Abstract
Healthcare social networks have a significant role in providing connected and personalized healthcare environment with realtime capabilities. However, building resilient, robust and technologydriven healthcare 5.0 has its own barriers. Especially with social media’s high susceptibility to rumors and fake news, these networks can harm the society. Many researchers have been investigating the process of information diffusion, and it has been one of the most intriguing issues in network analysis. Modeling rumor propagation is one of the prominent researched topics in recent years. Traditional models assume that rumor propagation happens only in one direction, where only supporters are supposed to be active, whereas, in a reallife situation, both supporters and deniers of the information operate simultaneously. In this paper, we introduce a model for the recovery of nodes in a setting where rumor propagation and rumor control happen simultaneously. We propose the SusceptibleInfectedRecoveredAntispreader model based on the notion of spreading of epidemics and also its applications to modeling the propagation of rumors and control of rumor. Our model assumes people have multiple forms of reactions to rumor, either posting it, deleting it or announcing the rumor as fake. This paper also suggests how the model can act as a simulation method to compare two node centrality algorithms where spreaders chosen from one centrality algorithm try to spread the rumor, and the antispreaders chosen from other centrality try to dispel the rumor and vice versa. We simulate the proposed algorithm on different weighted and unweighted realworld network datasets and establish that the experimental results agrees with the proposed model.
1 Introduction
The 5.0 revolution in healthcare is eventually digitally transforming society, plummeting the overload on emergency and critical care, and letting people to reinforce their resilience and health with a faster response time. The use of AI, the Internet of Medical Things (IoMT) and social media is offering number of advantages over traditional analytic and clinical decisionmaking techniques. In this view, many healthcare social networks (HSN) are evolving with the intent to augment patient care and awareness enticing the interest of clinical professionals and of the whole healthcare industry. Technological developments and digital platforms foster information diffusion possibilities to keep people safe, informed and connected. However, the same tools also expose possible perils of misinformation for patients. As the world combats with the outrageous and perilous coronavirus, the pandemic has demonstrated how the spread of misinformation, amplified on social media and other digital platforms, is proving to be as much a threat jeopardizing measures to control the pandemic.
Typically, rumors are information pieces that are unreliable, incorrect and spread on the network due to limited knowledge of the recipient nodes or wrong intentions (Qiu et al. 2021; Spohr 2017; Sivasankari and Vadivu 2021). Rumors are the unverified spread of societal concerns, issues by different methods, and create misunderstandings between individuals. Rumors on the Internet spread quicker and have a more extensive scope of an impact than wordofmouth rumors. These rumors spread by word of mouth over a small area and in vast geographical regions like a forest fire on social networks (Pfeffer et al. 2013), generally with malicious intent. Rumors alter and shape public opinion adversely affecting human behavior, causing multiple confusions (Lai et al. 2020) and instability. Many researchers have been investigating the process of information diffusion, and it has been one of the most intriguing issues in network analysis. Modeling rumor propagation is one of the prominent researched topics in recent years. The SIR (Harko et al. 2014) model is a traditional model that simulates information diffusion due to similarity in the field of epidemiology and social networks. It has been utilized by numerous researchers to model the process of information diffusion in social networks. A susceptible individual spreads data when influenced by a spreader and turns into a spreader. After some time, the spreader quits spreading the data and turns into a recovered node. The transformation parameters \(\beta \) and \(\gamma \) in the process are network characteristicsbased probabilities, and the state transformation of all nodes is equivalent. While the SIR model serves as an excellent benchmark for information propagation, it does not generalize well to rumor propagation. A notable difference arises because individuals may also be immunizing other nodes regarding the rumor and preventing them from getting infected. This phenomenon is very different from actual epidemics modeling because, in an epidemiologic setting, it is unlikely that a large portion of people who recover from the epidemic, make other people immune as well. However, in a social network, when a person propagated fake news intentionally or due to a lack of knowledge and after getting the truth, other persons can disseminate the fact about the misinformation. This makes their susceptible neighbors, directly recovered and immune to the rumor. We consider the above recovery aspects of information propagation in our paper and propose the SusceptibleInfectedRecoveredAntispreader (\({ SIRA}\)) model based on the notion of spreading of epidemics and its applications to modeling the propagation of rumors and control of rumor. Our model assumes people have multiple forms of reactions to rumor, either posting it, deleting it or announcing the rumor as fake. The contribution of our papers are as follows:

We establish mathematical model for the rumor propagation, rumor control and analyzing its equilibrium parameters and reproduction rate.

The proposed model introduces the notion of antispreaders that immune its susceptible neighbors from the rumors to minimize the effect of rumors in the network.

