Towards Effective Planning Strategies for Dynamic Opinion Networks
Authors: Bharath Chandra Muppasani
Protik Nag
Vignesh Narayanan
Biplav Srivastava
Michael Huhns
Venue: NeurIPS 2024 poster (2024)
Keywords: Opinion networks
Dynamic Planning
Misinformation Spread
Network dynamics.
Primary area: machine_learning_for_social_sciences
TL;DR: Containing misinformation spread through learning-based approaches using Graph Neural Networks

Abstract: In this study, we investigate the under-explored intervention planning aimed at disseminating accurate information within dynamic opinion networks by leveraging learning strategies. Intervention planning involves identifying key nodes (search) and exerting control (e.g., disseminating accurate/official information through the nodes) to mitigate the influence of misinformation. However, as the network size increases, the problem becomes computationally intractable. To address this, we first introduce a ranking algorithm to identify key nodes for disseminating accurate information, which facilitates the training of neural network (NN) classifiers that provide generalized solutions for the search and planning problems. Second, we mitigate the complexity of label generation—which becomes challenging as the network grows—by developing a reinforcement learning (RL)-based centralized dynamic planning framework. We analyze these NN-based planners for opinion networks governed by two dynamic propagation models. Each model incorporates both binary and continuous opinion and trust representations. Our experimental results demonstrate that the ranking algorithm-based classifiers provide plans that enhance infection rate control, especially with increased action budgets for small networks. Further, we observe that the reward strategies focusing on key metrics, such as the number of susceptible nodes and infection rates, outperform those prioritizing faster blocking strategies. Additionally, our findings reveal that graph convolutional network (GCN)-based planners facilitate scalable centralized plans that achieve lower infection rates (higher control) across various network configurations (e.g., Watts-Strogatz topology, varying action budgets, varying initial infected nodes, and varying degree of infected nodes).

Towards Effective Planning Strategies for Dynamic Opinion Networks
Bharath Muppasani, Protik Nag, Vignesh Narayanan, Biplav Srivastava, and Michael N. Huhns Al Institute and Department of Computer Science University South Carolina, USA (bharath@email., pnag@email., vignaræ, biplav. s0, huhns@)ac.edd edu
Abstract
In this study, we investigate the under-explored intervention planning aimed at disseminating accurate information within dynamic opinion networks by leveraging learning strategies. Intervention planning involves identifying key nodes (search) and exerting control (e.g.. disseminating accurate/official information through the nodes) mitigate the influence of misinformation. However, the network size increases, the problem becomes computationally intractable. To address this, we first introduce ranking algorithm identify key nodes for disseminating accurate information, which facilitates the training of neural network (NN) classifiers that provide generalized solutions for the search and planning problems. Second, we mitigate the complexity of label generation-whic becomes challenging as the network grows-by developing reinforcement learning (RL)-based centralized dynamic planning framework. We analyze these NN-based planners for opinion networks governed by two dynamic propagation models. Each model incorporates both binary and continuous opinion and trust representations. Our experimental results demonstrate that the ranking algorithm- based classifiers provide plans that enhance infection rate control, especially with increased action budgets for small networks. Further, we observe that the reward strategies focusing on key metrics, such the number of susceptible nodes and infection rates, outperform those prioritizing faster blocking strategies. Additionally, our findings reveal that graph convolutional network (GCN)-based planners facilitate scalable centralized plans that achieve lower infection rates (higher control) across various network configurations (e.g., Watts-Stroggen topology, varying action budgets, varying initial infected nodes, and varying degree of infected nodes).
Introduction
The spread of information across social networks profoundly impacts public opinion, collective behaviors, and socictal outcomes [ Especially during crises such as disease outbreaks or disasters, there is often too much information coming from different sources. Sometimes, the resultant flood of information unreliable or misleading, or spreads too quickly, which can have serious effects society and health [8) Online platforms such as Facebook, X, and WeChat, while essential for communication, significantly contribute to the rapid spread of misinformation (2] This has led to public confusion and panic in events ranging from the Fukushima disaster the COVID- 19 pandemic, and even the U.S. presidential elections information management these platforms (8)
demonstrating the need for effective
The first step in combating misinformation propagation to reliably detect them. However, detec- tion alone is insufficient to effectively mitigate its spread. It must be complemented with strategic
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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intervention planning to contain its impact. In this context, numerous studies have focused on rumor while comprehensive strategies for controlling misinformation are limited Existing research works on controlling misinformation emphasizes three primary strategies:
detection |5
47
[15]
node removal [ 45 29 edge removal [2 19, 44 and counter-rumor dissemination [ 12]
Node removal involves identifying and neutralizing key nodes using community detection methods, with dynamic models that adapt to changes in propagation. For example, [50] [6] present ranking algorithms that are critical for identifying influential nodes within complex networks, which can then be targeted to block, remove, or cascade information, to reduce the overall spread of misinformation. These algorithms rank nodes based on various metrics, determining their importance influence within the network. The second approach, edge removal, focuses on disrupting misinformation path- ways by strategically severing connections between nodes. For example, authors in [38] considered mitigating misinformation by identifying potential purveyors to block their posts. However, taking strong measures like censoring user posts may violate user rights.
The third strategy, counter-rumor dissemination, promotes the spread of factual information, leverag- ing user participation and 'positive cascades' to counteract misinformation [4]. For instance, authors in [13 developed a method that involves learning an intervention model to enhance the spread of true news, thereby reducing the influence of fake news. The effectiveness of such intervention planning methods relies on the ability of intervention models to identify key nodes and disseminate accu- rate/official information through these nodes to mitigate the influence of misinformation. Following this strategy, in [3], a search problem was formulated and sequential plans using automated planning techniques were generated for the targeted spread of information. Despite several model-based efforts to strengthen this approach (see Appendix A.I for a review of relevant literature), the existing research often overlooks key features of opinion propagation models, such as their rich network dynamics, asynchronous communication, and the impact of factors like the degree of infected nodes, action budget, and various reward models on the effectiveness of planners.
We address this gap by studying the intervention planning problem using two learning methodologies. In both these methodologies, we investigate three distinct cases of opinion network models, ranging from discrete to continuous representations of opinion and mutual trust. Specifically, in this paper, we propose novel ranking algorithm integrated with supervised learning (SL) framework to identify influential nodes using a robust feature set of network nodes and evaluate their performance in all three cases. Additionally, centralized planners that are suitable for larger networks. Furthermore, we develop comprehensive datasets with wide range of -trtos-Stroattz (or morldworldd) network topologies, varying degrees of initial infected nodes, action budgets, and reward models, e.g., from those requiring local network information those utilizing global real-time network states, and analyze both the developed methodologies. Next, we use an example to illustrate the intervention planning problem and highlight our contributions.
also develop a reinforcement learning (RI.)-based solution to design
Example: Consider a research community discussing the NeurIPS submission deadline. In this context, a topic is just a statement such 'NeurIPS submission deadline is on May 22'. An opinion of an agent on this topic is defined as the belief of the agent in the truthfulness of the statement [3. A positive (respectively negative) opinion value represents that the agent believes the statement is true (respectively false). In our study, we consider the opinion value of an agent on a topic lies in [-,, Our problem setup consists of network of connected agents (with opinion values on a given
Start State
Action: 7
Action:
Action:
time-step:
time-step: 2
time-step: 3
time-step:
Figure 1: Example of misinformation propagation and control choices at each timestep. Blue nodes: neutral (opinion value 0), red nodes: misinformed (opinion value 1)), green nodes: received accurate information (opinion value 1).

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topic) represented by a graph. We consider that subset of agents propagate misinformation to their neighbors. Our goal is to counteract this by disseminating accurate information selected agents at each time step. The agents receiving accurate information update their opinion to a level where they no longer believe the misinformation and, consequently, cease to propagate it. The process ends when no agents are left to receive the misinformation. red nodes represent the agents not believing about the NeurIPS deadline being May 22. When they interact with other agents, they share this misinformation, leading to spread of incorrect information throughout the network. The blue nodes represent neutral agents who are unaware of the deadline, and the green nodes represent agents who believe in the NeurlPS deadline being May 22. Initially, agents, ie., nodes {0, 1}. propagate misinformation to their neighbors. To control the spread of this misinformation, timely control actions are taken at each timestep disseminate accurate information to selected agents. In the example, with an action budget of 1, agents 7, 8, and 6 are sequentially chosen receive the accurate information, represented by the green nodes. Through these control actions, the example demonstrates how timely intervention at critical points can effectively mitigate misinformation spread, ensuring agents are accurately informed.
[iguur/Tiillutratee such a scenario where the
Our paper makes several significant contributions to intervention planning, focusing on the integra- tion of more realistic and complex modeling approaches, label generation techniques, and training methodologies: (a) We utilize a continuous opinion scale to model the dynamics of misinformation spread (instead of just binary scale as shown in the example), providing more realistic representation of opinion changes over time. (b) We develop a ranking algorithm for generating labels in networks with discrete opinions, addressing significant gap in efficient data preparation for SL algorithms in this domain. (c) We develop an RL methodology and perform comprehensive analysis. This allows for adaptive intervention strategies in response to evolving misinformation spread patterns, a critical improvement over traditional static approaches. (d) Utilizing graph convolutional networks (GCNs) with an enhanced feature set of opinion value, connectivity degree, and proximity misinformed node, we improve the training of models for selecting effective intervention strategies. This enhance- ment ensures scalability and generalizes well across various network structures, demonstrating the robust capabilities of GCNs in complex scenarios. Table highlights our contributions by providing a comparative overview of the key features and innovations in our work that of previous studies. The code and the datasets developed as part of the analysis presented in this paper can be found in [33].
2 Problem Formulation
In this section, we discuss our approach to modeling the opinion network environment, the dynamics of information propagation, and our strategy for containing misinformation.
2.1 Environment Description
An opinion network is formally represented as directed graph G = EV, where V denotes the set of nodes (agents), and denotes the set of edges (relationships or trust) between agents [15]. The graph structure we consider for our study is undirected, indicating that relationships between agents are bi-directional. Each node within the graph represents an individual agent, and each agent holds a specific opinion on a given topic. An edge between any two nodes signifies a direct connection or relationship between those agents, facilitating the exchange of opinions.
Opinion values are quantified within the range [-1, 1]. representing different levels of belief on a given topic, and the weight assigned to each edge quantifies the mutual trust level between connected agents, scaled within the interval [o, 1]. While existing works in the literature have explored only binary opinion and trust models, in computational social science, researchers have developed models with opinion and trust values as continuous variables. Investigation of planning strategies in continuous models remains under-explored. In this paper, we explore three distinct cases of opinion and trust values. Case-1 1 involves binary opinion values with binary trust, simplifying the network dynamics into discrete states. In Case-2, we use floating-point opinion values while maintaining binary trust, allowing for a more granular assessment of opinions while still simplifying trust dynamics. Finally, C features both Doating-point opinion values and matinng-poitt trust, representing more realistic opinion and trust relationships within the network capturing continuous variations.
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2.2 Propagation Model
In the analysis of opinion networks, it is essential to understand how opinions form and evolve, guided by the dynamics of trust among agents. In our analysis, the evolution and propagation of opinions within opinion networks are modeled using a linear adjustment mechanism (discrete linear maps), described by the following transition function
5x + = = =0)))) (xx(1)))))))))))))))))))))())( - xx((),, 1= 1,.....
Equation I models the dynamics of opinion evolution, where the opinion of agent at time + 1 depends on its current opinion and the influence exerted by a connected agent k, who is actively sharing some information with agent moderated by the trust factor jik- This asynchronous propagation model adapts differently across various experimental setups as detailed below.
In Case-1 and Case-2, where mutual trust values are discrete {0, 1}. the application of Equation results in immediate shifts in opinion. For example, an agent with a current opinion value of 0.5 on some topic is influenced by connected neighboring agent l with an opinion value of 1 on the same topie, agent i's opinion immediately shifts to -1 in the next timestep, reflecting a discrete transition. Conversely, in Case-3, which involves continuous range of opinion and trust values, changes are more gradual. Here, if agent holds an opinion of 0.5 and is influenced by a neighbor with opinion of 1 and moderate trust factor, the opinion of agent incrementally moves closer to 11 in subsequent timesteps. This reflects gradual shift towards a consensus opinion, depending on the magnitude of the trust level between agents and k.
To further enhance our understanding of opinion dynamics in networks with continuous trust relation- ships, we have also used the DeGroot propagation model [6] in Case-3. The propagation of opinions in this model is governed by the following equation:
7
(1 1)
Σ (a) 1= 1,.,....
2)
Equation 2 describes the opinion of agent at time + as a weighted average of the opinions of all the neighboring agents at time where the weights faa represent the trust agent i has in agent k. Often, in the DeGroot model, which is synchronous propagation model, the summation in is convex sum, i.e., the trust values add one so that we have Σ Mikk for each I,..., n. This normalization allows the DeGroot model to exhibit stable asymptotic behaviour.
At each timestep, the following processes occur: Nodes with opinion values lower than -0.95 are identified as sources of misinformation and transmit the misinformation to their immediate neighbors (referred to 'candidate nodes ') according to one of the propagation model detailed in Equations neighbors- cenabbor--cosstainnne by action budget-is selected to receive credible information from trusted source. This source is characterized by an opinion value of 1 and we vary trust parameter among 0.8, and 0.75. The process includes blocking mechanism where a node that exceeds positive opinion threshold of 0.95 is considered 'blocked', ceasing to interact with the misinformation spread or disseminate positive influence further. The simulation concludes when there are no viable "candidate nodes left to propagate misinformation. Our primary objective is to devise a learning mechanism that efficiently identifies and selects key nodes within the network disseminate accurate information at each time step.
[ and 2 Concurrently, an intervention strategy is applied where a subset of these
3 Methods
In this section, we will explain our methodologies, presenting an overview of the network architectures employed, including the GCN and ResNet frameworks. Additional details about the neural network architecture utilized for our experiments can be found in Appendix A.2 We detail our proposed ranking algorithm utilized in the SL process. Additionally, we elaborate on the implementation of the Deep Value Network (DVN) with experience replay for the proposed RL.-based planners. Furthermore, we provide an explanation of the various reward functions analyzed in our RL setup.

