A multiscale topological characterization approach for identifying vital nodes in complex networks

Influential nodes refer to the ability of a node to spread information in complex networks. Identifying influential nodes is an important problem in complex networks which plays a key role in many applications such as rumor controlling, virus spreading, viral market advertising, research paper views, and citations. Basic measures like degree centrality, betweenness centrality, closeness centrality are identifying influential nodes but they are incapable of largescale networks due to time complexity issues. Chen et al. [1] proposed semi-local centrality, which is reducing computation complexity and finding influential nodes in the network. Recently Yang et al. 2020 [2] proposed a novel centrality measure based on degree and clustering coefficient for identifying the influential nodes. Sanjay et al. 2020 [3] gave voterank and neighborhood coreness-based algorithms for finding the influenced nodes in the network. Zhiwei et al. 2019 [4] considered the average shortest path to discover the influenced node in the network. These are the few recent local,global and mixed centralities. In this paper, we show a broad view of recent methods for finding influential nodes in complex networks. It also analyzes the new challenges and limitations for a better understanding of each method in detail. The experimental results based on these methods show better performance compared with existing basic centrality measures.

Evaluating the importance of nodes in complex networks is an important topic in the research of network characteristics. Its relevant research has a wide range of applications, such as network supervision and rumor control. At present, many methods have been proposed to evaluate the importance of nodes in complex networks, but most of them have the deficiency of one-sided evaluation or too high time complexity. In order to break through the limitations of existing methods, in this paper a novel method of evaluating the importance of complex network nodes is proposed based on Tsallis entropy. This method takes into account both the local and global topological information of the node. It considers the structural hole characteristics and K-shell centrality of the node and fully takes into account the influence of the node itself and its neighboring nodes. To illustrate the effectiveness and applicability of this method, eight real networks are selected from different fields and five existing methods of evaluating node importance are used as comparison methods. On this basis, the monotonicity index, SIR (susceptible-infectious-recovered) model, and Kendall correlation coefficient are used to illustrate the superiority of this method and the relationship among different methods. Experimental results show that this method can effectively and accurately evaluate the importance of nodes in complex networks, distinguish the importance of different nodes significantly, and can show good accuracy of evaluating the node importance under different proportions of nodes. In addition, the time complexity of this method is <inline-formula><tex-math id="M2">\begin{document}$ O({n^2}) $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20210979_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="21-20210979_M2.png"/></alternatives></inline-formula>, which is suitable for large-scale complex networks.

Online environments have evolved from the early-stage technical systems to social platforms with social communication mechanisms resembling the interactions which can be found in the real world. Online marketers are using the close relations between the users of social networks to more easily propagate the marketing contents in their advertising campaigns. Such viral marketing campaigns have proven to provide better results than traditional online marketing, hence the increasing research interest in the topic. While the majority of the up-to-date research focuses on maximizing the global coverage and influence in the complete network, some studies have been conducted in the area of budget-constrained conditions as well as in the area of targeting particular groups of nodes. In this paper, a novel approach to targeting multi-attribute nodes in complex networks is presented, in which an MCDA method with various preference weights for all criteria is used to select the initial seeds to best reach the targeted nodes in the network. The proposed approach shows some symmetric characteristics—while the global coverage in the network is decreased, the coverage amongst the targeted nodes grows.

A novel method of evaluating key nodes in complex networks

Generalized walks-based centrality measures for complex biological networks

Identifying the vital nodes in networks is of great significance for understanding the function of nodes and the nature of networks. Many centrality indices, such as betweenness centrality (BC), eccentricity centrality (EC), closeness centricity (CC), structural holes (SH), degree centrality (DC), PageRank (PR) and eigenvector centrality (VC), have been proposed to identify the influential nodes of networks. However, some of these indices have limited application scopes. EC and CC are generally only applicable to undirected networks, while PR and VC are generally used for directed networks. To design a more applicable centrality measure, two vital node identification algorithms based on node adjacency information entropy are proposed in this paper. To validate the effectiveness and applicability of the proposed algorithms, contrast experiments are conducted with the BC, EC, CC, SH, DC, PR and VC indices in different kinds of networks. The results show that the index in this paper has a high correlation with the local metric DC, and it also has a certain correlation with the PR and VC indices for directed networks. In addition, the experimental results indicate that our algorithms can effectively identify the vital nodes in different networks.

