu4→f2(t) bmf1→u6(t) bmf1→u1(t) ⋆ mu1→f1(t) bmu6→f1(t) Fig. 1: Graph representation of the disinformation network, showing interactions between disinformation sources (s), fact-checkers (f) and users (u). The directed edges represent belief propagation, where messages (ms→u) are conveyed from sources to users, while fact-checkers counteract with corrective messages ( b mf→u). node updates its belief state based on incoming messages from its neighbours based on xj(t+ 1) = tanh X i∈St mi→j(t) + X k∈Ftb mk→j(t)1(t≥tf) +X u∈Ut ⋆ mu→j(t) + X q∈St,Ft ⋆ mq→j(t) +hj+Djxj(t)!.(13) As it can be readily seen the belief updates include four key terms, i.e., Pi∈Stmi→j(t)the messages from disinformation source ito user j,Pk∈Ftb mk→j(t)1(t≥tf)the messages from fact-checkers to user j, activated only if fact-checkers are deployed, Pu∈Ut ⋆ mu→j(t)the messages among users, where user jis influenced by connected users and Pl∈St,Ft ⋆ ml→j(t)the messages from users to both disinformation sources and fact-checkers. The external influences hjaffects how users change their belief and the self-reinforcement term Djxj(t)which represents memory effects and confirmation bias. For simplicity reasons we have kept hjtime-independent, ensuring that external interventions exert a consistent and uniform influence on user beliefs, avoiding additional complexity from time-varying effects that may introduce instability or require additional tuning parameters. In the following sub-sections we formalize the message passing update equations and provide a detailed iterative algorithm for belief evolution over time. 6.1 Messages from Disinformation Sources Disinformation sources act as fixed nodes in the network that continuously propagate false narratives. Unlike users, whose beliefs evolve dynamically, disinformation sources do not change their beliefs over time, thus we can safely set xi(t) = +1,∀i∈St. Their messages mainly attempt 5 to mislead network users into adopting false narratives, undermine fact-checking efforts and strengthening the beliefs of already disinformed users. Their messages can be formulated as mi→j(t+ 1) = tanh (βSwij(t)) ,(14) where i∈Stand j∈Utor j∈Ftand taking into account that xi(t) = 1,∀i∈St. These messages mi→k(t)do not alter fact-checkers’ beliefs but they weaken their credibility among users, reducing the effectiveness of interventions. 6.2 Messages from Fact-Checkers Fact-checkers play a corrective role in the network by countering disinformation and reinforcing verified information. Unlike users, their beliefs are fixed at xi(t) = −1,∀i∈Ft, since they do not change over time but continuously influence others. Fact-checkers interact with both users and disinformation sources, aiming to correct disinformation and reduce its influence. Fact-checkers provide corrective information to users who might be disinformed while they also attempt to counteract disinformation sources, reducing their influence. Similarly with the disinformation sources, they exchange one type of message, i,e., b mk→j(t+1) = tanh −βFwkj (t), k ∈Ft, j ∈Utor j∈St, (15) where βFis the fact-checking influence strength, controlling how aggressively fact-checkers correct disinformation. It is worth highlighting that if wkj(t)is large, then the user j receives a strong corrective signal from fact-checking node k. Their messages to users provide corrective influence, pushing beliefs toward factual accuracy, while their messages to disinformation sources attempt to reduce credibility and suppress disinformation impact. 6.3 Messages from Users The users in the network have a dynamic role, interacting with other platform users, disinformation sources and factcheckers. Unlike disinformation sources and fact-checkers, whose beliefs remain fixed, users update their beliefs over time based on the messages they exchange and their own biases. However, users also actively influence others, spreading, reinforcing or challenging both disinformation and factchecking narratives. Users generate messages that shape the evolution of beliefs in the network indicating peer influence, interactions with disinformation sources and engagement with fact-checkers. Through user-to-user messages, individuals reinforce or challenge each other’s beliefs, potentially forming echo chambers or corrective discussions depending on their connections and trust levels, thus forming the respective message as ⋆ mq→j(t+ 1) = tanh βUwqj (t)xq(t).