The Identification of Influential Nodes Based on Neighborhood Information in Asymmetric Networks

Abstract

Identifying influential nodes, with pivotal roles in practical domains like epidemic management, social information dissemination optimization, and transportation network security enhancement, is a critical research focus in complex network analysis. Researchers have long strived for rapid and precise identification approaches for these influential nodes that are significantly shaping network structures and functions. The recently developed SPON (sum of proportion of neighbors) method integrates information from the three-hop neighborhood of each node, proving more efficient and accurate in identifying influential nodes than traditional methods. However, SPON overlooks the heterogeneity of neighbor information, derived from the asymmetry properties of natural networks, leading to its lower accuracy in identifying essential nodes. To sustain the efficiency of the SPON method pertaining to the local method, as opposed to global approaches, we propose an improved local approach, called the SSPN (sum of the structural proportion of neighbors), adapted from the SPON method. The SSPN method classifies neighbors based on the h-index values of nodes, emphasizing the diversity of asymmetric neighbor structure information by considering the local clustering coefficient and addressing the accuracy limitations of the SPON method. To test the performance of the SSPN, we conducted simulation experiments on six real networks using the Susceptible–Infected–Removed (SIR) model. Our method demonstrates superior monotonicity, ranking accuracy, and robustness compared to seven benchmarks. These findings are valuable for developing effective methods to discover and safeguard influential nodes within complex networked systems.

1. Introduction

Network science, the employment of nodes to represent entities and edges to denote connections, serves as a fundamental tool for comprehending and modeling complex systems across various disciplines [1]. It effectively describes and elucidates dynamic relationships and behaviors within these systems. As we know, finding the critical nodes is often considered the first step when studying complex networks. For example, transportation networks are vulnerable to disruptions due to accidents, congestion, or unexpected events. Identifying and fortifying vital nodes helps improve the resilience of the overall network [2]. Similarly, the identification of key operatives within the Mafia network [3] has been proven to be an effective strategy, leading to the total disruption of the criminal network in the least number of steps. Hence, identifying critical nodes is often of paramount theoretical and practical importance when investigating complex network systems in the real world, making it a highly significant topic within network science.

Up to now, researchers have introduced diverse methods across disciplines for identifying influential spreaders in complex networks, broadly categorized into four subclassifications [4]. The first, structural centrality, identifies critical nodes based on the network’s topological structure, encompassing methods such as degree, betweenness, and coreness. The second, iterative refinement centralities, considers dynamic processes within a static topological framework, accounting for the mutual enhancement effect [5]. Notably, the PageRank [6], VoteRank [7], and variants like eigenvector centrality [8] and leader rank centrality [9], fall into this category. The third subtype quantifies nodal importance through node operation, with the CI method [10] as a classic example. The final category, dynamic–sensitive methods, examines how a node’s influence varies with dynamic processes under different parameters, including the time-aware method [11].

Nonetheless, no single method currently can dominate in identifying influential nodes in complex networks, making finding the most critical nodes a formidable challenge. This study is committed to comprehensively investigating approaches to quantifying key nodes using network structural features. Lü et al. [4] provide more methods for interested readers in this field. Many methods related to topological structures have been proposed for finding the most influential spreaders in networks. From the perspective of a structural scale, the existing methods can be classified into global, local, and semi-local structural methods [12]. The global-structure-feature-based methods, including closeness centrality [13], betweenness centrality [14], coreness [15] and HybridRank Centrality [7], rely on global information to characterize nodal importance.

In contrast, the local-information-based method, with its relatively low time complexity for calculating local structure features, is gaining attention in influential network node identification. Lü et al. [16] proposed an h-index, balancing coreness and degree, but it focuses on high-influence neighbors, overlooking low-influence nodes with small degree values. Improved extension methods include the local h-index [17], extended h-index centrality [18], and the weighted h-index [19]. Liu et al. [20] introduced a method based on local structural similarity, utilizing neighbor analysis. In addition, topology features like the clustering coefficient are integrated for accurate nodal influence quantification; for instance, Gao et al. [21], by introducing local structure centrality using a clustering coefficient, proposed a novel method called local centrality. Zhu et al. [22] superimposed node degree centrality with neighbor layer nodes, achieving more effective results than traditional centrality algorithms. Furthermore, semi-local structural methods, integrating local and global perspectives, include the entropy-based ranking measure (ERM) [23] and extended cumulative coefficient ranking method (ECRM) [24].

This study primarily focuses on methodologies for quantifying critical nodes, emphasizing network structural attributes, particularly leveraging local structural characteristics. Notably, a novel centrality method, SPON (sum of proportion of neighbors) [25], balances algorithmic efficiency and accuracy. Despite its lower time complexity for large-scale networks, SPON falls short of accuracy compared to the degree and h-index. To address this, we propose an improved approach, called the sum of the structural proportion of neighbors (SSPN), based on the SPON. The SSPN maintains the simplicity and efficiency of the original method while preserving its capability to identify highly influential nodes and enhancing its performance in identifying nodes critical for the propagation process. Our improvements incorporate the h-index and local clustering coefficients, resulting in a better alignment with the Susceptible–Infected–Removed (SIR) model, retaining monotonicity and robustness without added computational complexity.

