Identifying influential nodes through hierarchical k-shell and extended neighborhood integration

Abstract

The identification of influential nodes has extensive applications in complex network research. To address the challenge of balancing accuracy and computational efficiency in existing methods, this paper proposes an algorithm named HKEN that integrates hierarchical k-shell decomposition with extended neighborhood information. The approach incorporates both global structural features and local topological information of nodes. First, the coarse-grained hierarchical mechanism of the k-shell algorithm is optimized, and the initial weight of nodes is dynamically computed based on their degree and k-shell values. Second, the neighborhood range of nodes is extended, and a local clustering coefficient is introduced to establish a threshold for information transmission distance. Finally, an influence aggregation strategy based on the Jaccard similarity coefficient is proposed. The performance of HKEN was validated through comparative experiments on 10 real-world networks against 12 benchmark methods. Experimental results demonstrate that the proposed method achieves higher consistency with SIR model outcomes and enhances the propagation capability of top-ranked nodes, confirming its improved identification accuracy.

Introduction

With the deepening of research on complex networks, numerous scholars have applied complex network theory to various real-world networks^1–3, such as social networks^4–8, power grids^9,10, disease transmission networks^11,12, and transportation networks^13–15. In these networks, entities in the real world are often abstracted as nodes, and the relationships between entities are abstracted as edges, thereby constructing corresponding complex network models to address practical problems using complex network theory. Among the many research directions in complex networks, the identification of influential nodes holds significant importance, as such nodes exert a substantial impact on both the topological structure and functionality of networks. For example, in social networks, identifying influential individuals can facilitate the spread of beneficial information while curbing the dissemination of harmful content; in power grids, prioritizing the protection and monitoring of influential nodes helps implement timely countermeasures during failures to prevent cascading outages and avoid large-scale blackouts; In criminal networks, identifying influential nodes enables the timely implementation of control measures to contain the proliferation of illicit activities. Therefore, the identification of influential nodes in complex networks not only carries theoretical significance but also possesses substantial practical value^16–19.

Given the central position of influential nodes identification in the research of complex networks, this field has continuously received extensive attention from the academic community. Existing studies have fully demonstrated that the topological structure and its basic characteristics of networks play a decisive role in the process of identifying influential nodes^20–25. Early influential node identification methods can generally be divided into local-structure-based and global-structure-based approaches^26–28.

Local-structure-based methods mainly rely on neighborhood information, such as degree and clustering coefficient, and are computationally efficient. However, these methods often lack sufficient discrimination ability, especially when nodes share similar local structures. Global-structure-based methods incorporate broader topological information, including shortest-path-based measures and k-shell decomposition, and usually achieve more stable identification results. Nevertheless, many of these methods suffer from high computational complexity or coarse-grained ranking. To address these limitations, recent studies have attempted to integrate local and global information through hybrid frameworks^24,25. Although such methods improve performance to some extent, they may introduce additional parameters or rely heavily on specific structural assumptions, which limits their robustness across different networks. This motivates the development of a more balanced and efficient framework for influential node identification.

This paper integrates the advantages of local and global structures and proposes an influential nodes identification method that combines hierarchical k-shell and extended neighborhood (HKEN). Firstly, this method calculates the initial weight of each node according to its degree and k-shell value. Subsequently, based on the weight, a threshold for the information transmission distance is set to classify second-order neighborhood nodes, and different strategies are adopted according to different categories to calculate the initial influence of the nodes. Finally, the Jaccard similarity coefficient²⁹ is introduced to aggregate the influence contributions of one-hop neighbor nodes. Combining the importance of the nodes themselves, the comprehensive influence of the nodes in the network is ultimately determined. The novelty of HKEN lies not in individual structural measures, but in their mechanism-driven integration. By constraining neighborhood influence propagation through k-shell hierarchy and modulating the contribution of second-order inner neighborhood nodes based on similarity and clustering-based attenuation, HKEN provides a principled balance between global and local network characteristics. In summary, the main contributions of this algorithm are as follows:

1. Dynamic threshold strategy. Initial node weights are computed from local (degree) and global (k-shell) attributes, and a threshold is set for information transmission. Nodes are classified based on this threshold, with low-weight nodes downweighted and high-weight nodes amplified, reflecting realistic information spread.

2. Multi-layer propagation framework. The approach considers both local (direct neighbors) and semi-global (second-order neighbors) influence. Incorporating k-shell values from the global network mitigates biases from relying on a single structural layer.

3. Weighted similarity aggregation mechanism. This approach uses Jaccard similarity coefficient to weight neighbor contributions, allowing closely connected community nodes to reinforce each other’s influence—accurately reflecting propagation patterns in real networks.

The remainder of this paper is organized as follows: Section “Related work” systematically elaborates on and analyzes classical and recently proposed influential node identification methods. Section “Proposed algorithm” details the design rationale and algorithmic workflow of the proposed HKEN method, providing an in-depth explanation of the node influence calculation process with concrete network examples, along with a theoretical analysis of the algorithm’s time complexity. Section “Experiments” presents the simulation experiments, introducing the basic information of the datasets used, describing the evaluation metrics employed, and providing a detailed and objective analysis and discussion of the experimental results. Section “Conclusion” concludes the paper by summarizing the proposed method and elucidating its application scenarios and practical value.

Related work

To clearly position the proposed method, this section first reviews influential node identification algorithms from a conceptual and structural perspective. Subsequently, it elaborates in detail on the fundamental principles of the local clustering coefficient and the Jaccard similarity coefficient adopted in the proposed method, as well as their specific mechanisms for measuring node influence, providing a theoretical basis for the subsequent method design.

Classification of influential node identification algorithms

Existing influential node identification methods are categorized into global-structure-based, local-structure-based, and hybrid algorithms according to the scope of topological information they exploit. Representative approaches are discussed with emphasis on their core ideas and shared limitations, thereby revealing the research gap addressed in this work.

Global-structure-based algorithms

Global-structure-based algorithms evaluate node influence by leveraging macroscopic network topology. Classical methods include Betweenness Centrality (BC)³⁰, Closeness Centrality (CC)³¹, k-shell decomposition³², and Eigenvector Centrality (EC)³³.

BC and CC assess node importance through shortest-path information, effectively capturing global connectivity. However, both methods suffer from high computational complexity and limited scalability. The k-shell decomposition algorithm efficiently identifies core–periphery structures, yet its coarse-grained shell assignment limits discrimination among nodes within the same shell. Improved variants such as Extended Local K-Shell Sum (ELKSS)³⁴ incorporates neighborhood k-shell information, but still over-rely on neighbor contributions, neglecting intrinsic node characteristics. EC considers both node degree and neighbor importance, making it suitable for modeling long-term influence, but it tends to overemphasize high-degree nodes and requires iterative computation.

Overall, global-structure-based methods capture large-scale structural properties but generally exhibit either high computational cost or insufficient ranking resolution.

