2.1. Information deployment

In social networks, communities’ formation and detection depend on several factors like network attributes, temporal characteristics, and social attributes. The data stream in the organization influences every one of these variables contrastingly at various levels. In this manner, it is fascinating to see what the data stream means for local area recognition and informal communities. The elements influencing the informal communities can be extensively delegated (Das & Biswas, Reference Das and Biswas2021b):

1. 1. Network Properties. Topological and structural characteristics of the network.

   • • Edge Strength Measures. These measures, like connection strength, value strength, and tie strength, are used to assess the relationship between two vertices that share a common edge. The strength, however, can be measured by computing the number and the type of vertices impacted by the diffusion process.

   • • Belongingness Measures. For any vertex v, the belongingness in a specific local area c is controlled by having a place degree or having a place factor. Thinking about the viewpoint of data dissemination, the belongingness strength can be estimated by deciding the kind of vertices impacted during dispersion.

   • • Centrality Measures. The increasing popularity of social networks has led to an increase in the amount of information that is generated and made available to users. However, the ideas absorbed by the individuals highly depend on certain influential entities that can activate the inactive vertices through their influence. Such entities control the propagation of information in the network to a great extent and enhance the rate of diffusion. Some of the measures for identifying influential entities include betweenness centrality, degree centrality, and closeness centrality.

   • • Clustering Coefficient. During the time spent on data dissemination, the data is proliferated across the organization through the innate vertices that are available in the base conceivable length. Simultaneously, some topological properties of the organization are taken advantage of during the dissemination interaction. One such property is the grouping coefficient, which decides the degree to which the vertices bunch together and contribute fundamentally to local area identification.

   • • Structural Equivalence. Part of data is accessible in interpersonal organizations nowadays, yet while getting the data, the dependability of the data assumes a significant part for the client. Here impact makes an effect, that is, the data spread directed by the impact is more dependable and arrives at a bigger arrangement of vertices. Such an impact can be addressed by data not really set in stone from client credits, client demands, and other pertinent elements. The features that can measure the information diffusion probability by utilizing appropriate similarity/dissimilarity evaluation techniques. Thus, these methods are used to evaluate the structural equivalence. Structural equivalence measures include the Jaccard coefficient, cosine similarity, and Euclidean distance.

   • • Boundary Vertex. Users in social networks connect with other users who are either similar or known to them. Hence, the information propagation in the network is either controlled by the similarity or by the topology. In terms of topology, the extent to which the neighboring nodes tend to connect. Concerning similarity, it is the user’s attribute, interest, or characteristic that controls the connections. The diffusion of information in the network sometimes affects a pair of vertices differently as they may be exposed to two types of information that are different from each other. Such neighboring vertices of a vertex are referred to as boundary vertex.

2. 2. Social Facets. It quantifies the various aspects of social interaction.

   • • Contagion. In social networks, following a particular trend, is a usual practice. Hence, any activity of a user is affected by the neighbors’ actions. The inactive vertices get activated by the neighboring vertices during the information diffusion process. This influence of neighbors on the inactive vertex is called contagion. A higher degree of the vertex indicates higher chances of getting affected by the information propagated by the neighbors and vice versa.

   • • Common Neighbors. Formation of groups based on some similarity is a common characteristic of social networks. These groups exchange information during the process of diffusion. It is observed that there is a strong relationship among those users who have a huge number of common neighbors. Therefore, a dense sub-graph can depict the diffusion outcomes in such cases, and the community detection algorithms can utilize this facet to recognize inherent communities.

   • • Topic. Interest in the group and the dynamics of the diffusion network highly affect the speed at which the diffusion of a topic happens in a social network. For example, a politically inclined group will be affected more by a political campaign and will not be bothered about the advertisement of trekking equipment. A high diffusion speed indicates a higher affinity for that topic in the underlying network.

   • • Homophily. The users in social networks tend to connect with like-minded or similar users. This similarity is termed homophily which is exploited during the information diffusion process. The interaction pattern between similar users can be modeled through the independent cascade model. It is assumed that the diffusion initiated based on homophily affects many homophilic connections.

   • • Social Interaction Channels. Nowadays, users spend quite some time on social networks to interact with their peers and friends. Much information exchange happens on social networks, which opens up a kind of interaction channel through which people staying at physically different locations can also interact. This enhances and broadens the information diffusion process and increases the efficiency of the interaction channels.

   • • Influential Spreaders. Certain entities are popular enough and have one too many relations in the network. Such users are referred to as influential spreaders as they can initiate information diffusion in social networks by activating inactive users. However, the strength of information propagation depends upon the data (whether it meets the user’s requirement) and the popularity of the spreader in the network. The higher the impact of the influential spreader, the higher the probability of diffusion.