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3.1 Ranking Algorithm-base -baedd Supervised Learning
In this section, we propose ranking algorithm based SL model to classify the key nodes at each time step to disseminate accurate information. Our SL method utilizes GCN architecture.
Ranking Algorithm: We pose the ranking algorithm search problem where the objective is to find the optimal set of nodes that, when blocked, minimizes the overall infection rate. The network is represented as graph G, where nodes be infected, blocked, or possess opinion values within the range [-0.95, 0.95]. Initially, a simulation network S is created by setting the opinion values of infected nodes to -1 and removing blocked nodes. Let M denote the number of nodes in S that are neither infected nor blocked. Given an action budget K, select K nodes from M in (&) possible ways, forming the set C' of all possible combinations. For each subset e E C, we temporarily block the nodes in e by setting their opinion values to 1 and simulate the spread of misinformation within S. The resulting infection rates for each subset с are stored in the set R. We identify the subset E C' that yields the minimal infection rate, denoted as r", This subset c* is our target set. We then construct a target matrix '' E IR,x1, where N is the total number of nodes in the original network G. All entries of T are initialized O, and for each node E c", the i-th entry of set to This target matrix T is subsequently used to train the GCN-based model. A pseudocode for this ranking algorithm is presented in Algorithm iin Appendix A..
Overall Training Procedure: The training of our GCN-based model leverages the labels defined in the target matrix 7 E RR This matrix is compared with the model's output matrix O E R-x11, which estimates the blocking probability of each node. We evaluate training efficacy using the binary cross-entropy loss between T and O, which quantifies prediction errors. Model weight adjustments are implemented via standard backpropagation [23] based on this loss.
Each training iteration consists of several episodes, starting with the generation of random graph state G containing initially infected nodes. The GCN then processes this state to output matrix O using the graph's 's features and structure. Labels are generated, as detailed above using the ranking algorithm, generating the target matrix T. The binary cross-entropy loss between O and T is calculated for backpropagation. The environment updates by blocking predicted nodes, allowing the infection spread, and adjusting node attributes. The process repeats until misinformation spread is halted, with each episode refining the graph's state for subsequent iterations. The results of the planners for difference cases are summarized in the Appendix [
However, we note that, in the case of continuous opinion and continuous trust (case-3), the process of label generation becomes more complex. In such scenarios, agents do not change their opinion immediately but gradually, making it difficult to predict which agents will be misinformed based single propagation simulation. Therefore, simulations across multiple time steps are necessary to identify the optimal nodes to block. As the ranking algorithm uses a brute force method to determine optimal nodes, this approach becomes increasingly challenging with continuous opinion models.
3.2 Reinforcement baariigg-based Centralized Dynamic Planners
In SL, the process of generating labels can be costly and impractical as network size increases. This is evident while considering mitigating misinformation propagation in large networks, where identifying the optimal set of nodes for blocking requires combinatorial search that is computationally infeasible. Thus, RL emerges as a viable alternative.
Deep -eetwwrkks (DQNs) 32) using random exploration combined with experience replay have been demonstrated to effectively learn Q-values for sequential decision making with high-dimensional data. Unlike the classical DQN, where the network outputs a Q-value corresponding to each possible action, in our problem, which also deals with high-dimensional data, we develop a DVN, as the number of available actions at each time step need not be fixed. Consequently, the output layer consists of a single neuron that outputs the value for given input state. The agent's experiences at each time step are stored in replay memory buffer for the neural network parameter updates. The loss function for training is given by
[(st, St- s0- = (r: (S111) Vo(s)))
where / represents the current time step, 8t is the current state, 8141 denotes the subsequent state after action a, is taken, and is the reward received for taking ag in st. The parameters 0 denote
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the weights of the value network used to estimate the state value Vg(ss), while 0- represents the parameters of a target network, typically a lagged copy of used stabilize training. Here, 18 is the estimated value of the next state $141 according to the target network. The specific reward functions used in this study are discussed later in the section. Algorithm 2 in Appendix A.5
provides a detailed implementation of our DVN with experience replay.
3.2.1 Reward Functions for RL setup
The reward function is designed to encourage policies that effectively mitigate the spread of misinfor- mation. Specifically, the reward functions modeled for our study are: (1) Ro = -Ainnfectio rate), where Ainfection rate is defined as the change in infection rate resulting from taking action 0t- Specifi- cally, Ainfection rate infection rate at 8t+1 infection rate at st. This reward structure encourages the model to reduce the rate at which misinformation spreads by penalizing increases in the infection rate; (2) R₁ -(# candidate nodes), targets the number of immediate neighbors of infected nodes that are susceptible to becoming infected in the next timestep. thereby promoting strategies that mini- mize the potential for misinformation to spread; (3) R = (# candidate nodes) - (Ainfection rate), takes into account the previous two rewards, balancing the need to control both the number of steps
susceptible nodes and the overall infection rate; (4) R3 = 1 - Total time steps ). rewards quicker reso- lutions, providing higher rewards for strategies that contain misinformation rapidly and evaluating the effectiveness only the end of each episode; (5) R4 = (infection rate), directly penalizes the current infection rate, thus favoring actions that achieve lower overall infection rates; and finally, combined reward that incorporates elements of both R3 and Rj. Throughout the simulation, the agent continually receives rewards based on the number of candidate nodes, fostering strategies that limit the expansion of the infection network. As the simulation concludes, the agent receives an thereby reinforeing the
episodic reward calculated as (6) R₂ = -(# candidate nodes) Total time steps importance quick and efficient resolution misinformation spread. Note that all these reward structures, in addition to differing in how they represent the goal for the planners, also differ in the network information required to compute them.
#ime steps
3.3 Network Architectures
In our experiments, we utilized two neural network architectures. First, a GCN to model node features within network. Each node was characterized by three key features. These features were represented in matrix dynamic and evolves to reflect changes in the network. It includes the opinion value, the connectivity degree, which identifies nodes potentially susceptible to misinformation while excluding those already blocked or misinformed, and the proximity to misinformed node, which is calculated as the shortest path to the nearest infected node, assigning distance of infinity to unreachable nodes.
R.Nx3, where N denotes the total number of nodes. The feature matrix
We have also considered using Residual Network (ResNet) Architecture. The ResNet model imple- mented in our study is a variant of the conventional ResNet architecture [14]. The core component of our ResNet model is the addressing the vanishing gradient problem through skip connections. Each ResidualBloo ock consists of two sequences of convolutional layers (Conv2d), batch normalization (BatchNorm2d)) and sig- moid activations. Complete details about the model architectures used in our study are provided in Appendix A.2
which allows for the training of deeper networks by
4 Experiments
In this section, present the details about training data generation and configurations chosen for our SL and RL methodologies. We also explain the test data used for evaluating the trained models.
4.1 Training Setup
In the SL setup, we experimented with three distinct graph structures: Watts Strtts-Strogazz (with nearest neighbors k = 3 and rewiring probability p 0.4), Fidos-Renyl (with branching factor of 4), and Tree graphs (with branching factors randomly selected from the range [1, 4).. Each graph type facilitated training models to evaluate the influence of various structural dynamics on performance.

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We used the GCN model for the SL method. Due to the consistent performance of the trained models on the different graph topologies, we chose the sorll-world topology to present all the subsequent analyses and summarize the results in Appendix A.6.1 On the other hand, we trained centralized RL planners using both ResNet and GCN network architectures. Each trained configuration is represented as model-n-T-y, where model E [ResNet, GCN), n E (10,25,50) represents the network length (in terms of the number of nodes), æ E {1,2,3] represents the number of initial infected nodes, and y E {1,2,3) represents the action budget.
4.2 Test Data Generation
The datasets used in related works, such as 117. typically consist of network structures, and no real-time opinion propagation data could be found. Therefore, to evaluate our intervention strategies, we generated two sets of synthetic datasets using the Watts-Stroe Strogatz model with the training dataset's configurations. This approach allows us to simulate complex networks and control the structure, connectivity, and initial infected nodes to assess our models effectively.
Dataset v1 examines the effects of network size and the initial count of infected nodes on misinfor- mation spread. We generated data with network sizes of 10, 25, and 50 nodes with 1, 2, and 3 initially infected nodes, respectively, creating 9 unique datasets. Fach confi configuration has 1000 random network states with the opinion values of non-infected nodes uniformly distributed between -0.5 and 0.6.
Dataset v2 examines how the initial connections of infected nodes affect the spread of misinformation. Like Dataset v1, it includes networks of 10, 25, and 50 nodes. However, the initial number of connections (degrees of connectivity) for the infected nodes varies from to 4. Here by degree of connectivity, mean the number of candidate nodes present at the start of the simulation. This variation results in a total of 12 datasets for each configuration, with each dataset containing 1000 states. In these configurations, the number of initially infected nodes is randomly chosen from to 3.
5 Results and Discussion
In this section, evaluate the models, using the infection rate metric, trained using ranking- based SL and RL algorithms with various reward functions. We discuss the efficiency of these models using Dataset v2, particularly represents the most complex test dataset we generated. Similar evaluation results for other datasets can be found in the Appendix [A.6| Additionally, using directed and undirected real-world network models reported in the literature. These evaluations are presented in Table 2 The details of the hardware used for our experiments are provided in the Appendix A.4 Our empirical investigation yielded insightful results regarding the performance of our trained models under various training conditions. With comprehensive experimental evaluations, we were able to address the following research questions.
network of 50 nodes with connectivity degree of 4, as it
have also evaluated our planning algorithms
Objective and Research Questions: O: Identify the optimal combination of initially infected nodes and action budget parameters for training models to effectively control the spread of misinformation. RQ1: For reward functions that focus on the blocking time, does adding any other factor lead to better results? If yes, which factor? RQ2: Do reward functions that look at global graph information perform better than those considering local, neighboring information? RQ3: Does GCN offer better scalability and performance when compared with ResNet.
O: What is the best combination of initially infected nodes and action budget parameters for training the models to control the misinformation spread?
To examine this, we focused our analysis on the Mean Squared Error (MSE) loss plots obtained during the training phase. Figure in (opendix(A.6fillusttee
the comparison of training loss across
various network parameter settings for all considered reward types in Case-1, employing ResNet model trained on a network of 50 nodes. The trend in loss convergence across episodes was found to be consistent for both the ResNet and GCN models across all cases examined. The analysis revealed that reward functions exhibiting lower and more stable loss values correlate with improved model learning performance. Our findings highlight that increasing the number of initially infected nodes typically elevates the stabilization point of MSE loss, indicating a more challenging learning environment. Additionally, a higher action budget contributes to increased MSE variability, reflecting
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Table 1: Inference results on Dataset v2, with a degree of connectivity 4, featuring a network of 50
nodes. This table presents the average infection rates for different models, with ResNet trained on network of 50 nodes and GCN trained on networks of 10 nodes, under various methods (M.) tested with varying action budgets (A.) across different cases considered.
M.
Case-1
Case-2
Case-3
ResNet(50)
GCN(10)
ResNet(500)
GCN(10)
ResNet(550)
GCN(10)
RL+Ro
0.2334
0.2481
0.2496
0.2454
0.0449
0.0461
RL+R1
0.1917
0.1608
0.1965
0.1607
0.0449
0.0435
RL+R2
0.2427
0.1608
0.2148
0.1608
0.0451
0.0438
RL+Ra
0.2331
0.2958
0.2281
0.3381
0.0444
0,046
RL+A4
0.199
0.1593
0.2513
0.1596
0.045
0.0442
RL+AG
0.1607
0.1605
0.0439
SL+GCN(25)
0.304
0.2889
0.3715
RL+At
0.0974
0.1012
0.0992
0.1007
0.0398
0.04
RL+R1
0.0886
0.0842
0.0901
0.0843
0.0398
0.0398
RL+ R2
0.097
0.0842
0.0957
0.0843
0.0399
0.0398
RL+Rs
0.0959
0,0969
0.0962
0.1032
0.0398
0.04
RL+R4
0.0898
0,0842
0.1005
0.0842
0,0399
0.0398
RL+Rs
0.0842
0.0842
0.0398
SL+GCN(25)
0.1464
0.1032
0.3491
RL+Ro
0.0599
0.0599
0.0599
0.06
0.0397
0.0399
RL+R;
0.0599
0.0597
0.0598
0.0597
0.0398
0.0397
RL+A2
0.0598
0.0597
0.0599
0.0597
0.0398
0.0397
3
RL+Hs
0.06
0.0598
0.0601
0.0602
0.0397
0.0399
RI+Ra
0.0598
0.0597
0.06
0.0597
0.0398
0.0398
RL+Rs
0.0597
0.0597
0.0398
SL+GCN(25)
0.0488
0.0559
0.2526
the added complexity and generally poorer performance during training. Based on this analysis, we find ResNet-n-1-1 and GCN-n-1-1, n E (10,25,50), to be the best training configurations.
Table pressenss the average infection rate values across different cases considered for Dataset v2 with degree of connectivity 4, featuring a network of 50 nodes, detailing the average infection rates. the ResNet model, trained on a network of 50 nodes, with the GCN model, trained on network of 10 nodes, using the RL training algorithm across the different reward types, and the GCN model trained using SL on network of 25 nodes. Results on the additional datasets are provided in appendix[A.6
compares the performance
RQI: For reward functions that focus on blocking time, does adding any other factor lead to better result? If yes, which factor? Answer: Yes. #candidate nodes.
Reward function R3, which is formulated to minimize the number of time steps required to halt the spread of misinformation, might inadvertently not be the most effective strategy for minimizing the overall infection within the network. As the reward is solely based on the speed of response, does not directly account for the magnitude of the misinformation spread, that is, the number of nodes affected. Therefore, the agent may prioritize actions that conclude the propagation swiftly but do not necessarily result the most substantial reduction in the spread of misinformation However, our results indicate that under specific training configurations with action budget or initial infected nodes greater than 1, the reward function R3 outperforms others in maintaining lower infection rates, as shown in Figure X in Appendix [.6 This finding significant since R3, sparser reward type, requires less computational effort and is independent of network observability. As the action budget increases the propagation tends to conclude fewer timesteps thereby resulting in the RL agent receiving higher reward in the case of Rg. Figure 2 shows that the RL agent trained with the R3 reward function chooses actions that conclude propagation in the least time. Conversely, Figure illustrates the sequence of actions chosen by an RL agent trained with the Rí reward function on the same network. Although Rú requires more time steps than R3. results a lower infection rate. This can also be observed from Table I where the infection rate is higher for the R3 reward function than