Local-structure-based algorithms

Local-structure-based algorithms focus on immediate neighborhood information and are typically more computationally efficient. Representative approaches include ProfitLeader (PL)³⁵ and Multilayer Neighbor Node Gravity and Information Entropy(MNNGE)³⁶.

PL evaluates node influence through resource-sharing mechanisms among neighbors, using node degree and Jaccard similarity coefficient. Although effective in certain networks, it is highly sensitive to node degree, leading to biased evaluations in the presence of hub nodes. MNNGE integrates multilayer neighborhood gravity with information entropy to aggregate influence, but the entropy component mainly reflects weight dispersion rather than actual propagation capability, limiting interpretability.

In summary, local-structure-based methods offer efficiency and fine-grained local characterization, but their limited awareness of global topology restricts their effectiveness in complex networks.

Hybrid algorithms

Hybrid algorithms aim to integrate global and local information to overcome the limitations of single-perspective approaches.

The algorithm based on adaptive adjustment of voting ability (AAVA)³⁷ dynamically adjusts voting ability based on node similarity, enhancing local influence aggregation. The algorithm for identifying influential nodes from a global perspective (GIN)³⁸ combines node degree with distance-weighted global influence, achieving improved performance at the expense of increased computational complexity and parameter sensitivity. The algorithm for finding influential nodes in social networks based on the neighborhood correlation coefficient (ECRM)³⁹ refines node hierarchy using k-shell decomposition and correlation analysis, but remains strongly dependent on k-shell values. The algorithm for identifying influential nodes based on the k-shell index and neighborhood information (BM)⁴⁰ and the algorithm for measuring node influence based on structural and adjacency information (IC-SNI)⁴¹ further incorporate neighborhood structure, path information, and adjacency attributes to model influence propagation.

Although these methods alleviate some limitations of traditional centrality measures, they often rely on complex fusion mechanisms, restrictive path assumptions, or normalization strategies, which may introduce bias and reduce robustness.

Research gap and motivation

From the above review, existing influential node identification methods face a persistent trade-off among global structural awareness, local discriminability, and computational efficiency. Global methods lack fine-grained resolution, local methods ignore long-range structural effects, and hybrid approaches often introduce excessive complexity or parameter dependence.

To address these issues, this paper proposes an influential node identification method that integrates hierarchical k-shell structure with extended neighborhood information, leveraging the local clustering coefficient and Jaccard similarity coefficient to enhance influence discrimination while maintaining computational efficiency. This design provides a balanced and interpretable framework for identifying influential nodes in complex networks.

Local clustering coefficient

The local clustering coefficient is an important metric for measuring the degree of local clustering between a node and its adjacent nodes in complex networks⁴². It reflects the ratio of the actual number of connections among the neighbors of a node to the maximum theoretically possible number of such connections. Its mathematical expression is as follows:

1

In the above equation, NE_i represents the actual number of edges among the nodes within the neighborhood of node i, and n_i is the number of nodes within the neighborhood of node i. The local clustering coefficient is generally used to measure the degree of connection tightness of a node within its local domain. In this paper, this coefficient is introduced when calculating the initial influence of a node to evaluate the node’s ability to spread information outward.

Jaccard similarity coefficient

The Jaccard similarity coefficient²⁹ is used to measure the degree of similarity between two sets, which is obtained by calculating the ratio of the number of elements in the intersection of the two sets to the number of elements in the union of the two sets. In complex networks, besides being used to calculate the similarity of the neighborhood structures of two nodes, it can also be employed to measure the overlap degree between the sequences of influential nodes identified by two algorithms. In this paper, it is mainly applied in two aspects: algorithm design and experimental verification. During algorithm design, it is used to calculate the influence provided by the first-order neighbor nodes of a node. In the experimental verification stage, this coefficient is utilized to evaluate the overlap degree between the Top N node sets identified by different algorithms and the SIR model. Its mathematical definition is as follows:

2

In Eq. (2), S₁ and S₂ represent two sets. During the algorithm design phase, S₁ and S₂ are the sets of neighbor nodes of two nodes to be compared. In the experimental phase, S₁ and S₂ are the Top N node sets in the sequences of influential nodes identified by a certain algorithm and the SIR model, respectively.

Proposed algorithm

This study presents a method for identifying influential nodes, which combines the hierarchical k-shell algorithm with extended neighborhood information and mainly involves three stages. Firstly, the initial weight of each node is computed using the degree of the node, its k-shell value, and the degrees of other nodes in the network. Secondly, the threshold of information transmission distance is calculated, nodes are categorized according to their initial weights, and the clustering coefficient is utilized to calculate the initial influence of the nodes. Finally, relying on the Jaccard similarity coefficient, the contribution of neighboring nodes to the spread of the target node is calculated.

Algorithm description

(1) Calculation of initial node weights. The main computational procedure in this phase is as follows:

Definition 1

Average degree of nodes. During the network initialization phase, the degree of all nodes is counted and its average value is calculated. The equation is given as follows:

3

where D(i) denotes the degree of node i, and avgD represents the average degree of all nodes in the network.

Definition 2

Initial weight of nodes. The initial weight of each node is computed. The initial weight of node i is calculated as follows:

4

Here, IW(i) represents the initial weight of node i, and KS(i) denotes the k-shell value of node i. The k-shell value reflects the positional importance of a node within the network, though its identification results tend to be relatively coarse-grained. On the other hand, the node degree can also reflect influence within a local scope. However, since node degrees are generally large in magnitude and differ significantly in scale from k-shell values, the degree of each node is divided by the average degree of the network. This adjustment reduces the proportional weight of the degree value in the calculation of the initial node weight, thereby enabling a more balanced and reasonable comprehensive evaluation.

Definition 3

Threshold of information transmission distance. In the HKEN method, the threshold of information transmission distance for nodes is defined as half of the maximum initial weight value among all nodes in the network.

5

6

In Eqs. (5) and (6), maxIW denotes the maximum value of the initial weights among all nodes, and T represents the threshold of information transmission distance. To examine the robustness of the threshold setting, a parameter sensitivity analysis is conducted in the experimental section by varying the threshold factor over a wide range.

After determining the information transmission distance threshold, the information transmission distance of a node can be determined based on the relationship between the initial weight of the node and the threshold. When the initial weight of node j is greater than the information transmission distance threshold, its information transmission distance is referred to as a short distance, with a set value of 1. When the initial weight of node j is less than or equal to the information transmission distance threshold, its information transmission distance is referred to as a long distance, with a set value of 2.

(2) Calculate the initial influence of node i. This value is defined as the sum of the node’s own influence and the influence provided by all nodes within its two—hop range.

At this stage, different methods are used to calculate the influence provided by one—hop neighbor nodes and two—hop neighbor nodes. Node j is a one-hop neighbor of node i, and node k is a two-hop neighbor of node i. For neighbor nodes with different hop counts, the influence they provide to node i is calculated using different methods.