3.1. Information diffusion vulnerabilities

Information diffusion models can face many vulnerabilities if the common weaknesses are uncontrolled. As with the consistent advancement of networks over organizations, social media, etc., online protection has become a significant subject. Vulnerabilities of Information Diffusion mainly depend upon the model applied for link prediction. Some of the common vulnerabilities of Information Diffusion models are discussed below:

• • Node Paralysis. A lot of exploration is directed at node paralysis of motion in various models proposed in state-of-the-art approaches (Razaque et al., Reference Razaque, Rizvi, Almiani and Al Rahayfeh2019). A one-dimensional model begins with a random initial state. The intermittent limit conditions are dependent on different models. The likelihood of the resistance is 0.5, a big part of the endorsement and half resistance. Under various boundaries, the likelihood of developing the last attractors of ferromagnetism is likewise unique. The likelihood that the attractor gives up at a similar starting thickness fluctuates with the collection file. Under a similar collection file, the more noteworthy the likelihood of a similar beginning thickness keeps an eye on the attractor when the association likelihood becomes bigger; the expansion of the association likelihood will make the framework bound to look steady. When the accumulation record expands, something similar starting thickness will generally be more noteworthy in the likelihood of the attractor (i.e., the framework is bound to look reliable because of the expansion in the number of short-range associations). Such different circumstances prompt Vulnerabilities.

• • Continuously Changing.: The organization is in an implicit, persistent way. The method involved with adding additional associations proceeds until the number of huge distance affiliations and the number of short distance affiliations (i.e., affiliation probability) shows up at a particular worth. Like this, the typical neighbor number of each center point in the little world organization is $4(1 + x)$ . At the point when the amassing record (T) reaches 0, the theory of the little world organization has transformed into a little world association. Regardless, with T1, there are incredibly close center points. Such extraordinary minimal world associations are relative to standard associations. At the point when 0 $\le $ T $\le$ 3, because of the characteristics of extra-critical distance affiliations, this reach has rich characteristics. In any case, it creates a shortcoming that can be the justification for the attack (Razaque et al., Reference Razaque, Rizvi, Almiani and Al Rahayfeh2019).

• • Instability. Instability in the model significantly reduces the efficiency and performance of the model. Given the Instability of network geography, deciding the impact of joining and leaving the host is essential, particularly in a P2P organization. To more readily comprehend the effect of entering and leaving on the transmission of dynamic worms, the proportion of the pestilence and irresistible infection boundaries ought to be utilized. The aftereffects of several models show that the expansion and flight of hosts significantly affect the notoriety and spread execution of the topology-aware dynamic worms (Razaque et al., Reference Razaque, Rizvi, Almiani and Al Rahayfeh2019).

• • Harder Node Evaluation. To control the diffusion of viruses, certain models combine the minimum traffic with the time Markov chain features. One such model is the SIR model, wherein the infection and its removal can be described through this vulnerability. The traffic flow is affected in the congestion and free flow phases by the viral transmission, and to assess its dynamics, models like SIR may be used. The goal is to identify the effect on the traffic conditions and dynamics of the system due to virus diffusion, that is, in the congestion and free flow stages (Razaque et al., Reference Razaque, Rizvi, Almiani and Al Rahayfeh2019).

• • Limited Size. In social networks, limited size is one of the vulnerabilities. It can be studied considering the network construction. The complex process of the experimental methods leads to large size. Also, the data sets that are compiled are of varying and considerable sizes. The data that is obtained from multiple sources have sufficient subjective biases which is not compatible with existing social network size. However, this issue can be tackled by using the affiliation network as performed in the SIR model to study the state of the nodes in the network.

• • Data Reduction. In Serban et al. (Reference Serban, Sordoni, Bengio, Courville and Pineau2016), the authors have suggested that, in general, the distance between the source of every estimator and its closest tree network is constant. The number of sources can be estimated using the proposed algorithm, even if the real and existing sources are unknown. Collisions between the wireless intermediate nodes and transmission signals lead to a reduction in the throughput of end-to-end data transmission. Therefore, in certain models like SIR, there are increased chances of inviting vulnerabilities due to this reduction.

• • Abhorrent Types. It is difficult to assess the authenticity and truth value of the information flowing through social networks. When the information arrives at a node, it is the choice of the netizen whether he/she wants to forward it or not. Similarly, sending real or distorted information to the disseminator is also the choice of the netizen. The multi-agent and game theory concepts can well describe both situations, and it is utilized in the background to study the information dissemination in the networks. The model of Anshuman et al. (Chhabra et al., Reference Chhabra, Vashishth and Sharma2017) can be explored to identify the communication behavior of online users connected through social networks.

• • Independence of Values. Based on the user’s awareness of security, their behavior is affected. The quantifying factors for these behavioral changes have been analyzed by studying the propagation mechanism of the worms in the social networks (Seddiki & Frikha, Reference Seddiki and Frikha2012). The information transmission for multi-layer online networks follows the mechanism of information dissemination in social networks. This model is based on the idea of the evolutionary game model that is used in the information diffusion process established between government bodies and netizens. For example, the strategies executed by the government to control the harmful network group events that negatively impact the social order have been proven to be very effective.