The identification of critical nodes in the network is of great significance for the research of the invulnerability and structural stability of complex networks. Most of the existing studies evaluate the importance of nodes in networks from a single structural perspective. Considering the local and global characteristics of the network, an identification algorithm of critical nodes in complex networks based on comprehensive topological importance (CTI) is proposed. The algorithm uses the theory of multiple attribute decision making (MADM) to quantitatively analyze the influence of different attributes on nodes and determine the comprehensive topology importance of nodes, which represents their structural importance in complex networks. Based on two simulation networks and five classical real networks, the proposed algorithm is proved to be superior to DC, LLS, P and WL indexes based on local information and BC indexes based on global information through the change of network efficiency under critical nodes attacking.

Ranking influential nodes in complex network using edge weight degree based shell decomposition

The issue of mining influential nodes in complex networks is a topic of immense interest. Recently, many methods have been proposed, but they suffer from certain limitations. In this article, a novel centrality measure based on local fuzzy information centrality (LFIC) is proposed. LFIC puts forward the concept that the inner structure of a node’s box contains information about the node’s importance. LFIC uses the amount of information contained in the node’s box as a measure of its importance. In LFIC, the uncertainty of information contained in nodes’ boxes is measured by the improved Shannon entropy. Most importantly, fuzzy logic is applied to deal with the uncertainty of neighbor nodes’ contributions to the center node’s importance, which is neglected by most existing methods. To verify the effectiveness of our proposed method, six existing methods are used for comparison and five experiments are conducted using six real-world complex networks. The experimental results indicate that the influential nodes identified by LFIC can cause a wider scope of infection in networks and have a larger effect on the network connectivity, thereby proving the effectiveness and accuracy of LFIC. The correlation between nodes’ LFIC values and their real infection ability is highly positive according to Kendall’s tau coefficient, proving LFIC’s credibility and superiority. The extension of LFIC, namely the bi-directional local fuzzy information centrality, is also proposed to explore its feasibility in weighted directed complex networks.

It is well known that protein-protein interaction (PPI) networks are typical evolving complex networks. Identification of important nodes has been an emerging popular topic in complex networks. Many indexes have been proposed to measure the importance of nodes in complex networks, such as degree, closeness, betweenness, k-shell, clustering coefficient, semi-local centrality, eigenvector centrality. Based on multivariate statistical analysis, through integrating the above indexes and further considering the appearances of nodes in network motifs, this paper aims at developing a new measure to characterize the structurally dominant proteins (SDP) in PPI networks. Moreover, we will further investigate the evolution of the defined dominant nodes in temporal evolving real-world and artificial PPI networks. Our results indicate that the constructed artificial networks have some similar statistical properties as those of the real-world evolving networks. In this case, the artificial PPI networks can be used to further investigate the above evolution characteristics of the real-world evolving networks. Simulation results reveal that SDP in the yeast PPI networks are evolutionary conserved, however, the undominant nodes evolve rapidly. Furthermore, PPI networks are very robust against random mutations, while fragile yet with certain robustness to targeted mutations on SDP. Our investigations shed some light on the future applications of the evolving characteristics of bio-molecular networks, such as reengineering of particular networks for technological, synthetic or pharmacological purposes.

Identifying influential nodes in complex networks: Effective distance gravity model

Entropy has been widely measured the complexity of complex networks. At present, many measures about entropies were defined based on the directed connection of nodes. The similarity of nodes can better represent the relationship among all nodes in complex networks. In the definition of similarity of nodes, the importance of a node in the network depends not only on the degree of the node itself, but also on the extent of dependence of neighboring nodes on the node. In this paper, we proposed a new structure entropy based on nonextensive statistical mechanics and similarity of nodes. In the proposed method, the similarity of nodes and the betweenness of nodes are both quantified. The proposed method considers the extent of dependence between neighbouring nodes. For some complex networks, the proposed structure entropy can distinguish complexity of that while other entropies can not be. Meanwhile, we construct five ER random networks and small-world networks and some real-world complex networks such as the US Air Lines networks, the GD'01-GD Proceedings Self-Citing networks, the Science Theory networks, the Centrality Literature networks and the Yeast networks are measured by the proposed method. The results illustrated our method for quantifying the complexity of complex networks is effective.