For one—hop neighbor node j:

If its information transmission distance is short, the equation for calculating the influence FI(j) of node j on node i is as follows:

7

In Eq. (7), JCS(i, j) is the Jaccard similarity coefficient between node j and node i, sd is the information transmission distance of node j, hc is the hop count between node j and node i, and IW(j)/maxIW is the influence coefficient. Since nodes with different initial weights have different influences on other nodes, the influence of node i on node j is not the same as that of node j on node i. To reflect this asymmetry in influence propagation and highlight the advantages of nodes with higher initial weights during the propagation process, the normalization factor IW(j)/maxIW is introduced as the influence coefficient.

If its information transmission distance is long, the equation for calculating the influence FI(j) of node j on node i is as follows:

8

In Eq. (8), ld is the long information transmission distance between node j and node i.

For two-hop neighbor node k:

If its information transmission distance is short, the equation for calculating the influence SI(k) of node k on node i is as follows:

9

In Eq. (9), CC(k) is the local clustering coefficient of node k. Since information is not directly transmitted between node k and node i, there will be information loss. Therefore, when calculating the influence of node k on node i, the local clustering coefficient is introduced. Moreover, the larger the clustering coefficient of a node, the more difficult it is for information to spread; the smaller the clustering coefficient, the easier it is for information to spread. Thus, (1—CC(k)) is used as the weight factor.

If its information transmission distance is long, the equation for calculating the influence FI(k) of node k on node i is as follows:

10

In summary, the equation for calculating the initial influence NI(i) of node i is as follows:

11

where j represents a node in the one—hop neighborhood Nei(i) of node i, and k represents a node in the two—hop neighborhood SN(i) of node i.

(3) The final influence of node i. The final influence of a node is determined by its own initial influence and the initial influence contributed by its one-hop neighbor nodes. The final network influence of node i is denoted as FNI(i), and its calculation equation is as follows:

The calculation equation for the final influence FNI(i) of node i in the network is as follows:

12

In Eq. (12), Nei(i) represents the set of neighbor nodes of node i, and JCS(i, j) represents the Jaccard similarity coefficient between node i and node j. During the influence calculation process, if we only consider the sum of the influence of the node itself and the influence provided by all its neighbor nodes, the weight of the node’s own influence may be relatively low. In practical applications, although the influence of neighbor nodes plays a certain role, the influence of the node itself should occupy a more important position. Therefore, when calculating the contribution of each neighbor node, the influence of node i itself is simultaneously introduced into the calculation of the influence of its neighbors, so as to increase its weight in the evaluation of the final influence and enhance the rationality and stability of the evaluation results.

The detailed workflow of the proposed method is summarized as follows:

Algorithm 1.

Hierarchical k-shell and extended neighborhood method

Toy network analysis

This section illustrates the execution workflow of the HKEN algorithm using the network depicted in (Fig. 1) as an example.

Fig. 1.

A toy network.

Specifically, nodes 4 and 16 are selected to demonstrate the detailed computational process of their final influence. Based on the analysis of Fig. 1, the degree and k-shell values of each node are obtained. Nodes with a k-shell value of 3 are marked in deep red, those with a value of 2 in red, and those with a value of 1 in light red. The clustering coefficient of each node is computed according to Eq. (1), and the initial weight of each node is determined using Eq. (4). The detailed calculation results are presented as follows:

The toy network comprises 16 nodes and 20 edges. As indicated in Table 1, the average degree of nodes within the network is 2.5. The detailed computational procedure for determining the final influence is elaborated below using node 4 as an example.

Table 1.

Node degrees and k-shell values in the toy network.

Node	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Value
D(i)	6	4	5	5	2	3	2	1	1	1	2	3	1	1	2	1
KS(i)	3	3	3	3	2	2	1	1	1	1	2	2	1	1	2	1

The initial weight of node 4 is calculated using Eq. (4):

Following this method, the initial weights of the remaining nodes can be computed, with specific results provided in (Table 2).

Table 2.

Initial weights of nodes in the toy network.

Node	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
IW(i)	5.4	4.6	5.0	5.0	2.8	3.2	1.8	1.4	1.4	1.4	2.8	3.2	1.4	1.4	2.8	1.4

The maximum weight among all nodes is 5.4, with half of this value being 2.7. The one-hop neighbors of node 4 include nodes 1, 2, 3, 11, and 12, which are marked in dark green in (Fig. 2). The two-hop neighbors comprise nodes 5, 6, 7, 8, 9, 10, 15, and 16, indicated in light green in (Fig. 2).

Fig. 2.

Second-order neighborhood graph of node 4 in the toy network.

Taking node 1 in the one-hop neighborhood as an example, we calculate the influence it exerts on node 4. Since the initial weight of node 1 is 5.4, which exceeds the threshold of information transmission distance (2.7), its information transmission distance is classified as short-range. Therefore, Eq. (7) is applied for the calculation.

Following this procedure, the influences provided by nodes 2, 3, 11, and 12 to node 4 are calculated sequentially as: 11.2, 11.57, 2.07, and 2.37.

Additionally, taking node 6 in the two-hop neighborhood as an example, we calculate the influence it exerts on node 4. Since the initial weight of node 6 is 3.2, which exceeds the threshold of information transmission distance (2.7), its information transmission distance is classified as short-range. Therefore, Eq. (9) is applied for the calculation.

Following this approach, the influences provided by nodes 5, 7, 8, 9, 10, 15, and 16 to node 4 are calculated sequentially as: 0, 0.05, 0.03, 0.03, 0.03, 0.04, and 0.03, respectively.

Consequently, the initial influence of node 4 is computed according to Eq. (11). When calculating the influence provided by each node to node 4, apart from the initial influence of node 4 itself, the influence values of all other nodes must incorporate the initial influence of node 4 into consideration. The specific calculation is as follows:

Following the same procedure, the initial influence of all nodes can be calculated, with specific numerical values provided in (Table 3).

Table 3.

Initial influence of nodes in the toy network.

Node	1	2	3	4	5	6	7	8
NI(i)	116.91	93.63	96.48	110.77	34.46	37.29	18.11	10.19

Node	9	10	11	12	13	14	15	16
NI(i)	11.48	11.48	27.97	33.84	7.27	4.69	19.48	7.51

The final influence of node 4 is calculated by aggregating the initial influences of its first-order neighbors (nodes 1, 2, 3, 11, and 12) according to Eq. (12):

Following the method described above, the final influence of the remaining nodes can be calculated, as detailed in (Table 4).

Table 4.

Final influence of nodes in the toy network.

Node	1	2	3	4	5	6	7	8
FNI(i)	990.55	649.11	747.4	834.83	181.46	222.5	86.68	57.83

Node	9	10	11	12	13	14	15	16
FNI(i)	55.12	55.12	125.31	174.59	33.19	21.45	85.98	31.93

By sorting the nodes according to their final influence values, a descending sequence of node importance can be obtained, thereby allowing for the assessment of their relative significance.