The structure of this study is as follows: Firstly, we present our proposed method and benchmark methods in Section 2. Subsequently, Section 3 delineates our experimental setup and evaluation metrics. Section 4 presents the experimental results by calculating the monotonicity, accuracy, and robustness performance. Finally, Section 5 summarizes our primary findings and shortcomings and outlines potential directions for future work.

2. Methods

This section primarily elucidates the node centrality employed in this research. We express an asymmetric undirected network as a graph $G=\left(V,E\right)$, where $V$ signifies the node set ($\left|V\right|$ is the number of nodes in the network) and $E$ represents the edge set ($\left|E\right|$ presents the number of edges). The adjacency matrix $A$ serves as the representation of the network’s topological structure, reflecting the connections between nodes. If $a_{ij}$ is equal to 1, it indicates a connection between nodes $v_i$ and $v_j$, while $a_{ij}=0$ denotes the absence of a connection.

2.1. Benchmark Methods

2.1.1. Baseline Methods: Degree ($d$), K-Shell ($k^s$), and H-Index ($h$)

(1)

Degree ($d$)

Degree ($d$) is the primary method used to characterize node importance [26]. For node $v_i$, its degree defined as

$$d_i=\left|N_i^1\right|$$

(1)

where $N_i^1$ denotes the set of neighbors of $v_i$ in the one-hop neighborhood.

(2)

K-shell ($k^s$)

Kitsak et al. [15] introduced the coreness ($k^s$) method, which assigns nodes to specific layers, for identifying the node importance within a network. The $k^s$ value for each node is determined through a sequential pruning process, starting by removing nodes with a given degree value ($d_i=1$) from the network. Subsequently, higher k-shell values are successively removed until no nodes remain in the network. Consequently, each node is associated with a $k^s$ index, and the network can be stratified into k-shell layers.

(3)

H-index ($h$)

The h-index ($h$) was initially designed as a metric to assess individual researchers’ scholarly impact and productivity, primarily based on the citation counts of their publications [16]. Recently, its applicability has been extended to identify pivotal nodes. The h-index for node $v_i$ in a network with $k_i$ neighbors is defined as

$$h_i=\mathcal{H}\left(d_{j1}, d_{j2},\dots , d_{js},\dots , d_{j{k}_i}\right),j\in N_i^1$$

(2)

where the operator $\mathcal{H}$ determines the maximum integer $h$ for which there exist at least $h$ neighbors whose degrees are greater than or equal to $h$.

2.1.2. Extended Methods: Social Capital ($S^c$), Local H-Index ($L^h$), Neighborhood Coreness ($C^{n\mathrm{c}+}$)

(1)

Social Capital ($S^c$)

Zhou et al. [27] introduced the concept of “Social Capital” as a concise method for characterizing local centrality and identifying fast influencers within intricate network structures. Their premise revolves around the notion that a node assumes a pivotal role in a network if it exhibits a high degree and shares connections with numerous high-degree neighboring nodes. The formal definition of social capital ($S^c$) is articulated as

$$S_i^c=d_i+\sum _{j\in N_i^1}{d}_j.$$

(3)

(2)

Local h-index ($L^h$)

Liu et al. [17] overcome the weakness of the h-index model by putting forward a modified method named local h-index ($L^h$) centrality. The method simultaneously accounts for both the intrinsic influence of the node itself and that of its neighbors. This composite metric comprises two distinct components: the initial component signifies the node’s h-index, which gauges its influence concerning the number of high-quality neighbors. The $L^h$ value of $v_i$ is defined as

$$L_i^h=h_i+\sum _{j\in N_i^1}{h}_j.$$

(4)

(3)

Neighborhood coreness ($C^{nc}$)

The influence of a node in a network is often associated with the number of neighbors it has, as this connectivity reflects its potential to affect or be affected by various processes within the network. Hence, Bae et al. [28] developed a ranking method that simultaneously considers the degree and coreness of a node, and defines the neighborhood coreness $C^{nc}$ of node $v_i$ as

$$C_i^{nc}=\sum _{j\in N_i^1}{k}_j^s,$$

(5)

where $k_j^s$ is the $k$-shell index of its neighbor $v_j$.

Recursively, the extended neighborhood coreness $C^{nc+}$ of node $v_i$ is defined as

$$C_i^{nc+}=\sum _{j\in N_i^1}{C}_j^{nc}.$$

(6)

2.2. Proposed Methods

The recently proposed SPON method [25] concerning the sum of the proportion of neighbors only requires information within the three-hop neighborhood of the node. The SPON method involves a two-step calculation process. First, the neighbor proportion (i.e., $PN$) is defined as

$${PN}_i=\frac{\left|N_i^2\right|}{\left|N_i^1\right|},$$

(7)

where $N_i^1$ is the set of the first-order neighbors and $N_i^2$ is the set of the second-order neighbors.

Then, the summation of neighbor proportions, denoted as $SPON$, for node $v_i$ is defined as

$${SPON}_i=\sum _{j\in N_i^1}{PN}_j.$$

(8)

A higher $SPON$ value of one node represents its more significant influence and its importance. As mentioned earlier, a higher $PN$ value implies that the node can establish connections with a more significant number of second-order neighbors through a small number of first-order neighbors, indicating the potential ability of node $v_i$ through infecting its neighbor $v_j$. The influence of $v_i$ on the network is necessarily high when it and its neighboring hub are impacted and could spread to the other part of the network. The sum of $PN$ that is the $SPON$ can be considered the importance of $v_i$.