We consider the case where the information is not a rumor, and the proposed model simplifies to information propagation model for the influence maximization.

We discuss the novel approach of directly comparing two different node centrality algorithms for the selection of spreaders and antispreaders. The spreaders, chosen from one centrality algorithm, try to spread the rumor, and the others try to dispel the rumor and vice versa.
The paper organization is as follows. In Sect. 2, we present some traditional models used to study rumor propagation and clarify the enhancements that our model proposes. In Sect. 3, we describe the proposed SIRA model of rumor propagation and rumor control with mathematical modeling using the system of differential equations. This section also explores the analysis of the proposed model on homogeneous networks, calculation of reproduction number and how our model can be simplified to a general information diffusion model as the SEI model followed by the simulation of the model on a toy network. In Sect. 4, we discuss the different weighted and unweighted realworld datasets used in this work, and we prove the mathematical integrity of our model for information propagation, rumor propagation, rumor control and node centrality algorithm performance comparison on various datasets used. Finally, Sect. 5 concludes the paper.
2 Related work
Social networks are complex networks with nontrivial topology and intricate interactions between users. The effective analysis of complex systems is essential to model and comprehend the method of rumor propagation. The spreading of rumors on online social networks is very similar to epidemic spreading. Researchers from various disciplines have extensively studied models of rumor propagation in complex networks. The majority of the conventional models for investigating information diffusion depend on elementary models of epidemics, for example \(SI\) (SusceptibleInfected), \(SIS\) (SusceptibleInfectedSusceptible) and \(SIR\) (SusceptibleInfectedRecovered) (Britton 2010; Kumar et al. 2020).
Tripathy et al. Tripathy et al. (2010) claimed that the effect of rumor can be combated by injecting messages called antirumors, just like the rumor process. They studied two metrics, namely the belief time, which is the duration for which a person believes the rumor to be true, and point of decline, which is a point after which the antirumor process dominates the rumor process. Huang and Jin Huang and Jin (2011) modified the SIR model to prevent the rumor propagation by devising immunization strategies, namely the random immunization and the targeted immunization, to the rumor model on smallworld networks. Wan et al. Wan et al. (2015) introduced the SIERsEs model, which is an extension of the SIR spreading model with the addition of the elimination of the rumor process. Zhu and Ma argue that certain people may not believe in the rumor and do not spread it, a different state called a hesitant state added to the traditional SIR model and proposed SHIR model (Zhu and Ma 2018). This model also takes care of the varying degree of nodes, which commonly happens in online social networks. Indu and Thampi argue that the spread of rumors in social networks is similar to the spread of fire in a forest and identified various components to model the spread of rumors in online social networks (Indu and Thampi 2019). Wang et al. Wang et al. (2019) explored the propagation of rumor using an improved energy model applicable to multilayer social networks. The model also introduces negative energy to simulate the mitigation process of rumors. Zhu et al. Zhu et al. (2020) recently proposed a SIR like an epidemic rumor propagation model by employing forced silence function, time delay and network topology. Recently, Yang et al. Yang et al. (2019) have developed an ILSR rumor propagation model based on the notion of the degree of different nodes in networks. They introduced a new type of users known as lurkers users who have heard the rumors, but temporarily not spreading rumors in the system. Kaur et al. Kaur et al. (2020) proposed a fake news detection system using multilevel voting model by exploiting various feature extraction techniques and machine learning classifiers. Qui et al. Qiu et al. (2021) introduced the SIR rumorspreading model with an influence mechanism in social networks, named SIRIM, by integrating the number of current spreaders into the spreading probability and time function.
3 Proposed methodology
3.1 \(SIRA\) model description
We propose the SusceptibleInfectedRecoveredAntiSpreader (\(SIRA\)) model for the recovery of nodes in a setting where rumor propagation and rumor control happen simultaneously. The entire population is divided into four categories: Susceptible nodes (S) that are unaware of the rumor, Infected nodes (I) that are actively spreading the rumor, Recovered nodes (R) that are not spreading the rumor and antispreader nodes (A) that are dispelling the rumor. We consider people have multiple forms of reactions to rumor, either posting it, deleting it or announcing the rumor as fake. We introduce the notion of antispreader nodes, such nodes that get to know about the irrelevance of the information, and the piece of the information is a rumor, and tend to dispel the rumor by announcing the truth. Hence, the antispreaders nodes are immunizing their susceptible neighbor’s nodes from false information and play a crucial role in controlling the rumor.