(16) Users also interact with disinformation sources, either amplifying disinformation if they are already disinformed or attempting to discredit false narratives if they trust verified information, exchanging messages of the following form ⋆ mq→s(t+ 1) = tanh βUwqs(t)xq(t),(17) Algorithm 1: Message Passing Algorithm 1: Input: 2: Gt= (Vt, Et, Wt),xj(0),wij(0),tf,βS, βF, βU,hj,Dj, Tmax,ϵ,ϵp 3: Output: Final belief states xj(T) 4: Initialization: 5: for all users j∈Utdo 6: Initialize xj(0) randomly or set to neutral belief 7: end for 8: for all edges (i, j)∈Etdo 9: Initialize influence weights wij(0) 10: end for 11: for t= 1 to Tmax do 12: Step 1: Compute Messages from Disinformation Sources 13: for all j∈Utdo 14: Compute messages according to Eq (14) 15: end for 16: Step 2: Compute Messages from Fact-Checkers (If Deployed) 17: if t≥tfthen 18: for all j∈Utdo 19: Compute messages according to Eq (15) 20: end for 21: end if 22: Step 3: Compute User-to-User and Source Interactions 23: for all j∈Utdo 24: Compute messages, according to Eq. (16) 25: Compute user influence according to Eq. (17) 26: end for 27: Step 4: Apply Network Perturbation Due to Fact-Checking Interventions 28: if t≥tfthen 29: Update influence weight matrix with perturbation according to Eq. 2 30: Compute perturbation effect based on Eq. (3) 31: Modify network Laplacian according to Eq. (4) 32: Ensure spectral stability condition: 33: if max |e λk|<1then 34: Continue belief updates 35: else 36: Adjust perturbation strength ϵp 37: end if 38: end if 39: Step 5: Update Belief State for Each User 40: for all j∈Utdo 41: Compute belief update based on Eq. (13) 42: end for 43: Step 6: Check Convergence 44: if max |x(t+1) j−x(t) j|< ϵ then 45: Break: Convergence achieved 46: end if 47: end for 48: Return: Final belief states xj(T) 6 Fig. 2: Convergence of the average belief state E[x(t)] over T= 50 iterations for six scenarios combining different percentages of disinformation sources and fact-checkers under sparse and dense network connectivity. Solid lines correspond to sparse networks (p= 0.1), while dashed lines correspond to dense networks (p= 0.5). Model parameters include number of nodes n= 100, disinformation influence strength βS= 0.2, fact-checker influence strength βF= 0.6, peer-to-peer influence βU= 0.1, memory effect D= 0.05 and external influence h= 0.02. where β′ Ucontrols the user influence on disinformation sources. Similarly, users engage with fact-checkers, where some accept corrections and spread factual content, while others resist fact-checking efforts, reinforcing skepticism toward corrective interventions: ⋆ mq→k(t+ 1) = tanh βUwqk(t)xq(t).(18) 7 PERFORMANCE EVALUATION To assess the effectiveness of fact-checker deployment in mitigating disinformation, we conduct a series of simulations under varying conditions. Our evaluation focuses on key parameters that influence the suppression dynamics, including the ratio of disinformation sources to fact-checkers, the activation threshold θ, and the external influence h. We analyze how different intervention strategies affect the global belief state of the network. Through these analyses, we aim to determine optimal configurations for fact-checker activation, ensuring that interventions are both timely and effective in preventing disinformation from becoming selfreinforcing. The following subsections provide a detailed exploration of the simulation results, highlighting the impact of varying model parameters on disinformation suppression. The selection of the parameter values in the following simulations is primarily guided by a balance between theoretical relevance, computational feasibility and real-world applicability, since as mentioned in various other related scientific papers (e.g., [18]), there is a lack of appropritate datasets to accurately analyze the disinformation spread. The values are chosen to capture key dynamics observed in real-world disinformation networks while ensuring meaningful and interpretable results. More specifically, factcheckers tend to have a weaker influence compared to disinformation sources, as corrective information is often less engaging and spreads more slowly [19]. Setting βF> βS ensures that, at some point, fact-checkers have a stronger corrective effect, aligning with studies suggesting that sustained verification efforts eventually outweigh initial disinformation. The choice of a moderate βUreflects the fact that users influence each other through social interactions, but this effect is generally weaker than direct disinformation campaigns or fact-checking interventions. Moreover, the sensitivity parameter βgoverns how abruptly fact-checkers activate. A higher βensures a sharp transition in response to disinformation surges, mirroring social media platforms’ policies, which tend to apply fact-checking labels only after significant disinformation activity. A network with n= 100 users is selected for most experiments to allow meaningful statistical analysis while maintaining computational feasibility. Larger networks (e.g., N≈105) would increase realism but also significantly raise computational complexity. 