Taking Figure 1 as an illustrative example, we focus on node $v_i$, in whose vicinity are four pink nodes (i.e., $v_1$, $v_2$, $v_3$, and $v_4$), denoted as $N_i^1$, and orange nodes, which are its second-order neighbors (i.e., $N_i^2$). It is clear that $\left|N_i^1\right|=4$ and $\left|N_i^2\right|=8$, respectively. The value of $P{N}_i$ is 2 according to Equation (7). If the second-order neighbors of $v_i$ are disregarded, $v_i$ is directly connected to its first-order neighbors, and there is no indirect connection between these neighbors. These first-order neighbors of node $v_i$ are entirely identical, while the connection structure between the first-order neighbors and the second-order neighbors of $v_i$ is entirely different. Based on the discussions of Liu et al. [29] and Yu et al. [30], once a disease originates from node $v_i$ and propagates through one path, $v_i$ will be the initial infected node. Subsequently, node $v_1$ is infected, and the contagion may extend to the second-order neighbor nodes via $v_1$, to $v_5$ and $v_6$. A similar scenario may also occur within $v_2$ and its neighbors in $N_i^2$. However, due to the entirely distinct structures between nodes $v_{10}$, $v_{11}$, $v_{12}$, and $v_2$, the propagation effect of $v_1$ and $v_2$ are completely different. Nevertheless, the $PN$ neglects the difference, considering only the quantity of second-order neighbors.

To address this limitation, this study introduces an innovative and effective SSPN model by partitioning each second-order neighboring node into distinct regions and assigning weights to the regions based on their features. The proposed method relies on the h-index values of each node to subdivide the second-order neighbor set $N_i^2$ into subregions and allocate weights accordingly.

As illustrated in

Table 1. The ranking results of simple networks simulated using the SIR model and measured by SPON and SSPN.

Rank	SIR	SPON	SSPN
1	i	2	2
2	3	3	3
3	2	1	1
4	1	I	4
5	4	4	i
6	7	7	7
7	10	5, 6	5, 6
8	6	8, 9, 10, 11, 12	10, 11, 12
9	5		8, 9
10	12
11	11
12	9
13	8

, each node in the simple network exhibits its spreading ability based on the SIR model simulation. Notably, SPON only classifies the nodes into eight groups, aligning with our prior discussion. Upon examining the graph in Figure 1, we determine the epidemic threshold (i.e., ${\beta }^c$) to be 0.361, choosing a $\beta$ value (0.37) slightly larger than ${\beta }^c$ for subsequent comparison. In the computation results, we observe ${\tau }_{SPON}=0.733$ and ${\tau }_{SSPN}=0.755$, clearly indicating an improvement in our method over SPON.

However, we find that SSPN also attributes nodes to distinct ranking positions, but these ranking results are different from the SIR model results, as shown in Table 1. We attribute this phenomenon to the inherent randomness of SIR infection in small networks, diminishing its advantage result. Nevertheless, as the network expands, the advantages of our method will become more apparent, as substantiated by subsequent accuracy validations.

Initially, we utilize information from first-order neighboring nodes as the foundation for accurately delineating the subregions of second-order neighboring nodes. The proportion $P_p^i$, delineated by the first-order neighbors, is defined as

$$P_p^i=\frac{\left|N_{\left(1,p\right)}^i\right|}{\left|N_1^i\right|},p={\alpha }_1,{\alpha }_2,$$

(9)

where $\left|N_1^i\right|$ represents the number of first-order neighbors of node $v_i$, $\left|N_{\left(1,p\right)}^i\right|$ signifies the quantity of the subcategory $p$ within the first-order neighbors, the subcategory ${\alpha }_1$ comprises nodes with an h-index greater than or equal to the h-index of node $v_i$, and the subcategory ${\alpha }_2$ comprises nodes with an h-index less than that of node $v_i$.

Subsequently, the values of $P_p^i$ for two categories are mapped onto the set of second-order neighbors, with the quantity of the two sets being defined as

$$\left|N_{\left(2,p\right)}^i\right|=\left\{\begin{array}{cc}\left[P_p^i\times \left|N_2^i\right|\right]& p={\alpha }_1\\ \left|N_2^i\right|-\left|N_{\left(2,{\alpha }_1\right)}^i\right|& p={\alpha }_2\end{array}\right..$$

(10)

It is important to note that the computation of $P_p^i$ may result in decimals. A floor function, denoted by the square bracket (“[]”), is adopted to calculate the $\left|N_{\left(2,\alpha \right)}^i\right|$. In detail, the nodes belonging to the second-order neighbors $N_2^i$ are ranked according to their h-index and then segmented by the set sizes from Equation (10). The nodes with higher h-index values are grouped into $N_{\left(2,{\alpha }_1\right)}^i$, and the remaining nodes are collected in $N_{\left(2,{\alpha }_2\right)}^i$.