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Table 2: Average infection rate values from experiments conducted on 100 random instantiations for each real-world network, each starting with 1 random initially infected node, obtained using GCN model trained on synthetic networks of 10 nodes, under various reward structures (M.) tested with varying action budgets (A.). The network properties of # nodes (V), # edges (E), and Average Degree for each network are shown in the table.
A.
M.
Zachary's Karate
Facebook [Undi-
Email [Directed]
Cora
[Undi-
Club [Undirected]
rected] 25
24
rected] 31
[11]
Network Properties
V: 34, 78
V: 250, E: 1352
3000, E: 2358
2000, E: 2911
Avg. Deg.: 4.59
Avg. Deg.: 10.8
Avg. Deg.: 7.9
Avg. Deg.: 2,9
Ro+Ro
0.8579
0.4569
0.395
0,0603
RL+R; R
0.5279
0.4547
0.315
0.0095
RL+ Rs
0.5279
0.4547
0.3169
0.0095
RL+As
0,8468
0.511
0.3759
0.0609
RL+R4 0,5326
0.4589
0.3183
0,0096
RL+Hs
0,5276
0.4547
0.316
0.0096
RL+ Ro
0,2762
0.3621
0.2512
0.0166
RI+R:
0.1641
0.3022
0.0988
0.002
RL+R2
0,1641
0.3024
0,1062
0.002
RL+Rs
0.2562
0.3697
0.1838
0,0142
RL+R4 0.1582
0.3047
0.0966
0.002
RL+Rs
0.1641
0.3022
0.0988
0.002
RL+Ro
0.0926
0.233
0.103
0,0126
RL+R
0.0697
0.1649
0.0347
0.0017
RL+R2
0,0662
0.1647
0.0347
0.0017
RL+Rs
0.085
0.2039
0.0587
0.0052
RL+R4
0.0662
0.1633
0.0351
0.0017
RL+Hs
0.0662
0.164
0.0347
0.0017
Start State
Action:
Action: 6
time-slep:
time-step step::
time-step:
Figure 2: Sequence of actions chosen by the RL agent trained using reward function R3.
any other reward function. In order effectively implement this have considered combining this episodic reward along with R1, resulting in reward type Rs. This has shown a significant performance improvement when compared to the original version.
RQ2: Do reward functions functtfunntions that look at global gruph information perform better than those considering local, neighboring information ? Answer: Yes
Analysis the inference outcomes using Dataset v2, presented in Table [ shows that single factor reward functions, specifically R₁ = -(# candidate nodes) and R4 - (infection rate), consistently resulted lower infection rates across various settings compared to their more complex counterparts with R4 providing relatively better results compared to Rj. This trend was observed in both ResNet and GCN models, From a practical standpoint, R1 can be particularly advantageous because it does not require complete observability of the network, but just the immediate neighbors of infected nodes. Conversely, to compute R4, which reflects the infection rate of the network, complete understanding of the state of each node within the network is required. This requirement for total network observability could limit the practicality R4 situations where such detailed information unavailable or difficult gather.

---

Table 3: This table outlnnes the unique attributes of our approach, including the use of deep value network, network dynamicity across multiple cases, asynchronous communication, and the exploration of five different reward models.
Implications
Features of Our
Our Work
Previous Work
Citations
work
Action-Space
In- Deep Value Net- work
X
DQN
variant
Expressive Models
Network Dynamic-
Case 1, Case 2, Case
Only Case
[28-40]
ity
Realistic Communi-
Asynchronous Com- munication
X
cation Dynamics
Wider Applications Reward Models
variants studies
Typically
7
36
RQ3: Does GCN offer better scalability and performance when compared with ResNet? Answer: Yes
GCNs are hypothesized to outperform traditional convolution- based architectures like ResNet in tasks involving graph data due to their ability to naturally process the structural information of networks and their enhanced ability to represent complex feature sets [4]. This study compares the scalability and performance of GCN, which excels in node classification within graphs, to a ResNet model that, despite its success image recognition, may not scale as effectively to larger graph structures beyond the size it was initially trained on. Referring to Table I the GCN model, trained on only 10 node networks, consistently exhibits lower average infection rates across all the cases and under varying action budgets, when compared with the ResNet model trained on 50 node networks. The ability GCN to maintain lower infection rates even as network complexity increases underscores its robustness and scalability in complex network scenarios. This performance contrast highlights the suitability of GCN architectures for graph-based tasks.
Conclusions
This paper investigates scalable and innovative intervention strategies for containing the spread of misinformation within dynamic opinion networks. Our significant contributions include analysis using continuous opinion models, design of ranking algorithm for identifying key nodes facilitate SL-based classifiers, and the utilization of GCNs to optimize intervention strategies. Additionally, we design and study various reward functions for reinforcement learning, enhancing our approach to misinformation mitigation.
Despite significant progress, work has limitations. In the field of computational social science, often more complex agent models are being investigated. While we have made significant efforts to extend the understanding of planning strategies, especially in continuous opinion networks, exploring complex agent traits such as stubbornness and the representation of directed trust, and implementing topie-dependency in a multi-topic network along with distributed planners instead of centralized planners as in our work is a compelling future direction.
Broader Societal Impact: This work provides methods that can be used to exert control information spread. When used responsibly by authorized information providers, which the authors support, will help reduce prevalent infodemics in social media. But it may also be misused by adversary to wean control from an authorized party (e.g., information owner) and counter efforts to tackle misinformation. Overall, the authors believe more research efforts are needed understand opinion networks and information dissemination strategies in dynamic and uncertain environments in pursuit of long-term societal benefits.
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Acknowledgments and Disclosure of Funding
This material is based upon work supported in parts by the Air Force Office of Scientific Research under award number FA9550-24-1-0228 and NSF award number 2337998. Any opinions, findings, conclusions, or recommendations expressed here are those of the authors and do not necessarily reflect the views of the sponsors. The authors thank the anonymous reviewers for their insightful and constructive reviews.
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A Appendix
A.1
Related Works
This section reviews existing studies on controlling the flow of misinformation in networks. While most previous research has focused on detecting fake news through various features such as linguistic, demographic, or community-based indicators, there has been comparatively less work on mitigating misinformation. Mitigation strategies are typically categorized into three main approaches: removing critical nodes, severing essential connections, and countering rumors with factual information. In the following sections, will first discuss the literature on misinformation detection, followed by a review of studics aimed at mitigating the spread of misinformation. Finally, we will provide an overview of approaches utilizing Graph Neural Networks (GNN) and Reinforcement Learning (RL) to mitigate misinformation spread.
Misinformation Detection: Initial efforts in misinformation detection aimed at curbing rumor spread through strategic node blocking. Wu et al. 47 developed a community detection algorithm to segment network nodes, evaluate their influence, and block key nodes. Zheng and Pan (53) addressed the least cost rumor community blocking (LCRCBO) using community-centric influence model and a greedy algorithm to select optimal nodes for containment. However, both methods have raised concerns regarding cost-effectiveness and operational efficiency. Expanding on node-centric approaches, Ding et al. [7] developed a dynamic rumor propagation model with algorithms to identify and remove critical nodes and connections, introducing an 'outbreak threshold' to evaluate interventions. In contrast, Khalil et al. [9 and Yan et al. 49 advanced edge removal strategies under linear threshold (L.T) model, creating heuristic algorithms to manage misinformation spread effectively [18]. Tong and Du 43 used a hybrid sampling method, which could assign high weights to users susceptible to misinformation, to pinpoint users most vulnerable to fake news, while technique that leverages entropy to balance network diffusion efficiency. Nguyen et al. (35] applied network analysis to explore sentiment propagation in social networks, finding that sentiments often cluster within comment threads, suggesting that online forums may serve as echo chambers that reinforce uniform opinions among participants.
Zareie and Sakellariou (52) introduced passive edge-blocking
Misinformation Propagation Minimization: This research category promotes disseminating truthful information as a counter-rumoo measure. Lin et al. (26) suggested a crowdsourcing framework to enable users to select from various collaborative independent rumor control methods. Tong al. [41] analyzed the effectiveness of the peer-to-peer independent cascade (PIC) model in private social networks, where independent rumor agents create 'positive cascades', and demonstrated that such strategies are robust under Nash equilibria. Yan et al. [48) identified the rumor minimization challenge as monotonically decreasing and devised a two-stage process for selecting effective blocking candidates. Manouchehri et al. (30) addressed maximizing influence blocking (IBM) with considerations for timing and urgency, proposing an efficient, theoretically sound sampling method. Lastly, Srinivasan and LD 339 proposed a competitive cascade model that focuses on leveraging user opinions and the critical nature of rumors identify and activate influential nodes promoting positive information.
These studies highlight the complexity of misinformation propagation minimization and underscore the need for deeper analysis. Our study builds on these works by exploring how misinformation spreads under different environmental conditions and agent dynamics. We aim to propose more effective mitigation strategies, contributing to a nuanced understanding of misinformation dynamics and enhancing the resilience of information networks.
GNN Based Approaches: Graph Convolutional Networks (GCNs) are increasingly being utilized to detect the rumors and propagation patterns within social networks. For instance, Wei et al. [46] developed GCN-based model to analyze user stances and conversation content for better rumor detection. Ma et al. [28] created a tree kernel to compare similarities between subtrees in retweeting trees. Kumar and Carley 22 employed a multitask learning framework to extract representations from retweeting trees for rumor and stance detection. Similarly, Khoo et al. (20] explored various influences within a retweeting tree and utilized a transformer model to enhance rumor detection by
learning the interactions among these influences. Moreover, Liu al. [27] proposed structure-aware
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retweeting GNN that identifies rumor patterns based on retweeting behaviors. This method leverages both node and structural-Ievel data, suggesting that propagation paths offer distinct insights into the credibility of the disseminated information. Contrastingly, while the detection of rumors has been extensively studied, the use of GCNs for rumor minimization remains under-explored. Authors in [15] introduce an innovative approach for blocking rumors on social networks by integrating user opinions with confidence levels into a new model (CBOA) and employing a directed GCN (DGCN) to identify and block critical nodes capable of mitigating rumor spread.
However, there are opportunities to enhance their study. Our research investigates the scalability of GCNs and their performance across three different environments with varying agent dynamics. Additionally, we propose novel ranking algorithm for training GNNs. Furthermore, we identify sce- narios where supervised learning faces challenges and address these limitations using reinforcement learning techniques.
RL Goindani and Neville [3] develop social reinforcement learning approach to mitigate the spread of fake news through social networks. Their method involves learning an intervention model enhance the spread of true news, thereby reducing the influence of fake news. The authors in [TO] model the news diffusion process using a Multivariate Hawkes Process (MHP) and employ policy optimization learn intervention strategies. Ohi al. [36] investigate strategies mitigate the spread of pandemics using a reinforcement learning approach. The model is based on the SEIR (Susceptible-xppon Infectious-Recovveeed compartmental model. It allows for dynamic interaction where individuals move randomly, influencing the spread of the disease. The agent is trained to determine optimal movement restrictions (from no restrictions to full lockdowns) to minimize disease spread while considering economic factors.
Current research in this field largely focuses on identifying misinformation or removing nodes and edges to limit rumor propagation. While some studies, such as those by He et al. [[5]. investigate misinformation suppression methods, they often do not address complex environmental scenarios, which our study aims to explore. Our research specifically targets the minimization of misinformation spread after the detection of misinformed agents is done. We aim hinder their attempts to dissemi- nate false information by strategically blocking selective nodes with positive information. Our work begins with the implementation of GCN- based supervised learning model to detect misinformation. To overcome the limitations of supervised learning, we further incorporate reinforcement learning paradigm. This progression enables us develop and optimize more effective strategies within complex network environments, ultimately enhancing the robustness of misinformation control measures.
A.2
Model Details
A.2.1
Graph Neural Networks (GNNs)
Graph Neural Networks (GCNs) are advanced deep learning models tailored for handling data with graph structure. Such models are particularly adept at processing information within complex networks by learning to synthesize node representations. These representations are derived through the aggregation and transformation of information from neighboring nodes. Specifically, the represen- tation of a node v; in a graph is iteratively updated by integrating information from immediate neighbors, denoted as N(x)). This process employs propagation function S. parameterized by neural network weights W, and an activation function a. The updated representation of a node in multi-layered GCN, proposed in the literature, can be mathematically expressed as:
H(+1) =
= Hu+)==
Here, Ā= A IN represents the augmented adjacency matrix of the graph G. incorporating self-
connections by adding the identity matrix IN. The diagonal matrix D, Σ, Aiy facilitates the normalization of Á. The term W(( denotes the weight matrix at layer 1, and a(-) is the activation function. The matrix (1) E R-NDD encapsulates the node features layer l.
We engage with a graph comprising N nodes in the application discussed. Our GCN outputs a vector O E xx1, representing the likelihood of each node e; e V being pivotal for propagation through
16