7.1 Convergence Analysis Figure 2 illustrates the the convergence behavior of the average belief state, E[x(t)], under varying configurations of disinformation source and fact-checker deployment percentages across sparse and dense social network topologies. Each line traces the evolution of the network’s global belief over several iterations, highlighting how different agent distributions and network structures influence the suppression or dominance of disinformation spread. In sparse networks, configurations with balanced or higher percentages of fact-checkers achieve convergence to negative beliefs - indicating effective suppression. Conversely, scenarios with higher disinformation presence or fewer fact-checkers result in positive belief convergence, signaling the prevalence of disinformation. In dense networks, the effects of disinformation sources become more pronounced, as increased connectivity accelerates belief propagation. This leads to rapid convergence toward extreme belief values, either +1 or −1, depending on the dominant force in the network. These dynamics mirror real-world observations where disinformation spreads faster and more broadly in highly connected environments such as social media [20]. These results emphasize the need for proportional fact-checker deployment and strategic intervention in denser networks to counteract disinformation more effectively. 7.2 Effectiveness of Fact-Checker Deployment Figure 3 illustrates the temporal evolution of belief states under varying source-to-fact-checker ratios, offering a comparative analysis of how different configurations influence the suppression of disinformation. Each subplot represents a different ratio of disinformation sources (s) to fact-checkers (f), with values ranging from s/f = 0.1to s/f = 2.0. The x-axis represents time steps (t), and the y-axis represents the mean belief state (E[x(t)]), which varies between −1 (verified information dominance) and +1 (disinformation dominance). A notable trend observed across all subplots is that when the source-to-checker ratio is low (s/f = 0.1), the system initially experiences an increase in disinformation prevalence, but as fact-checkers are activated (at tf= 18), 7 Fig. 3: The evolution of the average belief state x(t)over time for different source-to-fact-checker ratios s/f. The network consists of n= 100 users with varying numbers of disinformation sources sand fact-checkers f, evolving over T= 60 time steps. The disinformation influence strength is set to βS= 0.2, while fact-checkers have an influence of βF= 0.6, and user-to-user interactions follow βU= 0.1. The self-reinforcement effect is given by D= 0.1, and external bias is set at h= 0.05. A fact-checking activation threshold of θ= 0.6with an activation sensitivity β= 5.0determines when interventions occur. The dashed black line represents the critical belief threshold θ, while the dotted red and purple lines denote the belief states associated with disinformation (x= +1) and verified information (x=−1), respectively. Vertical green dashed lines indicate the time step tfat which fact-checkers are deployed. The shaded regions represent confidence intervals based on multiple trials. the mean belief state quickly transitions toward verified information. The deployment time is relatively late compared to other cases, suggesting that when fact-checkers are significantly outnumbered, disinformation can spread more extensively before being mitigated [21]. Despite this delay, fact-checkers effectively suppress disinformation over time, demonstrating the resilience of their influence even under unfavorable initial conditions. As the source-to-checker ratio increases (s/f = 0.5and s/f = 1.0), we observe that disinformation initially spreads more aggressively, reaching higher peak values compared to the s/f = 0.1scenario. However, fact-checkers activate earlier (tf= 16 for s/f = 0.5and tf= 13 for s/f = 1.0), leading to a faster suppression of disinformation. The more rapid intervention is attributed to the system crossing the activation threshold earlier, triggering fact-checkers before the influence of disinformation becomes dominant. The decreasing deployment time as s/f increase,s suggests that systems with a higher proportion of sources relative to factcheckers are more sensitive to belief shifts, leading to earlier interventions. In the extreme case where the source-to-checker ratio reaches s/f = 2.0, the system behaves differently from the previous cases. The mean belief state stabilizes near the disinformation state (x≈+1), indicating that the factcheckers fail to reverse the spread of false information. Despite their activation at tf= 11, their influence is insufficient to counterbalance the overwhelming presence of disinformation sources. This outcome underscores a critical tipping point: when disinformation sources