As evident from Fig. 1, solid lines represent direct conversion of a node from one state to another, for examples \(S>A\), \(I>R\). However, the dotted line corresponds to indirect conversion from one state to other. For example, the conversion from A to S signifies the indirect effect of antispreaders on their susceptible neighbors. This makes susceptible neighbors transition to the recovered state with the rate \(\beta _2\) and becomes immune to the fake news. The steps of the algorithm are as follows:

Node population starts from certain infected nodes and certain antispreaders nodes that spread the rumor or dispel it, and all other nodes are susceptible

Infected nodes affect their susceptible neighbors and convert them into infected members that propagate the rumor. This conversion happens with a probability \(\beta _1\)

Probability that the infected members recover is at a rate of \(\gamma \)

Under these conditions recovered nodes can now become antispreaders at the rate of \(\delta \)

Nodes that become antispreaders affect \(\beta _2\) fraction of their susceptible neighbors, converting them directly into recovered state and immune to the rumor. Due to the influence of antispreaders nodes, some of their susceptible neighbors can directly go to a recovered state, and they will not be affected by the rumor.
Also, other than the rate of conversion \(\beta _1\) from \(S>I\), the nodes in susceptible state can only transit to an infected state, if they are direct neighbors of an infected node. Similarly, other than the rate of conversion \(\beta _2\) from \(S>R\), the susceptible nodes can directly transition into recovered nodes if they are direct neighbors of antispreader nodes.
At the end of simulation when there are no infected nodes, i.e., \(I(t) = 0\) and equilibrium is reached then some nodes can still remain in susceptible state other then antispreaders. It is not like that the final state is antispreader, and all the nodes should be in this state at the end; rather, it depends on the various conversion rates considered. Also, other than the rate of conversion \(\beta _1\) from \(S>I\), the nodes in susceptible state can only transit to an infected state, if they are direct neighbors of an infected node. Similarly, other than the rate of conversion \(\beta _2\) from \(S>R\), the susceptible nodes can directly transition into recovered nodes if they are direct neighbors of antispreader nodes.
3.2 System of differential equations
The population is divided into the 4 states Susceptible \(S\), antispreaders \(A\), Infected \(I\) and Recovered \(R\) and at any time t their sum is 1 as given in Eq. (1).
As evident from Fig. 1, some recovered nodes may become antispreaders at the rate \(\delta \) as per Eq. (2). Antispreaders are those nodes who know the ingenuity of the information. The node in the antispreader(A) state continues to be in the same state throughout.
Recovered nodes increase through multiple methods, either through the direct recovery of susceptible nodes due to influence of antispreaders (edge between S to R in Fig. 1) or by direct conversion from infected to recovered (edge between I to R in Fig. 1). Some recovered nodes may also become antispreader at the rate \(\delta \). The change in recovered nodes is presented in Eq. (3).
where k is the average degree of the nodes in the network. Change in infected nodes, Eq 4, arises from the conversion of susceptible nodes to infected nodes with the rate \(\beta _1\) and infected nodes go to recovered state with the rate \(\gamma \).
Initially, all the nodes are assumed susceptible except few seed nodes, and slowly, they decay into infected state with rate \(\beta _1\) and recovered state with rate \(\beta _2\) as given in Eq. (5). Now let us solve these equations by substituting \(R(t)\) with \(1  S(t)  A(t)  I(t)\) from Eq. (3), and hence, getting the system given by 6, 7, 8.
3.3 Calculation of reproduction number
The basic reproduction number (\(R_0\)) is an important parameter in epidemics which measures the propagation potential of contagious diseases. The reproduction number (\(R_0\)) is an epidemiologic measure used to characterize the contagiousness or transmissibility of infectious agents. We have applied a similar concept for the rumor propagation. Since for successful reproduction of the rumor, we require that initially \(\frac{d}{dt}I(t)>0\), this produces Eq. (9)
Thus, giving us Eq. (10),
\(R_t >1\) means that the rumor is active in the system. For the initial spread of rumor, \(R_0 > 1\), and since \(S(0) \cong 1\), we get Eq. (11)
Now, when \(R_t < 1\), there is no rumor in the system. When \(R_t < 1\) then \(\frac{d}{dt}I(t)=0\). This implies there is no infected node in the system, and hence, there is no rumor in the system.
3.4 Model analysis in a homogeneous network
We analyze the proposed \(SIRA\) model on a homogeneous network and prove the validity of the model at the stable equilibrium points. An equilibrium point of a dynamic mathematical framework produced by an independent system of ordinary differential equation (ODEs) is a solution point where the count of participating entities does not change with time, and hence, their rate of change becomes 0 (Roussel 2005). In the case of epidemics, the diseasefree equilibrium is asymptotically stable, and the infection asymptotically vanishes. Similarly, in the case of information propagation, rumorfree stable equilibrium is stable, and information is no longer being spread, and the network has no active rumor propagating nodes. We prove the stability of our differential equation at the equilibrium condition given by Eq. (12)
The Jacobian method (Izhikevich 2007), as given by Eq. (13), is a popular iterative technique used for determining the solutions for the system of linear equations, which is diagonally dominant.
