3. The proposed method: MCPR centrality

As previously stated, the PageRank centrality measure is the most effective method for locating appropriate immunization nodes. Thus, the PageRank measure is used as the primary metric for ranking nodes in multiplex network nodes. Because the virtual interactions layer serves as a platform for the spread of infection information, the importance of these nodes in the virtual layer cannot be overstated. Consequently, we should propose an approach that considers the importance of nodes in both the physical and virtual layers.

Since the PageRank algorithm begins with no knowledge of the nodes’ PageRank importance, the constant value of $\frac{1}{N}$ is applied to all nodes (while N is considered as the total number of the network nodes).

Equation (3) is used to calculate the PageRank of an undirected graph with high mathematical accuracy, where $\alpha =0.85$ (Zheng et al., Reference Wang, Guo, Sun and Xia2018).

(3) \begin{equation} x_{i}=\alpha _{A}\sum _{j}A_{ij}\frac{x_{j}}{g_{j}}+\left(1-\alpha _{A}\right)\frac{1}{N} \end{equation}

MPR treats the first layer ( $L_{1}$ ) as a layer with the neighborhood matrix ${a}_{ji}^{[L_{1}]}$ and applies classical PR to it.

(4) \begin{align} x_{i} = \mu {\sum }_{j=1}^{N}{a}_{ji}^{\left[L_{1}\right]}\frac{x_{j}}{\hat{k}_{j}}+ \omega \\[-30pt] \nonumber \end{align}

\begin{align*} \hat{k}_{j} =\max \left(\tilde{k}_{j}.1\right).\,{\rm with}\,\tilde{k}_{j}= {\sum }_{r=1}^{N}{a}_{jr}^{\left[L_{1}\right]} \end{align*}

Here $x_{i}$ denotes the PageRank score of node i in the first layer which later serves as the input for the second layer. The matrix element ${a}_{ij}^{[L_{1}]}$ represents the adjacency relation in the first layer (virtual layer) that equals to one if a link exists from node j to node i. The parameter $\mu \in (0,1)$ is the probability following an outgoing link, while $\omega =(1-\mu )/N$ is the teleportation term for ensuring ergodicity. The quantity $\tilde{k}_{j}$ is the degree of node j in $L_{1}$ and $\hat{k}_{j}=\max\{\tilde{k}_{j},1\}$ prevents division by zero.

Equation (5) extends this formulation to the second layer ( $L_{2}$ ) by incorporating node weights obtained from the first layer. The MPR in the second ( $L_{2}$ ) layer is then examined with the neighborhood matrix ${a}_{jr}^{[L_{1}]}$ by utilizing the $x_{i}$ rank obtained from this layer.

(5) \begin{equation} X_{i}\left(q.n\right)= \mu {\sum }_{j=1}^{N}\left(x_{i}\right)^{q}{a}_{ji}^{\left[L_{2}\right]}\frac{X_{j\left(q.n\right)}}{k_{j}}+\omega \left(\frac{x_{i}}{\left\langle x\right\rangle }\right)^{n} \end{equation}

\begin{equation*} k_{j}=\mathit{\max } \left({\sum }_{r=1}^{N}{a}_{jr}^{\left[L_{2}\right]}\left(x_{r}\right)^{q}.1\right) \end{equation*}

\begin{equation*} \omega =\frac{1}{N}\left[1-\mu +\mu \delta \left({\sum }_{r=1}^{N}{a}_{jr}^{\left[L_{2}\right]}\left(x_{r}\right)^{q}.0\right)X_{i}\left(q.n\right)\right] \end{equation*}

In this expression $X_{i}(q.n)$ represents the MPR score of node i in the second (physical) layer, where the ranking is influenced by the first-layer values $x_{j}$ . The adjacency matrix element ${a}_{jr}^{[L_{2}]}$ encodes connections in $L_{2}$ . The $(x_{i}/\langle x\rangle )^{n}$ is the teleportation term scaled by the average $\langle x\rangle$ of the first-layer PageRank. The normalization $\mathit{\max } ({\sum }_{r=1}^{N}{a}_{jr}^{[L_{2}]}(x_{r})^{q}.1)$ ensures stability by avoiding divisions by zero and $\delta (k_{i},0)$ is the Kronecker delta that accounts for isolated nodes. The parameters $q,n\in [0,1]$ regulate the relative influence of these functions.