After segmenting the second-order region, we evaluate the structural properties of different areas. This analysis is centered on measuring the structural features of various categorized regions, employing clustering coefficients for precision. The clustering coefficient is defined as

$${Cl}_p^i=\left\{\begin{array}{cc}0& \left|N_{\left(1,p\right)}^i\right|=1\\ \frac{2\left|E_{\left(2,p\right)}^i\right|}{\left|N_{\left(1,p\right)}^i\right|\left(\left|N_{\left(1,p\right)}^i\right|-1\right)}& \left|N_{\left(1,p\right)}^i\right|>1\end{array}\right.,p={\mathsf{\alpha }}_1,{\mathsf{\alpha }}_2,$$

(11)

where $\left|E_{\left(2,p\right)}^i\right|$ denotes the number of edges between nodes within the node set $N_{\left(2,p\right)}^i$.

Building upon the earlier discussion, we introduce a novel approach for quantifying the significance of nodes. This is achieved by emphasizing the uniqueness of various regions by summating the h-index values within different categorized regions. In alignment with this concept, we redefine a method known as $SPN$, the definition of which is as follows:

$${SPN}_i=\sum _{p\in {\mathsf{\alpha }}_1,{\mathsf{\alpha }}_2}\frac{\left|N_{\left(2,p\right)}^i\right|{\sum }_{j\in N_{\left(2,p\right)}^i}{h}_j}{\left|N_1^i\right|(1+{Cl}_p^i)}.$$

(12)

Lastly, neighborhood rules should effectively enhance the accuracy of identifying the most influential nodes. The influence of node $v_i$ extends beyond itself and encompasses the nodes in its vicinity. Along these lines, the $SSPN$ model can be regarded as an evolution of the $SPN$ model, defined through the aggregation of neighboring nodes, as

$${SSPN}_i=\sum _{j\in N_1^i}S{PN}_j.$$

(13)

To explore the core algorithm of SSPN, we use node $v_i$ in Figure 1 as an illustrative example. Node $v_i$ is surrounded by four neighboring nodes, which are nodes $v_1$, $v_2$, $v_3$, and $v_4$, constituting the set $N_1^i$. It is evident that $\left|N_1^i\right|=4$, and similarly that $\left|N_2^i\right|=8$. Next, we calculate the values of $P$ based on Equation (9), enabling us to categorize the neighborhood into two classes based on their h-index values. For instance, nodes with an h-index greater than or equal to $v_i$ are presented as $v_2$, $v_3$, and $v_4$, and grouped as $N_{(1,{\alpha }_1)}^i$. Hence, we obtain $\left|N_{(1,{\alpha }_1)}^i\right|=3$, and similarly, $\left|N_{(1,{\alpha }_2)}^i\right|=1$. Subsequently, we determine that $P_{{\alpha }_1}^i=3/4$, and $P_{{\alpha }_2}^i=1/4$. The eight nodes in the second-order neighborhood of $v_i$ are divided into two categories according to their value of $P$. The six nodes with a higher h-index are grouped into $N_{(2,{\alpha }_1)}^i$, which includes nodes $v_7$, $v_8$, $v_9$, $v_{10}$, $v_{11}$, and $v_{12}$. The rest of the nodes, $v_5$ and $v_6$, belong to the set $N_{(2,{\alpha }_2)}^i$. Afterwards, we must compute the clustering coefficients ($C{l}_p^i$) for the different categories. $N_{\left(1,{\alpha }_2\right)}^i$ contains only one node, resulting in ${Cl}_{{\alpha }_2}^i=0$. Since $\left|N_{\left(1,{\alpha }_1\right)}^i\right|$ > 1, we can calculate that ${Cl}_{{\alpha }_1}^i=\frac{2\times 4}{3\times (3-1)}=\frac{4}{3}$, where 4 is the number of edges between nodes $v_7$, $v_8$, $v_9$, $v_{10}$, $v_{11}$, and $v_{12}$ in set $N_{(2,{\alpha }_1)}^i$. Finally, based on the calculation above, we arrived at $SP{N}_i=\frac{6\times \left(3+3+3+2+2+2\right)}{4(1+\frac{4}{3})}+\frac{2\times \left(1+1\right)}{4(1+0)}=10.64$.

Observing the above algorithms, in comparison to SPON, our method has not introduced excessive complexity; rather, it simply involves an additional round of weighted classification. Specifically, the time complexity of SSPN is mainly divided into two stages. First, the calculation of the $P_p^i$ values involve the second-order neighbors of each node. If the average degree is used to represent the number of neighbors, denoted as $\overline{n}$, the time complexity of this stage is $O\left({\overline{n}}^2\right)$. Then, the second stage, concerning the categorized $P_p^i$ values of the immediate neighbors, has the time complexity of $O\left(n\right)$. Based on the two stages, the overall time complexity of SSPN is determined by the size of neighbors of each node, taking the value of $O\left({\overline{n}}^2\right)$.