Time complexity analysis of the algorithm

The proposed HKEN algorithm primarily comprises three stages. The first stage involves computing the foundational information required by the algorithm, including the k-shell values, degree values and their averages, as well as the initial weight of each node. Since these computations entail only linear operations over the node set, the time complexity of this stage is O(n), where n denotes the total number of nodes. The second stage employs a nested loop structure to calculate the initial influence of each node. The outer loop iterates over all nodes i; for each node i, the algorithm further traverses its first-order neighboring nodes j. For each j, it continues to traverse its first-order neighbors to acquire the second-order neighboring nodes k of i. Consequently, the time complexity of this stage is closely related to the average degree of the nodes, resulting in an overall complexity of O(n < k > ²), where < k > represents the average degree of the network. The third stage is mainly dedicated to computing the final influence of the nodes. Through a double-layer loop, the influence of each node is calculated, with a time complexity of O(n < k >).

Based on the above analysis, the overall time complexity of the HKEN algorithm is O(n < k > ²). In most real-world networks, the average degree O(n < k >) is generally much smaller than the total number of nodes n; hence, the algorithm exhibits high efficiency and scalability in practical applications.

Experiments

Experimental setup

To evaluate the performance of the HKEN algorithm, ten real-world network datasets with varying scales and functions were selected for testing: Karate, Dolphins, Euroroad, USAir, Protein, Citeseer, Ca-GrQc, Powergrid, Ca-Erdos, and Ca-Astroph. Their specific structural characteristics and data links are shown in (Table 5).

Table 5.

Statistical characteristics of real networks.

DataSets	N	M	< k >	Kmax	KSmax	< cc >	Data links
Karate	34	78	4.588	17	4	0.571	https://networkrepository.com/soc-karate.php
Dolphins	62	159	5.129	12	4	0.259	https://networkrepository.com/soc-dolphins.php
Euroroad	1174	1417	2.414	10	2	0.017	https://networkrepository.com/road-euroroad.php
USAir	1226	2408	3.928	34	4	0.068	https://tianchi.aliyun.com/dataset/7292/
Protein	1870	2203	2.356	56	5	0.067	https://networkrepository.com/ppi-yeast-uetz.php
Citeseer	2114	3672	3.474	99	7	0.171	https://networkrepository.com/citeseer.php
Ca-GrQc	4158	13422	6.456	81	43	0.557	https://snap.stanford.edu/data/ca-GrQc.html
Powergrid	4941	6594	2.669	19	5	0.080	https://networkrepository.com/inf-power.php
Ca-Erdos	5094	7515	2.951	61	7	0.082	https://networkrepository.com/ca-Erdos992.php
Ca-Astroph	18771	198050	21.102	504	56	0.631	https://snap.stanford.edu/data/ca-AstroPh.html

Among them, N represents the total number of nodes in the network, M denotes the number of edges, < k > indicates the average degree of the network nodes, K_max represents the maximum degree among all nodes, KS_max denotes the maximum k-shell value among all nodes, and < cc > represents the average clustering coefficient of the network; Data Links refer to the link addresses of the datasets.

Twelve mainstream algorithms were selected for comparison with the HKEN algorithm, namely BC, CC, EC, k-shell, ELKSS, PL, GIN, ECRM, MNNGE, BM, IC-SNI, and AAVA. The classic centrality methods such as BC, CC, EC, and k-shell are implemented through the NetworkX library. The source code of the AAVA algorithm was provided by the original author, and the PL algorithm was obtained from the link provided in the paper (https://github.com/ice-river/ProfitLeader), while the rest of the comparison algorithms were independently reproduced by us based on the pseudo-code and technical descriptions in their original literature. All experiments were conducted on a desktop computer with the following specifications: an Intel® Core™ i5-10,400 processor and 16 GB of RAM. All algorithms were implemented in Python 3.7, using NetworkX 2.3 for network manipulation.

Evaluation metrics

Multiple evaluation metrics are employed to conduct a comparative analysis between the proposed HKEN algorithm, the aforementioned twelve baseline algorithms, and the experimental results derived from the SIR model, thereby enabling a comprehensive assessment of its performance.

SIR model

The SIR model is a mathematical model utilized to describe the process of infectious disease propagation and is currently widely adopted in the evaluation of influential node identification algorithms in complex networks⁴³. In this model, nodes are categorized into three types based on their states: Susceptible (S), Infected (I), and Recovered (R). Specifically, Susceptible denotes that a node is in a state vulnerable to infection, with a certain probability of being infected; Infected indicates that a node is in an infected state and possesses the ability to transmit the disease; Recovered signifies that a node is in a recovered state and no longer participates in subsequent propagation processes. The core of this model involves characterizing the temporal evolution of the quantities of these three node categories through a set of differential equations, as detailed below:

13

In the network, infected nodes transmit the infection to susceptible nodes with an infection probability α, while simultaneously transitioning to the recovered state with a recovery probability β. Once recovered, nodes are immune to reinfection. By selecting different initial seed nodes to simulate the infection process, the infection capacity sequence for all nodes can be obtained.

The HKEN algorithm determines the initial influence of nodes by calculating their influence in the second-order neighborhood under different scenarios using Eqs. (8) and (9). This is validated through dynamic SIR simulation (see Eq. 13), thereby achieving consistency between static evaluation and dynamic propagation behavior³⁷.

Kendall’s tau correlation coefficient

Kendall’s tau correlation coefficient is primarily used to evaluate the degree of consistency between two sets of ranked data⁴⁴. In the context of complex network analysis, this metric quantifies the correlation by statistically comparing the ordering consistency of nodes in two ranking sequences. Consider two ranking sequences, L1 and L2. For any pair of nodes i and j, if the relative order is consistent across both sequences—that is, L1(i) > L1(j) and L2(i) > L2(j)—the pair is considered concordant. Conversely, if L1(i) > L1(j) but L2(i) < L2(j), the pair is deemed discordant. The Kendall’s tau correlation coefficient between sequences L1 and L2 is defined as:

14

where N denotes the total number of nodes in the network, N_C represents the number of concordant pairs, and N_D indicates the number of discordant pairs. A higher value of the coefficient signifies greater consistency between L1 and L2, while a lower value indicates weaker agreement.

Parameter sensitivity analysis

To evaluate the robustness of the threshold setting in Eq. (6), in this section, the threshold of the node information transmission distance is defined as T = λ × maxIW, where λ takes values in the interval [0.3, 0.7]. For each λ value, the Kendall’s tau correlation coefficient between the HKEN ranking results and the SIR model simulation propagation results is calculated under different information transmission probabilities α ∈ [0.01, 0.1] (with a step size of 0.01). Because the Ca-Astroph network is large and the running time is too long, the number of simulation iterations is set to 100, and the number of simulation iterations for the other networks is set to 1000. The experiments are conducted on six networks with different scales and topological features—Dolphins, Euroroad, Protein, Ca-GrQc, Powergrid, and Ca-Astroph—and the results are shown in (Fig. 3).