MPR’s value is dependent on two parameters, $q,n\in (0.1)$ , and results in PageRank multiplexing, multiplication, and combination with different values of zero and one.

In this study, we used the MCPR method to construct initial knowledge about nodes based on their importance in the virtual interactions layer and obtained it from the layer’s centrality measures. Unlike the MPR method, which assumes that all layers have the same centrality, the PR method (the best centrality measure for the physical layer) is used to determine the best centrality measure for the virtual layer in the physical layer. Equation (6) introduces the MCPR, which integrates context-specific centralities from both layers:

(6) \begin{align} x_{i}=\alpha _{A}\sum _{j}A_{ij}\frac{x_{j}}{g_{j}}+\left(1-\alpha _{A}\right)y_{i} \\[-30pt] \nonumber \end{align}

\begin{equation*} g_{j}=\max \left\{{\sum }_{r=1}^{N}A_{jr},1\right\} \end{equation*}

This equation defines the MCPR score for node i in the physical layer. The first term on the right-hand side corresponds to a PageRank-style update: $A_{ij}$ denotes the adjacency matrix entry of the physical layer ( $L_{2}$ ) and $g_{j}$ represents the degree of node j lower-bounded by one to avoid division by zero. This term is weighted by the damping parameter $\alpha _{A}\in (0,1)$ controls the probability of following network links. The second term incorporates the prior score $y_{i}$ , derived from the virtual (awareness) layer, and weighted by $1-\alpha _{A}$ . In practice $y_{i}$ can be computed using a context-appropriate centrality measure (e.g., closeness centrality), thereby injecting awareness-layer importance into the infection-layer ranking.

Because each node receives its initial value from its neighbors in the original nature of PageRank centrality, when the initial value of the nodes is taken from another layer, the value of the nodes in this layer affects the final value of a node in the physical layer and the value of its neighbors.

Each node’s value is proportional to the value of the same node and its neighbors in the previous layer (using any type of centrality measure), the node’s degree, the node’s neighbors’ PageRank, and the node’s neighbors’ inverse degree in the current layer.

4. Simulation results

By employing a free-scale network of 2,000 nodes with a P (k) ∼ K ^–2.5 degree distribution, the simulations are represented as a layer of physical interactions. By adding 800 additional random links to this layer of the network, a virtual interaction layer is created (Pan and Yan, Reference Wang, Andrews, Wu, Wang and Bauch2018; Zweig, Reference Zheng, Xia, Guo and Dehmer2016). The parameters of the data set are listed in Table 4.

Table 4.

Structural features of the applied data

We used network epidemic size to evaluate the efficacy of immunization strategies; for example, a smaller epidemic size indicates a more effective immunization approach. The primary seeds of infected nodes were chosen at random from a pool of 0.2 network nodes. The model presented in Section 2.1 was used to simulate the transmission of infection and awareness within a network. Finally, the epidemic size was calculated using an average of more than 20 runs of the infection and awareness models (Morone and Makse, Reference Zheng, Xia, Guo and Dehmer2015; Santoro and Nicosia, Reference Tortosa, Vicent and Yeghikyan2020).

Notably, the type of infection affects the model parameter values. We selected two infections with varying parameters for our simulations. Table 5 summarizes the infections and their associated reproduction numbers (R ₀ ) used to calculate the infection rate ( $\beta $ ^U ) for the proposed model. The infection rate was calculated using Equation (1) (Hartvigsen et al., Reference Granell, Gómez and Arenas2007; Shams and Khansari, Reference Wang, Guo, Sun and Xia2013).

Table 5.

Data on multiple epidemic samples and calculating their corresponding infection rate in the target network

Knowing the network’s topology and flow enables the appropriate centrality measure for determining the nodes in the virtual interaction layer. To better understand this issue, we compared and analyzed five different measures for each node, using the rank obtained from each measure to calculate the physical interactions layer’s PR. To validate the MCPR technique’s results, we compared it to the single-layer PR method and the multiplex MPR method.

