Collective Influence Algorithm to find influencers via optimal percolation in massively large social media

Abstract: We elaborate on a linear time implementation of the Collective Influence (CI) algorithm introduced by Morone, Makse, Nature 524, 65 (2015) to find the minimal set of influencers in a network via optimal percolation. We show that the computational complexity of CI is O(N log N) when removing nodes one-by-one, with N the number of nodes. This is made possible by using an appropriate data structure to process the CI values, and by the finite radius l of the CI sphere. Furthermore, we introduce a simple extension of CI when l is infinite, the CI propagation (CI_P) algorithm, that considers the global optimization of influence via message passing in the whole network and identifies a slightly smaller fraction of influencers than CI. Remarkably, CI_P is able to reproduce the exact analytical optimal percolation threshold obtained by Bau, Wormald, Random Struct. Alg. 21, 397 (2002) for cubic random regular graphs, leaving little improvement left for random graphs. We also introduce the Collective Immunization Belief Propagation algorithm (CI_BP), a belief-propagation (BP) variant of CI based on optimal immunization, which has the same performance as CI_P. However, this small augmented performance of the order of 1-2 % in the low influencers tail comes at the expense of increasing the computational complexity from O(N log N) to O(N^2 log N), rendering both, CI_P and CI_BP, prohibitive for finding influencers in modern-day big-data. The same nonlinear running time drawback pertains to a recently introduced BP-decimation (BPD) algorithm by Mugisha, Zhou, arXiv:1603.05781. For instance, we show that for big-data social networks of typically 200 million users (eg, active Twitter users sending 500 million tweets per day), CI finds the influencers in less than 3 hours running on a single CPU, while the BP algorithms (CI_P, CI_BP and BDP) would take more than 3,000 years to accomplish the same task.

An Analysis of the Collective Influence Algorithm for Optimal Percolation in Social Networks

The paper presented focuses on the implementation and analysis of the Collective Influence (CI) algorithm, aimed at determining the minimal set of influential nodes within complex networks using optimal percolation methods. It builds upon previous work by Morone and Makse (Nature 524, 2015), emphasizing improvements in computational efficiency and exploring extensions of the algorithm that incorporate global network information.

Computational Framework of the CI Algorithm

The CI algorithm, designed for influence maximization in networks, benefits from a computational complexity of $O(N \log N)$, where $N$ represents the number of nodes. This is achieved through a max-heap data structure that efficiently processes CI values by organizing them into a hierarchy, facilitating swift access to the maximum CI value for iterative node removal. The algorithm calculates the CI values using a sphere of finite radius $\ell$ centered around each node, allowing for the systematic dismantlement of the giant component.

Extensions to the CI Algorithm: CI Propagation and CI Belief Propagation

The paper introduces two algorithmic extensions: CI Propagation ($CI_{\rm P}$) and Collective Immunization Belief Propagation ($CI_{\rm BP}$). While $CI_{\rm P}$ considers the global optimization of influence via message passing, thereby identifying a smaller fraction of influencers than the original CI framework, it suffers from increased computational complexity, $O(N^2 \log N)$. Similarly, the $CI_{\rm BP}$ algorithm, inspired by the Susceptible-Infected-Recovered (SIR) model, struggles with quadratic time complexity, rendering it impractical for large-scale networks typical in contemporary social media platforms.

Practical Implications and Theoretical Considerations

The advancements proposed in this paper highlight a trade-off between computational efficiency and the accuracy of influencer detection. Despite achieving more precise identification of influential nodes in small networks, $CI_{\rm P}$ and $CI_{\rm BP}$ are hindered by prohibitive running times in large datasets. Conversely, the original CI algorithm provides a practical solution for large networks, executing in linear time and accommodating networks with up to hundreds of millions of nodes, a critical requirement given the scale of platforms like Twitter.

Theoretical implications of this work suggest opportunities for future research in network theory, particularly concerning network integrity, percolation theory, and the optimization of large networks beyond social media.

Future Directions

Looking forward, the development of algorithms that marry computational efficiency with optimal performance remains a challenge. Approaches blending theoretical insights from non-backtracking matrices with practical machine learning innovations might hold promise. Moreover, exploring parallel computation or advanced data structures could further improve execution times for extensions like $CI_{\rm P}$ and $CI_{\rm BP}$.

In summary, the paper offers a detailed exploration of the CI algorithm, stressing its utility in efficiently locating key influencers within expansive social networks while acknowledging the limitations of more computationally intensive algorithmic variants. This study sets a robust foundation for continued exploration into network analysis and its applications in digital communication landscapes.