4.1. Influence maximization

Identification of influential word-of-mouth promoters is a challenge in influence maximization (IM), which aims to increase product uptake. Kempe et al. (Reference Kempe, Kleinberg and Tardos2003) was the first to present an optimization model for the IM problem under the linear threshold and independent cascade models of classical diffusion. They have proposed a simulation-based greedy solution to identify seed users with an approximation guarantee $1-1/e-\epsilon$ . The greedy algorithm utilizes the information diffusion capacity of each individual in the network to evaluate the user’s influence spread. The algorithm finds influential seeds iteratively to avoid overlapping avoidance. The main limitation of the greedy solution is its time efficiency due to the time-consuming Monte Carlo (MC) simulation. It is because of the stochastic nature of the diffusion process that a new set of users is activated at each iteration. The influence maximization algorithms have been categorized based on the context of the product, and network (Singh et al., Reference Singh, Srivastva, Verma and Singh2021) as shown in Table 2. The basic framework of an influence maximization algorithm is depicted in Figure 3 (Arora et al., Reference Arora, Galhotra and Ranu2017; Singh et al., Reference Singh, Srivastva, Verma and Singh2021). The first component experimental setup of the IM framework configures the diffusion process with influence and activation probabilities. The seed selection process identifies the seed users corresponding to the seeding strategy and diffusion model. Finally, the algorithm’s performance is evaluated regarding influence spread and efficiency. The characteristic summary of the influence maximization method is present in Table 2.

Table 3. The comparison of influence maximization algorithm based on information dissemination model (Singh et al., Reference Singh, Srivastva, Verma and Singh2021)

Figure 3. The influence maximization framework under information diffusion model (Singh et al., Reference Singh, Srivastva, Verma and Singh2021)

4.1.6. Profit maximization

Profit maximization (PM) extends the classical IM problem, which considers actual product adoption rather than social influence. Profit maximization integrates the costs of advertisers and network service providers. There are some efforts (Lu & Lakshmanan, Reference Lu and Lakshmanan2012; Tang et al., Reference Tang, Tang and Yuan2017; Li et al. Reference Li, Liu, Chen, Qu, Fang and Ko2017; Tang et al., Reference Tang, Tang and Yuan2018; Weersink & Fulton, Reference Weersink and Fulton2020) have been made in this direction. The authors of Lu and Lakshmanan (Reference Lu and Lakshmanan2012) utilize the extension of traditional diffusion models to estimate actual product adoption over social influence. The traditional diffusion models IC and LT are modified by adopting new states and active and inactive states. These states help to compute actual product adoption, which drives profit and revenue computation. The authors also proved that the profit function is submodular under these diffusion models.

Table 4. The comparison of link prediction algorithm based on information dissemination model (Kumar et al., Reference Kumar, Singh, Singh and Biswas2020)

4.2. Link prediction

Link prediction (LP) (Singh et al., Reference Singh, Srivastva, Kumar and Srivastava2022; Mishra et al., Reference Mishra, Singh, Kumar and Biswas2022; Singh et al., Reference Singh, Muhuri, Mishra, Srivastava, Shakya and Kumar2024) licenses to gather absent or future associations in an organization are termed Link Prediction. The organization association characterizes how data spreads through the hubs. Thus, the spreading might actuate changes in the associations and accelerate the organization’s development. The development of web-based media has drawn little consideration from specialists and organizations. New stages are ceaselessly arising, for example, Facebook and Flickr (2004), YouTube (2005), Twitter (2006), and Miniature blog (2009), among others. Given the significance of various spaces and regions, research points, for example, Connection Prediction (LP). Furthermore, data diffusion has gotten considerable consideration in perplexing social network regions during the last few years. Notwithstanding, they are points, for the most part, examined in isolation, even though their outcomes are applied in comparative areas, such as viral promoting, political missions, and business process displaying.

The issue of suggesting joins has a few applications, such as recommending absent and likely associations in cloud data or, on the other hand, robust hub identification. Specifically, the expectation of future connections is helpful in comprehending the organization and correspondence evolution. For instance, in web-based media stages, promising associations that do not exist yet can advance commitment and collaboration among users, influencing the organization structure. Thus, the organizational structure impacts the correspondence or the spread of data. The link prediction working framework is shown in Figure 4.