significantly outnumber fact-checkers, suppression mechanisms become ineffective, and the network remains trapped in a disinformation-dominated state. Figure 4 illustrates the impact of the source-to-factchecker ratio on the final average belief state, providing critical insights into the phase transition behavior of disinformation suppression. The x-axis represents the ratio of disinformation sources to fact-checkers, while the yaxis depicts the final average belief state after the network reaches stability. Three distinct curves are shown, each corresponding to different activation thresholds θ. The inside plot further clarifies the activation probability, Pf(t), as a function of the average belief state, E[x(t)], which governs when fact-checkers are activated based on the prevailing disinformation level. A key observation from the figure is the presence of a sharp phase transition in the final belief state as the source-to-fact-checker ratio increases. For small values of s/f, where fact-checkers outnumber or are comparable to disinformation sources, the final belief state remains at or near −1, indicating that disinformation has been successfully suppressed and fact-checkers have stabilized the 8 Fig. 4: Impact of the source-to-fact checker ratio on final belief states. Different curves correspond to activation thresholds θ={0.3,0.5,0.8}. The inside plot illustrates the activation probability Pf(t)as a function of the average belief. The simulation was conducted with n= 100 users, f= 5 fact-checkers, T= 100 time steps, and model parameters: βS= 0.2,βF= 0.6,βU= 0.1,D= 0.05, h= 0.02 and τ= 0.7. network towards truthful information. However, as the ratio increases beyond a critical threshold, a sudden jump occurs, where the final belief state abruptly shifts toward 1, signifying a dominant dissemination of false narratives. This transition occurs at different points depending on the activation threshold θ, with lower thresholds triggering fact-checkers earlier and requiring fewer fact-checkers for suppression. The inside plot provides additional context for the activation mechanism of fact-checkers described in Eq. (11). The probability of activation follows a sigmoid function, meaning that fact-checkers are more likely to intervene when the average belief state approaches the disinformation domain (E[x(t)] >0). The different threshold values affect the steepness and onset of this activation. Lower values of θ (e.g., 0.3) lead to a higher probability of early intervention, ensuring suppression even with moderate numbers of factcheckers. In contrast, higher values (e.g., 0.8) require the disinformation level to grow significantly before activation, leading to delayed suppression and requiring a lower source-to-fact-checker ratio for stability. The produced results highlight a fundamental trade-off between early activation of fact-checkers and disinformation resilience. A low threshold θleads to earlier intervention and a smaller critical source-to-fact-checker ratio, meaning fewer fact-checkers are needed to prevent disinformation from taking over the network. In contrast, a high threshold allows disinformation to spread longer before intervention, requiring a substantially lower s/f ratio to maintain suppression. This trade-off suggests that setting θtoo high may reduce the effectiveness of fact-checkers [22], making suppression more difficult unless a very large number of fact-checkers are deployed. The sharpness of the transition reveals that fact-checker deployment follows a nonlinear behavior, i.e., a small inFig. 5: Heatmap of fact-checker effectiveness illustrating the relationship between disinformation strength βSand activation threshold θ. The color scale represents the mean final belief state E[x]after 20 trials. Blue regions indicate successful suppression of disinformation, while red regions correspond to higher disinformation retention. The transparency overlay captures variability across trials. Parameters used are number of βSvalues = 70, number of θ values = 70, fact-checking influence strength βF= 0.3, peer influence strength βU= 0.1, memory effect D= 0.05, external influence h= 0.01. crease in disinformation sources can lead to a dramatic collapse of truth stabilization beyond a critical point. This underscores the importance of proactive intervention strategies [23], where fact-checkers should be deployed before the belief state reaches a dangerously high disinformation level. These findings suggest that regulatory policies or algorithmic interventions should focus on maintaining an optimal balance between sources and fact-checkers while setting activation thresholds at a level that ensures timely responses without unnecessary overuse of fact-checking resources. The heatmap in Figure 5 illustrates the effectiveness of fact-checkers