3. Evaluation Methods

3.1. Single Seed SIR Model

To objectively assess the SSPN method used in this study, we adopt the widely applied Susceptible–Infectious–Recovered (i.e., SIR) model for simulating the dynamics of infectious disease propagation. It stratifies the nodes into three distinct categories: susceptible (S), infected (I), and recovered (R). According to Moreno [31], only $v_i$ will be set as the $I$ state at the beginning of the simulation, while other nodes are in $S$ states. In each simulating round, the $I$-state node infect its $S$-state neighbors with probability $\beta$. The infected nodes are recovered as $R$-state with probability $\gamma$ and cannot be infected in the subsequent rounds. This process continues until the number of infected nodes does not increase. The number of recovered nodes (N) in the whole network is the influence strength of the seed $v_{\mathrm{i}}$ (i.e., the first infected node). The numerical simulation is performed $T$ times, and the average over this period of simulation is analyzed. The infected probability $\beta$ will be set slightly higher than the epidemic threshold ${\beta }_C$ established by Bae and Kim [28]. The dynamic equations governing the SIR model are defined as

$$\left\{\begin{array}{c}dS/d\left(t\right)=-\beta SI\\ dI/d\left(t\right)=\beta SI-\gamma I\\ dR/d\left(t\right)=\gamma I\end{array}\right.,$$

(14)

where the disease transmission rate is symbolized as $\beta$, characterizing the ease with which the disease spreads from one individual to another. In parallel, the recovery rate is $\gamma$, signifying the velocity at which infected individuals recover and remain immune to the disease.

The spreading influence of node $v_i$ is evaluated as

$$N^a=\frac{nI+nR}{N^{it}},$$

(15)

where $N^{it}$ denotes the number of iterations, $nI$ represents the number of infected nodes, and $nR$ represents the number of nodes that have undergone recovery. It is pertinent to mention that we have established 1000 independent runs with 1000 iterations and $\gamma =1$, which can acquire suitable values for this calculation.

It is worth mentioning that the infection probability $\beta$ is larger than the epidemic criticality ${\beta }_C$ in our simulation of the SIR model. An epidemic’s critical threshold [31] is defined as

$${\beta }_C=\frac{〈k〉\lambda }{〈k^2〉}, \lambda =1.0,$$

(16)

where $\langle k\rangle$ denotes the average degree, 〈$k^2$〉 denotes the average of the square of the degree.

Inherent dynamics inform this deliberate choice of infectious disease. A higher $\beta$ensures a more contagious pathogen, leading to rapid transmission and sustained prevalence. This prioritization acknowledges the need for proactive measures to effectively curb the impact of highly infectious diseases, emphasizing the importance of strategies that outpace the epidemic’s critical threshold.

3.2. Evaluation Indicators

A series of indicators, including monotonicity, accuracy, and robustness, are arranged to evaluate the proposed SSPN method compared to various benchmark methods. The results obtained from the SIR model are deemed the standard values.

3.2.1. Monotonicity ($M$)

Centrality should be able to uniquely determine the node’s importance rather than roughly categorizing it. This study employs $M$ [28] to measure the monotonicity of the node’s importance ranking, which is defined as

$$M_{\mu }^R={\left[1-\frac{{\sum }_{r_i\in R}{n}_{r_i}(n_{r_i}-1)}{\left|R\right|\left(\left|R\right|-1\right)}\right]}^2,$$

(17)

where $R$ denotes the ranking list generated by the ranking method $\mu$, and $n_{r_i}$ represents the count of nodes at the rank $r_i$ in the ranking list. The $M$ ranges from 0 to 1, where an $M$ value closer to $1$ indicates a higher monotonicity.

3.2.2. Complementary Cumulative Distribution Function ($CCDF$)

The $CCDF$ is employed to elucidate the ranking distribution of various measuring approaches [32]. It quantifies the probability of a random variable, denoted as $x$, exceeding a specified threshold and is defined as

$${CCDF}_x={Prob}_{X>x}=1-{CDF}_x,$$

(18)

where ${CDF}_x$ signifies the cumulative distribution function, denoting the probability that a random variable is less than or equal to $x$. Generally, a more significant number of nodes sharing the same rank leads to a steeper decrease in the $CCDF$ plot. In brief, if a curve exhibits a rapid rate of decrease, the method employed to compute this curve might have many nodes that are challenging to distinguish effectively, indicating a lower quality of the measurement method.

3.2.3. Kendall Correlation Coefficient ($\tau$)

The Kendall correlation coefficient ($\tau$) [33] is a quantitative measure for assessing the correlation between two ordinal ranking lists. It provides a numerical representation within the interval $[-1, 1]$, with a higher $\tau$ score indicating a stronger resemblance between the two ranking lists (i.e., $R_1$ and $R_2$), while a lower score signifies a more significant dissimilarity. The coefficient ($\tau$) is defined as

$$\tau \left(R_1,R_2\right)=2\frac{n_c-n_d}{\left|R_1\right|\left(\left|R_1-1\right|\right)},$$

(19)

where $n_c$ represents the count of concordant pairs and $n_d$ represents the count of discordant pairs in $R_1$ and $R_2$.

3.2.4. Robustness ($R$)

The connectivity test represents a superior method for assessing the model’s ability to identify important nodes and edges. In this approach, nodes are systematically removed based on their ranking, and the size of the most significant connected component (MSCC) in the remaining network is recorded after each removal. A noticeable reduction in the MSCC size indicates rapid network structure degradation, demonstrating the method’s effectiveness in efficiently identifying influential nodes. The robustness metric $R$, a widely accepted measure of network connectivity [34], is consistently employed and is defined as follows:

$$R=\frac{1}{\left|N\right|}\sum _{v_i\in N}\frac{{|N}_{v_i}|}{\left|N\right|},$$

(20)

where $N$ represents the set of nodes in the network, and $N_{v_i}$ denotes the MSCC after removing the node $v_i$ from $N$. The ratio ${|N}_{v_i}|/|N|$ represents the robustness measure $r_i$ concerning node $v_i$. The robustness metric $R$ quantifies the sum of $r$ values obtained during the node removal process. The normalization factor $1/\left|N\right|$ ensures that $R$ values can be compared across networks of different sizes.