Figure 4. The link prediction framework under information diffusion model (Kumar et al., Reference Kumar, Singh, Singh and Biswas2020)

LP techniques gauge the new edges as indicated by some association systems, similar to the distance and briefest ways among hubs, the triangles or Triassic conclusion, and the likeness with shared neighbors, among others. These primary components are fundamental in deciphering network development. For instance, renowned and persuasive clients will generally acquire associations, making traffic-based alternate ways and working on the productivity of data spreading on the network. Thus, investigating the dispersion interaction can assist with understanding the effect of clients’ cooperation, for example, what re-posting a message means for the spread of images, recordings, or fake news (bits of hearsay) on the networks. Clients in web-based interpersonal organizations make new companions and look for and share data. When a client shares a message, his/her contacts can be affected to re-post that data, driven by the homophily property that creates a diffusion cycle. Link Prediction Approaches are broadly categorized into:

• • Similarity-based Approaches

• • Probabilistic and Maximum Likelihood-based Approaches

• • Dimensionality Reduction-based Approaches

• • Information Diffusion-based Link Prediction

4.3. Community detection

Several community detection (Kumar et al. Reference Kumar, Mishra, Singh and Biswas2024; Mishra et al. Reference Mishra, Singh, Mishra and Biswas2024) algorithms already exist concerning the information diffusion procedure. We can broadly divide these procedures into four categories which are classical techniques, based on network centrality, overlapping in nature, and depending on stochastic optimization. Figure 5 illustrates the framework of community detection algorithms. In the following subsections, we have discussed all of them in detail.

Figure 5. The community detection framework under information diffusion model (Das & Biswas, Reference Das and Biswas2021b)

4.3.4. Stochastic optimization

There are several optimized community detection methods proposed for efficient information diffusion. In one of the popular methods, Hajibagheri et al. (Reference Hajibagheri, Alvari, Hamzeh and Hashemi2012), each node of the network is considered an agent. The interactions among the agents with their neighbors are optimized based on the proposed utility function. In Hajibagheri et al. (Reference Hajibagheri, Hamzeh and Sukthankar2013), the particle swarms method is used in network neighborhoods for optimizing objective functions. In another work, an approach is developed to discover information diffusion processes from online social networks based on modularity maximization (Obregon et al., Reference Obregon, Song and Jung2019). Another technique based on a genetic algorithm is utilized in all types of networks for optimized community formation (Mozafari & Hamzeh, Reference Mozafari and Hamzeh2015). Recently, a multi-objective optimization model utilizing key node identification has been introduced (Kalantari et al., Reference Kalantari, Ghazanfari, Fathian and Shahanaghi2020). Through the study mentioned above, we have shown that most of the diffusion techniques, irrespective of their discovering nature, heavily depend on the network properties for high accuracy and efficiency.

In Table 5, we have compared all the community detection algorithms based on the information dissemination model. Four diffusion models—the linear threshold (LT), independent cascade (IC), threshold model (TM), and time-aware model (TAM)—have been taken into account. We have defined the time complexity of each algorithm, solving perspective, social facets, and network properties individually. The comparison mentioned above among the algorithms gives us a clear idea about the impact of the techniques in this domain.

Table 5. The comparison of community detection algorithm based on information dissemination model (Das & Biswas, Reference Das and Biswas2021b)

5.1. Influence maximization evaluation metrics

Four matrices quality, efficiency, memory footprint, and robustness analyze the performance of IM algorithms.

1. 1. Quality. The influence spread estimates the quality of an IM approach, that is, the number of product adoption or users who a piece of information, news, idea, innovation, and product has influenced. Most of the traditional approaches ignore the effectiveness of seed users. Therefore, some of the recent works (Barbieri et al., Reference Barbieri, Bonchi and Manco2012; Zhuang et al., Reference Zhuang, Sun, Tang, Zhang and Sun2013; Wang et al., Reference Wang, Zhu, Ming, Zou, Han, Sun, Jing, Peng and Lu2017; Singh et al., Reference Singh, Kumar, Singh and Biswas2019a) focus on contextual features to improve the effectiveness of seed users.

2. 2. Efficiency. The efficiency of IM algorithms has been measured by the running time of algorithms to produce the desired seed users. In general, running time will increase with the seed size. However, there are some exceptions like TIM (Tang et al., Reference Tang, Xiao and Shi2014) and IMM (Tang et al., Reference Tang, Shi and Xiao2015). The simulation-based approaches don’t have a better performance in terms of efficiency. Therefore some heuristic-based (Reference Singh, Singh, Kumar and BiswasSingh et al., 2019b ), meta-heuristics (Singh et al., Reference Singh, Kumar, Singh and Biswas2019a; Singh et al., Reference Singh, Singh, Kumar and Biswas2019b) and sampling (Jiang et al., Reference Jiang, Song, Cong, Wang, Si and Xie2011; Goyal et al., 2011b) methods have been introduced to overcome the limitations of simulation-based methods.

3. 3. Memory Footprint. The memory footprint is also a factor in addition to the time efficiency of an approach to measure the scalability of an algorithm. The memory footprint is evaluated by analyzing the memory required to run a seeding strategy for seed identification. The large size of the dataset leads to sampling methods (Jiang et al. Reference Jiang, Song, Cong, Wang, Si and Xie2011; Goyal et al. 2011b) to take a sample of datasets for identifying seed users based on structural and topological features.

4. 4. Robustness. Most IM algorithms are not robust as a slight modification in diffusion models leads to major changes in inactivation results. These IM algorithms only focus on quality, efficiency, and scalability while ignoring robustness. However, few works (Jung et al., Reference Jung, Heo and Chen2012; Galhotra et al., Reference Galhotra, Arora and Roy2016; He & Kempe, Reference He and Kempe2016; Mehmood et al., Reference Mehmood, Bonchi and Garca-Soriano2016) have been presented in the literature to ensure robustness on diffusion analysis.

