[Paper Review] Maximizing Social Influence in Nearly Optimal Time

Christian Borgs, Michael Brautbar+2 more|arXiv (Cornell University)|Dec 4, 2012

Complex Network Analysis Techniques4 citations

TL;DR

This paper presents a quasi-linear time algorithm for influence maximization under the independent cascade model, achieving a $(1 - \frac{1}{e} - \epsilon)$-approximation in $O((m+n)k\epsilon^{-2}\log n)$ time. It is runtime-optimal up to logarithmic factors and supports early termination with a proportional approximation guarantee.

ABSTRACT

Diffusion is a fundamental graph process, underpinning such phenomena as epidemic disease contagion and the spread of innovation by word-of-mouth. We address the algorithmic problem of finding a set of k initial seed nodes in a network so that the expected size of the resulting cascade is maximized, under the standard independent cascade model of network diffusion. Runtime is a primary consideration for this problem due to the massive size of the relevant input networks. We provide a fast algorithm for the influence maximization problem, obtaining the near-optimal approximation factor of (1 - 1/e - epsilon), for any epsilon > 0, in time O((m+n)k log(n) / epsilon^2). Our algorithm is runtime-optimal (up to a logarithmic factor) and substantially improves upon the previously best-known algorithms which run in time Omega(mnk POLY(1/epsilon)). Furthermore, our algorithm can be modified to allow early termination: if it is terminated after O(beta(m+n)k log(n)) steps for some beta < 1 (which can depend on n), then it returns a solution with approximation factor O(beta). Finally, we show that this runtime is optimal (up to logarithmic factors) for any beta and fixed seed size k.

Motivation & Objective

• Address the need for scalable influence maximization algorithms in massive networks with billions of edges.
• Develop a runtime-efficient algorithm that achieves near-optimal approximation for influence spread under the independent cascade model.
• Provide theoretical guarantees on both runtime and approximation quality, even under early termination.
• Establish a lower bound proving that the algorithm's runtime is optimal up to logarithmic factors.
• Bridge the gap between heuristic, fast methods and provably accurate approximation algorithms in influence maximization.

Proposed method

• Leverage a randomized Monte Carlo approach to estimate influence spread with high probability using $O(\epsilon^{-2}\log n)$ samples per node.
• Use a greedy selection framework with fast influence estimation via sub-sampling and parallel simulation.
• Apply concentration inequalities (e.g., Chernoff bounds) to ensure high-probability approximation guarantees.
• Design a query-efficient algorithm that maintains approximation quality even when terminated early after $O(\beta(m+n)k\log n)$ steps.
• Construct a lower-bound family of graphs using Yao’s Minimax Principle to prove runtime optimality.
• Introduce a robust graph construction with small-weight edges to ensure the lower bound holds under various network assumptions.

Experimental results

Research questions

• RQ1 Can we achieve a $(1 - \frac{1}{e} - \epsilon)$-approximation for influence maximization in quasi-linear time?
• RQ2 What is the minimum runtime required to achieve a constant-factor approximation in the adjacency list model?
• RQ3 Can the algorithm be modified to support early termination while maintaining a predictable approximation factor?
• RQ4 Is there a theoretical lower bound on the number of queries required to achieve a given approximation ratio?
• RQ5 How does the algorithm’s performance scale with network size and seed budget $k$?

Key findings

• The proposed algorithm achieves a $(1 - \frac{1}{e} - \epsilon)$-approximation for influence maximization in $O((m+n)k\epsilon^{-2}\log n)$ time, which is nearly optimal.
• The algorithm supports early termination: after $O(\beta(m+n)k\log n)$ steps, it returns a solution with approximation factor $O(\beta)$.
• A lower bound of $\Omega(m+n)$ is proven for any algorithm achieving a constant approximation, showing the algorithm is optimal up to logarithmic factors.
• The runtime is optimal even under the assumption of uniform node sampling and adjacency list representation.
• The lower bound construction holds for both sparse and dense networks, with a general bound of $\Omega(nd / (T \min\{k,T\}))$ for $m = nd$ edges.
• Theoretical analysis confirms that no randomized algorithm can achieve better than $\beta$-approximation with fewer than $\Omega(\beta(m+n)/\min\{k,1/\beta\})$ queries with high probability.