[Paper Review] Influence Diffusion Dynamics and Influence Maximization in Social Networks with Friend and Foe Relationships

Yanhua Li, Wei Chen+2 more|arXiv (Cornell University)|Nov 21, 2011

Opinion Dynamics and Social Influence24 references4 citations

TL;DR

This paper proposes a signed voter model to study influence diffusion in social networks with both friend (positive) and foe (negative) relationships, extending the classic voter model to signed digraphs. It characterizes influence dynamics in terms of graph balance, identifies three structural regimes—balanced, anti-balanced, and strictly unbalanced—and develops efficient algorithms for influence maximization that outperform unsigned counterparts by up to 58% in influence coverage, especially in short-term cascades when edge signs are leveraged.

ABSTRACT

Influence diffusion and influence maximization in large-scale online social networks (OSNs) have been extensively studied, because of their impacts on enabling effective online viral marketing. Existing studies focus on social networks with only friendship relations, whereas the foe or enemy relations that commonly exist in many OSNs, e.g., Epinions and Slashdot, are completely ignored. In this paper, we make the first attempt to investigate the influence diffusion and influence maximization in OSNs with both friend and foe relations, which are modeled using positive and negative edges on signed networks. In particular, we extend the classic voter model to signed networks and analyze the dynamics of influence diffusion of two opposite opinions. We first provide systematic characterization of both short-term and long-term dynamics of influence diffusion in this model, and illustrate that the steady state behaviors of the dynamics depend on three types of graph structures, which we refer to as balanced graphs, anti-balanced graphs, and strictly unbalanced graphs. We then apply our results to solve the influence maximization problem and develop efficient algorithms to select initial seeds of one opinion that maximize either its short-term influence coverage or long-term steady state influence coverage. Extensive simulation results on both synthetic and real-world networks, such as Epinions and Slashdot, confirm our theoretical analysis on influence diffusion dynamics, and demonstrate the efficacy of our influence maximization algorithm over other heuristic algorithms.

Motivation & Objective

• To model influence diffusion in social networks that include both positive (friend) and negative (foe) relationships, which are often ignored in existing studies.
• To extend the classic voter model to signed digraphs, where negative edges reverse the opinion of the neighbor.
• To characterize the short-term and long-term dynamics of influence diffusion based on structural properties of signed networks.
• To develop efficient algorithms for influence maximization that select initial seed sets to maximize either short-term or long-term influence coverage.

Proposed method

• Extends the classic voter model to signed digraphs by having each node adopt the opinion of a random outgoing neighbor if the edge is positive, or the opposite opinion if the edge is negative.
• Introduces a mathematical framework to analyze the dynamics of two-opinion diffusion on signed networks, focusing on steady-state behavior.
• Classifies signed digraphs into three types based on structural balance: balanced, anti-balanced, and strictly unbalanced, each exhibiting distinct dynamic behaviors.
• Proposes the Signed Voter Influence Maximization (SVIM) algorithm to select initial seed sets that maximize influence spread under the signed voter model.
• Employs simulation on synthetic networks and real-world datasets (Epinions, Slashdot) to validate theoretical findings and compare performance against unsigned heuristics.
• Evaluates influence coverage under three scenarios: original signed graphs, unsigned graphs with only positive edges, and graphs with ignored edge signs, to quantify the value of sign information.

Experimental results

Research questions

• RQ1 How does the presence of negative (foe) relationships alter the dynamics of opinion diffusion in social networks compared to models that only consider positive (friend) relationships?
• RQ2 What are the structural conditions in signed networks—specifically balanced, anti-balanced, and strictly unbalanced graphs—that lead to distinct long-term steady-state behaviors in influence diffusion?
• RQ3 To what extent does incorporating edge signs improve influence maximization performance in terms of both short-term and long-term influence coverage?
• RQ4 Can the proposed signed voter model accurately predict influence dynamics in real-world signed networks such as Epinions and Slashdot?
• RQ5 How does the convergence behavior of the signed voter model differ across various signed network topologies, especially in unbalanced or weakly connected graphs?

Key findings

• Influence maximization using the signed voter model (SVIM) outperforms unsigned heuristics by up to 58% in influence coverage on synthetic weakly connected signed digraphs for t ≥ 6.
• On synthetic balanced digraphs, incorporating edge signs yields 5%–16% higher influence coverage for t ≥ 6 compared to unsigned models.
• In the Epinions dataset, which is strictly unbalanced, the benefit of using edge signs is most pronounced in the short term, with up to 38% and 21% more influence generated at step 1 in the full dataset and largest strongly connected component, respectively.
• The signed voter model identifies a new class of anti-balanced digraphs that exhibit oscillating steady-state behavior, not previously covered by classical structural balance theory.
• The performance gap between signed and unsigned models is most significant in short-term influence spread, indicating that edge sign information is critical for early cascade growth.
• Theoretical analysis confirms that convergence times for balanced and anti-balanced ergodic digraphs are equivalent to those in unsigned networks, while convergence in strictly unbalanced and weakly connected graphs remains an open problem.