Fig. 3.

Sensitivity analysis of threshold parameter λ on HKEN performance.

As shown in Fig. 3, all trendlines exhibit a generally horizontal trajectory, indicating that the Kendall’s tau correlation coefficient remains relatively stable across different λ values. This phenomenon is consistent across various infection probabilities. The results demonstrate that HKEN maintains stable performance under multiple threshold settings with only minor fluctuations, further validating the method’s robustness in threshold T selection. Based on the above analysis, λ = 0.5 is determined as a reasonable and robust default parameter setting.

Experimental results and analysis

This section evaluates the HKEN algorithm through multiple approaches: analyzing the infection scale to examine the spread of high-impact nodes, using the Kendall’s tau correlation coefficient to verify the consistency of the ranking consistency with SIR, and employing auxiliary indicators such as the Jaccard similarity coefficient of the Top N nodes and the infection capability of the Top 10 nodes. This strategy adheres to the standards of complex network analysis⁴⁵, overcomes the limitations of a single indicator, and ensures a comprehensive assessment from both static and dynamic perspectives.

Spreading ability

To validate the effectiveness of the HKEN algorithm in identifying influential nodes, the node ranking sequences obtained by the HKEN algorithm and various baseline algorithms were compared with the sequence generated by the SIR model, thereby acquiring the spreading capacity of the corresponding nodes in the SIR model. If the node sequence identified by an algorithm matches that produced by the SIR model, the recognition performance should be reflected as a smooth and monotonically decreasing curve; otherwise, fluctuations will be observed in the curve. To present algorithmic performance more clearly and intuitively, the infection probability in the SIR model was uniformly set to 0.05 in this experiment, and the probability of recovery was set to 1. Because the Ca-Astroph network is large and the running time is too long, the number of simulation iterations is set to 100, and the number of simulation iterations for the other networks is set to 1000. Figure 4 illustrates the spreading effects of node sequences obtained by the thirteen algorithms across ten datasets under the SIR model.

Fig. 4.

Comparison of infection capacity of node sequences identified by different algorithms in SIR model across various networks.

Due to the relatively small number of nodes in the Karate and Dolphins networks, their spreading effects are displayed using a linear coordinate scale. The other eight networks are presented in logarithmic scale (base 10).

As illustrated in Fig. 4, in the Karate network, the HKEN algorithm demonstrates favorable identification performance along with eight other algorithms, with the exception of CC, k-shell, PL and AAVA. In the Dolphins, USAir, and Protein networks, the HKEN algorithm performs well, with BM, MNNGE, ECRM, and ELKSS also achieving commendable results; IC-SCI exhibits moderate performance, while the remaining algorithms show relatively weaker effectiveness. Within the Euroroad, Citeseer, and Powergrid networks, HKEN maintains strong performance, and IC-SCI, BM, MNNGE, ECRM, and ELKSS also display certain advantages, whereas other algorithms perform poorly. In the Ca-GrQc network, although the overall performance of all algorithms is suboptimal, HKEN, BM, MNNGE, ECRM, GIN, ELKSS, and k-shell retain a degree of effectiveness among the top-ranked nodes. For the Ca-Erdos network, BM, MNNGE, and ECRM outperform HKEN; nevertheless, HKEN still holds a comparative advantage over other methods. In the Ca-Astroph network, which contains a larger number of nodes, despite observable fluctuations in the curve, HKEN, BM, MNNGE, ECRM, GIN, and ELKSS demonstrate certain stability, while the rest of the algorithms exhibit inferior performance.

This experiment is primarily employed to demonstrate the distribution of consistency and inconsistency between the node sequences obtained by each algorithm and those derived from the SIR model. However, it does not allow for a quantitative evaluation of the degree of consistency. Therefore, further quantitative analysis using the Kendall’s tau is required to assess the agreement rigorously.

Ranking consistency

In this part, the Kendall’s tau correlation coefficient is utilized to assess the consistency level between the node ranking sequences acquired by each algorithm and those derived from the SIR model. A larger coefficient value implies a higher degree of consistency between the node ranking determined by the algorithm and the outcomes of the SIR model, suggesting better algorithm performance. During the experiment, because the Ca-Astroph network is large and the running time is too long, the number of simulation iterations is set to 100, and the number of simulation iterations for the other networks is set to 1000; the infection probability of the SIR model was set to 10 different values within the interval of [0.01, 0.1], with an increment of 0.01, and the probability of recovery to 1. Node sequences under various infection probabilities were obtained. Then, by comparing these sequences with those generated by each algorithm, the corresponding Kendall’s tau values were computed using Eq. (13). Figure 5 shows the Kendall’s tau of the thirteen algorithms for the ten datasets.

Fig. 5.

Comparison of kendall’s tau correlation coefficients between nodes equences identified by various algorithms and the SIR Model across different networks within the [0.01, 0.1] range.

Experimental results indicate that in the Karate network, the HKEN algorithm achieves the maximum Kendall’s tau at infection probabilities of 0.01 and 0.04, demonstrating its optimal identification performance. The AAVA, MNNGE, BM, and IC-SNI algorithms also exhibit commendable identification capabilities, whereas the BC and k-shell algorithms show relatively inferior performance, with the remaining algorithms performing moderately. In the Dolphins, Euroroad, Powergrid, and Ca-Astroph networks, the HKEN and AAVA algorithms delivers the best identification performance within the infection probability range of 0.05–0.1. The PL, ELKSS, ECRM, MNNGE, and BM algorithms also achieve favorable results, while the EC algorithm performs poorly. In the USAir and Citeseer networks, the HKEN and AAVA algorithms have relatively good recognition performance, but the AAVA algorithm shows a downward trend starting from an infection probability of 0.06. At lower infection probabilities (0.01–0.03), the PL algorithm also demonstrates good performance. Within the range of 0.07–0.1, the HKEN algorithm demonstrates superior identification effectiveness, and the ELKSS, ECRM, MNNGE, and BM algorithms also achieve relatively ideal results. In contrast, the EC and BC algorithms perform inadequately. In the Protein network, the HKEN and AAVA algorithms consistently delivers optimal identification performance across all infection probabilities. The ELKSS, ECRM, MNNGE, and BM algorithms also perform commendably, whereas the BC, k-shell, and PL algorithms show weaker capabilities, and the remaining algorithms exhibit moderate performance. In the Ca-Erdos network, the HKEN algorithm performs effectively when the infection probability is within the range of 0.04 to 0.06, while the AAVA algorithm achieves the best recognition effect when the infection probability is between 0.01 and 0.03. At an infection probability of 0.09, the ELKSS and BM algorithms demonstrate better performance, while at 0.1, the ELKSS, BM, and IC-SNI algorithms achieve the most optimal identification results. The BC, k-shell, and PL algorithms, however, exhibit relatively poor performance.