We conducted simulations with $\varphi$ = 0.8 and α = 0.3 and then examine the effect of changing these parameters on the simulation results.

First, we ran a simulation to compare the new MCPR technique to the classical method, randomly selecting 0.2 network nodes as infected nodes solely based on their physical interaction layer relevance. In comparison to the classical technique, this method alters the priority of node selection. These changes in priority were necessary because vaccinated nodes are effectively discarded from the physical interaction layer while remaining effective as nodes that spread awareness in the virtual interaction layer. The network epidemic size was determined after discarding these nodes from the network with an average of more than 20 infection and awareness model implementations. This procedure was repeated for different immunization percentages. As discussed in Section 2.2, simulation results indicated that immunization against classical PageRank centrality is more efficient than immunization against other centralities. As a result, we compared the proposed method to the classical method based on PageRank centrality to determine the slightest difference between the two methods.

Due to the immunization of nodes at the physical interaction layer, it is possible to segment the network, thereby limiting infection spread. As a result, it is critical to have a physical layer. When the differences between nodes are minor, however, information from other layers can be used to make more informed decisions, especially when resources are limited. As a result, the network can be isolated by securing a specified number of nodes at the physical interaction layer. It is possible to incorporate data from additional layers, such as the virtual interaction layer.

To propagate awareness as a flow, we must determine which degree of centrality is the most appropriate for use in the virtual interaction layer. To discern this, we used the MCPR approach with five different centralities to rank the virtual interaction layer.

The simulation results for measles and smallpox were shown in Figures 2 and 3. We have the smallest epidemic size possible if we use close proximity to identify potential awareness nodes. According to the method used to determine the infection transmission rate for knowledgeable individuals, the higher the quality of information spread, the lower the infection transmission rate for aware individuals (Equation 2). As the awareness index increases, the quality of knowledge deteriorates. That is, the more information an individual receives, the higher the likelihood that the information will spread and the less likely the individual will respond to the infection. As a result, the awareness gained by taking a shorter path and passing it on less frequently is critical. A person can become conscious of something in a variety of ways. As a result, the evaluation function updates the individual awareness index with the lowest index value whenever an individual’s awareness reaches a path, as illustrated in Figure 1.

Figure 2.

MCPR method vs. conventional method for measles, with the corresponding parameter values in Table 5 for an epidemiological study duration = 100, number of implementations = 20, γ = 0.1, $\varphi$ = 0.8, and α = 0.3.

Figure 3.

MCPR method vs. conventional method for smallpox, with the corresponding parameter values in Table 5 for an epidemiological study duration = 100, number of implementations = 20, γ = 0.1, $\varphi$ = 0.8, and α = 0.3.

If the flow follows the shortest paths, the proximity measure can be used. There are two types of processes Relevant to the proximity measure: those that flow along the shortest paths and those that flow along all feasible paths concurrently (Borgatti, Reference Bockholt and Zweig2005).

Given the importance of information quality in our research, the highest level of awareness as a process follows the shortest paths and may be followed simultaneously along with any number of possible paths.

On the other hand, we demonstrated that selecting candidate nodes for knowledge propagation based on the special vector’s centrality does not produce satisfactory results. The centrality of the special vector was calculated by subtracting the importance of each node’s neighbors. Assume that an unimportant node is given the same ranking as its neighbor simply because it is adjacent to an important node. If the simulation is followed, this node is unsuitable for knowledge spread because awareness and its spread are purely haphazard. If an unimportant node is aware of an important node in its proximity, it has the option of sharing its knowledge with the neighboring node. If it shares the information with a critical neighbor, that neighbor can share it with others. However, this insignificant individual or node may prevent it from reaching its critical neighbor for whatever reason. It is possible that such centrality is not the optimal location for locating someone to disseminate information.

Additionally, awareness has a more significant effect on nodes with a higher centrality than on nodes with a lower centrality. Because higher centrality nodes are preferred in centrality calculations, it is predicted that the suggested method will exhibit a greater epidemic drop than the immunization method based on physical interaction layer information at lower immunization percentages. We understand that low immunization rates are critical due to limited resources.