5.2. Link prediction evaluation metrics

There are numerous conventional, straightforward, and fundamental connection expectation metrics used to evaluate Link Prediction. Mainly, it can be divided into two major parts, which are quality and accuracy as defined in Table 6.

5.3. Community detection evaluation metrics

The evaluation metrics of community detection are used to define the quality of the detected communities. Mostly, it can be divided into two major parts, which are quality and efficiency as defined in Table 6.

5.3.1. Quality

For any community detection algorithm, quality denotes the accuracy of the algorithm with respect to the ground truth. Also, it denotes the strength of the communities. The following parameters are used to measure the quality of the community detection techniques.

1. 1. Modularity. Modularity (Muhuri & Mukhopadhyay, Reference Muhuri and Mukhopadhyay2021) measures the robustness of the partitions in a network. Modularity finds the density of connections between the nodes of the same community rest of the communities. The modularity Q of a graph with m number of edges can be defined as

   (4) \begin{equation} Q=\frac {1}{2m} \sum_{vw} \left[A_{vw}-\frac{k_v k_w}{2m}\right]\delta(c_v,c_w)\end{equation}
   $A_{vw}$ represents the connectivity of node i and j, $k_v$ denotes the sum of the weights of the edges incident with node v, $c_v$ is the community of node $v, \delta(c_v,c_w)$ is equals to 1 when v and w are in the same community, 0 otherwise.

2. 2. f-Modularity. It is a fuzzy measure that finds the mutual information among communities (Guo et al., Reference Guo, Huang, Kong and Wang2021). The value decreases with the contraction of the community structures. f-modularity of a graph G can be denoted as

   (5) \begin{equation} f_{mod} (G)=max_{D\in C} \sum_{u,v} \left(\sigma f(D_{u,v})F_{u,v}-f^* (\sigma f (D_{u,v}))J_{u,v}\right)\end{equation}
   Here D distinguishes between frequency matrix F and random matrix J. C is a constraint set.

3. 3. ANUI. Average normalized unifiability and isolability (ANUI) analyses both unifiability and isolability to denote the quality of an overlapping community (Das & Biswas, Reference Das and Biswas2021b). A graph G with k communities, unifiability of community $C_i$ to community $C_j$ is denoted as

(6) \begin{equation} unifiability (C_i)=\sum_{i=1}^k Unifiability (C_i,C_j)\end{equation}

(7) \begin{equation} isolability(C_i)=\frac{{(u,v)|u \in C_i (v)}}{{(u,v);(u,w)|u \in C_i (v) \& w \notin C_i}}\end{equation}

ANUI is denoted as

(8) \begin{equation} ANUI (C_i)= \frac{unifiability (C_i)+isolability (C_i)}{2}\end{equation}

1. 4. Conductance. It finds structural cohesiveness by incorporating external connections of the communities. Conductance of forecast community C is denoted as

   (9) \begin{equation} f(w)=\frac{E_w^{out}}{2.E_w^{in}+E_w^{out}}\end{equation}
   where $E_w^{out}$ and $E_w^{in}$ denote the number of edges that are outside and inside of the community C, respectively.

2. 5. Permanence. Permanence (Chakraborty et al., Reference Chakraborty, Srinivasan, Ganguly, Mukherjee and Bhowmick2014) is a measure to find the pull of the neighborhood node on a particular node v. The internal clustering coefficient $c_in$ is also utilized to find the permanence perm(v).

   (10) \begin{equation} perm(v)=[\frac{I(v)}{E_{\max}(v)}\times \frac{1}{D(v)}] -[1-c_{in}(v)]\end{equation}
   where I(v) is internal connections, $E_{\max}(v)$ is the strength of internal community, and D(v) is the degree of node v.

5.3.2. Efficiency

For any community finding technique, efficiency denotes the accuracy of the algorithm with respect to the computation time. Also, it depends on the size of the network. Some parameters are utilized to measure the efficiency of the community detection algorithms.

1. 1. Purity. It assigns the detected community to the most frequent baseline (Lin et al., Reference Lin, Ohsuga, Liau and Hu2005).

   (11) \begin{equation} Purity(\Omega, C)=\frac{1}{N}\sum_k \max_j|\omega_k,c_j|\end{equation}
   where $\Omega$ is the set of ground truth communities.

2. 2. NMI. Normalized Mutual Information (NMI) finds the mutual dependence among the communities (Xie et al., Reference Xie, Kelley and Szymanski2013). NMI detects the similarity and quality between two divisions. Let the two communities are C and C ^′. Then NMI can be represented as

   (12) \begin{equation} NMI(C,C')=\frac {2I(C,C')}{H(C)+H(C')}\end{equation}
   where $H(.)$ represents entropy function and the mutual information $I(C, C')=H(C)+H(C')-H(C,C').$ No similarity and maximum similarity of two communities are denoted as the NMI value 0 and 1, respectively.