Jaccard similarity coefficient within the top N influential nodes

In real-world networks, the number of highly influential nodes is generally limited. These nodes possess stronger propagation capabilities compared to others and play crucial roles in the network. The ability to accurately identify such nodes is an important metric for evaluating algorithm performance. Typically, as the scale of a network increases, the absolute number of influential nodes also rises. In small-scale networks, the total number of nodes is limited, resulting in a relatively small number of influential nodes. In medium-scale networks, the number of influential nodes increases along with the growth in node count. In large-scale networks, although the proportion of influential nodes relative to the total number of nodes may be small, their absolute quantity remains significant. The top 5% of nodes play a critical role in the connectivity and functionality of the entire network, being sufficient to significantly disrupt it⁴⁶. Studies have shown that while the most influential nodes typically constitute less than 10% of a network³², the critical subset for maintaining robustness is often narrower, encompassing the top 1% to 5% of nodes^47,48. Therefore, when calculating the Jaccard similarity coefficient between the sets of influential nodes identified by each algorithm and those derived from the SIR model, this study considers the relatively small sizes of the Karate and Dolphins networks. For these, the computation range is set to 5 to 10 nodes. For the remaining eight datasets, the Jaccard similarity coefficient is calculated within the range of 1 to 5% of nodes.

Due to the minor differences in the identification of influential nodes among various algorithms in the Karate and Dolphins networks, line charts fail to clearly demonstrate the distinctions. Therefore, bar charts are employed to more clearly present the variation in performance across the algorithms. For the remaining eight networks, given the larger number of nodes involved in the comparison, bar charts exhibit limitations in effectively visualizing extensive data. Hence, line charts are utilized to more effectively reflect the distribution trends of identification performance.

Figure 6 illustrates the Jaccard similarity coefficient between the sets of influential nodes identified by various algorithms and those identified by the SIR model. In the Karate network, due to its small scale, the performance of all algorithms except CC and k-shell is relatively comparable. In the Dolphins network, the PL algorithm demonstrates the best performance, followed by the HKEN and AAVA algorithms, while BC, CC, and k-shell exhibit suboptimal results; the remaining algorithms show moderate performance. In the Euroroad, USAir, Citeseer, and Powergrid networks, HKEN, AAVA and BM algorithms perform similarly well, whereas BC, CC, EC, k-shell, and GIN yield unsatisfactory results. The remaining algorithms achieve moderate performance. In the Protein and Ca-Erdos networks, the HKEN and AAVA algorithms outperforms others, while CC, EC, k-shell, and GIN perform poorly. The remaining algorithms exhibit relatively good performance. In the Ca-GrQc network, ELKSS, MNNGE, BM, and ECRM demonstrate superior performance, with HKEN and AAVA following closely. GIN and EC show satisfactory performance for up to 60 nodes, but their efficacy gradually declines beyond this point. BC, CC, and IC-SNI algorithms perform poorly. In the Ca-Astroph network, the PL and AAVA algorithms achieves the best results, followed by HKEN, while BC, EC, and k-shell exhibit relatively inferior performance.

Fig. 6.

Jaccard similarity coefficient of Top-N influential nodes identified by each algorithm and the SIR Model. Bar charts are used for the Karate and Dolphins networks (a,b) to elucidate subtle differences in node identification due to their small scale. In contrast, line charts are employed for the larger networks (c–j) to better illustrate the distribution trends of performance across a greater number of nodes.

Top 10 nodes by propagation capacity

While the Jaccard similarity coefficient reflects the quantity of overlapping influential nodes identified by both the algorithms and the SIR model, it does not capture the consistency in node ranking, thus presenting limitations in evaluating identification coherence. Furthermore, existing studies indicate that the top 10 influential nodes typically reside in the core region of the network and play a significant role in information propagation⁴⁵. Controlling these nodes can reduce the information dissemination scope by 42–58%³⁷.

To more intuitively illustrate the disparities between the algorithmic results and the SIR model, this section compares the top 10 nodes identified by each algorithm with those identified by the SIR model in tabular form. Due to space constraints, only the comparative results for the Dolphins, Euroroad, and Powergrid networks are presented, as shown in (Tables 6, 7, and 8), respectively. In these tables, the top 10 nodes identified by the SIR model are marked in red. Nodes identified by other algorithms that exactly match both the identity and order of the SIR model results are similarly marked in red. Nodes that match in identity but differ in order are marked in blue. Nodes that do not coincide with the SIR model results remain unmarked.

Table 6.

Top 10 nodes in the Dolphins network identified by various algorithms.

BC	CC	EC	k-shell	ELKSS	PL	GIN	ECRM	MNNGE	BM	IC-SNI	AAVA	HKEN	SIR
37	37	15	1	15	15	38	15	15	38	21	15	15	15
2	41	38	11	38	46	15	38	38	15	37	38	46	38
41	38	46	15	46	38	46	46	46	46	15	46	38	46
38	21	34	16	34	34	41	34	34	34	41	34	34	34
8	15	51	41	41	52	34	41	41	41	38	52	52	52
18	2	30	43	51	58	21	51	30	51	46	30	30	41
21	29	52	48	30	21	37	30	52	21	30	21	21	19
55	8	17	2	52	14	51	52	51	30	34	41	41	21
52	34	41	18	21	30	19	21	21	44	51	58	19	30
58	9	22	20	17	18	30	17	17	17	19	51	58	39

Table 7.

Top 10 influential nodes in the Euroroad network identified by various algorithms.

BC	CC	EC	k-shell	ELKSS	PL	GIN	ECRM	MNNGE	BM	IC-SNI	AAVA	HKEN	SIR
402	401	7	2	7	107	401	7	7	7	7	7	284	284
284	402	43	3	43	284	7	43	107	284	43	284	7	7
277	403	454	17	107	7	402	107	43	39	411	107	107	107
453	432	411	4	284	39	411	181	284	499	401	43	137	499
452	1019	39	855	181	499	453	499	181	107	454	39	39	137
403	253	453	6	499	43	454	284	39	137	107	137	499	39
401	452	8	7	39	137	232	39	499	181	499	499	43	181
404	404	42	880	401	181	43	454	401	43	39	181	401	236
837	232	881	8	137	236	253	411	137	236	181	401	181	43
836	284	23	22	236	141	400	137	236	401	23	236	236	401

Table 8.

Top 10 influential nodes in the Protein network identified by various algorithms.

