Description:[001] The present invention discloses a method and system for identifying influential nodes in directed weighted networks using Pythagorean fuzzy sets. Node Pack Fuzzy Information Centrality (NPFIC) quantifies the significance of a node by assessing the information content within its pack, which is calculated by the improved Havrda and Charavat entropy.
BACKGROUND OF THE INVENTION
[002] In recent studies, numerous real-world systems are effectively modeled as complex networks, including social networks, the internet, power grids, and various online networks that significantly impact our daily lives. The analysis of complex networks enables us to comprehend the complexities of unpredictability and forecast the evolution of systems. Within these complex networks, specific nodes exhibit notable influence and have a great impact on the overall structure and functionality. If a node has greater influence, then it plays crucial roles in facilitating information exchange, and their removal can substantially alter the network dynamics. Consequently, the identification of these influential nodes remains a focal point in the ongoing research on complex networks.
[003] Numerous centrality metrics predominantly address unweighted undirected networks, often overlooking the significance of edge weights and directions. When dealing with directed networks, the weights of connections between the nodes can hold crucial information and should be considered. In many real-world scenarios, the strength, intensity, or some other quantitative measures associated with the relationship between nodes can greatly influence the network's behavior and characteristics. Directed weighted networks offer a more effective means of accurately depicting the relationships among individual nodes compared to simple networks. The inclusion of edge weight information proves invaluable in gaining a more precise understanding of the structure and functionality of the network. Consequently, detecting crucial nodes in networks with directed weights holds significant promise in research domains such as the propagation of influence in social networks, strategic planning for transportation infrastructure, disease spread and epidemiology, as well as ranking in scientific collaborations among others. The centrality metrics employed in complex networks with directed and weighted edges can effortlessly extend beyond those originally formulated for simple networks.
[004] L. Zadeh introduced Fuzzy Sets (FSs), a concept widely employed across various domains, including uncertainty measurement, multiple attributes decision making, analyzing and supporting decision-making, information granules, and similarity measures. Unlike Boolean logic, Fuzzy Sets excel at quantifying uncertain information, making them well-suited for complex networks. Pythagorean membership grades for multicriteria decision making were first proposed by Yager. Zhang and Xu expanded the application of the Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) to incorporate Pythagorean and hesitant fuzzy sets in the context of multiple criteria decision-making. More recently, Xue et al. presented the Pythagorean fuzzy LINMAP method, incorporating entropy for effective decision making in railway project investments. Shannon entropy, a concept in information theory, plays a crucial role in evaluating the anticipated information within a message. Its applications extend to the analysis of complex networks, particularly in identifying influential spreaders. Zareie et al. introduced the Entropy-Based Ranking Measure (ERM), emphasizing the concept that influential spreaders are characterized by substantial and evenly distributed degrees. ERM assesses the entropy of the immediate neighbors of a node those at the second order and concentrating on local information.
[005] However, it exclusively considers the node's immediate and secondary neighbors. Deng and Wen applied Shannon entropy in their study to evaluate node importance through the introduction of the LID model. Even though LID employs Shannon entropy for quantifying information within a node box, it does not investigate the internal structure of these boxes. Traditional centrality metrics, including Degree Centrality(DC), Betweenness Centrality (BC), Closeness Centrality (CC), Eigenvector Centrality (EC), and PageRank (PR), have been developed to assess the importance of nodes in a network by considering the factors like node distance and the number of connections. These traditional approaches focus on various characteristics of complex networks. Wang et al. investigated the detection of pivotal nodes within directed biological networks, relying on the characterization of node importance derived from instances observed in diverse networks with 2, 3, and a subset of 4 nodes.
[006] Additionally, Sheng et al. introduced the concept of influential nodes in complex networks. Wang et al. introduced a novel approach for discerning influential nodes in complex networks through a semi-local measure. Panfeng implemented a voting methodology to recognize pivotal nodes within social networks. CC is only applicable to undirected networks, while PR and EC are generally used for directed networks. Garas and colleagues presented a technique for k-shell decomposition in weighted networks. It can identify the nodes with the greatest impact in a network with weights by splitting a network into k-shell structure. PageRank asserts that a node's significance in web page ranking is tied to the quantity and quality demonstrated by the neighboring nodes.
[007] However, these approaches overlook the wider configuration of networks. In addressing this issue, Betweenness Centrality (BC) examines the centrality of a node by assessing the number of shortest paths that traverse through it. On the other hand, Closeness Centrality (CC) posits that a node with the minimal average distance to other nodes wields greater influence. Although BC and CC prove to be effective, they are hindered by computational complexity, leading to suboptimal performance in complex networks.