in suppressing disinformation as a function of the disinformation strength βSand the activation threshold θ. From this figure, we observe a clear transition in the final belief state as βSand θvary. In regions where βSis low and θis high, the network maintains a strong suppression of disinformation, leading to a more verifiable belief state (blue region). Conversely, when βSincreases and θis low, the network struggles to suppress disinformation, resulting in a persistent disinformation state (red region). This result aligns with theoretical expectations, as a stronger disinformation influence (βS) naturally makes suppression more difficult, while a lower activation threshold (θ) means fewer fact-checkers are deployed, further exacerbating the problem. A key insight from the heatmap is the presence of a transition region, where the suppression of disinformation is highly sensitive to small changes in βSand θ. This transition suggests a critical threshold beyond which disinformation becomes self-reinforcing, making fact-checker intervention less effective. The presence of this threshold highlights the importance of proactive fact-checking strate- 9 gies, where maintaining a sufficiently high θcan prevent the network from entering an irreversible disinformationdominated state. This behavior is representative of the realworld information ecosystems, where the spread of disinformation depends not only on its intrinsic strength (e.g., virality, engagement) but also on the responsiveness of factchecking mechanisms [24]. In online social networks, platforms with stricter content moderation policies (analogous to high θ) are better at preventing disinformation proliferation. Conversely, when fact-checkers respond passively or with high latency (low θ), disinformation can rapidly gain traction, making subsequent suppression efforts significantly harder. The figure also suggests an optimal operating regime where fact-checkers can be most effective. For moderate values of βS, increasing θsignificantly improves suppression. However, in extreme cases of high βS, even aggressive fact-checking may not be sufficient, indicating the need for complementary interventions such as platform-level content regulation, user awareness campaigns, or reducing the initial spread of disinformation. From a policy making perspective, this figure reinforces the notion that timely and widespread fact-checking interventions are imperative in limiting disinformation propagation. Additionally, the presence of a sharp transition emphasizes the importance of staying ahead of the critical threshold, as allowing disinformation to take hold in a network may render subsequent suppression efforts ineffective [25]. 8 CONCLUSION This paper introduced a novel perturbation-theoretic framework for mitigating disinformation in complex social networks, emphasizing the role of adaptive fact-checker deployment in stabilizing belief dynamics. Unlike conventional approaches that indiscriminate removal of disinformation sources, our method leverages network perturbations to dynamically adjust fact-checker influence, ensuring targeted suppression of disinformation spread. The proposed perturbation-theoretic approach enables precise interventions ensuring that disinformation does not become self-reinforcing. In contrast with traditional static fact-checking models, our framework dynamically adjusts fact-checker influence based on belief evolution, optimizing suppression efforts while minimizing resource allocation. Additionally, we reveal a phase transition threshold beyond which disinformation becomes dominant, highlighting the critical conditions required for effective suppression. The findings of this work suggest several promising directions for future research. Exploring heterogeneous user behaviors, such as varying susceptibility to disinformation, could enhance the adaptability of the fact-checking intervention strategy. Another promising direction is the integration of reinforcement learning techniques to optimize fact-checker placement dynamically, allowing the system to learn from evolving disinformation trends and improve intervention timing. Finally, extending the framework to multilayer networks, where disinformation propagates across multiple platforms or communication channels, would provide a more realistic model of modern (dis)information ecosystems. ACKNOWLEDGMENT This work has received funding from the European Union’s Horizon Europe framework programme through AVALANCHE and FERMI projects under grant agreement No. 101168393 and 101073980 respectively. The content reflects only the authors’ views and the European Commission is not responsible for any use that may be made of the information it contains. REFERENCES [1] S. 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