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the network. This mechanism enables the model to figure out significance of each node in mitigating information spread within the graph structure.
GCN Architecture Overview Our model is structured as follows: It comprises three graph convolutional layers followed by linear layer. The architecture is designed to progressively transform the input node features into a space where the final classification (e.g., determining the likelihood of node being pivotal in propagation minimization).
The model begins with a graph convolutional layer (GCNConV), taking input_size features and transforming them into hidden representation of size hidden_sidd size. This layer is followed by a ReLU activation function.
Initial Graph Convolution:
Hidden Graph Convolutional Layers: After the initial layer, the architecture includes four GCNConv layers, all utilizing hidden_ sidden_size units. These layers are designed to iteratively process and refine the features extracted from the graph's structure. Each of these layers is followed by a ReLU activation to introduce non-linearity.
The model employs repeated application of the GCNConv layer with hidden_ _szee units, demonstrating the capacity to deepen the network's understanding of the graph's topology through successive transformations.
Output Graph Convolution: A final GCNConv layer reduces the hidden representation to the desired num_clmmmaaaaaaammm cam_ccasses, preparing the features for the prediction task.
Linear Layer and Sigmoid Activation:
The architecture concludes with a linear transforma- tion (nn Linear) directly mapping the output of the last GCNConv layer num_clast classes. A sigmoid activation function is applied to this output, producing probabilities for each class in a binary classification scenario.
In this specific implementation, the input_size is set to 3, indicative of the initial feature dimen- sionality per node. The hidden_sied size is configured at 128, providing substantial feature space for intermediate representations. Lastly, the num_ classes is established at 1, signifying only one numerical output for each nodes.
The development of our supervised learning models, particularly those utilizing graph convolutional networks, leveraged several Python packages instrumental in defining, training, and evaluating our models. Below is a list of these packages and a brief description of their roles in implementation:
Packages Used
torch: Serves as the foundational framework for constructing and training various neural network models, including those for graph-based data.
torch-geometric: An extension of Py PoTorc tailored for graph neural networks. It provides efficient data structures for graphs and collection of methods for graph convolutional operations, making it essential for implementing graph convolutional neural networks (GCNs).
networkx:
Utilized for generating and manipulating complex networks. In our project, networkx primarily used for creating synthetic graph data and for preprocessing tasks that require graph analysis and manipulations before feeding the data into the neural network models.
A.2.2 Residual Network (ResNet)
The Residual Network (ResNet) model implemented in our study is variant of the conventional ResNet architecture. The core component of our ResNet model the ResidualBlock. which allows for the training of deeper networks by addressing the vanishing gradient problem through skip connections. Each ResidualBlock consists of two sequences of convolutional layers (Conv2d), batch normalization (BatchNorm22)) and sigmoid activations. A distinctive feature is the adaptation of the skip connection to include a convolutional layer and batch normalization if there discrepancy in the input and output channels the stride is not equal to one.
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ResNet Architecture Overview
Initial Convolution: Begins with a convolutional layer applying 64 filters of size 3x3, followed by batch normalization and sigmoid activation.
Residual Blocks: Three main layers (layer1, layer2, 1ayer3) constructed with the _make__ layer method. Each layer contains sequence of ResidualBlocks, with channel sizes 32, 64, and 128, respectively. The number of blocks per layer is determined by the num_ blocks parameter.
Each ResidualBlock implements two sequences of convolutional operations, batch normalization, and sigmoid activation, with an optional convolution in the shortcut connection for channel or stride adjustments.
Adaptive Input Reshaping: Inputs are dynamically reshaped to a square form based on the square root of the second dimension, ensuring compatibility with different input sizes.
Pooling and Output Layer: dimensions, followed by a fully connected layer mapping 128 features single output, thus producing the final prediction.
Concludes with an average pooling layer to reduce spatial
Training Details Our model is trained using the PyTorch library, leveraging its comprehen- sive suite of neural network tools and functions. The optimizer of choice is the Adam optimizer (torch. optim aptim.Adam), selected for its adaptive learning rate properties, which helps in converging faster. The learning rate was risk of overshooting minimal loss values. The training process involved the iterative adjustment of weights through backpropagation, minimizing the loss calculated the output layer. This procedure was executed repeatedly over batches of training data, with the model parameters updated in each iteration to reduce the prediction error.
0.0005, balancing the trade-off between training speed and the
Packages Used The implementation of our ResNet model and the training process was facilitated by the following Python packages:
torch: Provides the core framework for defining and training neural networks.
torch. nn: A submodule of Py/Toch that contains classes and methods specifically designed for building neural networks.
torch. mn functional: Offers functional interfaces for operations used in building neural networks, like activations and pooling functions.
torch. optim: Contains optimizers such as Adam, which are used for updating model parameters during training.
A.3 Metries
Training Loss
SL: In the Supervised Learning (SL) framework, for our study, we employed the Binary Cross Entropy (BCE) loss to train the Graph Convolutional Network (GCN) model. Here, we delineate the iterative process used during the training phase under this setting. The model is fed with new graph state at the beginning of each iteration. The GCN produces an output vector O E [0, 1] Nx where N the number of nodes in the graph. Each component of O, denoted as O,. represents the probability that node i should be blocked to minimize the propagation rate in the network. A greedy algorithm is employed to ascertain the optimal node to block. For each node i, temporarily set the node as blocked. We compute the propagation rate of the network with node blocked. Then, we revert the blockage of node and proceed to evaluate the next node. We select the node that, when blocked. results in the lowest propagation rate across the network. Upon determining the node j. which yields the minimum propagation rate when blocked, target vector T 0, Nx1 is constructed such that T3 = 1 (indicating the target node to block), and T, = 0 for all * j.
The BCE loss between the output vector O and the target vector T is computed as follows:
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BCE Loss 1 N
[ log(O,) + (1 TT) log(1 0+)]
The Binary Cross Entropy (BCE) loss is particularly useful in this context because it measures the performance of the model in terms of how effectively it can predict the binary outcome (block/no block) for each node in the graph. Unlike our reinforcement learning setup, which utilizes batch updates across multiple states or episodes, the supervised learning approach updates the model weights based on the loss calculated from single graph state per iteration.
The training loss for our model is computed using the Mean Squared Error (MSE) metric. In each episode, the loss is calculated across a batch of samples drawn from the replay buffer. Specifically, it measures the squared difference between the predicted value function of the current state from the policy network and the Bellman target - the observed reward plus the value of the subsequent state as estimated by the target network. Executing this calculation over batches of experiences allows the policy network to learn from a diverse set of state transitions, thereby refining its predictions to better approximate the true expected rewards through temporal difference learning.
RL:
Evaluation Metric Infection Rate The infection rate measures the proportion of the network that is infected over time and is a crucial metric for evaluating the spread of misinformation within a simulated environment. is calculated as the ratio of infected nodes the total number of nodes within the network at given timestep:
Number of Infected Nodes
Infection Rate
Total Number of Nodes
This metric serves not only as means to understand the dynamics of misinformation spread during simulations but also as a vital testing metric for evaluating model performance on test datasets. Models that effectively contain or reduce the infection rate are considered to have performed well, as they demonstrate the ability to mitigate the spread of misinformation across the network.
A.4 Hardware details
We have used two servers to run our experiments. One with 48-core nodes each hosting V100 32G GPUs and 128GB of RAM. Another with 256-cores, eight A 100 40GB GPUs, and 1TB of RAM. The processor speed 2.8 GHz.
A.5
Training Details
A.5.1 SL
Training Methodology: Our SL setup is coupled with a ranking algorithm which is shown in Algorithm We GCN with an input size of (opinion value, degree of node, proximity to source node), a hidden size of 128, and an output size of 1. The model was trained using the Adam optimizer with learning rate of 0.001 and binary cross-entropy loss function. The training process involved 1000 epochs. where in each epoch, a graph with 25 nodes was generated. During each epoch, the model iteratively minimized the infection rate by selecting nodes block based on GCN output until no uninfected nodes remained. The loss was calculated and backpropagated, and the weights were updated accordingly. The total loss for each epoch was averaged over iterations.
A.5.2 RL
Training Algorithm: Shown in Algorithm 2
In the DQN framework with experience replay, two neural networks are utilized: the target network Qe- and the policy network Qa. The target network is periodically updated by copying weights from the policy network, while the policy network is optimized continuously through the replay memory D. which stores agent's past experiences. During training, the agent samples random mini-batches of transitions from D to minimize the loss function via gradient descent, enabling the policy network
19

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Algorithm Ranking Algorithm
1: Input: Graph G, infected nodes blocked nodes B, action budget
2: Output: Target matrix 7 e xx
3: S < initialize catiiiiiiimluationnnm simulation_networ((....
4: .M get ret_uninfected_ and suublocked_nodes((
combinations(M, K)
6: min_rate +
7: terget_set +- None
8:
9: for c
10: temp_S +
11: block nodes(temp S.c)
> Temporarily block nodes e by setting their opinion values to 1
12: rate simulate simulate__ropaaeeeeeeeeeeeee gamulate_ropaggaioonteeey propagation(temp_5))
Infection Rate Number of
if rate < min rale then
14: min rale
miierrate <- rale
15: largel set <-
16: end if
end for
18:
19: T [0]
20: for i E target_set do
21: T
22: end
23:
24: return 7''
estimate optimal actions. This strategy helps the agent break the temporal correlation inherent in sequential data, enhancing the stability and convergence of the learning process. We use the Mean Squared Error (MSE) metric to calculate the loss value between the policy network and the target network.
Training Methodology: The Neural Network model is trained using a variant of Q-learning, with replay buffer approach to stabilize the learning process, aimed learning the value function. Training commences with a fully exploratory policy (epsilon 1) and transitions to an exploitation-focused strategy as epsilon decays over time. The learning rate is set (MSE) loss is utilized to measure the prediction quality.
10 10-4, and mean squared error
At each episode, 200 random initial states are generated, with a selected parameter of the number of initially infected nodes, and actions are determined through an epsilon- epsilon-greedy method, balancing between exploration and exploitation. The agent performs actions by selecting nodes the network to effectively contain misinformation spread. For each action, the network's 's new state and corresponding reward are observed, which are then stored the replay buffer.
Batch updates are carried out by sampling from this buffer, ensuring that learning occurs across a tuples. We have used batch-size of 100 across the
diverse set of state-action- wewttoo-rrward-eex----ee experiments. The policy network parameters are optimized using the Adam optimizer, and the target network's parameters are periodically updated to reflect the policy network, reducing the likelihood of divergence.
The training process continues for 300 number of episodes, with the epsilon parameter decaying after each timestep within an episode, encouraging the model to rely more on learned values rather than random actions training progresses. The duration of each episode and overall training, along with average rewards and loss, are logged for post-training analysis. The model parameters yielding the best performance on the validation
preserved for subsequent evaluation phases.
A.6 Inference Results
A.6.1 SL
We have conducted comprehensive testing our SL. model across various topographies and environ- ments, examining the performance under different conditions. The overall performance comparison
20

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Algorithm 2 DVN with experience replay
Input: states per episode n,, batch size m, action budget k. parameter update interval T", max number
episodes Cmax
2: Output: V network Vg
Initialize experience replay memory
4: Initialize policy network V with random weights
Initialize target network V with weights
6: Initialize e-decay to and anneal 0.1 with training
7: for 1 Emax do
8: t+-
8t with initial infected nodes 10: Calculate candidate node set C for 8,
Generate random
11:
while any |C] > 0 do Initialize blocker set B, <- 0
12:
13:
Randomly sample number from uniform distribution (4(0, if e then
15:
Randomly sample it candidates from C as blocker set B else
16:
17:
Initialize the infection prediction set K for all node of candidate set do infection number K, using V=(8t+1), where 8441 is state resulting after taking action u in state 8t Append K,, to K end for Select the nodes with the least infection prediction from K as the blocker
18:
19:
Calculate
20:
21:
22:
23:
24:
Block the nodes in the blocker set B,
25:
Update the statc 81+1
26:
Update the candidate set
27:
Update the reward Fz & < ++1
28:
29:
for 0 1 do
30:
Store the transition (si Bi, ri. in
31:
for
32:
Sample random minibatch of m transitions (#j ay, Ty. 85+1) from if episode terminates at step j +
33:
7 Vg+ otherwise
34:
Calculate loss using mean squared error between and Vo(s;,, set gradients to zero, perform backpropagation, and update weights using the Adam optimizer if mod T" 0 then
35:
36:
Update target network 0 <- 0
37:
end
38:
end while
39: end for
21