The equilibrium points exist if I(t) = 0, and under this condition, we have Eq. (14)
The Jacobian matrix is given by Eq. (15):
We will calculate the eigenvalues of \(J_E\) as given in Eq. (16):
Expanding on the third row, we arrive at the following characteristic equation in Eq. (17):
As evident, \((\gamma , 0)\) are eigenvalues and also the solutions of \(\lambda ^2 + (\delta  \beta _2 kA)\lambda  \beta _2 k \delta (A + S)\) which we find and analyze as follows using the Quadratic Formula:
Here, since \((\delta + \beta _2 kA)^2 \ge 0 \because (x^2 > 0, \forall x \in R)\) and also \(S, k, \beta _2, \delta \ge 0\), we have Eq. (19):
Thus, the eigenvalues are given by Eq. (20):
Now, for a stable equilibrium, real eigenvalues must be \(\le 0\), and since \(\gamma , \delta , \beta _2 > 0\) and \(S, I, R, A \ge 0\), we know that eigenvalues satisfy these properties in the assumptions of our system for equilibrium. The first eigenvalue is 0 and \(\gamma \) and \((\delta  \beta _2 kA)  \sqrt{D})/2\) are \(\le 0\) as given by Eq. (21).
Thus, rearranging and squaring both sides, we get to Eq. (22):
Hence, Eq. (22) holds true \(\forall A, S \ge 0\), and at this point, the mathematical stability of the system is proved quantitatively by finding the eigenvector related to every eigenvalue. An equilibrium point is hyperbolic if none of the eigenvalues have zero real part. One of our eigenvalues is zero. Thus, the equilibrium is nonhyperbolic. If all eigenvalues have a negative real part, the equilibrium is steady, as given by the Lyapunov stability theorem (Lyapunov 1992). As evident in Eqs. (20–22), the corresponding eigenvalues are \(\le 0\), and the equilibrium for our model is stable.
3.5 The simplification to information propagation SEI model
When the information is nonrumor, the model can directly be simplified to an information propagation model that leads to influence maximization achieved through antispreaders.
The roles of the nodes, however, remain almost the same. Antispreaders are still the carriers of correct information and the recovered nodes still are the dormant nodes that know about the information but are not spreading it. Similarly, susceptible nodes still remain the nodes that are unaware of the information and can likely be turned into recovered nodes by antispreaders. This effect can be seen when we plug in \(I(t) = 0, \frac{d}{dt}I(t) = 0\) into the model (Eqs. 1, 2, 3 and 5) and we get the following system of equations, namely Eqs. (2324, 25 and 26).
For the purpose of an analogy between these states, we rename the states for direct correspondence with the SEI model as follows:

1.
Antispreader nodes (A) become source spreaders or seed nodes (I)

2.
Recovered nodes (R) are renamed as exposed nodes (E)

3.
Susceptible nodes remain in the same naming convention (S)
These seed nodes (I) affect \(\beta _2\) fraction of their neighbors, and they convert into exposed (E). Exposed users can become spreaders of the information with the rate \(\delta \).
Thus, the proposed \(SIRA\) model for the rumor propagation and rumor control simplifies to information spreading \(SEI\) (Guihua and Zhen 2004) model for influence maximization. This conversion is depicted in Fig. 2
3.6 Simulation of the proposed model on a toy network
A toy network, as shown in Fig. 3, is chosen as a simulation network. The most influential node in the network is chosen as the spreader, and \(\beta _1 = 0.5, \beta _2 = 1, \gamma = 1, \delta = 1\) are assumed without loss of generality. Initially, all the nodes, except the one chosen as the influential node based on outdegree, are susceptible, and the given node is in an Infected state. As time progresses, random neighbors of this node become infected, and the node recovers. After recovery, it is dormant, and other nodes continue spreading information. The recovered node decides to dispel the rumor and makes itself an antispreader. This effect makes all its susceptible neighbors instantly recover. The spreading continues till all the nodes either are against the rumor or have never heard of the rumor or its dispel. We describe the spread in Fig. 3 as a change of state in an eventdriven simulation.