3. 3. ONMI. Overlap-Normalized Mutual Information (ONMI) is proposed in Van Lierde et al. (Reference Van Lierde, Chen and Chow2019) for overlapping communities. The true and the predicted partitions are denoted by $C_l$ and $\overline{C_l}$ , respectively, where $1\leq l \leq K,\overline{K} $ . The number of true and predicted partitions are K and $\overline{K}$ , respectively. Then ONMI can be represented as:

   (13) \begin{equation} ONMI(X,\overline{X})=1-\frac{1}{2} (H(X|\overline{X})+H(\overline{X}|X))\end{equation}
   where $X_l$ = 1 denotes the event that a node belongs to $C_l$ and $\overline{X_l}$ = 1 denotes the event that a node belongs to $\overline{C_l}$ . Here normalized entropy of all $X_k$ given all $\overline{X_l}$ is
   (14) \begin{equation} H(X|\overline{X})= \frac{1}{k} \sum _{k} \frac{H(X_k|\overline{X})}{H(X_k)}\end{equation}
   High ONMI value is expected for the overlapping communities.

4. 4. F-Measure. it is the harmonic mean of both types of purity measures or between Precision and recall (Chakraborty et al., Reference Chakraborty, Dalmia, Mukherjee and Ganguly2017).

   (15) \begin{equation} F-measure=\frac{2. Purity(\Omega,C). Purity(C,\Omega)}{Purity(\Omega,C)+Purity(C,\Omega)}\end{equation}

5. 5. F1-score. It measures the similarities between two overlapping communities (Chakraborty et al., Reference Chakraborty, Dalmia, Mukherjee and Ganguly2017).

   (16) \begin{equation} F1=\frac{1}{2} (\frac{1}{\phi}\sum_{\phi_i \in \phi}F1(\phi_i,C_{g(i)}+ \frac{1}{C}\sum_{c_i \in C}F1(\phi_g,(i),c_i)))\end{equation}
   It represents the average of the F1-score of the best-detected partition and ground truth communities and the F1-score of the ground truth communities and best-detected partition.

6. 6. ENMI. Extended normalized mutual information (ENMI) compares the partitions with the baseline (Sun et al., Reference Sun, Wang, Sheng, Yu and Shao2018). ENMI is defined as

   (17) \begin{equation} ENMI(X|Y)=1-[H(X|Y)+H(Y|X)]/2\end{equation}
   where $H(X|Y)$ is denoted as
   (18) \begin{equation} H(X|Y)=1-\frac{1}{C'}\sum \frac{H(X_k|Y)}{H(X_k)}\end{equation}
   where X and Y denotes the random variables connected with the communities C and C’, respectively. $H(X|Y)$ is the normalized entropy of community X given community Y.

Table 7. Illustration of information diffusion challenges over different spplications (Kumar et al., Reference Kumar, Singh, Singh and Biswas2020; Das & Biswas, Reference Das and Biswas2021b; Singh et al., Reference Singh, Srivastva, Verma and Singh2021)

6.1. Influence maximization

The computation of approximate influence spread is dependent on the diffusion process, which is stochastic. Therefore, it activates a different set of users in each diffusion round. Few efforts have been made to tackle these uncertainties, illustrated as follows.

• • Diffusion Process Modeling. The first challenge of information diffusion analysis corresponding to the IM problem is how to adapt the diffusion process for viral marketing. It is hard to accommodate user behavior about product adoption as the diffusion process is random. Therefore, some efforts (Kempe et al., Reference Kempe, Kleinberg and Tardos2003; Guille et al., Reference Guille, Hacid, Favre and Zighed2013) have been made to tackle the adaption of the diffusion process for the IM problem by incorporating and extending traditional diffusion models.

• • Complexity and Optimality. Kempe et al. (Reference Kempe, Kleinberg and Tardos2003) have proved that the IM problem is NP-hard under classical diffusion models. The influence spread computation under these models is computationally complex. Due to this, most simulation-based IM algorithms are inefficient and have some approximation guarantee to the optimal solution. However, heuristics and sampling methods are more time-efficient but less optimal. It is tough to find a near-optimal scalable solution. Therefore, most of the algorithms (Chen et al., Reference Chen, Wang and Wang2010; Singh et al., Reference Singh, Kumar, Singh and Biswas2019a; Singh et al., Reference Singh, Singh, Kumar and Biswas2019b) focus on the trade-off between optimality and scalability.

• • Stochasticity of Diffusion Process. Most of the diffusion models are stochastic. Therefore, IM algorithms produce a different set of active users in each iteration from seed users due to the randomness of the diffusion process. So it is very tough to collect the final product adoption. However, some strategies like the mean and median of repeated iteration outcomes are applied to tackle the IM problem. For example, simulation-based methods (Kempe et al., Reference Kempe, Kleinberg and Tardos2003; Leskovec et al., Reference Leskovec, Krause, Guestrin, Faloutsos, VanBriesen and Glance2007) perform $r=10\,000$ simulations while sampling methods (Cheng et al., Reference Cheng, Shen, Huang, Zhang and Cheng2013; Ohsaka et al., Reference Ohsaka, Akiba, Yoshida and Kawarabayashi2014) iterate multiple times on different sample episodes.