[008] In our discussions so far with different approaches one problem identified is that all the models rely on observing the entire network and due to computational complexity, implementing this approach becomes impractical for large-scale social networks. Additionally, the center node receives the most significant contributions from its neighboring nods and their edge weights, are considered as an important concept in the directed weighted networks. It is a very typical task to identify vital nodes by considering all these features of individual nodes in directed weighted networks. Many current centrality metrics only consider the overall count of neighbors connected to a node and neglect an in-depth analysis of the local network structure, leading to inaccuracies.
[009] Here, we introduce NPFIC as an approach to assess the significance of nodes within directed weighted networks by considering two factors: in-weights and out-weights. Based on the above discussions, the motivating factors behind our study can be succinctly summarized as follows: i) to account for the variability in centrality values within a specific node due to uncertainty in edges, utilize Pythagorean fuzzy relations. ii) to explore and analyze the inner structure of a node's pack within directed weighted networks and ascertain the amount of information it encompasses. iii) to address the uncertainty linked with contributions from neighboring nodes to their central node, by leveraging Pythagorean fuzzy sets. iv) to enhance the efficiency of Havrda and Charavat entropy and to optimize its utilization in directed weighted networks.
SUMMARY OF THE PRESENT INVENTION
[010] The present invention introduces a novel method and system for identifying influential nodes in directed weighted networks, addressing the limitations of existing centrality measures. Centrality, which assesses the importance of nodes in complex networks, is crucial for various applications such as social network analysis and disease spread prediction. However, traditional centrality metrics often overlook the internal structure of nodes' neighborhoods and fail to account for uncertainty in node importance.
[011] To overcome these challenges, the invention proposes a new centrality metric termed "Node Pack Fuzzy Information Centrality" (NPFIC). NPFIC leverages Pythagorean Fuzzy Sets to capture the uncertainty associated with the contributions of neighboring nodes to the centrality of the center node. By considering the internal structure of a node's pack, NPFIC quantifies the significance of a node based on the information content within its neighbourhood
[012] Key components of the proposed method include the calculation of weighted degrees of membership for neighboring nodes within a specific layer, determination of non-membership and indeterminacy values, and computation of Pythagorean fuzzy entropy for each node. The method further ranks nodes based on their fuzzy membership values, thereby identifying influential nodes within the network.
[013] The effectiveness of NPFIC is demonstrated through experiments conducted on a real-world directed weighted complex network. Comparative analysis with four established centrality measures, including Degree Centrality, H-index Centrality, Betweenness Centrality, and PageRank, highlights the superiority of NPFIC in identifying crucial nodes that significantly influence network connectivity. Furthermore, the invention applies the susceptible-infected (SI) model to assess the dissemination capability of the identified influential nodes. Results indicate that nodes identified using NPFIC exhibit a larger impact on disease spread compared to nodes identified by alternative centrality metrics.
[014] In conclusion, the proposed method provides a comprehensive approach to identifying influential nodes in directed weighted networks, integrating principles from information theory with network science. Its application extends beyond traditional network analysis to domains such as e-commerce, where identifying central nodes is crucial for enhancing network performance and business outcomes.
BRIEF DESCRIPTION OF THE DRAWINGS
[015] when considering the following thorough explanation of the present invention, it will be easier to understand it and other objects than those mentioned above will become evident. Such description refers to the illustrations in the annex, wherein:
FIGS. 1-7, illustrates systematic diagrams related to a method and system for identifying influential nodes in directed weighted networks using pythagorean fuzzy sets, in accordance with an embodiment of the present invention.
DETAILED DESCRIPTION OF THE INVENTION
[016] The following sections of this article will provide various embodiments of the current invention with references to the accompanying drawings, whereby the reference numbers utilised in the picture correspond to like elements throughout the description. However, this invention is not limited to the embodiment described here and may be embodied in several other ways. Instead, the embodiment is included to ensure that this disclosure is extensive and complete and that individuals of ordinary skill in the art are properly informed of the extent of the invention.
[017] Numerical values and ranges are given for many parts of the implementations discussed in the following thorough discussion. These numbers and ranges are merely to be used as examples and are not meant to restrict the claims' applicability. A variety of materials are also recognised as fitting for certain aspects of the implementations. These materials should only be used as examples and are not meant to restrict the application of the innovation.