On the network mentioned above, the generalized SIR-UA model was used. We used 100-time units as the length of the epidemic research period and changed the model’s fixed parameters, as shown in Table 5. The infection transmission and recovery rates were altered based on the infection type. The epidemic size was estimated using a generalized SIR-UA model.

The MPR technique, which prioritizes nodes in both the physical and virtual interaction layers, was then used to compare the proposed method to a more recent method.

L1 was used to represent the virtual interaction layer in this work, while L2 represented the physical interaction layer. To compare the MCPR and MPR methods, we used both layers as the initial layer of MPR computations. We simulated the most relevant centrality measure, namely proximity centrality, for the virtual interaction layer.

According to the findings, when the proximity centrality measure for the virtual interaction layer is used, MCPR was more efficient than MPR for network immunization with limited resources. Additionally, it reduced the size of epidemics by 20% compared to the MPR technique (Figure 4).

Figure 4.

MCPR method vs. conventional method for smallpox (left) and measles (right) with the corresponding parameter values in Table 5 for an epidemiological study duration = 100, number of implementations = 20, γ = 0.1, $\varphi$ = 0.8, and α = 0.3.

In this study, we chose hybrid MPR because it is more efficient than the other two methods and significantly reduced the size of the epidemic.

5. Empirical validation on the real-world multiplex network

To transcend the controlled environment of synthetic networks and evaluate the MCPR strategy’s performance under realistic conditions, we conducted an empirical validation using a well-documented, real-world multiplex dataset. This step is critical to demonstrate the robustness and practical applicability of our proposed method in the face of the structural complexities and emergent behaviors inherent in genuine human social systems.

5.2. Performance analysis and comparative results

We benchmarked the performance of our proposed MCPR strategy, which uses Closeness centrality for the awareness layer, against the baseline of a classical, single-layer PageRank immunization. The results, presented below, provide compelling empirical support for the advantages of a context-aware, multiplex approach.

5.2.1. Measles simulation

The simulation revealed that for a rapidly spreading pathogen like Measles, the MCPR strategy offers a consistent yet modest enhancement in containment. As illustrated in Figure 5, with a 10% immunization budget, the classical PageRank strategy resulted in a final epidemic size of approximately 1,100. In contrast, our MCPR method reduced this to nearly 1,060, a performance gain of ∼2.2%. This finding is significant as it underscores the utility of incorporating the virtual layer’s structure even when the temporal window for awareness to influence behavior is narrow.

Figure 5.

Comparative performance of immunization strategies for the measles epidemic simulation on the Copenhagen networks study dataset.

5.2.2. Smallpox simulation

The strategic advantage of the MCPR method becomes markedly more pronounced when simulating a pathogen with slower transmission dynamics, such as Smallpox. The results, presented in Figure 6, demonstrated a substantial mitigation of the outbreak. With a 10% immunization budget, where classical PageRank contained the epidemic to 1,020 cases, our MCPR method curtailed the final size to approximately 950. This constitutes a significant improvement of 7% over the baseline. This higher performance highlights a critical insight: when disease progression is slower, the awareness dynamics propagating on the virtual layer have a greater opportunity to influence physical behavior, thus amplifying the efficacy of a context-aware immunization strategy.

Figure 6.

Comparative performance of immunization strategies for the smallpox epidemic simulation on the Copenhagen networks study dataset.

The empirical validation on the Copenhagen Networks Study dataset provided powerful evidence for the central thesis of this research: the efficacy of an immunization strategy is significantly enhanced by adopting a context-aware, multiplex perspective.

The performance differential observed between the Measles and Smallpox simulations is particularly revealing. It suggests that the value derived from the awareness layer is directly correlated with the temporal dynamics of the pathogen. For slower diseases like Smallpox, the awareness propagation process has a sufficient temporal window to permeate the social network and enact behavioral change. This makes Closeness centrality on the virtual layer a highly effective heuristic for identifying nodes whose immunization yields a dual benefit: direct removal from the infection network and maximized disruption of awareness flow to their contacts. Conversely, for fast-moving epidemics, this effect is diminished but, critically, not eliminated. These findings confirm that MCPR is not merely a theoretical construct effective on idealized networks but a robust and practical method applicable to the inherent complexity of real-world human interaction systems.