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for each model under these varied conditions is illustrated in Figure 3 This figure provides com- prehensive view of how the model performs across the different environments and topographical scenarios.
Time
Case Eriox Rengl
Wetre
Nethods
Nathoda
Mathoda
randort
random
random
max so SBBC
max tur Static
max_Aegrez, Stati
dynames
degges, Synamic
Irfectinn Rate
63
Thee
Case 2: Erdos Renyl
Watts Stogatz
Mechods
Methoda
0.1
gam
gm
gm
max dagres
degrez stalic
maac deggee dynamic
maxe deg we Syname
max degree Synamic
Irfection Rate
Dudge
Dudget
Sudget
Erdos Renel
Watss
Neiheds
Nathods
Methods
raidorrs
random
maxe static
static
degres, static
deggee Synamic
Vrfecdan Rate
Figure 3: Comparative analysis of the GCN-Based SL Model Against Baseline Models Across Different Network Types and Budgets. Each subfigure represents one of the three cases (1, 2, and 3), organized by rows, for three different types of networks: Tree, Erdös-Rényt Rényi, and WattssStroogta organized by columns. Within each panel, the infection rate is plotted for four methodologics. SL based on GCN (bluc), random node selection (orange), static selection of maximum degrees (green), and dynamic selection of maximum degrees (red) across three levels of budget (1, 2, and 3). These results underscore the variability in performance with changes in network structure and budget allocation, highlighting the superior effectiveness the GCN model in simpler cases and under increased budget conditions, with diminishing returns in more complex environments.
Dataset v2 Testing In addition the initial dataset, have also tested model using dataset v2. For more granular analysis, we have compiled the test results into three distinct cases:
Case 1: Detailed in Figure
22

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Nothoda
Infectin Sare
Sources: 2, Budget: 1
Sources: 2, Budget: 2
Sources: 2, Budget:
Infec
02
83
Seerage Degres
Average Degiee
Auwrage Degres
Budget:
Budget:
Budget:
Inflcc on RaFE
Degies
Figure 4: Case-1: Comparative Mean Infection Rate across different parameter settings for GCN- based model trained on 25-node dataset tested on dataset v2 consisting of nodes
23

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Case 2: Detailed in Figure 5
Sources: Budget:
Sources: Budget:
Sources: Budget:
Nethods
211
14
Infoct Ratt
04
Sources: Budget:
Sources: 2. Budget: 2
Sources: 2, Budget:
Infect Raite
02
Avenae Degie
Average Begine
Aueeage Degree
Sources: Budget:
Sources: Budget:
Sources: Budget:
04
Frfctinn Rart
08
Degres
Degiee
Figure 5: Case-2: Comparative Mean Infection Rate across different parameter settings for GCN-
based SL model trained 25-node dataset and tested dataset v2 consisting of 50 nodes
24

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Case 3: Detailed in Figure 6
Sources: Budget:
Sources: Budget:
Sources: Budget:
Methock
Inlect Ratts
04
Degine
Sources: Budget:
Sources: 2. Budget:
Sources: 2, Budget:
Infect Rait
Avenag Degie
Average Begine
Aueeage Degree
Sources: Budget:
Sources: Budget:
Sources:
Budget:
Hrletton Rart
0.8
Degres
Degiee
Figure 6: Case-3: Comparative Mean Infection Rate across different parameter settings for GCN-
based SL model trained 25-node dataset and tested dataset v2 consisting of 50 nodes
25

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A.6.2 RL
Comparison of MSE loss across different reward functions for Case-1: Figure) 7
10"
17"
2.
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0 10
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9
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51
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121 15"
101 201 201 271
Figure 7: Case-1: Comparative MSE loss across different reward functions for a ResNet model trained on a 50-node dataset. Columns represent an increase in action budget during training, while rows indicate a rise in the number of initial infected nodes.
Comparison of Mean Infection Rate across different reward functions, showcasing that the reward Rs performs better in model configurations with higher action budget and higher initial infected nodes: Figure g
Mean infectinn Rate 58 roets lifecto bopre aarite
Model
Medel
Figure 8: Case-1: Comparative Mean Infection Rate across different reward functions for ResNet model trained 50-node dataset tested Dataset v2 of 50 nodes with degree of connectivity 3.
Case-1
Type: Binary Opinion and Binary Trust.
Opinion Dynamic Model: Discrete Switching.
26

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RO
The loss plot is presented in Figure 9 Dataset v1 Inference Result: 50 Nodes Figure 10
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27
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Figure 9: Case-1 I using RO: Training MSE loss evolution for RI. policy using ResNet model across networks of 10 (blue). 25 (orange), and 50 (green) nodes, for varying initial misinformation sources (Inf.) and action budgets (Act.. Plotted on logarithmic scale, the loss decreases over episodes, indicating improved policy performance and adaptation across network sizes.
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Figure 10: Case-l using RO: Performance comparison of the RL policies trained using ResNet model for 50-node network. The barplot displays the mean infection rate for datasets d1, d2, and d3 of Dataset vI type, differentiated by the number of initial misinformation sources (Inf.) and action budgets (Act.: a2, a3). Each subplot illustrates the performance of policy trained with the parameters indicated in its title. Lower infection rates indicate more effective policy learning and misinformation containment.
27

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R1 The loss plot is presented in Figure
Dataset v1 Inference Result: 50 Nodes Figure 12
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Figure 11: Case-1 using R1: Training MSE loss evolution for RL policy using ResNet model across networks of 10 (blue), 25 (orange), and 50 (green) nodes, for varying initial misinformation sources (Inf.) and action budgets (Act.). Plotted on logarithmic scale, the loss decreases over episodes, indicating improved policy performance and adaptation across network sizes.
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Figure 12: Case-1 using RI: Performance comparison of the RL policies trained using ResNet model for a 50-node network. The barplot displays the mean infection rate for datasets d1, d2, and d3 of Dataset vI type, differentiated by the number of initial misinformation sources (Inf.) and action budgets (Act.: al, a2, a3). Each subplot illustrates the performance of policy trained with the parameters indicated in title. Lower infection rates indicate more effective policy learning and misinformation containment.
28

---

R2
The loss plot is presented in Figure 13 Dataset v1 Inference Result: 50 Nodes Figure 14
Piete cwaet 1,002
1,
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127 T8T 211 241
127 T81 211 241 277
03 01 12T T8T 211 241 277
Figure 13: Case-1 using R2: Training MSE loss evolution for RL policy using ResNet model across networks 10 (blue), 25 (orange), and 50 (green) nodes, for varying initial misinformation sources (Inf.) and action budgets (Act.). Plotted on logarithmic scale, the loss decreases over episodes, indicating improved policy performance and adaptation across network sizes.
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Figure 14: Case-1 using R2: Performance comparison of the RL policies trained using ResNet model for 50-node network. The barplot displays the mean infection rate for datasets d1, d2, and d3 of Dataset vI type, differentiated by the number of initial misinformation sources (Inf.) and action budgets (Act.: al, a2, a3). Each subplot illustrates the performance of policy trained with the parameters indicated in its title. Lower infection rates indicate more effective policy learning and misinformation containment.
29

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R3
The loss plot is presented in Figure 15 Dataset v1 Inference Result: 50 Nodes Figure 16
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Figure 15: Case-1 using R3: Training MSE loss evolution for RL policy using ResNet model across networks 10 (blue), 25 (orange), and 50 (green) nodes, for varying initial misinformation sources (Inf.) and action budgets (Act.). Plotted on logarithmic scale, the loss decreases over episodes, indicating improved policy performance and adaptation across network sizes.
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Figure 16: Case-1 using R3: Performance comparison of the RL policies trained using ResNet model for 50-node network. The barplot displays the mean infection rate for datasets d1, d2, and d3 of Dataset vI type, differentiated by the number of initial misinformation sources (Inf.) and action budgets (Act.: al, a2, a3). Each subplot illustrates the performance of policy trained with the parameters indicated in its title. Lower infection rates indicate more effective policy learning and misinformation containment.
30

---

R4
The loss plot is presented in Figure
Dataset v1 Inference Result: 50 Nodes Figure 18
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01 12" 153 211 241 277
47
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Figure 17: Case-1 using R4: Training MSE loss evolution for RL policy using ResNet model across networks 10 (blue), 25 (orange), and 50 (green) nodes, for varying initial misinformation sources (Inf.) and action budgets (Act.). Plotted on logarithmic scale, the loss decreases over episodes, indicating improved policy performance and adaptation across network sizes.
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Figure 18: Case-1 using R4: Performance comparison of the RL policies trained using ResNet model for 50-node network. The barplot displays the mean infection rate for datasets d1, d2, and d3 of Dataset vI type, differentiated by the number of initial misinformation sources (Inf.) and action budgets (Act.: al, a2, a3). Each subplot illustrates the performance of policy trained with the parameters indicated in its title. Lower infection rates indicate more effective policy learning and misinformation containment.
31

---

Dataset v2 Results
Degree of connectivity 50 Nodes
Figure 19
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Rocee St ie:: LAAe:
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6000
600
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Figure 19: Case-1 inference on Dataset v2: Performance comparison of the RL policies trained using ResNet model for a 50-node network. The barplot displays the mean infection rate for different reward functions on Dataset misinformation sources (Inf.) and action budgets (Act.: al, a2, a3). Each subplot illustrates the performance policy trained with the parameters indicated in title. Lower infection rates indicate more effective policy learning and misinformation containment.
Degree of connectivity 1, differentiated by the number of initial
32

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Degree of connectivity 50 Nodes Figure 20
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Figure 20: Case-1 inference on Dataset v2: Performance comparison of the RL policies trained using ResNet model for a 50-node network. The barplot displays the mean infection rate for different reward functions on Dataset v2 of Degree of connectivity 2, differentiated by the number of initial misinformation sources (Inf.) and action budgets (Act.: al, a2, a3). Each subplot illustrates the performance of policy trained with the parameters indicated in its title. Lower infection rates indicate more effective policy learning and misinformation containment.
33

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Degree of connectivity 50 Nodes Figure 21
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Figure 21: Case-1 inference on Dataset v2: Performance comparison of the RL policies trained using ResNet model for a 50-node network. The barplot displays the mean infection rate for different reward functions on Dataset v2 of Degree of connectivity 3, differentiated by the number of initial misinformation sources (Inf.) and action budgets (Act.: al, a2, a3). Each subplot illustrates the performance of policy trained with the parameters indicated in its title. Lower infection rates indicate more effective policy learning and misinformation containment.
34

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Degree of connectivity
50 Nodes Figure 22
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Figure 22: Case-1 inference on Dataset v2: Performance comparison of the RL policies trained using ResNet model for a 50-node network. The barplot displays the mean infection rate for different reward functions on Dataset v2 of Degree of connectivity 4, differentiated by the number of initial misinformation sources (Inf.) and action budgets (Act.: al, a2, a3). Each subplot illustrates the performance of policy trained with the parameters indicated in its title. Lower infection rates indicate more effective policy learning and misinformation containment.
Case-2
Type: Floating Point Opinion and Binary Trust.
Opinion Dynamic Model: Linear Adjustment.
35

---

RO
The loss plot is presented in Figure 23 Dataset v1 Inference Result: 50 Nodes Figure 24
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10-
13
10.
10
247
Figure 23: Case-2 using RO: Training MSE loss evolution for RL policy using ResNet model across networks 10 (blue), 25 (orange), and 50 (green) nodes, for varying initial misinformation sources (Inf.) and action budgets (Act.). Plotted on logarithmic scale, the loss decreases over episodes, indicating improved policy performance and adaptation across network sizes.
hodox LAbd:!
Nodex Acz1
hocks Int. 1,6d23
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Figure 24: Case-2 using RO: Performance comparison of the RL policies trained using ResNet model for 50-node network. The barplot displays the mean infection rate for datasets d1, d2, and d3 of Dataset vI type, differentiated by the number of initial misinformation sources (Inf.) and action budgets (Act.: al, a2, a3). Each subplot illustrates the performance of policy trained with the parameters indicated in its title. Lower infection rates indicate more effective policy learning and misinformation containment.
36

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R1
The loss plot is presented in Figure 25 Dataset v1 Inference Result: 50 Nodes Figure 26
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33
13
INF-2.M ACD1
1/
10
13
247
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Figure 25: Case-2 using R1: Training MSE loss evolution for RL policy using ResNet model across networks 10 (blue), 25 (orange), and 50 (green) nodes, for varying initial misinformation sources (Inf.) and action budgets (Act.). Plotted on logarithmic scale, the loss decreases over episodes, indicating improved policy performance and adaptation across network sizes.
hodox LAdd:!
Nodex Infil Ii:1,
hocks lnt 1,4d23
ACL
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Nedac SL AC1
Noces 2,M23
Figure 26: Case-2 using RI: Performance comparison of the RL policies trained using ResNet model for 50-node network. The barplot displays the mean infection rate for datasets d1, d2, and d3 of Dataset vI type, differentiated by the number of initial misinformation sources (Inf.) and action budgets (Act.: al, a2, a3). Each subplot illustrates the performance of policy trained with the parameters indicated in its title. Lower infection rates indicate more effective policy learning and misinformation containment.
37