• • Stability and Robustness. Most of the IM methods are based on the assumption that the network structure is fixed while the network structure keeps on changing in the real world. Therefore, a slight change in the network can lead to a significant update in seed users. The authors of He & Kempe (Reference He and Kempe2014) also stated that noisy influence probabilities could generate the inaccurate seed. Therefore some efforts (Jung et al., Reference Jung, Heo and Chen2012; Galhotra et al., Reference Galhotra, Arora and Roy2016; He & Kempe, Reference He and Kempe2016; Mehmood et al., Reference Mehmood, Bonchi and Garca-Soriano2016) have been made to find a robust solution for IM problem.

• • Incorporating Group Norms. Most of the literature considers only individual influence under the IC diffusion model and avoids group norms. However, people are also influenced by acquaintances and group norms such as trust, conformity, and context. People who share common interests, backgrounds, cultures, etc., are more inclined to each other’s thoughts. Therefore, some efforts (Li et al., Reference Li, Chen, Wang and Zhang2011; Tang et al., Reference Tang, Wu and Sun2013; Zhang et al., Reference Zhang, Tang, Zhuang, Wing-ki Leung and Li2014) have been made to incorporate group norms with diffusion models. There are few possible future directions with group norms that can consider user profile information with group characteristics such as trust and conformity.

• • Adoption of Weaker Submodularity. The evaluation function is strict with the definition in general submodularity of approximate influence. This reduces the number of MC simulations under classical diffusion models for the IM problem. However, some of the advances (Li et al., Reference Li, Chen, Wang and Zhang2011; Gionis et al., Reference Gionis, Terzi and Tsaparas2013; Galhotra et al., Reference Galhotra, Arora and Roy2016) in the IM problem states that strictness of submodularity leads to no approximation guarantee of the solution. Because these advances are based on opinion formation and change. Therefore, it is required to adopt weaker submodularity in influence diffusion function to ensure some theoretical guarantee of the solution. There are some efforts (Das & Kempe, Reference Das and Kempe2011) have made to weaken the submodularity of influence diffusion.

• • Incorporating Heterogeneous Diffusion Process. Most of the algorithms under the IM2 and MIM2 framework assume that every layer, relationship, or network has used the same diffusion models. However, in reality, each network has its diffusion method. Therefore, one possible future work can be the consideration of different diffusion processes across the layers and networks to achieve a more realistic scenario. However, some efforts (Zhan et al., Reference Zhan, Zhang, Wang, Yu, Xie, Cao, Lim, Zhou, Ho, Cheung and Motoda2015; Kuhnle et al., Reference Kuhnle, Alim, Li, Zhang and Thai2018) have been made to ensure heterogeneity across layers for the IM problem.

• • Considering Contextual Diffusion Models. Most of the classical IM algorithms ignore context-aware diffusion models by incorporating conformity, semantics, location, etc., with diffusion models for novel viral marketing applications. However, some efforts (Lee & Chung, Reference Lee and Chung2015; Bozorgi et al., Reference Bozorgi, Samet, Kwisthout and Wareham2017; Su et al., Reference Su, Li, Cheng and Sun2018) have been made to consider contextual features with diffusion models. Still, most of these contextual features are unexplored.

• • Incorporating Conformity. Conformity is a crucial factor in how social influence spreads throughout societies. It describes how people tend to conform to the attitudes, beliefs, and behaviors of a majority group or societal norms. This phenomenon can occur through different mechanisms, such as informational conformity, where individuals conform because they believe others have relevant knowledge, or normative conformity, where individuals conform to gain social approval or avoid rejection. It supports processes of influence diffusion, where the adoption of certain behaviors or attitudes by a few individuals within a social network can lead to their widespread acceptance and adoption by others. Gaining insight into conformity is essential for understanding the development and evolution of social norms, as well as their influence on individual and collective behavior. Thus, to enhance the efficiency of the seed nodes, the authors in Li et al. (Reference Li, Chen, Wang and Zhang2011) made some attempts, yet many of these aspects remain unexplored.

6.2. Link prediction

This section discusses the challenges and efforts made to tackle these challenges, illustrated as follows.

• • Efficiency with Link Prediction. Because of the huge measure of information in interpersonal organizations, efficient and reliable link prediction addresses a critical test. To resolve this issue, researchers have proposed two calculations for interface expectation. The proposed calculations tackled the productivity issue by embracing low-rank factorization models while demonstrating extremely proficient contrasts with different techniques. This way, the review addresses a huge advance forward in the test of fostering a productive connection forecast model. Another significant issue that should be tended to worries the precision of connection forecast draws near. The test discoveries uncovered that the proposed approach beat numerous different techniques as far as exactness and versatility, and the related runtime were not exactly as seen in past examinations. Be that as it may, albeit the proposed approach improved connection forecast precision, it stays untested on large network information, which might see its proficiency become sabotaged (Fire et al., Reference Fire, Tenenboim-Chekina, Puzis, Lesser, Rokach and Elovici2014).