The change of state in an eventdriven simulation are as follows:
t1: Node F is infected, chosen because of maximum outdegree, all other nodes are currently susceptible.
t2: Node F has recovered and has infected its neighbors with a uniform probability of \(\beta _1 = 0.5\), thus randomly infecting nodes E and J.
t3: Node F is now an antispreader, all the infected nodes recover at the rate of \(\gamma = 1\), and their neighbors are infected at the rate \(\beta _1\).
t4: Node F has recovered its susceptible neighbor H at rate \(\beta _2\), all recovered nodes became antispreaders at the rate of \(\delta = 1\).
t5: Antispreaders are beginning to take more portion of the network. However, there is still an infection going on in the network.
t6: No neighbors of the current infected nodes are susceptible.
t7: All reachable nodes in the network have seen the information. However, there is still a node G, which has never heard of the rumor or antirumor.
t8: All nodes that saw the rumor or the truth are now antispreaders. Other nodes like G are still susceptible. After the equilibrium point, either all nodes will end as susceptible, or antispreaders and no nodes are spreading the rumor if the parameters \(\beta _1, \beta _2, \gamma , \delta \) are coherent to the reproduction number.
In the above toy network simulation in Fig. 3, the value of various probabilities considered is \(\beta _1 = 0.5, \beta _2 = 1, \gamma = 1, \delta = 1\), which are relatively very high values. The conversion rate from \(S>I\) is 0.5%, i.e., an infected node can randomly infect 50% of its susceptible neighbors. Similarly, each of the conversion rates from \(I>R\), \(S>R\) and \(R>A\) is 1. It implies the probability of conversion from susceptible to recovered, infected to recovered and recovered to antispreaders is 100%. At the end of the simulation, all the nodes are in the antispreader state except one node in the susceptible state. The node ‘G’ is nonreachable from all other nodes, and hence, it is not influenced by any nodes during the simulation process and its color remains unchanged.
4 Result and analysis
In this section, we simulate our proposed methodology and analyze the different findings related to rumor propagation and rumor control using realworld various weighted and unweighted networks given in Subsection 4.1. Table 1 depicts the brief description used datasets in this paper. We use the spreading characteristics defined by \(F(t)\) giving the count of the susceptible, infected, antispreaders and recovered nodes as a function of time. Infection scale, i.e., F(t), is an important evaluation parameter. When information propagates in a network then at any time t, F(t) denotes the fraction of nodes in the network which have received the particular information. A higher value of F(t) indicates that information spreads rapidly in the network. Section 4.2 discusses the propagation of nonrumors, similar to the traditional \(SEI\) information propagation model, using the proposed \(SIRA\) model. Section 4.3 discovers the role of antispreaders in containing the harmful effects of the rumor. In section 4.4, a novel method for comparison of centralities is proposed. Here, we compare the recently introduced NCVoteRank (Kumar and Panda 2020) centrality algorithm for unweighted networks and WVoteRank Kumar and Panda (2022) centrality on weighted networks for finding the initial spreaders. These centrality algorithms are simulated in comparison to traditional algorithms, DegreeRank Freeman (1978) for unweighted networks, and its weighted version WDegreeRank for weighted networks.
4.1 Datasets used
4.2 Information propagation on realworld networks
As highlighted in Section 3.5, the proposed algorithm reduces to a SusceptibleExposedInfected (\(SEI\)) model where susceptible nodes remain susceptible (S), recovered nodes act as exposed (E) nodes and antispreader nodes are the carrier of the correct information and act as infected (I) nodes. The information diffuses based on two parameters, namely the rate of infection as \(\beta _2\) and rate of recovery as \(\delta \), leading to a simplified model of information propagation. As evident, the model has similar characteristics to those of the \(SEI\) model with similar parameters and has expected behavior on the realworld weighted and unweighted networks, as shown in Fig. 4. We use NCVoteRank centrality (Kumar and Panda 2020) and to identify one top spreader as infected node on the unweighted networks like Brightkite and EmailEnron. Similarly, we utilize WVoteRank centrality (Kumar and Panda 2022) to identify one top spreader as infected node to start the information diffusion process on the weighted networks like USAirports and PowerGrid. Notice that all the graphs behave similarly to the \(SEI\) epidemic model. As evident from Fig. 4a, the spreader on weighted network USAirports is able to infect around \(25\%\) (\(750\) nodes of the network receive the information at the end of the simulation, out of the total population of \(3000\)) of the network. In Fig. 4b, the spreader on Brightkite is able to infect around 50% of the network and ended up having a stable equilibrium. In Fig. 4c, the spreader on EmailEnron is able to infect more than \(50\%\) of the network, notice that the peak reaches an early time stamp due to high connectivity. On the weighted PowerGrid network, notice that the change in the numbers is gradual, and very few nodes eventually become exposed as shown in Fig. 4d.