4. Experimental Results

In this section, we assess the performance of eight identification methods, including the proposed SSPN method, three baseline methods ($d$, $k^s$, $h$), three extended methods $S^c$, $L^h$, $C^{nc+}$), and the SPON method, by using empirical networks in different domains. They include Dolphins [35] (a network portraying the social interactions among 62 dolphins), Netsci [36] (a co-authorship network consisting of 379 scientists collaborating on publications related to network analysis), Fb-pages-foods [37] (Datasets of Facebook pages from November 2017, encompassing diverse categories represented by blue-verified page networks), Euroroad [38] (an extensive infrastructure network predominantly covering Europe), Facebook (A social network representing user interactions worldwide [39]), and power grid (an integral infrastructure network spanning the western United States, ensuring vital regional electricity distribution [40]). The networks mentioned above are extensively utilized in the literature for critical node identification, showcasing a level of representativeness in our network selection. For instance, the power grid network serves as a prominent example; influential articles on the identification of critical nodes in complex networks, such as those by Wu et al. [41] and Ai et al. [42], published in Physica A, have employed this network. Additionally, the diverse structural characteristics of these networks, as they are part of infrastructure networks, substantiate the generality of our approach. The details of these networks are summarized in

Table 2. Topological characteristics of six real-world networks and their parameters including their total number of nodes ($\left|N\right|$), number of edges ($\left|E\right|$), epidemic threshold (${\beta }^c$), average degree (〈k〉), maximum degree ($k^{max}$), mean shortest path length ($〈l〉$), maximum $k$-shell ($k^{s, max}$), clustering coefficient ($c$), network diameter ($D$), and assortative matches ($A$).

Network	$\left|\mathit{N}\right|$	$\left|\mathit{E}\right|$	${\mathit{\beta }}^{\mathit{c}}$	$〈\mathit{k}〉$	${\mathit{k}}^{\mathit{m}\mathit{a}\mathit{x}}$	$〈\mathit{l}〉$	${\mathit{k}}^{\mathit{s},\mathit{m}\mathit{a}\mathit{x}}$	c	D	A
Dolphins	62	159	0.1470	5.1290	12	3.3570	6	0.2590	8	−0.044
Netsci	379	914	0.1247	4.8230	34	6.0420	22	0.7410	17	0.0817
Fb-pages-foods	620	2091	0.0502	6.7450	132	5.0890	33	0.3310	17	−0.032
Euroroad	1039	1305	0.3240	2.5120	10	18.395	7	0.0190	62	0.1267
Facebook	4039	88234	0.0094	43.691	1045	3.6930	374	0.6060	8	0.0636
Power grid	4941	6594	0.2583	2.6690	19	18.989	20	0.0801	46	0.0035

.

4.1. Monotonicity

We first evaluate the monotonicity of SSPN and the other methods through the $M^R$ listed in

Table 3. The monotonicity $M_{\mu }^R$ measured by the $\mu$ method, including the proposed SSPN ($SSPN$), degree ($d$), k-shell ($k^s$), h-index ($h$), social capital ($S^c$), local h-index ($L^h$), neighborhood coreness ($C^{nc+}$), and SPON model ($SPON$). The bold text denotes the best result, and the underlined text denotes the second-best result.

Network	${\mathit{M}}_{\mathit{d}}^{\mathit{R}}$	${\mathit{M}}_{{\mathit{k}}^{\mathit{s}}}^{\mathit{R}}$	${\mathit{M}}_{\mathit{h}}^{\mathit{R}}$	${\mathit{M}}_{{\mathit{S}}^{\mathit{c}}}^{\mathit{R}}$	${\mathit{M}}_{{\mathit{L}}^{\mathit{h}}}^{\mathit{R}}$	${\mathit{M}}_{{\mathit{C}}^{\mathit{n}\mathit{c}+}}^{\mathit{R}}$	${\mathit{M}}_{\mathit{S}\mathit{P}\mathit{O}\mathit{N}}^{\mathit{R}}$	${\mathit{M}}_{\mathit{S}\mathit{S}\mathit{P}\mathit{N}}^{\mathit{R}}$
Dolphins	0.8312	0.5576	0.6841	0.9675	0.9592	0.9406	0.9968	0.9979
Netsci	0.7642	0.7026	0.6825	0.9694	0.9513	0.9712	0.9943	0.9951
Fb-pages-foods	0.8152	0.7587	0.7681	0.9865	0.9760	0.9789	0.9978	0.9989
Euroroad	0.4442	0.3312	0.2534	0.8621	0.7165	0.8058	0.9504	0.9892
Facebook	0.9739	0.9721	0.9665	0.9995	0.9990	0.9997	0.9999	0.9999
Power grid	0.5927	0.3713	0.3930	0.9048	0.8262	0.8267	0.9866	0.9981
Average	0.7369	0.6156	0.6246	0.9483	0.9047	0.9205	0.9876	0.9965