BC	CC	EC	k-shell	ELKSS	PL	GIN	ECRM	MNNGE	BM	IC-SNI	AAVA	HKEN	SIR
26	26	181	171	26	26	26	181	26	26	26	26	181	181
78	78	592	172	181	19	78	26	181	181	78	181	26	26
58	58	26	173	78	78	58	78	78	78	58	78	78	78
181	77	604	174	58	124	77	58	58	58	60	58	58	58
3	686	268	175	124	402	402	60	124	124	181	124	60	60
270	374	291	177	402	181	686	124	60	60	79	60	124	124
124	400	608	1058	60	58	405	179	402	402	124	402	270	402
60	969	602	176	405	179	60	558	405	405	59	19	16	558
79	1353	598	1149	558	405	969	402	558	558	270	270	79	270
592	60	610	1150	179	244	400	270	179	179	232	558	19	79

As shown in Table 6, in the Dolphin network, the first node identified by the HKEN algorithm matches the result identified by the SIR model. A total of 8 nodes appear in the top 10 nodes identified by the SIR model, but their rankings are inconsistent; hence, they are marked in blue. The top 5 nodes identified by the AAVA algorithm exhibit exact matches with those identified by the SIR model in both identity and order. The top 4 nodes identified by the EC, ELKSS, ECRM, and MNNGE algorithms exhibit exact matches with those identified by the SIR model in both identity and order. Among the top 10 nodes identified by the PL algorithm, one node matches the SIR model result in both identity and ranking. Although GIN, BM, and IC-SNI algorithms do not have nodes that fully coincide with the SIR model in both identity and sequence, each of them has more than 7 nodes appearing in the top 10 nodes identified by the SIR model, demonstrating relatively good identification performance. In contrast, the BC, CC, and k-shell algorithms exhibit comparatively inferior efficacy in identifying influential nodes.

As shown in Table 7, in the Euroroad network, the top three nodes identified by the HKEN algorithm completely match the results obtained by the SIR model. Although the remaining seven nodes are ranked differently, they all appear in the identification results of the SIR model, demonstrating relatively good identification performance. Although none of the other algorithms achieves complete consistency with the SIR model, the PL, MNNGE, BM and AAVA algorithms each have more than eight nodes appearing in the top 10 node sequence identified by the SIR model. The ELKSS, GIN, ECRM, and IC-SNI algorithms also have more than six nodes present in the node sequence identified by the SIR model, showing relatively ideal identification effectiveness. In contrast, the BC, CC, EC, and k-shell algorithms demonstrate relatively poorer identification performance, with fewer nodes matching the SIR model’s results, lower consistency, and comparatively weaker identification capability.

As shown in Table 8, in the Protein network, both the ECRM and HKEN algorithms demonstrate superior identification performance, with their top six nodes exhibiting exact matches with the SIR model’s results in both node composition and ranking order. Additionally, among the subsequent four nodes, the ECRM algorithm still shows three nodes overlapping with the SIR model’s results, while the HKEN algorithm has two overlapping nodes. Although the BC, ELKSS, MNNGE, IC-SNII and AAVA algorithms do not achieve complete consistency with the SIR model in the top 10 node sequence, each has eight or more nodes appearing in the top 10 nodes identified by the SIR model, demonstrating relatively satisfactory identification performance. The PL and GIN algorithms show moderate identification effectiveness. In contrast, the CC, EC, and k-shell algorithms exhibit relatively inferior performance, with fewer nodes overlapping with the SIR model’s results, lower consistency, and comparatively weaker identification capability.

After identifying the top 10 influential nodes in the network, this study simulates their propagation capability based on the SIR model. During the experimentation, the number of iterations was set to 100, the propagation rounds were set to 30, and the infection probability for all nodes across networks was uniformly configured at 0.05. To comprehensively and intuitively demonstrate the propagation effects of the top 10 influential nodes identified by various algorithms within the SIR model, the relevant data were visualized using graphical representations, with specific results presented in (Fig. 7).

Fig. 7.

Infection capacity of top 10 influential nodes in the SIR model.

As illustrated in Fig. 7, the HKEN algorithm exhibits the optimal performance in identifying the top 10 influential nodes across most networks, with only marginal inferiority to certain baseline algorithms in specific networks. In the Karate, Dolphins, Euroroad, USAir, Protein, Citeseer, and Powergrid networks, the top 10 influential nodes identified by HKEN demonstrate strong propagation capacity, with the diffusion process rapidly stabilizing, the AAVA algorithm also has good propagation performance. In the Ca-GrQc network, although HKEN’s propagation capacity is inferior to that of the BC and CC algorithms, it surpasses other methods and still maintains commendable identification performance. For the Ca-Erdos network, the HKEN algorithm achieves stability within the first 10 propagation rounds and outperforms all comparative algorithms in terms of propagation capacity. In the Ca-Astroph network, the propagation capacities of the top 10 influential nodes identified by various algorithms exhibit close proximity. While HKEN does not achieve optimal performance during the initial 9 propagation rounds, it attains the highest propagation capacity starting from the 10th round, demonstrating significant superiority.

The results of the above four experiments indicate that HKEN performs poorly in Ca-Erdos and Ca-GrQc. The reason for this is that for networks like Ca-Erdos, which exhibit nearly random connection patterns, the discriminative ability of structural similarity and hierarchical neighborhood information weakens, thereby limiting the advantages of HKEN. In the Ca-GrQc network, the highly clustered core structure and dense local connections may lead to redundant phenomena affecting the spread, and at this time, the clustering-based attenuation mechanism will suppress the edge diffusion gain. These observations suggest that HKEN is particularly effective in networks with rich and heterogeneous structural information, but its advantages may not be as significant in random-like or extremely dense core networks.

Statistical variability analysis of SIR simulation results

To quantify the transmission variability introduced by randomness in the SIR process, four representative networks—Karate, USAir, Citeseer, and Powergrid—were selected for experimentation. In the SIR model, the transmission probability α was set within the range [0.01, 0.1] with a step size of 0.01. Each α value was independently simulated 1,000 times to calculate the mean infected scale (Mean_Infected), standard deviation of infected scale (Std), and coefficient of variation (CV). Mean_Infected represents the average number of infected nodes across 1,000 simulations, used to evaluate the average diffusion capacity of the seed node set under specific transmission conditions. Std reflects the dispersion of simulation results, where a larger value indicates greater deviation from the mean, suggesting higher instability in the transmission process. CV is defined as the ratio of standard deviation to mean infected scale (CV = Std / Mean_Infected). Normalization was applied to eliminate dimensional effects, enabling more accurate comparison of relative volatility under different α conditions.

As shown in Table 9, as α gradually increases from 0.01 to 0.1, the average infection rate of each network shows a significant upward trend. At the same time, the standard deviation and coefficient of variation also increase simultaneously, indicating that under stronger transmission intensity, the random variability of the SIR dynamics is amplified. Although dense networks (such as Citeseer) exhibit higher absolute infection scales, their relative volatility (CV: 0.14–0.20) is similar to that of sparse networks (such as Powergrid, CV: 0.09–0.20), suggesting that network topology mainly affects the breadth of transmission, while the relative random intensity of the transmission process is consistent across networks. All CV values are below 0.22, and combined with the stable statistical indicators obtained from 1,000 simulations, they confirm that the observed performance differences are not due to random fluctuations. Despite the inherent randomness of the SIR process, the coefficient of variation and standard deviation remained at low levels across different transmission probabilities and network conditions, indicating that the observed performance differences are statistically stable.