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R2
The loss plot is presented in Figure 27 Dataset v1 Inference Result: 50 Nodes Figure 28
Pioke CAM02
137
3
10"
Hos N4
10"
247
101
271
Figure 27: Case-2 using R2: Training MSE loss evolution for RL policy using ResNet model across networks 10 (blue), 25 (orange), and 50 (green) nodes, for varying initial misinformation sources (Inf.) and action budgets (Act.). Plotted on logarithmic scale, the loss decreases over episodes, indicating improved policy performance and adaptation across network sizes.
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Figure 28: Case-2 using R2: Performance comparison of the RL policies trained using ResNet model for 50-node network. The barplot displays the mean infection rate for datasets d1, d2, and d3 of Dataset vI type, differentiated by the number of initial misinformation sources (Inf.) and action budgets (Act.: al, a2, a3). Each subplot illustrates the performance of policy trained with the parameters indicated in its title. Lower infection rates indicate more effective policy learning and misinformation containment.
38

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R3
The loss plot is presented in Figure 29 Dataset v1 Inference Result: 50 Nodes Figure 30
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Figure 29: Case-2 using R3: Training MSE loss evolution for RL policy using ResNet model across networks of 10 (blue), 25 (orange), and 50 (green) nodes, for varying initial misinformation sources (Inf.) and action budgets (Act.). Plotted on logarithmic scale, the loss decreases over episodes, indicating improved policy performance and adaptation across network sizes.
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Nodex Infil Inc: Acc
hoces lnt: 1,4d23
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Figure 30: Case-2 using R3: Performance comparison of the RL policies trained using ResNet model for 50-node network. The barplot displays the mean infection rate for datasets d1, d2, and d3 of Dataset vI type, differentiated by the number of initial misinformation sources (Inf.) and action budgets (Act.: al, a2, a3). Each subplot illustrates the performance of policy trained with the parameters indicated in its title. Lower infection rates indicate more effective policy learning and misinformation containment.
39

---

R4
The loss plot is presented in Figure 31 Dataset v1 Inference Result: 50 Nodes Figure 32
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Figure 31: Case-2 using R4: Training MSE loss evolution for RL policy using ResNet model across networks 10 (blue), 25 (orange), and 50 (green) nodes, for varying initial misinformation sources (Inf.) and action budgets (Act.). Plotted on logarithmic scale, the loss decreases over episodes, indicating improved policy performance and adaptation across network sizes.
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Figure 32: Case-2 using R4: Performance comparison of the RL policies trained using ResNet model for 50-node network. The barplot displays the mean infection rate for datasets d1, d2, and d3 of Dataset vI type, differentiated by the number of initial misinformation sources (Inf.) and action budgets (Act.: al, a2, a3). Each subplot illustrates the performance of policy trained with the parameters indicated in its title. Lower infection rates indicate more effective policy learning and misinformation containment.
40

---

Dataset v2 Results
Degree of connectivity 50 Nodes
Figure 33
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Rocee St ie:: LAAe:
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6000
600
600
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Figure 33: Case-2 inference on Dataset v2: Performance comparison of the RL policies trained using ResNet model for a 50-node network. The barplot displays the mean infection rate for different reward functions on Dataset v2 of Degree of Connectivity differentiated by the number of initial misinformation sources (Inf.) and action budgets (Act.: al, a2, a3). Each subplot illustrates the performance policy trained with the parameters indicated in title. Lower infection rates indicate more effective policy learning and misinformation containment.
41

---

Degree of connectivity
50 Nodes Figure 34
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Figure 34: Case-2 inference on Dataset v2: Performance comparison of the RL policies trained using ResNet model for a 50-node network. The barplot displays the mean infection rate for different reward functions on Dataset v2 of Degree of Connectivity 2, differentiated by the number of initial misinformation sources (Inf.) and action budgets (Act.: al, a2, a3). Each subplot illustrates the performance of policy trained with the parameters indicated in its title. Lower infection rates indicate more effective policy learning and misinformation containment.
42

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Degree of connectivity
50 Nodes Figure 35
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Figure 35: Case-2 inference on Dataset v2: Performance comparison of the RL policies trained using ResNet model for a 50-node network. The barplot displays the mean infection rate for different reward functions on Dataset v2 of Degree of Connectivity 3, differentiated by the number of initial misinformation sources (Inf.) and action budgets (Act.: al, a2, a3). Each subplot illustrates the performance of policy trained with the parameters indicated in its title. Lower infection rates indicate more effective policy learning and misinformation containment.
43

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Degree of connectivity
50 Nodes Figure 36
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Nodee MESS ReL
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Figure 36: Case-2 inference on Dataset v2: Performance comparison of the RL policies trained using ResNet model for a 50-node network. The barplot displays the mean infection rate for different reward functions on Dataset v2 of Degree of Connectivity 4, differentiated by the number of initial misinformation sources (Inf.) and action budgets (Act.: al, a2, a3). Each subplot illustrates the performance of policy trained with the parameters indicated in its title. Lower infection rates indicate more effective policy learning and misinformation containment.
Case-3
v1
Type: Floating Point Opinion and Floating Point Trust.
Opinion Dynamic Model: Linear Adjustment.
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R0
The loss plot is presented in Figure 37 Dataset v1 Inference Result: 50 Nodes Figure 38
Loss Piote candket- #0 1,6d.2
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121 101
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Figure 37: Case-3 v1 using RO: Training MSE loss evolution for RL policy using ResNet model across networks of 10 (blue), 25 (orange), and 50 (green) nodes, for varying initial misinformation sources (Inf.)) and action budgets (Act.). The loss decreases over episodes, indicating improved policy performance and adaptation across network sizes.
hodoc 58 1.Ad211
Nodex Inf:: Acc
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Nodeis ACL
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4.50
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Figure 38: Case-3 v1 using RO: Performance comparison of the RL policies trained using ResNet model for 50-node network. The barplot displays the mean infection rate for datasets d1, d2, and d3 of Dataset v1 type, differentiated by the number of initial misinformation sources (Inf.) and action budgets (Act.: al, a2, a3). Each subplot illustrates the performance of policy trained with the parameters indicated in its title. Lower infection rates indicate more effective policy learning and misinformation containment.
45

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R1
The loss plot is presented in Figure 39
Dataset v1 Inference Result: 50 Nodes Figure 40
Loss Pitta conclet-
3
(M1 141.8MM (unA JUIMM (TT))) AUU
10
10"
16
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Figure 39: Case-3 v1 using R1: Training MSE loss evolution for RL policy using ResNet model across networks 10 (blue), 25 (orange), and 50 (green) nodes, for varying initial misinformation sources (Inf.) and action budgets (Act.). Plotted on logarithmic scale, the loss decreases over episodes, indicating improved policy performance and adaptation across network sizes.
hodoc 1.Ad:!
Nodax Inf:l. Acc
1,Ad:3
Pootes 2.Art11
ACL
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Dubeet Nodia INTAA ACT-2
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Figure 40: Case-3 v1 using RI: Performance comparison of the RL policies trained using ResNet model for 50-node network. The barplot displays the mean infection rate for datasets d1, d2, and d3 of Dataset v1 type, differentiated by the number of initial misinformation sources (Inf.) and action budgets (Act.: al, a2, a3). Each subplot illustrates the performance of policy trained with the parameters indicated in its title. Lower infection rates indicate more effective policy learning and misinformation containment.
46

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R2
The loss plot is presented in Figure 41 Dataset v1 Inference Result: 50 Nodes Figure 42
Pitta condlet-
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247 271
Figure 41: Case-3 v1 using R2: Training MSE loss evolution for RL policy using ResNet model across networks of 10 (blue), 25 (orange), and 50 (green) nodes, for varying initial misinformation sources (Inf.) and action budgets (Act.). Plotted on logarithmic scale, the loss decreases over episodes, indicating improved policy performance and adaptation across network sizes.
hodoc 1.Ad:!
Nodax Inf: Acc:
1.kt:3
Podtes S.Art1
ACL
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Indee 68 lef: J.bar1
Nodio NNT
Noces 3.0013
Figure 42: Case-3 v1 using R2: Performance comparison of the RL policies trained using ResNet model for 50-node network. The barplot displays the mean infection rate for datasets d1, d2, and d3 of Dataset v1 type, differentiated by the number of initial misinformation sources (Inf.) and action budgets (Act.: al, a2, a3). Each subplot illustrates the performance of policy trained with the parameters indicated in its title. Lower infection rates indicate more effective policy learning and misinformation containment.
47

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R3 The loss plot is presented in Figure 43 Dataset v1 Inference Result: 50 Nodes Figure
Loss Pitte condlet- d) 1,6d22
ABL1
10'
16"
13
13
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13
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13
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160
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10
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1et. 2
16"
150
10
93
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10
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247
271
121 101
271
191 101
Epwodds
Figure 43: Case-3 v1 using R3: Training MSE loss evolution for RL policy using ResNet model across networks of 10 (blue), 25 (orange), and 50 (green) nodes, for varying initial misinformation sources (Inf.) and action budgets (Act.). The loss decreases over episodes, indicating improved policy performance and adaptation across network sizes.
hodux (.Ad2!1
Nodox: Inf:l Acc
hocks Ini: 1,Ad:3
3.0
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3.54
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Nodic SL INTA AC:
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Figure 44: Case-3 v1 using R3: Performance comparison of the RL policies trained using ResNet model for 50-node network. The barplot displays the mean infection rate for datasets d1, d2, and d3 of Dataset v1 type, differentiated by the number of initial misinformation sources (Inf.) and action budgets (Act.: al, a2, a3). Each subplot illustrates the performance of policy trained with the parameters indicated in its title. Lower infection rates indicate more effective policy learning and misinformation containment.
48

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R4
The loss plot is presented in Figure 45
Dataset v1 Inference Result: 50 Nodes Figure 46
Los Pitta condlct-
13
10"
3 13
10"
10"
13
www.
10
Eshodd
Figure 45: Case-3 v1 using R4: Training MSE loss evolution for RL policy using ResNet model across networks 10 (blue), 25 (orange), and 50 (green) nodes, for varying initial misinformation sources (Inf.) and action budgets (Act.). Plotted on logarithmic scale, the loss decreases over episodes, indicating improved policy performance and adaptation across network sizes.
hodic 1.Ad:!
Nodax Inf: Acc:
1.Ad:3
Pootes
4.0
Detual Indee 68 left J.bar1
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Figure 46: Case-3 vl using R4: Performance comparison of the RL policies trained using ResNet model for 50-node network. The barplot displays the mean infection rate for datasets d1, d2, and d3 of Dataset v1 type, differentiated by the number of initial misinformation sources (Inf.) and action budgets (Act.: al, a2, a3). Each subplot illustrates the performance of policy trained with the parameters indicated in its title. Lower infection rates indicate more effective policy learning and misinformation containment.
49

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Dataset v2 Results
Degree of connectivity 50 Nodes
Figure 47
Cauabot idlcter hodes Dusasinegg..
Rocee St ie:: LAAe:
Hedax 20, IE:: Acti2
Vodar ACL
6000
600
600
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Roundi
Figure 47: Case-3-v1 inference on Dataset v2: Performance comparison the RL policies trained using ResNet model for a 50-node network. The barplot displays the mean infection rate for different reward functions on Dataset v2 of Degree of Connectivity differentiated by the number of initial misinformation sources (Inf.) and action budgets (Act.: al, a2, a3). Each subplot illustrates the performance policy trained with the parameters indicated in title. Lower infection rates indicate more effective policy learning and misinformation containment.
50

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Degree of connectivity
50 Nodes Figure 48
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Hedex
Teder 2.
6:0
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600
609
Figure 48: Case-3-v1 inference on Dataset v2: Performance comparison of the RL policies trained using ResNet model for a 50-node network. The barplot displays the mean infection rate for different reward functions on Dataset v2 of Degree of Connectivity 2, differentiated by the number of initial misinformation sources (Inf.) and action budgets (Act.: al, a2, a3). Each subplot illustrates the performance of policy trained with the parameters indicated in its title. Lower infection rates indicate more effective policy learning and misinformation containment.
51

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Degree of connectivity
50 Nodes Figure 49
Cauainn Hikii Ras for hooee butseroig
Vedex H4:2 AAL
"edes 20
Pocee 53 Inr 3.Ac1:1
Vedex 50, ME-3 ALL:2
Todm 30, PLLLAA
6:8
Figure 49: Case-3-v1 inference on Dataset v2: Performance comparison of the RL policies trained using ResNet model for a 50-node network. The barplot displays the mean infection rate for different reward functions on Dataset v2 of Degree of Connectivity 3, differentiated by the number of initial misinformation sources (Inf.) and action budgets (Act.: al, a2, a3). Each subplot illustrates the performance of policy trained with the parameters indicated in its title. Lower infection rates indicate more effective policy learning and misinformation containment.
52

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Degree of connectivity
50 Nodes Figure 50
Caudion bikil bu hodel Taaworogg
600
Hedex AAL
Vedas 21
Romen
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Vedey 50, ME-3 ALL:2
Tedm 30, FL-LAA
6.6
Figure 50: Case-3-v1 inference on Dataset v2: Performance comparison of the RL policies trained using ResNet model for a 50-node network. The barplot displays the mean infection rate for different reward functions on Dataset v2 of Degree of Connectivity 4, differentiated by the number of initial misinformation sources (Inf.) and action budgets (Act.: al, a2, a3). Each subplot illustrates the performance of policy trained with the parameters indicated in its title. Lower infection rates indicate more effective policy learning and misinformation containment.
53