Efficiently identifying multiple influential nodes is crucial for maximizing information diffusion and minimizing rumor spread in complex networks. Selecting multiple influential seed nodes requires consideration of both their individual influence potential and their spatial dispersion within the network topology to avoid overlapping propagation ranges ("rich-club effect"). Traditional VoteRank method suffer from two key limitations: (1) they assume uniform voting contributions from a node to all its neighbors, neglecting the impact of topological similarity (structural homophily) on voting preferences observed in real-world scenarios, and (2) they employ a homogeneous voting attenuation strategy which inadequately suppresses the propagation range overlap among selected seed nodes. To address these shortcomings, this paper proposes IMVoteRank, an improved VoteRank algorithm featuring dual innovations. First, we introduce a Structural Similarity-Driven Voting Contribution Mechanism. Recognizing that voters (nodes) are more likely to support candidates (neighbors) with whom they share stronger topological ties, we decompose a neighbor's voting contribution into two components: a Direct Connection Contribution and a Structural Similarity Contribution (quantified using common neighbors). A dynamic weight parameter θ, adjusted based on the candidate node's degree, balances these components, significantly refining vote allocation accuracy. Second, we devise a Dynamic Group Isolation Strategy. During each iteration, after selecting the highest-scoring seed node <i>v<sub>max</sub></i>, we identify and isolate a tightly-knit group (OG) centered around it. This involves: (i) forming an initial group based on shared neighbor density with <i>v<sub>max</sub></i>, (ii) expanding it by incorporating nodes with more connections inside the group than outside, and (iii) isolating this group by setting the voting capacity (Va) of all its members to zero and virtually removing their connections from the adjacency matrix. Neighbors of <i>v<sub>max</sub></i> not in OG have their <i>Va</i> halved. This strategy proactively enforces spatial dispersion among seeds. Extensive simulations using the Susceptible-Infected-Recovered (SIR) propagation model on nine diverse real-world networks (ECON-WM3, Facebook-SZ, USAir, Celegans, ASOIAF, Dnc-corecipient, ERIS1176, DNC-emails, Facebook-combined) demonstrate the superior performance of IMVoteRank. Compared to seven benchmark methods (Degree, K-shell, VoteRank, NCVoteRank, VoteRank++, AIGCrank, EWV), IMVoteRank consistently achieves significantly larger final propagation coverage (infected scale) for a given number of seed nodes and transmission probability (<i>β</i>=0.1). Furthermore, seeds selected by IMVoteRank exhibit a consistently larger average shortest path length (<i>Ls</i>) among themselves across most networks, confirming their effective topological dispersion. This combination of high individual influence potential (optimized voting) and low redundancy (group isolation) directly translates to more efficient global information spread, as evidenced by the SIR results. Tests on LFR benchmark networks further validate these advantages, particularly at transmission rates above the epidemic threshold. IMVoteRank effectively overcomes the limitations of traditional voting models by integrating structural similarity into the voting process and employing dynamic group isolation to ensure seed dispersion. It provides a highly effective and physically well-grounded approach for identifying multiple influential nodes in complex networks, optimizing the trade-off between influence strength and spatial coverage. Future work will focus on enhancing computational efficiency for large-scale networks and exploring the impact of meso-scale community structures.

A novel potential edge weight method for identifying influential nodes in complex networks based on neighborhood and position

Symmetries, due to their fundamental importance to dynamical processes on networks, have attracted a great deal of current research. Finding all symmetric nodes in large complex networks typically relies on automorphism groups from algebraic-group theory, which are solvable in quasipolynomial time. We articulate a conceptually appealing and computationally extremely efficient approach to finding and characterizing all symmetric nodes by introducing a structural position vector (SPV) for each node in networks. We establish the mathematical result that symmetric nodes must have the same SPV value and demonstrate, using six representative complex networks from the real world, that all symmetric nodes in these networks can be found in linear time. Furthermore, the SPVs not only characterize the similarity of nodes but also quantify the nodal influences in propagation dynamics. A caveat is that the proved mathematical result relating the SPV values to nodal symmetries is not sufficient; i.e., nodes having the same SPV values may not be symmetric, which arises in regular networks or networks with a dominant regular component. We point out with an analysis that this caveat is, in fact, shared by the known existing approaches to finding symmetric nodes in the literature. We further argue, with the aid of a mathematical analysis, that our SPV method is generally effective for finding the symmetric nodes in real-world networks that typically do not have a dominant regular component. Our SPV-based framework, therefore, provides a physically intuitive and computationally efficient way to uncover, understand, and exploit symmetric structures in complex networks arising from real-world applications.