6. Analysis of algorithm parameters

To investigate the effect of knowledge transmission rate and information quality on infection size changes, we conducted comprehensive parameter sensitivity analyses using both synthetic networks and real-world datasets. This dual method validates our findings across different network topologies and provides robust evidence for the MCPR method’s effectiveness.

6.1. Synthetic network parameter analysis

Using the synthetic scale-free networks described in Section 4, we first examined the fundamental parameter sensitivities. We varied the values of knowledge transmission rate and information quality in discrete intervals to understand their individual and combined effects on epidemic control.

Figures 7 and 8 illustrate the effect of changes in awareness transmission rate (α) for smallpox and measles, respectively. As demonstrated, the total number of infected individuals decreases as the rate of information transmission increases throughout the infection. For both diseases, higher α values consistently improve epidemic control, with the effect being more pronounced for smallpox due to its slower transmission dynamics.

Figure 7.

Results of modifying the rate of information transfer for smallpox in order to reduce the size of the epidemic, using the corresponding parameter values from Table 5 for an epidemiological study duration of =100, number of implementations = 20, γ = 0.1, and $\varphi$ = 0.8.

Figures 9 and 10 present the results of altering the information quality parameter ( $\varphi$ ) for smallpox and measles, respectively. The epidemic size decreases more rapidly at larger $\varphi$ values, with the reduction being greatest at $\varphi$ = 0.9 for both diseases. As previously stated, an individual susceptible to infection who is adjacent to an infected individual and has an awareness level of i will contract the infection at a rate of $(1-\varphi^i)\beta^U$ . When high-quality information is provided, infection incidence can be minimized through preventive measures.

6.2. Parameter analysis on realistic network

To validate our parameter analysis findings on realistic network structures, we conducted comprehensive testing using the Copenhagen Networks Study dataset introduced in Section 5. This real-world validation provides crucial evidence that our synthetic network findings translate effectively to actual social network topologies.

6.2.1. Information quality analysis

Figures 11 and 12 present the information quality parameter analysis for measles and smallpox on the Copenhagen dataset, respectively. The real-world network demonstrates consistent trends with synthetic results while revealing the practical implications of network heterogeneity.

Figure 8.

Results of modifying the rate of information transfer for measles in order to reduce the size of the epidemic, using the corresponding parameter values from Table 5 for an epidemiological study duration of =100, number of implementations = 20, γ = 0.1, and $\varphi$ = 0.8.

For Measles Classical PageRank achieves approximately 1,100 infections at 10% immunization, while MCPR with $\varphi$ = 0.9 reduces this to 1,070 cases, representing a 2.7% improvement. The relatively modest gains reflect the rapid transmission dynamics characteristic of measles, where the temporal window for awareness-driven behavioral change remains limited even in real social networks.

Figure 9.

Results of altering the information quality rate for smallpox to reduce the epidemic size, with an epidemiological study duration = 100, number of implementations = 20, γ = 0.1, α = 0.3, and the corresponding parameter values in Table 5.

Figure 10.

Results of altering the information quality rate for measles to reduce the epidemic size, with epidemiological study duration = 100, number of implementations = 20, γ = 0.1, α = 0.3, and the corresponding parameter values in Table 5.

For Smallpox MCPR with $\varphi$ = 0.9 achieves 940 infections compared to Classical PageRank’s 1,030 at 10% immunization, yielding an 8.7% reduction. This enhanced performance demonstrates how slower disease progression allows higher-quality information to more effectively influence protective behaviors, with the effect being amplified in realistic network structures.

Figure 11.

Results of altering the information quality parameter ( $\varphi$ ) for measles to reduce the epidemic size on the Copenhagen networks study dataset, with epidemiological study duration = 100, number of implementations = 20, γ = 0.1, α = 0.3, and the corresponding parameter values in Table 5.

Figure 12.