Table 9.

Statistical results of SIR simulation for top 10 propagation capability nodes on representative networks. (a) Karate, (b) USAir, (c) Citeseer, (d) Powergrid.

α	Mean_Infected	Std	CV
(a) Karate
0.01	10.396	0.606	0.058
0.02	10.810	0.880	0.081
0.03	11.273	1.104	0.098
0.04	11.679	1.304	0.112
0.05	12.023	1.422	0.118
0.06	12.622	1.566	0.124
0.07	12.843	1.633	0.127
0.08	13.397	1.791	0.134
0.09	13.852	1.823	0.132
0.1	14.383	2.054	0.143
(b) USAir
0.01	12.631	1.724	0.136
0.02	15.427	2.560	0.166
0.03	18.436	3.408	0.185
0.04	22.081	4.214	0.191
0.05	25.975	5.006	0.193
0.06	30.428	6.193	0.204
0.07	34.956	6.912	0.198
0.08	40.736	8.428	0.207
0.09	47.273	9.523	0.201
0.1	55.004	12.018	0.218
(c) Citeseer
0.01	13.540	1.912	0.141
0.02	17.446	2.949	0.169
0.03	21.918	3.899	0.178
0.04	26.630	4.937	0.185
0.05	32.021	6.015	0.188
0.06	38.046	7.013	0.184
0.07	44.272	8.574	0.194
0.08	52.041	9.449	0.182
0.09	59.943	10.597	0.177
0.1	68.875	12.222	0.177
(d) Powergrid
0.01	11.025	0.983	0.089
0.02	12.232	1.491	0.122
0.03	13.442	2.049	0.152
0.04	14.569	2.293	0.157
0.05	16.110	2.923	0.181
0.06	17.309	3.096	0.179
0.07	18.985	3.596	0.189
0.08	20.323	3.950	0.194
0.09	22.127	4.213	0.190
0.1	23.854	4.719	0.198

Actual runtime comparison

We measured the actual running time of the HKEN algorithm and eight representative methods, including BC and CC on networks of different scales. All algorithms were run in a unified software and hardware environment, and the reported running time data are the average values of multiple independent runs. For those that did not produce results within one hour, we marked them as "-".

As shown in Table 10 on the smaller-scale networks of Karate and Dolphins, the running time of each algorithm was relatively short and the effect was good. However, as the network scale increased, the running time of BC, CC, GIN, and AAVA algorithms increased significantly. Especially in the Ca-Astroph network, BC, CC, GIN, and AAVA did not produce results within one hour. Although the running time of HKEN was longer than that of PL and BM, it had a better recognition effect. Therefore, the HKEN algorithm achieved a better balance between accuracy and computational efficiency.

Table 10.

Running time of HKEN and representative algorithms on 10 networks.

Algorithms	BC	CC	PL	GIN	ECRM	BM	IC-SNI	AAVA	HKEN
DataSets
Karate	0.003	0.003	0.001	0.004	0.001	0.001	0.008	0.019	0.005
Dolphins	0.010	0.009	0.003	0.015	0.005	0.004	0.017	0.068	0.010
Euroroad	3.046	2.243	0.017	3.359	0.170	0.015	0.094	37.599	0.085
USAir	4.421	3.549	0.054	5.341	0.309	0.028	0.430	57.57	0.208
Protein	6.951	4.450	0.045	6.73	0.384	0.029	0.309	156.68	0.181
Citeseer	14.30	10.35	0.084	15.84	1.045	0.050	0.641	293.66	0.314
Ca-GrQc	72.49	53.71	0.351	91.97	6.556	0.184	3.690	1872.61	1.253
Powergrid	81.77	54.47	0.085	83.79	2.627	0.073	0.533	2939.88	0.376
Ca-Erdos	88.33	55.53	0.359	92.51	3.487	0.150	3.353	3224.36	1.271
Ca-Astroph	-	-	28.27	-	699.84	9.86	786.55	-	129.39

Conclusion

This paper comprehensively takes into account the local and global structural characteristics of nodes and proposes an influential nodes identification method that integrates the hierarchical k-shell algorithm and extended neighborhood information. This method calculates the initial weight of each node by combining the node degree, k-shell value, and the average degree of nodes in the network. To some extent, it alleviates the problem of insufficient node discrimination ability caused by the coarsening of the traditional k-shell algorithm. Subsequently, the scope of influence propagation of nodes is extended to the second-order neighborhood. Based on the maximum value of the initial weights of all nodes, the information transmission distance threshold is determined, and then the neighbor nodes within the second-order neighborhood are classified. On this basis, the local clustering coefficient is introduced and integrated with the idea of the gravity model to quantify the influence contribution of neighborhood nodes to the target node. Then, combined with the initial weight of the node itself, the initial influence of the node is calculated. Finally, the Jaccard similarity coefficient is used to perform weighted aggregation on the initial influence of the one-hop neighbors of the target node and fuse it with the initial influence of the node itself, thereby obtaining the final influence of the node. This stage further highlights the leading role of core nodes in the propagation process. Experimental results show that the HKEN algorithm can more accurately identify influential nodes in complex networks through a weighted hierarchical mechanism, an extended neighborhood influence range, and a multi-source influence aggregation strategy. It provides more robust and adaptable solutions for practical application scenarios such as the discovery of opinion leaders in social networks and the control of the source of major disease spread.

Despite its effectiveness, HKEN has certain limitations. In networks with near-random connectivity and weak hierarchical structure, the advantage of k-shell–based constraints may be less pronounced. In highly clustered networks, redundant neighborhood information may reduce the marginal benefit of similarity-weighted aggregation. Moreover, HKEN relies solely on static structural information and does not account for temporal or semantic factors. These limitations suggest potential directions for future extensions.

Author contributions

F. W. proposed the methodology and drafted the original manuscript; Z. S. curated the data; G. W. designed the visualizations; H. H. reviewed and edited the manuscript; X. S. performed validation; and S. Z. conducted the investigation. All authors reviewed the final manuscript.

Funding

The research is supported by the Key Scientific Research Projects of Colleges and Universities in Henan Province under Grant Nos. 25A520040, 26A520032, and 26B520037, as well as the Science and Technology Research Project of the Department of Science and Technology in Henan Province under Grant No. 252102210028.

Data availability

The datasets generated and analysed during the current study, as well as the source code for all algorithms, are available in the GitHub repository: https://github.com/wffei123/HKEN.

Declarations

Competing interests

The authors declare no competing interests.