Profit Maximization in Closed Social Networks

Diffusion of information, innovation, and ideas is an important phenomenon in social networks. Information propagates through the network and reaches from one person to the next. In many settings, it is meaningful to restrict diffusion so that each node can spread information to only a limited number of its neighbors rather than to all of them. Such social networks are called closed social networks. In recent years, social media platforms have emerged as an effective medium for commercial entities, where the objective is to maximize profit. In this paper, we study the Profit Maximization in Closed Social Networks (PMCSN) problem in the context of viral marketing. The input to the problem is a closed social network and two positive integers $\ell$ and $B$. The problem asks to select seed nodes within a given budget $B$; during the diffusion process, each node is restricted to choose at most $\ell$ outgoing links for information diffusion; and the objective is to maximize the profit earned by the seed set. The PMCSN problem generalizes the Influence Maximization problem, which is NP-hard. We propose two solution approaches for PMCSN: a sampling-based approximate solution and a marginal-gain-based heuristic solution. We analyze the sample complexity, running time, and space requirements of the proposed approaches. We conduct experiments on real-world, publicly available social network datasets. The results show that the seed sets and diffusion links chosen by our methods yield higher profit than baseline methods. The implementation and data are available at \texttt{https://github.com/PoonamSharma-PY/ClosedNetwork}.

💡 Research Summary

The paper introduces the Profit Maximization in Closed Social Networks (PMCSN) problem, which extends the classic Influence Maximization problem by imposing a per‑node out‑degree limit ℓ on information diffusion. The underlying social network is modeled as a directed weighted graph G(V,E,P) where each edge carries an activation probability. Each user also has an associated cost C(u) for being selected as a seed and a benefit b(u) earned if the user becomes influenced. The goal is to (i) construct a diffusion subgraph GD(V,E′) in which every vertex has at most ℓ outgoing edges, and (ii) choose a seed set S whose total cost does not exceed a budget B, so that the expected net profit ϕ(S)=β(S)−C(S) is maximized. Here β(S) is the expected sum of benefits over all nodes activated under the Independent Cascade (IC) model, averaged over all possible diffusion subgraphs (each equally likely).

Because PMCSN generalizes Influence Maximization, it is NP‑hard. The authors propose two solution strategies.

1. Sampling‑Based Approximate Algorithm – From the exponential set of feasible diffusion graphs α(G), a sample of size x is drawn uniformly at random. For each sampled graph, a seed set SG is obtained using a standard influence‑maximization routine (e.g., CELF) while respecting the budget. The profit ϕ(G) is evaluated, and the best sample is returned. The algorithm runs in O(BC_min·(m+n)·x) time, where C_min is the minimum seed cost, and requires O(m+n) memory for online sampling or O(x·(m+n)) for offline sampling. Using Hoeffding’s inequality, the authors derive a bound on x that guarantees, with probability 1−δ, that the estimated profit deviates from the true optimum by at most ε.

2. Marginal‑Gain Heuristic (HEU) – First, for each node with out‑degree ≥ℓ, the algorithm selects ℓ high‑degree neighbors to form a diffusion graph GD. Then, while budget remains, it repeatedly samples n·k·log(1/ε) candidate nodes (k≈B/C_min) from the non‑seed pool, computes the marginal influence gain per unit cost (σ(S∪{v})−σ(S))/C(v), and adds the node with the highest ratio to the seed set. The heuristic runs in O(n²·(m+n)·log(1/ε)) time and uses O(n²) space in the worst case.

Experimental evaluation on several public social‑network datasets (e.g., LiveJournal, Orkut) shows that both proposed methods outperform baseline influence‑maximization algorithms in terms of net profit, especially when ℓ is small and the budget is tight. The heuristic is notably faster (2–3×) than the baselines, while the sampling approach provides theoretical approximation guarantees at the cost of higher runtime.

The paper’s contributions are: (i) formalizing profit maximization under per‑node diffusion limits, (ii) providing two algorithmic frameworks with rigorous complexity analysis, (iii) deriving sample‑size bounds for the probabilistic approach, and (iv) validating the methods empirically. Limitations include a relatively shallow sensitivity analysis for ℓ and B, and reliance on a simple high‑degree neighbor selection in the heuristic stage. Future work could explore adaptive ℓ selection, multi‑product profit models, or reinforcement‑learning‑based seed selection.