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59


Reviews:
[{"review_id": "TGl2rzQOJ0", "reviewer": "Reviewer_Z3oY", "summary": "This study explores intervention strategies aimed at curbing the spread of misinformation in dynamic social networks. The authors propose a novel ranking algorithm to identify influential nodes for disseminating accurate information and a RL-based framework to address the computational complexity associated with label generation. The paper concludes that by integrating more realistic and complex modeling approaches, label generation techniques, and training methodologies, the proposed strategies can significantly mitigate the impact of misinformation in social networks.", "questions": "Why use ResNet and GCN as backbones?", "limitations": "The authors adequately addressed the limitations and broader impacts in the conclusion section.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The studied problem is of great importance and has considerable practical significance.\n- The paper introduces a novel ranking algorithm that effectively identifies key nodes within a network for the dissemination of accurate information. This algorithm is integrated with a supervised learning framework, providing a robust method for training neural network classifiers that can scale and generalize across different network structures.\n- By employing a reinforcement learning-based dynamic planning framework, the paper addresses the computational challenges associated with large networks. This RL methodology allows for the development of adaptive intervention strategies that can respond in real-time to the evolving patterns of misinformation spread, offering a significant improvement over traditional static approaches.\n- The authors have made the code available and provided detailed hyperparameter settings and computational resource information, enhancing the reproducibility of the study.", "weaknesses": "- While the proposed models and algorithms demonstrate efficacy in controlled settings, there may be concerns regarding their scalability when applied to real-world scenario.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for acknowledging the technical contributions and novelty of our work and for providing valuable feedback.\n\n**Remark 1: Real-world network models**  \nIn our work, we develop planning algorithms and analyze their efficacy using synthetic data in controlled settings. The synthetic data used in our study are generated using the Watts-Strogatz (or small-world) network model. As demonstrated in various studies, this network model captures the characteristics of common social and community networks [1]. Additionally, we have also evaluated our planning algorithms using directed and undirected real-world network models reported in the literature. These evaluations are included in the rebuttal PDF (Table $2$). These additional experiments confirm the major findings reported in our paper (Research Questions in Section 5).\n\nFurthermore, the Graph Convolutional Networks (GCNs)-based planners developed in our work perform effectively when applied to dynamic network models with more nodes (as illustrated in Table $1$ of our paper and Table $2$ of rebuttal PDF) even when they are trained using data generated from networks with fewer nodes (RQ4 from Section 5). Testing with GCNs on larger networks is significantly simpler and less computationally intensive than training, making it a practical solution for scaling our approach to handle extensive network sizes efficiently. Finally, we would like to emphasize that existing research work [eg., citation 14 from our manuscript] has analyzed opinion networks with a maximum of $1005$ nodes with an opinion propagation model that considers only a binary opinion and trust values (Case 1 in our paper in Section 2.1). In contrast, we have also considered more expressive opinion network models (Case 2 and Case 3 -- Section 2.1) in our study. \n\n**Question 1: Usage of ResNet and GCN**  \nGraph Convolutional Networks (GCNs) and Residual networks (ResNet) have been widely used for modeling graph-structured data. Since opinion dynamic models generate graph-structured data, we incorporated these neural network models in our study to develop planners. In our analysis, we found that GCNs offer better scalability and performance when compared with ResNet (RQ4 from Section 5).\n\n[1]. Watts, D.J. and Strogatz, S.H., 1998. Collective dynamics of ‘small-world’networks. nature, 393(6684), pp.440-442.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "5wHPvpbbdN", "reviewer": "Reviewer_jjEE", "summary": "This paper studies strategic planning for disseminating credible information within dynamic opinion networks. The main two contributions are (1) a ranking algorithm to identify influential nodes to spread accurate information, and (2) an RL-based framework for adaptive intervention strategies.", "questions": "Please see the detailed comments that include some questions.", "limitations": "To some extent. Again, please see the questions above as they pose some challenges and limitations.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Paper is well-written and easy to follow.\n2. The problem at hand is interesting.", "weaknesses": "1. One of the main building blocks of the paper is a proposed ranking algorithm to mitigate the computationally intractable issue of intervention planning, while the algorithm itself is just a brute force search, and computationally infeasible. The authors then propose an RL solution to address the problem. My question is what did the authors achieve here?  \n\n2. My other concern is about the scalability of the solution as mentioned several times throughout the paper, making it one the main contributions of the paper, without providing further analysis that proves the proposed framework is indeed scalable. The largest network size used in the paper is 50, which makes it hard to believe the method is scalable, especially since their ranking algorithm is brute force (can we quantify scalability other than network size by the way?). I would recommend increasing network size to more than 50, while providing detailed statistics of the graphs (e.g., # edges, diameter of the graph, etc.). Furthermore, I assume the 1000 states used correspond to the dynamic nature of their input graphs. This has to be clarified in the paper. \n\n3. Since the proposed method is claimed to be scalable, it is necessary to include analysis on how varying the number of states (or any other dynamic features of the input graphs) might affect the inference results.\n\n4. Normally, one would expect the R2 reward function to achieve better results as it minimizes the # candidate nodes and infection rate, but as stated by RQ3, this isn’t the case. This requires a better explanation and justification other than just showing numbers in Table 1, especially because according to Table 1, R1,2 and 4 are quite close as far as I can see. Also, in general, except for R0 and R3, differences between others are insignificant. Some kind of significance test is thus recommended to clarify this better.   \n\n5. One last question is: does adding directions to the input graphs change anything in the paper? This needs to be shown in the paper using some further analysis and experiments.\n\n6.  While the paper is very well-written, it still needs a thorough proof-read to fix some typos and mistakes (e.g., in the RQ3 section, Hypothesis RQ4 needs to be RQ3).\n\n7. I’m not sure if I understood how the Infection rate=0.7 for figure 2 was calculated and compared to that of figure 1. I would recommend adding the infection rate for figure 1 somewhere and explain how to calculate this using an example.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your valuable feedback. Thank you for pointing out that the paper is well written and for your valuable suggestions for improving our paper.\n\nWith regards to the infection rate calculation - it is calculated as the ratio of the number of infected nodes to the total number of nodes within the network at a given timestep, as mentioned in Section A.3. We will clarify this in the main paper, and add this to Figure 1 for consistency.\n\nWe will also thoroughly proof-read our paper to avoid typographical errors.\n\n**Remark 1: Building Blocks**  \nOur paper focuses on two methodologies: Supervised Learning (SL) and Reinforcement Learning (RL). We aimed to extend SL to dynamic network scenarios (Case 2 and 3), which previous research works have not explored. To achieve this, we introduced a ranking algorithm based label generation approach within SL. As discussed in the paper (Section 3.1), while the ranking algorithm provided improvements, it proved computationally infeasible when training on larger networks. To address this, we investigated RL-based solutions (Section 3.2). Based on our analysis we observed that RL-based solution using GCN model performed better in generating effective plans for networks with more nodes even when they were trained with data generated from networks with fewer nodes (RQ4). \n\nIn summary, we start by extending the static ranking algorithm based on the classical Breadth First Search to the problem of centralized planning in dynamic opinion networks. Supporting supervised learning-based planners, this algorithm can help planners when the network size is small and the global network information is available. To understand planning strategies for large networks, we also investigated reinforcement-learning-based centralized planners. We have reported comprehensive analyses in our work that were helpful in addressing specific research questions (Section 5 RQ1 to RQ4), which were not previously reported in the literature.\n\n**Remark 2, 3 & 5: On Scalability and Directed graphs**  \nFollowing the reviewers' suggestions, we have included additional analysis in the attached PDF. These include results on four real-world network models used in previous works to analyze intervention planning. These include both directed and undirected networks with the number of nodes ranging from 34 to 2000. These experimental results align with the major findings reported in our paper (Research Questions in Section 5).\n\nThe synthetic data used in our study is based on Watts-Strogatz (or small-world) network topology, which, as demonstrated in various studies, captures the characteristics of many common social and community networks [1]. The network statistics for this model are governed by the Watts-Strogatz network parameters, which are given in Section 4.2 of our paper. The network statistics of the four real-world networks considered are reported in Table 2 of the attached PDF. Further, we have considered two dataset versions (Section 4.2) that account for varying number of infected nodes and varying degree of initial infected nodes. In total we have 21 unique datasets and 1000 samples were randomly generated in each of these datasets for testing.\n\nWe have analyzed the performance of the developed planners as the number of opinion network nodes increase. In this context, the GCN-based planners developed in our work perform effectively when applied to dynamic network models with more nodes (as illustrated in Table 1 of our paper and Table 2 of rebuttal PDF) even when they are trained using data generated from networks with fewer nodes (RQ4 from Section 5). We would like to emphasize that existing research work [eg., citation 14 from our manuscript] has analyzed opinion networks with a maximum of 1005 nodes with an opinion propagation model that considers only binary opinion and trust values (Case 1 in our paper in Section 2.1). In contrast, we have also considered more expressive opinion network models (Case 2 and Case 3 -- Section 2.1) in our study. Due to the richness of the dynamic models in Cases 2 and 3, intervention planning with these expressive opinion propagation models was computationally more demanding than Case 1.\n\nThus, with regard to scalability, in our work, we mainly focus on the performance of centralized planners when the number of nodes increases. Varying the number of node features or the dimension of the opinion propagation model will introduce factors such as multiple topics and topic-dependencies in opinion networks. Though this is an important problem, this is beyond the current scope and will be investigated in our future study.\n\n**Remark 4: On Reward Models**  \nThank you for highlighting this intriguing result, which challenges the intuitive expectation that the R2 reward function would outperform others by minimizing candidate nodes and infection rate.\n\n*Global vs. Local Information:* As highlighted in RQ3 of our paper, our findings show that reward functions utilizing global information do not necessarily benefit from the addition of local information. This is evident with R2, which incorporates both global and local data but does not significantly outperform others using solely global metrics. The same is evident from Figure 7 in our appendix, which compares the MSE loss during training across different reward functions. **R2 (Green)** shows more variance and higher MSE loss, suggesting less stability and efficiency in learning compared to R1, R3, and R4.\n\n*Reward Richness and Application:* Although the differences in results for R1, R2, and R4 may appear insignificant, each reward type's computation and informational needs are distinct. While R4 requires the observability of the entire network, R1 only requires the observability of neighboring candidate nodes of the infected nodes. This diversity allows us to cater to various real-world applications, underscoring the necessity of evaluating multiple reward models to gain a comprehensive understanding.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "uq0bgbO16U", "reviewer": "Reviewer_VE7T", "summary": "The paper investigates intervention planning aimed at disseminating accurate information within dynamic opinion networks using learning strategies. It introduces a novel ranking algorithm to identify key nodes for disseminating accurate information and develops a Reinforcement Learning (RL)-based dynamic planning framework. The framework is tested on networks governed by two propagation models, incorporating both binary and continuous opinion and trust representations. The experimental results indicate that the proposed strategies can enhance infection rate control, especially with increased action budgets.", "questions": "1.How does the proposed RL-based dynamic planning framework compare with other existing state-of-the-art methods for misinformation control in terms of both performance and computational efficiency?\n2.Can the proposed methodologies be scaled effectively to much larger networks, and what are the computational requirements for such scaling?", "limitations": "Highlight Novelty: Clearly articulate the novel contributions of the paper. It would be beneficial to focus on what sets this work apart from existing research in the same area.", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1.Timely Topic: The paper addresses the significant issue of misinformation spread in social networks, which is a pressing problem in today’s digital age.\n2.Thorough Analysis: The paper presents a solid and detailed analysis of the proposed methods, providing in-depth insights into their effectiveness and behavior under various conditions.", "weaknesses": "1.Lack of Innovation: The primary issue with the paper is the lack of significant innovation. The methodologies proposed are incremental improvements over existing approaches rather than groundbreaking new techniques.\n2.Comparison with Existing Methods: The paper does not sufficiently compare its methods with existing state-of-the-art techniques. This makes it challenging to assess the true novelty and effectiveness of the proposed approaches.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for pointing out the novelty of the ranking algorithm and the thoroughness of the analysis presented in the paper. In the following, we address the specific concerns on novelty, comparisons with existing literature, and the question on scalability.\n\n**Remarks 1 & 3: Novelty**  \nExisting research on planning problems in the context of opinion networks often overlooks key features of opinion propagation models, such as their rich network dynamics, asynchronous communication, and the impact of factors like the degree of infected nodes, action budget, and various reward models on the effectiveness of planners. Our work addresses this gap by analyzing three distinct cases of opinion network models (Section 2.1)—ranging from binary to continuous spectra of opinion and trust values—and asynchronous communication (Section 2.2). This approach introduces greater richness and expressiveness to opinion dynamics models, making the model-based analysis of opinion propagation more reflective of real-world scenarios. Furthermore, we develop comprehensive datasets with a wide range of Watts-Strogatz (or small-world) network topologies (Section 4.2), varying degrees of initial infected nodes, action budgets, and reward models—from those requiring local network information to those utilizing global real-time network states (Section 3.2.1). \n\nFrom an algorithmic perspective, we extend the static ranking algorithm based on the classical BFS to the problem of centralized planning in dynamic opinion networks. Supporting supervised learning-based planners, this algorithm can help planners when the network size is small and the global network information is available. To understand planning strategies for large-networks, we also investigate reinforcement-learning-based centralized planners. In particular, our study explores five distinct reward structures, enhancing our understanding of reward dynamics and their computability (based on access to local vs global network information) that may be applicable to diverse applications depending on network observability and size. Further, the use of Deep Value Networks (DVN) proves more appropriate for our chosen application compared to the traditional and popular architecture of Deep Q-Networks (DQN), which has been investigated to study planning problems for opinion propagation models with simple binary state and trust parameters. DVN-based planners studied in our work are better suited for environments where the number of intervention nodes (action budget) is not fixed and varies dynamically. This flexibility allows DVN to handle dynamic and large-scale networks with greater efficiency as shown in our work  [see citation 14 from the manuscript]. \n\nThis makes our work a significant and non-trivial contribution to the existing literature on this topic.\n\n**Remark 2 & Question 1: Comparison with existing methodologies**  \nMost prior research has only addressed static network scenarios (primarily Case 1), without considering the rich dynamics of Cases 2 and 3 (see Section 2.1 for the distinction). This is the first study to incorporate such a rich dynamic nature of opinion networks in the study of intervention planning strategies. We have reported a comparative analysis of our Ranking Algorithm based Supervised Learning with three other planning strategies (random, max_degree_static, max_degree_dynamic, see Figures in Appendix section A.6.1) not just for Case 1 but also for Cases 2 and 3. In addition, we have also reported a comprehensive analysis (by varying degrees of initial infected nodes and action budgets) of our Reinforcement Learning-based Centralized Dynamic Planner across various reward models for all three cases. This includes results of static network scenarios and candidate nodes as reward models which are considered in the existing work [citation 14 from our paper].\n\n**Question 2: Scalability**  \nWe have analyzed the performance of the developed planners as the number of opinion network nodes increase. In this context, the GCN-based planners developed in our work perform effectively when applied to larger dynamic network models (as illustrated in Table 1 of our paper and Table 2 of rebuttal PDF) even when they are trained using data generated from smaller networks (RQ4 from Section 5). We would like to emphasize that existing research work [eg., citation 14 from our manuscript] has analyzed opinion networks with a maximum of 1005 nodes with an opinion propagation model that considers only binary opinion and trust values (Case 1 in our paper in Section 2.1). In contrast, we have also considered more expressive opinion network models (Case 2 and Case 3 -- Section 2.1) in our study. \n\n**_Computational Requirements_**\nTo answer the computational requirement for scalability we consider the computational intensity of our algorithms. The Ranking Algorithm (Algorithm 1 in Appendix Section A.5.1) has a time complexity of $O(V \\times (V + E))$, which increases quadratically with the number of vertices and edges, making it computationally intensive for larger networks. The training of DVN (Algorithm 2 in Appendix Section A.5.2) has a time-complexity of $O(e_{\\text{max}} \\times (V \\times (V + E) + nm))$, where $e_{\\text{max}}$​ is the maximum number of episodes, $n$ is the batch size, and $m$ represents the computational complexity of operations within the neural networks. This indicates substantial computational requirements due to the need for simulating network dynamics across multiple training episodes and updating complex neural network parameters repeatedly.\n\nAs specified earlier, we find that GCNs trained with data generated from smaller networks are useful for intervention planning even when size of the opinion network increases. Due to the richness of the dynamic models in Cases 2 and 3, intervention planning with these expressive opinion propagation models were computationally more demanding than Case 1 (which are commonly studied in the literature).", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "MmGnKBteQs", "reviewer": "Reviewer_BrMA", "summary": "This manuscript investigates intervention planning, a critical issue in complex social networks. To address this challenge, the authors introduce a novel ranking algorithm and a reinforcement learning-based dynamic planning framework. Three cases of opinion and trust values are considered, enhancing the method's applicability in real-world scenarios. The performance of the developed method is demonstrated through comprehensive simulations on synthetic networks.", "questions": "Additional questions and suggestions:\n1.\tThe authors mention that intervention planning involves two parts. However, it is not entirely clear how “exerting control” is considered in this work. Does it only mean adding green nodes in Figure 1? Please explain this more clearly.\n2.\tOnly synthetic networks are considered to evaluate the developed method's performance, which could limit its applicability. The authors should consider some real-world networks, even without real-time opinion propagation data, to better demonstrate the method's superiority.\n3.\tIn future work, real-time opinion propagation data should be considered, which could further enhance this study. This is not a critique of the current work but a suggestion for future research.\n4.\tA significant limitation is the size of networks analysed. The largest network contains only 50 nodes, which could impact the evaluation. Real-world networks typically have more than one million users (or at least more than one thousand users). The authors should explain how this method can be applied to real-world scenarios and justify why only networks with fewer than 50 nodes were analysed.\n5.\tWhat is the time complexity of this method? Is it difficult to analyse large-scale networks? Please provide an analysis of its time complexity and the resources needed for simulation.\n6.\tCan this method be compared with SOTA methods?\n7.\tIs Figure 19 correct? All bars and subfigures appear to be the same. Please verify this.", "limitations": "The authors have adequately addressed the limitations of their work.", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "This manuscript is well-written and engaging. The contributions of this work and the rationale for considering different models and factors are clearly illustrated. Various scenarios are accounted for to demonstrate the method's superiority and applicability. Four research questions are reasonably addressed in the manuscript. Overall, this work is novel and interesting and deserves to be published in NeurIPS.", "weaknesses": "One concern I have is the limited size of networks analysed in the simulation. As stated by the authors, “as network size increases, the problem becomes computationally intractable” (Line 6). Why do the authors only consider synthetic networks with fewer than 50 nodes in the simulation? Does this limitation affect the potential application of this method in real-world scenarios? Please provide reasonable explanations for this point. Additionally, the performance should be compared with state-of-the-art (SOTA) methods.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for recognizing our contributions and acknowledging our effort to cover various scenarios in our evaluation. We are also thankful for the constructive feedback and suggestions for future work. \n\nYes, Figure 19 is correct. The barplot displayed in Figure 19 depicts the mean infection rate for different reward functions on Dataset v2 of Degree of connectivity 1, differentiated by the number of initial misinformation sources (Inf.) and action budgets (Act.: a1, a2, a3). We observe an identical average infection rate in all cases because each model successfully stops the spread of misinformation in the first timestep. This can be attributed to low degree of connectivity (one) of the initial infected node.\n\n**Remark 1, Question 2 & Question 4: Scalability and results from real-world networks**  \nIn our work we develop planning algorithms and analyze their efficacy using synthetic data in controlled settings. The synthetic data used in our study are generated using Watts-Strogatz (or small-world) network model. As demonstrated in various studies, this network model captures the characteristics of common social and community networks [1]. Additionally, we have also evaluated our planning algorithms using directed and undirected real-world network models reported in the literature. These evaluations are included in the rebuttal PDF (Table 2). These additional experiments confirm the major findings reported in our paper (Research Questions in Section 5).\n\nFurthermore, the Graph Convolutional Networks (GCNs)-based planners developed in our work perform effectively when applied to larger dynamic network models (as illustrated in Table 1 of our paper and Table 2 of rebuttal PDF) even when they are trained using data generated from smaller networks (RQ4 from Section 5). Testing with GCNs on larger networks is significantly simpler and less computationally intensive than training, making it a practical solution for scaling our approach to handle extensive network sizes efficiently. Finally, we would like to emphasize that existing research work [eg., citation 14 from our manuscript] has analyzed opinion networks with a maximum of 1005 nodes with an opinion propagation model that considers only binary opinion and trust values (Case 1 in our paper in Section 2.1). In contrast, we have also considered more expressive opinion network models (Case 2 and Case 3 -- Section 2.1) in our study. \n\n**Question 1: Details of World Model**  \nIn our work, \"exerting control\" refers to applying interventions (disseminating accurate information) for a selected node within the network to combat misinformation. Specifically, this involves adding an external input to the propagation model. For example, when agent $k$ communicates with agent $i$, agent $i$’s opinion is updated using the propagation model given in Equation 1 from our paper. If the planner chooses agent $i$ as the candidate node for intervention, exerting control implies an external input applied to this model and it will be governed by the following discrete map -\n\n$x_i(t+1) = x_i(t) + \\mu_{ik}(x_k(t) - x_i(t)) + \\mu_{ij}(x_j(t) - x_i(t))$\n\n$\\mu_{ij}$ is the trust value associated with the external source $j$ sharing official information. $x_j$ is the associated opinion value.\n\n**Question 5: Time Complexity Analysis**  \nThe time complexities for the two main components of our method—the Ranking Algorithm and the Deep Value Network (DVN)—are as follows:\n\nRanking Algorithm (Algorithm 1 in Appendix Section A.5.1): The time complexity is $O(V \\times (V + E))$, where $V$ represents the number of vertices and $E$ is the number of edges in the network. \n\nDeep Value Network (DVN) (Algorithm 2 in Appendix Section A.5.2): The overall complexity of the DVN with experience replay is $O(e_{\\text{max}} \\times (V \\times (V + E) + nm))$, where $e_{\\text{max}}$ is the maximum number of episodes, $n$ is the batch size, and $m$ represents the computational complexity of operations within the neural networks. \n\nAnalyzing large-scale networks with our Ranking Algorithm and Deep Value Network (DVN) can be computationally demanding. The quadratic dependency on the number of nodes $V$ and the involvement of all edges $E$ in the Ranking Algorithm, and the complex operations of the DVN, make the method resource-intensive. As specified earlier, we find that GCNs trained with data generated from smaller networks are useful for intervention planning even when the size of the opinion network increases. Due to the richness of the dynamic models in Cases 2 and 3, intervention planning with these expressive opinion propagation models were computationally more demanding than in Case 1 (which are commonly studied in the literature).\n\n**Question 6: Comparison with existing methodologies**  \nMost prior research has only addressed static network scenarios (primarily Case 1), without considering the rich dynamics of Cases 2 and 3 (see Section 2.1 for the distinction). This is the first study to incorporate such a rich dynamic nature of opinion networks in the study of intervention planning strategies. We have reported a comparative analysis of our Ranking Algorithm based Supervised Learning with three other planning strategies (random, max_degree_static, max_degree_dynamic, see Figures in Appendix section A.6.1) not just for Case 1 but also for Cases 2 and 3. In addition, we have also reported a comprehensive analysis (by varying degrees of initial infected nodes and action budgets) of our Reinforcement Learning-based Centralized Dynamic Planner across various reward models for all three cases. This includes results of static network scenarios and candidate nodes as reward models which are considered in the existing work [citation 14 from our paper].", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We thank the reviewers for their valuable feedback and suggestions. In this rebuttal, we have tried to address all the specific concerns and comments of individual reviewers. To support our response to the reviewers' comments, we include two additional tables in the attached PDF that are not part of the manuscript. These tables are intended to reinforce key findings, provide supporting evidence for our claims, and help clarify some of the reviewers' comments addressed in our rebuttal. 