. The monotonicity ($M$) of these networks varies with different methods. The $k^s$ exhibits the poorest performance among the six real networks examined. This is primarily attributable to the possibility of multiple nodes within the $k^s$ having identical $d$ values, leading to a suboptimal monotonicity performance in most networks. The $d$ and $h$ share a similar characteristic in terms of $M$. The extended methods (i.e., $L^h$, $S^c$, and $C^{nc+}$) exhibit better performances than the baseline methods, effectively distinguishing differences in node importance. In addition, the SPON method displays a distinct advantage, demonstrating superior monotonicity in most networks. The proposed SSPN method outperforms the SPON method, surpassing other models or sharing the top position. The SSPN method maintains a stable performance across networks of various scales, achieving absolute or near-absolute uniqueness. Regarding uniqueness rankings, all methods exhibit an inferior performance in the Euroroad network. This performance discrepancy can be attributed to the structural attributes of the nodes within this network, as Euroroad exhibits the smallest values of $\langle k\rangle$, $k^{max}$, and $c$ among all networks, indicating that many nodes in this network possess similar local structures. Consequently, models relying on nonidentical structural attributes across these nodes struggle to distinguish the significance of the nodes, resulting in lower uniqueness rankings. Additionally, compared to the other seven methods, our SSPN method attains the highest average monotonicity, approaching one and surpassing the second-ranked SPON method by 0.0089. Therefore, our proposed SSPN method is a high-resolution measurement approach, with a verified, significant performance across most networks and showcasing superior node influence discrimination capabilities compared to other methods.

To elucidate the ranking distribution of the different methods and the superiority of the SSPN method, in addition to the aforementioned monotonicity measurements, we have also employed $CCDF$ to illustrate the comparison depicted in Figure 2. These $CCDF$ plots showcase the performance of various methods across the six networks. According to $CCDF$ principles, when multiple nodes occupy the same rank, the $CCDF$ plot exhibits a rapid decline in slope. Conversely, the $CCDF$ plot slope becomes gentler when nodes have varying ranks. A gentler slope to the $CCDF$ plot implies a better performance in the ranking distribution and a more discernible rank allocation. Among the $CCDF$ plots, the three baseline methods (i.e., $d$, $k^s$, and$h$) exhibit relatively inferior performances. In contrast, the extended methods (i.e., $S^c$, $L^h$, $C^{nc+}$) yield a more diverse array of measurement outcomes, with their $CCDF$ trends approaching zero, displaying similar slopes. Notably, in the contexts of Dolphins, Fb-pages-foods, and Facebook, the $CCDF$ lines of the SSPN method consistently follow diagonal lines with slower slopes than other methods, signifying a more competitive performance. The performance of the SPON method is slightly inferior to that of the SSPN. Among the other seven methods, SPON achieved the second-best performance. Across these six real networks, the SSPN demonstrates superior performance in both monotonicity and distribution characteristics. Notably, for the Facebook dataset, the $M_{SSPN}^R$ value of the SSPN method demonstrates near-absolute uniqueness ($M_{SSPN}^R=0.9999$). The SSPN demonstrates its outstanding ability to identify pivotal nodes within information propagation.

4.2. Accuracy

To illustrate the superiority of our proposed method’s accuracy, we compare the spreading capacity of nodes simulated by the SIR model with the ranking results calculated by different methods and assess their accuracy through the Kendall correlation coefficient $\tau$. A higher $\tau$ indicates a more accurate ranking of influential nodes. As shown in

Table 4. The accuracy performance of various methods applied to real-world networks, including the degree ($d$), k-shell ($k^s$), h-index ($h$), social capital ($S^c$), local h-index ($L^h$), neighborhood coreness ($C^{nc+}$), SPON model ($SPON$), and SSPN model ($SSPN$). $\beta$ is the infection rate, ${\beta }^c$ is the epidemic threshold. The bold text denotes the best result, the underlined text denotes the second-best result.

Network	${\mathit{\beta }}^{\mathit{c}}$	$\mathit{\beta }$	$\mathit{d}$	${\mathit{k}}^{\mathit{s}}$	$\mathit{h}$	${\mathit{S}}^{\mathit{c}}$	${\mathit{L}}^{\mathit{h}}$	${\mathit{C}}^{\mathit{n}\mathit{c}+}$	$\mathit{S}\mathit{P}\mathit{O}\mathit{N}$	$\mathit{S}\mathit{S}\mathit{P}\mathit{N}$
Dolphins	0.147	0.34	0.75	0.662	0.811	0.776	0.813	0.718	0.746	0.862
Netsci	0.125	0.24	0.489	0.458	0.495	0.716	0.587	0.581	0.592	0.772
Fb-pages-foods	0.0502	0.12	0.497	0.533	0.535	0.698	0.68	0.663	0.608	0.880
Euroroad	0.333	0.50	0.46	0.509	0.518	0.617	0.582	0.577	0.526	0.699
Facebook	0.009	0.02	0.635	0.681	0.669	0.777	0.748	0.728	0.467	0.801
Power grid	0.258	0.36	0.463	0.405	0.486	0.645	0.572	0.566	0.501	0.680

, one can observe that the proposed SSPN method, which is highly correlated with the size of the infected population, outperforms the other methods. SSPN obtains the maximum value in all network cases, outperforming the baseline methods ($d$, $k^s$, and $h)$, and extended methods ($S^c$, $L^h$, $C^{nc+}$), as well as SPON. This demonstrates the advantage of our proposed method. The value of SSPN is 20% higher than that of $S^c$ in the Fb-pages-foods network, and we are even pleasantly surprised to discover that our SSPN method outperforms the comparison method of SPON for all $\tau$ values. Thus, it can be observed that the further consideration of network structure characteristics in the SSPN method based on SPON is effective in improving the accuracy of identifying the critical nodes. Furthermore, $S^c$ exhibits a stable performance since it produces the second-best $\tau$ in almost all networks. In contrast, our method exhibits a more stable performance and further advantages at the given $\beta$ value.

