# Computing Influential Nodes Using the Nearest Neighborhood Trust Value and PageRank in Complex Networks

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## Abstract

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## 1. Introduction

**Our Contribution:**In this research work, we propose a nearest neighborhood trust PageRank (NTPR) method which considers not only neighborhood trust values at the one-neighborhood level but also the nearest neighbors of trust values up to the second-neighborhood level. In this work, the degree ratio is also defined, along with next neighbor levels, instead of adjacent nodes. Considering the greater number of neighbors, the degree ratio and trust value capture information spread more accurately. We notice first-level neighbors and second-level neighbors’ information plays a crucial role in the influence of a node. Additionally, our method uses the trust value, which includes a degree ratio with neighbors, the similarity ratio, and the second-level neighbors’ information. We proposed an enhanced version of trust–PageRank, which is a nearest neighborhood trust PageRank measure to compute the influential nodes in complex networks [29]. By using the proposed centrality, we computed the influential nodes in various real-world networks. We found the maximum influence of influential nodes by using the SIR, independent cascade, and greedy methods. In [29], we defined the NTPR measure. In this paper, we provide results and comparisons with other basic centralities. We also compare our centrality measure with existing basic centrality measures, and it produces greater influence than the others which are discussed in the coming sections.

#### 1.1. Related Work

## 2. A Centrality Measure Using Second-Level Neighborhood Trust Values

**Similarity Ratio:**The similarity ratio of a vertex ${v}_{i}$ and an adjacent neighbor ${v}_{j}$ is the similarity of ${v}_{i}$ and ${v}_{j}$ divided by the addition of the similarity between the adjacent neighboring vertex ${v}_{j}$ and its adjacent neighbor vertices ${v}_{l}$. The similarity ratio can be measured as follows:

**Degree ratio:**The degree ratio of a vertex ${v}_{i}$ and an adjacent vertex ${v}_{j}$ is the ratio of the degree of vertex ${v}_{i}$ to the sum of the degrees of adjacent vertices of ${v}_{j}$. For normalization of this degree ratio, we use the sum of the degrees of the neighbors of ${v}_{j}$ and the sum of the degrees of the second-level neighbors of ${v}_{j}$. The degree ratio can be defined as:

**Trust value:**Trust values of vertices ${v}_{i}$ and ${v}_{j}$ are defined by the similarity ratios and degree ratios of ${v}_{i}$ to ${v}_{j}$. Trust value is calculated as follows:

Algorithm 1: Computing $NTP{R}_{t}$ for every vertex of graph G |

Input: Graph $G=(V,E)$ with vertices and edgesOutput: $NTP{R}_{t}$ for every vertex of graph G |

#### Time Complexity of NTPR

## 3. Details on Implementation

**Susceptible–Infected–Recovered Model:**In this work, we investigated the dynamics of information spreading by using the SIR simulation model [36,37]. The SIR model is commonly used for how much information is spread with in the network. This model is used to understand the dynamics of the spreading of diseases and to find a total number of infected nodes at different infection probabilities. The SIR model is divided into 3 components, susceptible, infected, and recovered. The susceptible part tells us that no infection has taken place. The term “infected” refers to infections spread over the network by others. Finally, recovered means the cured individuals, and they do not infect after a certain number of rounds. At the starting stage, seed nodes will be infected, which can help us to find the spreading capability. These infected nodes later in each iteration infect their neighbors with a certain probability in the network. The number of infected nodes grows over time until it reaches a stable state. The SIR model is one of the methods used to estimate centrality measurements in terms of network performance.

**Independent Cascade Model (IC):**The IC model is a stochastic technique in which data are transferred from one node to another depending on probabilistic criteria [3]. Li et al. [38] categorized the diffusion models into two types, predictive and explanatory models. The IC model is a predictive model that uses specific parameters to estimate the forecast information diffusion process and influence maximization in social networks. The IC model’s information diffusion operates as follows: The network’s nodes can be in one of two states: active or inactive. If a node accepts the information being circulated in the network, it is deemed active; if it does not have information, it is considered inactive. Thus, an independent cascade model, which is a form of an epidemic model, proposes that an individual will achieve innovation with a specific probability if at least one of its neighbors has done so. We compared this model with the SIR model.