4.3 Numerical simulations for basic reproduction number and the equilibrium point
In this section, we discuss the equilibrium characteristics of the proposed method near the rumorfree equilibrium point, along with endemic equilibrium characteristics and the role of antispreaders nodes to minimize the effect of rumor on various network datasets. On reaching equilibrium on realworld networks, the rumor vanishes in the network only after the complete network has no more rumorspreading nodes. The equilibrium is stable, only if the conditions are such that \(R_t < 1\), as given by calculation in Eq. (10). We assume the values of the parameters \(\beta _1, \beta _2\), and \(\gamma \) based on the reproduction number and network topology. The value of \(\delta \) shall indicate how likely an informed (recovered) node is to dispel the rumor by becoming an antispreader. The value of \(\delta \) depends on the preference of the given group and the severity of the information. As evident from Fig. 5a, the spreaders on weighted network USAirports can infect around \(35\%\) of the network in the beginning, whereas around \(50\%\) nodes of the network recovers and becomes antispreader during the simulation and minimizing the spread of the rumor. In Fig. 5b, the spreader on Brightkite is able to infect around \(15\%\) of the network, and about \(20\%\) of the nodes became antispreaders. Due to the influence of antispreader, approximately \(80\%\) of the nodes remains susceptible means unaffected by the rumor. Hence, the roles of antispreaders nodes become vital to confine the rumor in the system. In Fig. 5c, the spreader on EmailEnron is able to infect about \(32\%\) of the network initially, and \(43\%\) nodes are antispreaders. It is clear that due to the impact of antispreaders about \(70\%\) of nodes remains susceptible. On the weighted PowerGrid network, the spreader is able to infect about \(15\%\) of the network initially, and \(22\%\) nodes are antispreaders, which enables approximately \(70\%\) nodes are susceptible and are unaffected by the rumor as depicted in Fig. 5d. Similarly, in the case of GrQc and p2p Guntella08 networks, as evident from Fig. 5e and f, respectively, the antispreader nodes are able to control the rumor, and most of the nodes remain susceptible in the system after reaching the equilibrium.
Here, in case of the results obtained on reallife network datasets in Fig. 5, results are obtained over 100 independent simulation of SIRA model with the value of various probabilities as \(\beta _1 = 0.1, \beta _2 = 0.1, \gamma = 0.2, \delta = 0.9\). The probability of conversion from \(S>I\) is 0.1, i.e., an infected node can randomly infect 10% of its susceptible neighbors. The probability of conversion \(S>I\) is 0.1, i.e., an infected node can randomly infect 10% of its susceptible neighbors. The probability of conversion from \(S>R\) is 0.1, which implies due to the influence of antispreader nodes, their 10% susceptible neighbors go directly in recovered state. Similarly, the rate of conversion from infected state to recovered state is 20%, and the rate of conversion from recovered state to antispreader state is 90%. Hence, in this case, we have performed the simulation on relatively lower conversion rates as compared to the simulation performed in the toy network as shown in Fig. 3. This results in a significant number of nodes that are in the susceptible state apart from in antispreader state when the equilibrium is reached in the system. The increase in the value of the conversion rate, especially \(\beta _1\) for \(S>I\) and \(\beta _2\) for \(S>R\), results in less number of susceptible nodes at the end of the simulation when equilibrium is reached. Also, many reallife networks like USAirports, Brightkite and social networks are sparse networks where many nodes exist with relatively less degree. Here, degree means the number of direct neighbors or number of friends in social networks, and some of such nodes may not come in contact with either infected or antispreader nodes in the system and remains in the susceptible state or unaware of the information.
The endemic equilibrium state is the state where the disease or rumor cannot be totally eradicated but remains in the population (Ugwa et al. 2013). For the disease or rumor to persist in the population, the antispreader class, the susceptible class, the infectious class and the recovered class must not be zero at the equilibrium state, i.e., (S, I, R, A) \(\ne \) (0, 0, 0, 0). In this case the equilibrium is stable, only if the conditions are such that \(R_t > 1\), as given by calculation in Eq. (10). The results obtained in Fig. 6 on the various used datasets confirm that there exit an endemic equilibrium point and its stability in the system.