[018] Referring now to the drawings, these are illustrated in FIGS. 1-7,
The innovative method proposed in this study, known as Node Pack Fuzzy Information Centrality (NPFIC), addresses the challenge of identifying influential nodes in complex networks, particularly focusing on directed weighted networks. Traditional centrality measures often overlook the intricate internal structure of a node's pack, leading to limitations in accurately assessing node importance. NPFIC introduces a novel approach that leverages Pythagorean Fuzzy Sets to account for the uncertainty associated with the contributions of neighboring nodes to the centrality of the center node. By incorporating Pythagorean Fuzzy Sets, NPFIC offers a more nuanced perspective on node significance within a network.
[019] At the heart of NPFIC is the concept of evaluating a node's significance by examining the information content within its pack. This is achieved through the calculation of improved Havrda and Charavat entropy, tailored to suit the characteristics of real-world directed weighted networks. By quantifying the importance of a node based on the internal structure of its pack, NPFIC provides a comprehensive centrality metric that captures the nuances of node influence more effectively than traditional approaches.
[020] The methodology of NPFIC involves several key steps. Firstly, the directed weighted network dataset is acquired and preprocessed as needed. The pack size, crucial for NPFIC calculations, is determined based on the network's characteristics. Then, the weighted degree of membership of nodes within a fuzzy set is computed, emphasizing the count of directed edges connecting neighboring nodes. Non-membership and indeterminacy values are also calculated to refine the centrality assessment.
[021] To illustrate the effectiveness of NPFIC, experiments are conducted on real-world directed weighted complex networks. These experiments involve comparing NPFIC with four established centrality measures: Degree Centrality (DC), H-index Centrality (HC), Betweenness Centrality (BC), and PageRank (PR). The outcomes of these experiments demonstrate the superior performance of NPFIC in identifying crucial nodes that significantly influence network connectivity.
[022] Furthermore, the study extends its analysis to evaluate the dissemination capability of nodes through the susceptible-infected (SI) model. NPFIC consistently outperforms other centrality measures across various time steps, indicating a larger number of infected nodes compared to alternative methods.
[023] Additionally, the study assesses the network quality by selectively removing top nodes identified by different centrality measures. The results show that nodes identified by NPFIC exhibit greater influence within the network compared to those identified by alternative metrics.
[024] Finally, the application of NPFIC in e-commerce business scenarios is discussed, highlighting its potential to enhance outcomes for online ventures. By considering factors such as node connectivity and self-weight, NPFIC offers a promising approach for identifying influential nodes in e-commerce networks.
[025] In conclusion, NPFIC represents a significant advancement in the field of complex network analysis, offering a refined and comprehensive method for identifying influential nodes in directed weighted networks. Its effectiveness is validated through experimental validations and comparative analyses, highlighting its superiority over traditional centrality measures.

, Claims:1. A method for identifying influential nodes in directed weighted networks, comprising:
a) Acquiring a directed weighted network dataset;
b) Preprocessing the dataset to obtain relevant network parameters;
c) Calculating the significance of each node using Pythagorean fuzzy sets based on the internal structure of its pack; and
d) Ranking the nodes based on their significance to determine influential nodes within the network.
2. A system for identifying influential nodes in directed weighted networks, comprising:
a) Means for acquiring a directed weighted network dataset;
b) Means for preprocessing the dataset to obtain relevant network parameters;
c) Means for calculating the significance of each node using Pythagorean fuzzy sets based on the internal structure of its pack; and
d) Means for ranking the nodes based on their significance to determine influential nodes within the network.
3. The method as claimed in claim 1, wherein the Pythagorean fuzzy sets are utilized to address the uncertainty associated with the contributions of neighboring nodes to the centrality of the center node.
4. The method as claimed in claim 1, wherein the significance of each node is quantified by assessing the information content within its pack using improved Havrda and Charavat entropy.
5. The method as claimed in claim 1, wherein the influential nodes identified by the method significantly influence network connectivity.
6. The system as claimed in claim 2, wherein the means for acquiring a directed weighted network dataset comprises a data acquisition module configured to collect network data from external sources.
7. The system as claimed in claim 2, wherein the means for preprocessing the dataset includes a data preprocessing module configured to clean and format the acquired network data.
8. The system as claimed in claim 2, wherein the means for calculating the significance of each node includes a computation module implementing algorithms for Pythagorean fuzzy set operations.
9. The system as claimed in claim 2, wherein the means for ranking the nodes comprises a ranking module configured to assign centrality scores to each node based on their significance.