**Greedy Model:**The greedy algorithm implements the problem-solving strategy of choosing the locally best option at every stage [3]. A greedy strategy may not generate an optimal result in many cases, but it can produce locally optimized solutions that resemble globally optimal solutions in acceptable time frames. Greedy algorithms are usually less computationally efficient than other techniques, such as dynamic programming, but they often compromise the quality of the solution to achieve speed. This algorithm was used to find top-ten seed nodes. The greedy model evaluates the incremental spread of each node separately rather than as a whole. It determines the maximum information spread for all remaining candidate nodes before selecting the node with the highest spread. The calculation of the spread for all nodes uses iterations and selects the top 10 nodes in terms fo influence. We compare the maximum influence with the greedy approach with that found by our centrality measure.

**Kendall’s tau ($\tau $):**This is used to determine how closely two ranking lists rate the same set of items [39,40]. Kendall’s tau ($\tau $) measures how many concordant and discordant ranking pairs there are in each of the two lists. Kendall’s tau ($\tau $) [41] is defined as:

## 4. Results

#### 4.1. Correlations of the NTPR Method with Various Centrality Measures

#### 4.2. Cumulative Infected Nodes for the Proposed Centrality and Basic Centralities

#### 4.3. Results on Spreading Information Rate vs. Centrality Value of a Vertex

#### 4.4. Maximum Influence at Various Infection Probabilities with Various Centrality Methods

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

NTPR | Nearest Neighborhood Trust Page Rank |

TPR | Trust–PageRank |

SIR | Susceptible Infected Recovered |

DC | Degree Centrality |

SC | Semi-local Centrality |

PR | PageRank |

CC | Closeness Centrality |

SR | Similarity Ratio |

BC | Betweenness Centrality |

DR | Degree Ratio |

TV | Trust value |

## Appendix A

#### Appendix A.1. Cumulative Infected Nodes for Proposed Centrality with Basic Centralities

**Figure A1.**Cumulative infected nodes of the SIR model according to NTPR and other centralities for email-univ network (top-10 seed nodes).

**Figure A2.**Cumulative infected nodes of the SIR model according to NTPR and other centralities for euroroad network (top-10 seed nodes).

**Figure A3.**Cumulative infected nodes of the SIR model according to NTPR and other centralities for powergrid network (top-10 seed nodes).

**Figure A4.**Cumulative infected nodes of the SIR model according to NTPR and other centralities for web-polblogs network (top-10 seed nodes).

**Figure A5.**The top-10 seed nodes are initial infected nodes, identified by NTPR and a greedy algorithm using the SIR model for four datasets.

#### Appendix A.2. Maximum Influence at Various Infection Probabilities with Various Centrality Methods

**Figure A6.**Normalized maximum influence levels of the top-ten most influential nodes of networks with various infection probabilities using the SIR model (email-univ and web-polblogs).

**Figure A7.**Average information spread of the top-ten most influential nodes of networks with various infection probabilities using the independent cascade model.

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**Figure 2.**Degree ratio $DR({v}_{i},{v}_{j})$, similarity ratio $SR({v}_{i},{v}_{j})$, and trust value $TV({v}_{i},{v}_{j})$ for every pair of nodes of the graph in Figure 1.

**Figure 3.**Correlation $\left(\tau \right)$ between NTPR with DC, BC, CC, SC, PR, and TPR and top nodes.

**Figure 4.**Centrality value with infection rate according to SIR simulations in four networks, column-wise.

**Figure 5.**Cumulative infected nodes of the SIR model according to NTPR and other centralities for four real-world networks (top-10 seed nodes).

