The random walk-based gravity model to identify influential nodes in complex networks

Jie Zhao, Tao Wen, Hadi Jahanshahi, Kang Hao Cheong

Abstract: The identification of influential nodes in complex networks has been a topic of immense interest. In most cases, the local approach represented by degree centrality performs well but has limitations when dealing with the bridge nodes. In order to solve the problem of being trapped in the locality, researchers have proposed many useful methods. The gravity model is an emerging research direction among them. However, such models have to exhaust the shortest distance between all nodes, which renders them impractical and difficult to run over large graphs. In order to address this issue, we propose a random walk-based gravity model to identify influential spreaders. Our proposed model decreases the time complexity of calculating the shortest distance—a critical step in the conventional gravity models, from O(|V|2)<math><mrow is="true"><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><mo stretchy="false" is="true">|</mo><mi is="true">V</mi><msup is="true"><mrow is="true"><mo stretchy="false" is="true">|</mo></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup><mo stretchy="false" is="true">)</mo></mrow></math> to O(|V|*γ*lr(l-r))<math><mrow is="true"><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><mo stretchy="false" is="true">|</mo><mi is="true">V</mi><mo stretchy="false" is="true">|</mo><mspace width="0.35em" height="0.8ex" is="true"></mspace><mo is="true">*</mo><mspace width="0.35em" height="0.8ex" is="true"></mspace><mi is="true">γ</mi><mo is="true">*</mo><mspace width="0.35em" height="0.8ex" is="true"></mspace><mfrac is="true"><mrow is="true"><mi is="true">l</mi></mrow><mrow is="true"><mi is="true">r</mi><mo stretchy="false" is="true">(</mo><mi is="true">l</mi><mo is="true">-</mo><mi is="true">r</mi><mo stretchy="false" is="true">)</mo></mrow></mfrac><mo stretchy="false" is="true">)</mo></mrow></math>, and reduces space complexity of O(|V|2)<math><mrow is="true"><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><mo stretchy="false" is="true">|</mo><mi is="true">V</mi><msup is="true"><mrow is="true"><mo stretchy="false" is="true">|</mo></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup><mo stretchy="false" is="true">)</mo></mrow></math> to O(<K>2|V|)<math><mrow is="true"><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><mo is="true">&lt;</mo><mi is="true">K</mi><msup is="true"><mrow is="true"><mo is="true">&gt;</mo></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup><mo stretchy="false" is="true">|</mo><mi is="true">V</mi><mo stretchy="false" is="true">|</mo><mo stretchy="false" is="true">)</mo></mrow></math>, where <K>2≪|V|<math><mrow is="true"><mo is="true">&lt;</mo><mi is="true">K</mi><msup is="true"><mrow is="true"><mo is="true">&gt;</mo></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup><mo is="true">≪</mo><mo stretchy="false" is="true">|</mo><mi is="true">V</mi><mo stretchy="false" is="true">|</mo></mrow></math> and γ*lr(l-r)≪|V|<math><mrow is="true"><mi is="true">γ</mi><mo is="true">*</mo><mfrac is="true"><mrow is="true"><mi is="true">l</mi></mrow><mrow is="true"><mi is="true">r</mi><mo stretchy="false" is="true">(</mo><mi is="true">l</mi><mo is="true">-</mo><mi is="true">r</mi><mo stretchy="false" is="true">)</mo></mrow></mfrac><mo is="true">≪</mo><mo stretchy="false" is="true">|</mo><mi is="true">V</mi><mo stretchy="false" is="true">|</mo></mrow></math>. Some random walk properties are also investigated to support our model. In order to demonstrate the feasibility of the proposed gravity centrality, we have verified its spreading ability and convergence speed under different random walk strategies. Experimental results indicate that our method performs far better than most gravity models.