• • Sparsity of Data. One more issue identified with interface forecast in marked SNs concerns the sparsity of information. To reduce this issue, researchers have proposed some effective methods, for instance, an imaginative methodology that is fit for investigating the client’s character utilizing web-based media. The acquired test results demonstrated an integral relationship between the marked connection forecast issue and character data. An intriguing cutting-edge technique for working with connect expectation is the utilization of worldly consistency in relational correspondence to focus on weighted edges in network diagrams. Another approach used is contrasted and different techniques, this strategy predicts interfaces regardless of whether there is a shortage in the quantity of edges required for investigation. A differentiating similitude-based connection expectation strategy is based on fuzzy connection significance. The technique performed well, utilizing two methodologies to accomplish its targets. Initially, for the determination of the neighbor, the distance between hubs was utilized. Besides, the fluffy connection significance was utilized to track down the pertinent connection. By utilizing these systems, the technique has acquired sound outcomes. The challenge for a sparse network is now been addressed but still needs a lot of attention as the network is getting sparse exponentially (Beigi et al., Reference Beigi, Ranganath and Liu2019).

• • Trusted Link Prediction. With the improvement of the Internet, an ever-increasing number of people or associations will more often than not discuss and communicate on the organizational stage. Through friendly stages, individuals can not just offer their sentiments about various items yet, in addition, express their perspectives on others, which extraordinarily improves individuals’ social exercises. Nonetheless, the fast improvement of social stages has filled them with many futile or bogus data and records. To rapidly and effectively peruse the substance for intrigued, clients as a rule add clients who have normal interests to the ‘trust list.’ simultaneously, to try not to peruse the substance not intrigued, clients generally add clients with inverse interests to the ‘doubt list.’ For instance, and have comparative interests and have some contention in a specific region. In this way, Twitter may follow and add to the boycott. We can assemble a marked interpersonal organization by catching the trust and doubt connections between clients. Through the trust/doubt connections in the marked informal community, we can not just know which clients the objective client has social associations with yet additionally know what sort of mentality the objective client embraces toward these clients. In any case, the social connections in an interpersonal organization are excessively inadequate, which truly upsets the extension of the client’s group of friends and further improves the social stage. Thus, it has become important to assist clients with finding all the more new companions or confided in clients. Luckily, clients have left a lot of verifiable conduct information (e.g., client’s appraising and remarks) on friendly stages, which gives great conditions to assessing the trust connections and comparability of inclinations between various clients. Notwithstanding, the way to deal with observing likely companions through shared interests actually faces many difficulties (Fan et al., Reference Fan, Xiong, Zhao and Yu2019).

• • Level-wise Link Prediction. In complex networks, level-wise link prediction is used as it breaks the network into smaller networks and then predicts the link for each small network. The biggest challenge is dividing the complex network into smaller networks and then combining the link prediction of all networks and concluding it. The networks are so dynamic, and thus, predicting the link is a challenge! Trust, efficiency, and accuracy have to be maintained at every level as the next level depends on the previous link prediction. Thus it becomes a very challenging task (Kumar et al., Reference Kumar, Singh, Singh and Biswas2020).

6.3. Community detection

There are several challenges associated with community detection concerning information diffusion. The challenges are mostly divided into five categories discussed here in detail.

• • Meaningful Communities. In an online social network, all the detected communities are not meaningful at all. The community detection methods might create some partitions that have no importance from the application point of view. In Shen et al. (Reference Shen, Song, Yang and Zhang2010), the problem has been addressed.

• • Running Time Optimization. Running time optimization is another important challenge in community detection as discussed in Das and Biswas (Reference Das and Biswas2021b). Though game-theoretic methods have performed better theoretically, their computational complexity should be reduced to achieve more efficiency.

• • Noise Removal. All the community detection algorithms are sensitive to noise. Different noise removal filters can be deployed to increase the technique’s accuracy. The problem has been discussed in Chen (Reference Chen2011) concerning a decentralized network.

• • Model Generalization. Researchers are very much focused on generalizing the model for different types of complex networks, including the independent cascade, biological, linear threshold, and epidemics models. In Li et al. (Reference Li, Zhang and Tan2015) and Devi and Tripathi (Reference Devi and Tripathi2020), authors have discussed the issue for encouraging futuristic models.

• • Sparse Network. If in a network, the number of 0 is greater than the number of 1, or the network consists of a huge number of isolated vertex. The community detection algorithms have failed to generate efficient communities as shown in Alvari et al. (Reference Alvari, Hajibagheri and Sukthankar2014). This challenge should be addressed for the refinement of the existing algorithms.