4.4 Spreader and antispreader seeding: comparing centrality algorithms
The proposed \(SIRA\) model can compare two different node centrality algorithms for evaluating the spread and control of information to model the reallife situation. In the framework, we can change the comparing centrality algorithms. Node centrality methods intend to assign some score to each node of the network based on its significance and different parameters (Kumar et al. 2022a, b) In this case, spreaders are chosen from one centrality algorithm spread the rumor, and the antispreaders are chosen from the second algorithm dispel the rumor and vice versa. We apply \(F(t)\), the infected scale, as a function of time, as the performance measure for this comparison. We employ the top 10 selected spreaders and top 10 selected antispreaders using the specified centrality algorithm for performing the comparison study. We intend to develop a generic framework where any two node centrality can be taken to compare the performance. In the case of unweighted networks, we consider NCVoteRank and Degree centrality as the choice of node centrality algorithms. However, in the case of the weighted networks, WVoteRank and WDegree centrality is viewed as the choice of node centrality algorithms. During the simulation, due to a nonzero value of \(\delta \), the count of antispreaders increases, and also newly became antispreaders can immune their neighbors as per the dynamics of the model.
The obtained results of comparing centrality is depicted in Fig. 7 for various considered datasets by using the parameters \(\beta _1 = 0.4, \beta _2 = 0.3, \gamma = 0.3, \delta = 0.5\). From the results, it can be inferred that initially, the count of infected nodes increases due to the influence of selected seed nodes. However, shortly the initial seed and the converted antispreader nodes tend to slow down the conversion of susceptible nodes to the infected node and rapidly decrease the count of spreaders by converting them to recovered nodes based on the parameters considered. PowerGrid network is a weighted network, and we use WVoteRank centrality and WDegreeRank centrality for the selection of spreaders and antispreaders, respectively, and vice versa. WVoteRank as the choice for spreaders and WDegreeRank as the choice for antispreaders can get \(350\%\) more infected users than WDegreeRank as the method for selecting spreaders and WVoteRank for the antispreaders, as evident in Fig. 7d. CAGrQc is an unweighted network, and hence, NCVoteRank and DegreeRank are applicable for comparison. NCVoteRank as the choice for spreaders and DegreeRank as the choice for antispreaders are able to get \(80\%\) more infected users than DegreeRank as the method for the selection of spreaders and NCVoteRank for the antispreaders, as evident in Fig. 7e. Hence, we can conclude that NCVoteRank, on unweighted networks, and WVoteRank, on weighted networks, act as better node selection algorithms at both the spreading, and the control of the rumor and fake news as compared to degree on the unweighted and weighted degree on the weighted networks.
5 Conclusion
The healthcare social networks present potential risks for patients due to the possible distribution of poor quality or wrong information. In this paper, we proposed the \(SIRA\) rumor propagation model to investigate the process of rumor propagation and control in healthcare social networks. We added a new state to the model called the antispreader that spread the truth, to prevent the harmful effects of the rumor. Considering the variety in people’s reactions to rumors, we divided nodes into four categories, namely susceptible, antispreaders, recovered and infected. We modeled the system of differential equations mathematically and determined the reproduction number to find the equilibrium point and also confirmed the consistency of the proposed model on realworld datasets. We generalized our model into an information propagation model similar to the traditional models for modeling information diffusion to achieve influence maximization. In both weighted and unweighted networks, we compared two different centrality algorithms: one for choosing the spreaders and the other for choosing the antispreaders and vice versa and the amount of information spread. The studies showed that our model is coherent with the spread of rumors in social network topology, and our proposal gives a great reference to simulating and controlling the spread of rumors and fake news on the network. The proposed model is not applicable for heterogeneous networks. In the near future, we can extend our model to heterogeneous networks.
Data availability
Enquiries about data availability should be directed to the authors.