Results of altering the information quality parameter ( $\varphi$ ) for smallpox to reduce the epidemic size on the copenhagen networks study dataset, with epidemiological study duration = 100, number of implementations = 20, γ = 0.1, α = 0.3, and the corresponding parameter values in Table 5.

Figure 13.

Results of modifying the awareness transmission rate (α) for measles to reduce the epidemic size on the Copenhagen networks study dataset, with epidemiological study duration = 100, number of implementations = 20, γ = 0.1, $\varphi$ = 0.8, and the corresponding parameter values in Table 5.

Figure 14.

Results of modifying the awareness transmission rate (α) for smallpox to reduce the epidemic size on the Copenhagen networks study dataset, with epidemiological study duration = 100, number of implementations = 20, γ = 0.8, $\varphi$ = 0.8, and the corresponding parameter values in Table 5.

6.2.2. Awareness transmission rate analysis

Figures 13 and 14 examine the effects of varying awareness transmission rate (α) values on the Copenhagen dataset for measles and smallpox, respectively. These results reveal even stronger parameter sensitivity than observed in synthetic networks.

For Measles MCPR with α = 0.9 achieves approximately 950 infections compared to 1,020 infections for Classical PageRank at 10% immunization, representing a 6.9% improvement. The steeper gradient observed with α variations suggests that the rate of awareness dissemination has a more immediate impact than information quality for rapidly spreading diseases, particularly in realistic social structures.

For Smallpox the improvement is even more substantial, with MCPR α = 0.9 reducing infections to 760 compared to 840 infections for Classical PageRank, yielding a 9.5% improvement. The convergence of MCPR curves at higher immunization percentages indicates that awareness transmission becomes less critical when sufficient immunization coverage is achieved.

7. Conclusions and future work

This study underscores the importance of the context functional to multiplex networks, which, in turn, shapes the identification of key nodes for immunization. Leveraging a two-layer network with disease spreading and awareness propagation features, we showed that each layer requires a distinct centrality measure: PageRank works best in the infection (physical) layer, gripping its potential in minimizing epidemic size, while closeness centrality performed better for supporting awareness flow in the virtual layer.

With that insight in mind, we proposed the MCPR method to inject centralities from both layers into a composite score to rank nodes for immunization. Simulation results incorporating measles and smallpox scenarios found that MCPR performed better than classical single-layer PageRank immunization and other multiplex ranking methods (e.g., MPR), particularly at lower immunization rates a crucial scenario when resources are scarce.

The effectiveness of our method was further validated through empirical testing on the Copenhagen Networks Study dataset, demonstrating that the advantages observed in synthetic networks translate effectively to real-world social structures with even more pronounced benefits. Specifically, on the real-world dataset, MCPR achieved a 2.2% improvement for measles and a substantial 7% improvement for smallpox compared to classical PageRank at 10% immunization coverage, with optimal parameter configurations (α = 0.9, $\varphi$ = 0.9) yielding even greater performance gains of up to 9.5% for awareness transmission rate optimization alone. Of particular significance was that the greater the rate of transmission of information, and the greater the quality of information (reflected in a larger awareness-index parameter), the greater the benefit of our method, illustrating the synergistic effect of combining infection and awareness dynamics.

These results demonstrate that MCPR outperforms both single-layer PageRank immunization and the existing MPR method, effectively reducing epidemic size by 30–37% over single-layer strategies and 13–19% over MPR implementations in limited immunization budgets. The real-world validation confirms these findings with measles epidemic reductions from 1,100 to 950 infections and smallpox reductions from 1,020 to 760 infections under optimal parameter settings, demonstrating practical applicability beyond synthetic network assumptions. These findings highlight the need for proper context-based centrality selection and for awareness layers in addressing immunization strategies on multiplex networks.

Incorporating layer-specific contexts in centrality definitions lays the groundwork for targeted immunization. Future research may focus on additional layers, inter-layer interactions, and finer behavioral assumptions, and eventually, real-world datasets once they become available. We hope that the context-centric method to multiplex network analysis described here inspires further developments in both theoretical modeling and practical strategies for epidemic containment.