**Figure 6.**Average information spread among NTPR with other centralities for the four networks (top-10 seed nodes) using the independent cascade model.

**Figure 7.**Normalized maximum influence levels of the top-ten most influential nodes of networks with various infection probabilities using the SIR model.

**Figure 8.**Normalized maximum information spread of the top-ten most influential nodes of networks with various infection probabilities using the independent cascade model.

Local Centrality | Author and Year |
---|---|

Degree | Freeman et al., 1978 [6] |

Semi-Local | Chen et al., 2012 [9] |

Local Centrality with Coefficient | Zaho et al., 2017 [22] |

Clustering Coefficient | Beralmand et al., 2018 [18,19] |

Normalized Local Centrality | Zhao et al., 2018 [20] |

Local Neighbor Contribution | Dai et al., 2019 [21] |

PageRank | Xing et al., 2004 [27] |

Trust–PageRank | Sheng et al., 2020 [28] |

Nearest Neighborhood Trust Value | Proposed in this paper and Hajarathaiah et al., 2021 [29] |

Real Networks | Vertices | Edges | Maximum Degree | Average Degree | Avg. Cluster Coeff. |
---|---|---|---|---|---|

email-univ | 1133 | 5451 | 71 | 9 | 0.220 |

euroroad | 1174 | 1417 | 10 | 2 | 0.017 |

powergrid | 4941 | 6594 | 19 | 2 | 0.080 |

web-polblogs | 643 | 2280 | 165 | 7.09 | 0.232 |

**Table 3.**Correlation ${\tau}_{(\mathit{NTPR},X)}$, where X is TPR, PR, SC, CC, BC, or DC centrality.

${\mathit{\tau}}_{(\mathit{NTPR},\mathit{X})/}$ Networks | ${\mathit{\tau}}_{(\mathit{NTPR},\mathit{DC})}$ | ${\mathit{\tau}}_{(\mathit{NTPR},\mathit{BC})}$ | ${\mathit{\tau}}_{(\mathit{NTPR},\mathit{CC})}$ | ${\mathit{\tau}}_{(\mathit{NTPR},\mathit{SC})}$ | ${\mathit{\tau}}_{(\mathit{NTPR},\mathit{PR})}$ | ${\mathit{\tau}}_{(\mathit{NTPR},\mathit{TRP})}$ |
---|---|---|---|---|---|---|

email-univ | 0.92 | 0.73 | 0.67 | 0.73 | 0.92 | 0.95 |

euroroad | 0.65 | 0.43 | 0.05 | 0.24 | 0.89 | 0.94 |

powergrid | 0.78 | 0.52 | 0.12 | 0.38 | 0.84 | 0.92 |

web-polblogs | 0.82 | 0.66 | 0.47 | 0.51 | 0.81 | 0.83 |

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**MDPI and ACS Style**

Hajarathaiah, K.; Enduri, M.K.; Anamalamudi, S.; Subba Reddy, T.; Tokala, S. Computing Influential Nodes Using the Nearest Neighborhood Trust Value and PageRank in Complex Networks. *Entropy* **2022**, *24*, 704.
https://doi.org/10.3390/e24050704

**AMA Style**

Hajarathaiah K, Enduri MK, Anamalamudi S, Subba Reddy T, Tokala S. Computing Influential Nodes Using the Nearest Neighborhood Trust Value and PageRank in Complex Networks. *Entropy*. 2022; 24(5):704.
https://doi.org/10.3390/e24050704

**Chicago/Turabian Style**

Hajarathaiah, Koduru, Murali Krishna Enduri, Satish Anamalamudi, Tatireddy Subba Reddy, and Srilatha Tokala. 2022. "Computing Influential Nodes Using the Nearest Neighborhood Trust Value and PageRank in Complex Networks" *Entropy* 24, no. 5: 704.
https://doi.org/10.3390/e24050704