Table 1, in the attached Rebuttal PDF, highlights the key features of our work and their implications, emphasizing aspects not covered in previous studies related to planning in the context of opinion networks available in the literature. Existing research on planning problems in the context of opinion networks often overlooks key features of opinion propagation models, such as their rich network dynamics, asynchronous communication, and the impact of factors like the degree of infected nodes, action budget, and various reward models on the effectiveness of planners. Our work addresses this gap by analyzing three distinct cases of opinion network models (Section 2.1)—ranging from binary to continuous spectra of opinion and trust values—and asynchronous communication (Section 2.2). This approach introduces greater richness and expressiveness to opinion dynamics models, making the model-based analysis of opinion propagation more reflective of real-world scenarios. Furthermore, we develop comprehensive datasets with a wide range of Watts-Strogatz (or small-world) network topologies (Section 4.2), varying degrees of initial infected nodes, action budgets, and reward models—from those requiring local network information to those utilizing global real-time network states (Section 3.2.1). The small-world network topology subsumes some of the common social and community networks as shown in various studies [1]. As a result, experiments on four of the real-world network models (Rebuttal PDF - Table 2) (directed and undirected), previously studied in the literature, align with the major findings reported in our manuscript (Section 5 - Research Questions). This makes our work a significant and non-trivial contribution to the existing literature on this topic.

From an algorithmic perspective, we extend the static ranking algorithm based on the classical Breadth First Search to the problem of centralized planning in dynamic opinion networks. Supporting supervised learning-based planners, this algorithm can help planners when the network size is small and the global network information is available. To understand planning strategies for large-networks, we also investigated reinforcement-learning-based centralized planners. In particular, our study explores five distinct reward structures, enhancing our understanding of reward dynamics and their computability (based on access to local vs global network information) that may be applicable to diverse applications depending on network observability and size. Further, the use of Deep Value Networks (DVN) proves more appropriate for our chosen application compared to the traditional and popular architecture of Deep Q-Networks (DQN), which has been investigated to study planning problems for opinion propagation models with simple binary state and trust parameters. DVN-based planners studied in our work are better suited for environments where the number of intervention nodes (action budget) is not fixed and varies dynamically. This flexibility allows DVN to handle dynamic and large-scale networks with greater efficiency as shown in our work [see citation 14 from the manuscript]. 

[1]. Watts, D.J. and Strogatz, S.H., 1998. Collective dynamics of ‘small-world’networks. nature, 393(6684), pp.440-442.

Meta-review: The paper studies strategic planning for disseminating accurate information within dynamic opinion networks. The authors propose a novel ranking algorithm and a reinforcement learning-based dynamic planning framework. The reviewers recognize the importance of mitigating misinformation spread in social networks and appreciate the thorough analysis of the results and clear paper presentation. The reviewers are unanimously positive after the rebuttal. The authors should incorporate the discussion and new results from the rebuttal into the final version. I recommend acceptance of the paper.
Decision: Accept (poster)