References
Britton T (2010) Stochastic epidemic models: a survey. Math Biosci 225(1):24–35
Cho E, Myers SA, Leskovec J (2011) Friendship and mobility: friendship and mobility: user movement in locationbased social networks. In: ACM SIGKDD international conference on knowledge discovery and data mining (KDD)
Colizza V, PastorSatorras R, Vespignani A (2007) Reactiondiffusion processes and metapopulation models in heterogeneous networks. Nat Phys 3:276–282
Freeman LC (1978) Centrality in social networks conceptual clarification. Soc Netw 1(3):215–239
Guihua L, Zhen J (2004) Global stability of an \(SEI\) epidemic model. Chaos, Solitons Fractals 21(4):925–931
Harko T, Lobo FSN, Mak MK (2014) Exact analytical solutions of the SusceptibleInfectedRecovered (SIR) epidemic model and the SIR model with equal death and birth rates. Appl Math Comput 236(2014):184–194
Huang J, Jin X (2011) Preventing rumor spreading on smallworld networks. J Syst Sci Complex 24(3):449–456
Indu V, Thampi SM (2019) The natureinspired approach based on the forest fire model for modeling rumor propagation in social networks. J Netw Comput Appl 125:28–41
Izhikevich EM (2007) Equilibrium. Scholarpedia 2(10):2014
Kaur S, Kumar P, Kumaraguru P (2020) Automating fake news detection system using multilevel voting model. Soft Comput 24(12):9049–9069
Kijkuit B, Ende J (2007) The organizational life of an idea: integrating social network, creativity, and decisionmaking perspectives. J Manage Stud 44(6):863–882
Kumar S, Panda BS (2020) Identifying influential nodes in social networks: neighborhood coreness based voting approach. Stat Mech Appl, Phys A 553:124215
Kumar S, Panda A (2022) Identifying influential nodes in weighted complex networks using an improved WVoteRank approach. Appl Intell 52:1838–1852
Kumar S, Saini M, Goel M, Aggarwal N (2020) Modeling information diffusion in online social networks using SEI epidemic model. Procedia Comput Sci 171:672–678
Kumar S, Mallik A, Khetarpal A, Panda BS (2022) Influence maximization in social networks using graph embedding and graph neural network. Inf Sci. https://doi.org/10.1016/j.ins.2022.06.075
Kumar S, Gupta A, Khatri I (2022) CSR: A community based spreaders ranking algorithm for influence maximization in social networks. World Wide Web, 1–20
Lai K, Xiong X, Jiang X, Sun M, He L (2020) Who falls for rumor? Influence of personality traits on false rumor belief. Pers Individ Differ 152:109520
Leskovec J, Kleinberg J, Faloutsos C (2007) Graph evolution: densification and shrinking diameters. ACM Trans Knowl Discov Data (ACM TKDD) 1(1):2
Leskovec J, Lang K, Dasgupta A, Mahoney M (2009) Community structure in large networks: natural cluster sizes and the absence of large welldefined clusters. Internet Math 6(1):29–123
Lyapunov AM (1992) The general problem of the stability of motion. Int J Control 55(3):531–534
O’Keeffe GS, ClarkePearson K (2011) The impact of social media on children, adolescents, and families. Pediatrics 127(4):800–804
Pfeffer J, Zorbach T, Carley KM (2013) Understanding online firestorms: negative wordofmouth dynamics in social media networks. J Mark Commun 20(1–2):117–128
Qiu L, Jia W, Niu W, Zhang M, Liu S (2021) SIRIM: SIR rumor spreading model with influence mechanism in social networks. Soft Comput 25(22):13949–13958
Ripeanu M, Foster I, Iamnitchi A (2002) Mapping the gnutella network: properties of largescale peertopeer systems and implications for system design. arXiv preprint cs/0209028
Roussel MR (2005) Nonlinear Dynamics, lecture notes. Stability analysis for ODEs. University Hall, Canada
Sivasankari S, Vadivu G (2021) Tracing the fake news propagation path using social network analysis. Soft Comput 19
Spohr D (2017) Fake news and ideological polarization. Bus Inf Rev 34(3):150–160
Tripathy RM, Bagchi A, Mehta S (2010) A study of rumor control strategies on social networks. In: Proceedings of the 19th ACM International Conference on Information and Knowledge Management  CIKM ’10. Presented at the 19th ACM international conference, 1817–1820
Ugwa KA, Agwu IA, Akuagwu NA (2013) Mathematical analysis of the endemic equilibrium of the transmission dynamics of tuberculosis. Int J Technol Enhanc Emerg Eng Res 2(12):263–9
Wan YP, Zhang DG, Ren QH (2015) Propagation and inhibition of online rumor with considering the rumor elimination process. Acta Phys Sin 64(24):240501
Wang C, Wang G, Luo X, Li H (2019) Modeling rumor propagation and mitigation across multiple social networks. Phys A 535:122240
Watts DJ, Strogatz SH (1998) Collective dynamics of smallworld networks. Nature 393:440–442
Yang A, Huang X, Cai X, Zhu X, Lu L (2019) ILSR rumor spreading model with degree in a complex network. Phys A 531:121807
Zhu H, Ma J (2018) Analysis of SHIR rumor propagation in random heterogeneous networks with dynamic friendships. Phys A 513:257–271
Zhu L, Liu W, Zhang Z (2020) Delay differential equations modeling of rumor propagation in both homogeneous and heterogeneous networks with a forced silence function. Appl Math Comput 370:124925
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Kumar, A., Aggarwal, N. & Kumar, S. SIRA: a model for propagation and rumor control with epidemic spreading and immunization for healthcare 5.0. Soft Comput 27, 4307–4320 (2023). https://doi.org/10.1007/s0050002207397x
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DOI: https://doi.org/10.1007/s0050002207397x
Keywords
 Complex network
 Healthcare social networks
 Information diffusion
 Node centrality
 Rumor control
 Rumor propagation