Influence Diffusion Dynamics and Influence Maximization in Social Networks with Friend and Foe Relationships

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.

💡 Research Summary

This paper pioneers the study of influence diffusion and influence maximization in online social networks that contain both positive (friend/trust) and negative (foe/distrust) relationships, which are naturally modeled as signed directed graphs. The authors extend the classic voter model—a stochastic opinion‑exchange process—by incorporating edge signs: when a node selects a neighbor through a positive edge it adopts the neighbor’s current opinion, whereas selection via a negative edge forces the node to adopt the opposite opinion. Formally, for a signed graph G = (V,E,A) with weighted adjacency matrix A (positive entries for friends, negative for foes) they define the signed transition matrix P = D⁻¹A, where D is the diagonal matrix of absolute out‑degrees, and the negative‑edge influence vector g⁻ = D⁻¹A⁻¹. The opinion distribution after t steps is given exactly by

 xₜ = Pᵗx₀ + (∑_{i=0}^{t‑1} Pⁱ) g⁻,

where x₀ is the initial white‑opinion probability vector. This closed‑form expression enables precise short‑term analysis.

For long‑term behavior, the authors classify signed graphs into three structural families, borrowing from structural balance theory:

1. Balanced graphs – every cycle has a positive sign product; the network eventually reaches consensus (all nodes share the same opinion).
2. Anti‑balanced graphs – every cycle has a negative sign product; the network settles into a bipartite pattern where the two partitions hold opposite opinions.
3. Strictly unbalanced graphs – neither of the above; nodes converge to a probabilistic mixture, each retaining a non‑trivial probability of being white or black.

These classifications hold for strongly connected components and extend to weakly connected or disconnected graphs by analyzing each strongly connected component separately.

The influence maximization problem is defined under two objectives: (a) maximize the expected number of white nodes after a fixed finite horizon t (short‑term), and (b) maximize the steady‑state white proportion (long‑term). For the short‑term case, the linearity of the expression for xₜ implies that the marginal gain of adding a seed node can be computed directly; a greedy algorithm that repeatedly selects the node with the largest marginal gain is provably optimal.

The long‑term case requires different treatment for each graph family. In balanced and anti‑balanced graphs the optimal seeds are those with the highest normalized centrality within each partition, because they most efficiently drive the whole component toward the desired opinion. In strictly unbalanced graphs the authors derive the steady‑state gain vector (I − P)⁻¹g⁻, which quantifies the expected increase in the white proportion caused by seeding each node. Selecting the top‑k entries of this vector yields the optimal seed set. All algorithms run in O(|E|) time using sparse matrix operations and iterative power methods to approximate (I − P)⁻¹ when necessary, making them scalable to networks with millions of nodes and edges.

Extensive experiments on real signed social networks (Epinions and Slashdot) and on synthetic graphs with varying sizes, densities, and negative‑edge ratios validate the theory. Compared with baselines such as unsigned PageRank, random seeding, highest‑degree seeding, and unsigned voter‑model greedy methods, the proposed algorithms achieve 15‑30 % higher influence coverage in anti‑balanced and unbalanced settings, and converge faster in balanced settings. Runtime measurements confirm linear scalability.

In summary, the paper makes four major contributions: (1) a novel signed voter model that captures opposite‑opinion adoption over negative ties; (2) a rigorous structural classification of signed graphs and corresponding steady‑state analyses; (3) optimal seed‑selection algorithms for both short‑term and long‑term objectives, with provable optimality and linear‑time complexity; and (4) empirical evidence that accounting for foe relationships dramatically improves influence‑maximization performance in real‑world signed social networks. The work opens avenues for future research on topic‑dependent sign dynamics, temporal signed networks, and multi‑opinion extensions.