We further vary the $\beta$ in the SIR model with an extensive range to assess the stability of its accuracy performance. As shown in Figure 3, SSPN, marked by red diamonds, exhibits a higher accuracy compared to the other seven methods in most cases, especially when the $\beta$ is larger than the epidemic threshold (i.e., ${\beta }^c$). In other words, the ranking list generated by SSPN is similar to the ranking list generated by the SIR model. SSPN consistently demonstrates a clear variation trend in its performance and maintains a prominent position across various parameters. In the Euroroad and Fb-pages-foods networks, the correlation values between $\mathrm{S}\mathrm{S}\mathrm{P}\mathrm{N}$and the spreading ability measured by the SIR model are always at the top of all the curves, while in the other real-world networks, SSPN also performs better over a wide range of $\beta$. However, in specific instances, certain improved methods demonstrate similar trends and, in some cases, attain the highest accuracy within a localized range of $\beta$. For instance, the $\tau$ measured by $S^c$ is the largest when $0.25<\beta <0.30$ in the Power grid network. However, it loses its advantage when $\beta \ge 0.30$, which is where SSPN performs very competitively. The baseline methods (i.e., $d$, $k^s$, and $h$) exhibit less reliability within the studied networks, particularly the $h$; their curves consistently reside at the bottom of almost all the methods, such as in the Dolphins, Power grid, and Netsci networks, demonstrating the worst performances.

We further investigate the diversity of the eight methods by computing the Kendall correlation coefficient (Kendall’s Tau ($\tau$)) between the ranking lists obtained by proposed method and the others. From Figure 3, it is evident that SSPN, along with the extended methods ($S^c$, $L^h$, $C^{nc+}$), typically ranks above other methods. Similar conclusions can be drawn from Figure 4. The $\tau$ values between SSPN and the extended methods exceed 0.7, and in some networks, even surpass 0.8. This indicates a strong similarity in the ranking of node centrality between SSPN and the extended methods. Because they count cumulative neighbor influences, these methods demonstrate improved accuracy. However, among all metrics, SPON exhibits a higher correlation with baseline methods ($d$, $k^s$, and $h$) that have moderate identification accuracy. In contrast, the correlation value between SSPN and SPON is average across all indicators. It is evident that methods with more pronounced identification abilities demonstrate higher correlations.

4.3. Robustness

To evaluate the robustness of the proposed SSPN method, we conducted a comprehensive examination pertaining to a system or network’s capacity to maintain stability and functionality amidst disturbances, attacks, or variations. Our assessment involved monitoring changes in network robustness during the node removal process and evaluating the magnitude of the robustness metric, denoted as $R$. In the experiment, nodes within the empirical network were scored using the SSPN and other seven methods. Nodes were then ranked based on their scores (i.e., importance), with those possessing higher values occupying the top positions. Subsequently, nodes were removed in order, according to their ranks, and the network robustness was calculated based on the remaining nodes.

Figure 5 illustrates the robustness performance of the proposed SSPN approach across six real-life networks. It showcases the network collapse process within these networks as nodes are successively removed, employing both the SSPN method and conventional node identification methods. The graphical depiction vividly illustrates the consistent positioning of the robustness curve attributed to SSPN, which is consistently at lower levels across all six networks. Notably, the area bound by this curve and the axis exhibits the most constrained dimensions, underscoring the better efficacy of SSPN in fostering heightened network resilience. This evidence shows that after further modification, the SSPN still retains the advantage of SPON under network destruction and performs well on various networks, with the ability to find relatively important key nodes.

5. Conclusions

When modeling practical problems such as epidemic management, social information dissemination optimization, and transportation network security enhancement as complex networks, identifying and ranking critical nodes has emerged as one of the most crucial and challenging issues of network science.

In this study, we attribute the relatively lower accuracy of the newly developed SPON method for finding influential nodes to its neglecting of the structural differences of its neighboring nodes. This study introduces the sum of the structural proportion of neighbors (SSPN), aimed at bolstering the efficacy of node identification. We consider the structural characteristics of categorized regions by leveraging weighted neighborhood proportions and employing the h-index to categorize neighborhoods. The SSPN exhibits superior performance, including monotonicity, accuracy, and robustness, outperforming seven benchmark models across various performance evaluation methods and mimicking the identification results calibrated by the recognized SIR model.

Although our proposed method performs better in terms of identifying influential nodes, there is still room for improvement. The experimental results are only arranged with respect to the unweighted and undirected networks used in this study. The judgment and modification of weighted undirected networks will be one of the main directions for future research. Moreover, we will continue to study large networks to further improve the SSPN method. More testing networks with various structures and characteristics will help further validate the accuracy and robustness of the proposed method.