Profit Maximization in Closed Social Networks

• Profit Maximization in Closed Social Networks (PMCSN) is defined as the strategic selection of seed nodes to maximize net profit by balancing user activation benefits, selection costs, and diffusion constraints.
• It employs advanced combinatorial, approximation, and learning-based algorithms to tackle the NP-hard challenges inherent in closed diffusion settings.
• Key methods such as sampling-based approximations, marginal-gain heuristics, and adaptive seeding strategies achieve near-optimal results under strict budget and propagation limits.

Profit maximization in closed social networks (PMCSN) concerns the selection of nodes (seed set) and the structuring of information diffusion for the purpose of maximizing net profit, accounting for both the monetary benefits of activating users and the costs associated with seed incentives and limited propagation capacities. A closed social network is typically defined as one in which the diffusion is restricted—either by explicit limits on the number of outgoing links each node can use for propagation or by community boundaries that limit external connectivity. The challenging nature of PMCSN arises from the interplay between submodular influence/diffusion processes, combinatorial budget and propagation constraints, and additional economic constraints such as pricing, inequity aversion, or profiting under uncertainty. PMCSN generalizes the classic influence maximization problem and is generally NP-hard, requiring advanced combinatorial, approximation, and, more recently, learning-based algorithms for practical solution (Sharma et al., 1 Feb 2026, Sharma et al., 1 Feb 2026, Guo et al., 2020, Gu et al., 2022, Lu et al., 2012, Tang et al., 2017, Amanatidis et al., 2016).

1. Formal Model and Problem Definition

In the canonical PMCSN setting, a social network is given as a (possibly directed) graph $G = (V, E, P)$, where $V$ is the set of users, $E$ the set of social ties, and $P: E \to (0,1]$ assigns propagation probabilities to edges. Each node $u \in V$ has an associated selection cost $C(u)$ and a benefit $b(u)$ realized if $u$ becomes activated. A total budget $B$ is available for seed incentives, with the requirement that $C(S) = \sum_{u \in S} C(u) \leq B$ for the seed set $S$.

The diffusion process is typically a variant of the Independent Cascade or Linear Threshold model, but in PMCSN, an additional structural constraint applies: each node can propagate influence to at most $\ell$ of its neighbors per diffusion realization (the "closedness" parameter), or further limitations may apply at community or pricing levels (Sharma et al., 1 Feb 2026, Amanatidis et al., 2016, Guo et al., 2020). The expected benefit from a seed set $S$ is given by

$$\beta(S) = \mathbb{E}_{G_D} \left[ \sum_{u\in\sigma_{G_D}(S)} b(u) \right],$$

where $G_D$ is a possible diffusion subgraph satisfying the closure constraint, and $\sigma_{G_D}(S)$ is the final influenced set.

The objective is to maximize the profit function

$\phi(S) = \beta(S) - C(S)$

subject to the constraints

$$C(S) \leq B, \;\; \deg^+_{G_D}(u) \leq \ell,\; \forall u \in V.$$

Extended models incorporate pricing vectors, multi-phase seeding, community constraints, and robust or partial-observation variants (Lu et al., 2012, Tang et al., 2017, Sharma et al., 2022).

2. Complexity and Theoretical Properties

PMCSN is NP-hard, subsuming influence maximization and discrete profit maximization as strict special cases (Sharma et al., 1 Feb 2026, Gu et al., 2022). Even for fixed diffusion graphs and simple profit functions, maximizing $\phi(S)$ is at least as hard as monotone submodular maximization under knapsack constraints, and tight hardness-of-approximation bounds ($(1-1/e) - \varepsilon$ unless $\text{P} = \text{NP}$) are established. With additional closure constraints, the feasible space becomes further restricted, often breaking monotonicity and complicating algorithm design (Gu et al., 2022, Guo et al., 2020).

Submodularity is generally preserved when the propagation process is monotone and the profit function is a nonnegative linear combination of activated nodes, but adding costs or more complex economic objectives (e.g., pricing, inequity aversion) introduces non-monotonicity and in some settings destroys submodularity altogether (Lu et al., 2012, Tang et al., 2017, Sharma et al., 22 Jan 2026).

For profit functions that are differences of submodular functions (e.g., benefit minus cost), algorithms based on iterative pruning and modular-modular approximations can provide empirical and sometimes theoretical guarantees (Tang et al., 2017).

3. Algorithmic Approaches

Several classes of algorithms have been developed for PMCSN:

3.1 Sampling-Based Approximation

Sampling approaches construct multiple diffusion graphs respecting closure constraints, solve profit maximization on each, and output the best observed solution. For the submodular part of the objective, greedy maximization yields a $(1-1/e)$-approximation per sample. Uniform convergence over samples yields a $(1-1/e-\varepsilon)$-approximate solution with high probability, though at substantial computational cost for large $n, m$ (Sharma et al., 1 Feb 2026, Guo et al., 2020, Gu et al., 2022).

3.2 Marginal-Gain Heuristics

Marginal-cost/benefit-ratio greedy heuristics build the seed set incrementally, selecting at each step the node with highest expected marginal profit per cost under the current diffusion graph. Fast variants select a sample of candidates at each step to accelerate the search. These methods are effective for large graphs and empirically provide near-optimal performance despite lacking worst-case approximation ratios (Sharma et al., 1 Feb 2026, Sharma et al., 22 Jan 2026, Sharma et al., 2022).

3.3 Lattice-Based Double Greedy and Pruning

When the objective is (possibly non-monotone) DR-submodular over the integer lattice, double greedy algorithms (including binary search and iterative pruning enhancements) provide a ½-approximation and have near-optimal worst-case guarantees. Pruning steps shrink the feasible domain to ensure non-negative starting points for effective application of double greedy methods (Gu et al., 2022, Guo et al., 2020).

3.4 Two-Phase and Adaptive Seeding

Two-phase approaches split the seed budget and allow for adaptive re-optimization after partially observing the first-phase diffusion. The first-phase seeds are chosen to explore the graph, and the second-phase seeds exploit new information. The associated objective is subadditive but generally neither monotone nor submodular. In empirical studies, profit gains of 18–40% over single-phase greedy have been observed (Sharma et al., 22 Jan 2026, Sharma et al., 2022).

3.5 Deep Learning Approaches

DeepPM and other learning-based frameworks embed the seed-selection problem into a latent space parameterized by neural networks (e.g., GCNs), with a teacher–student architecture learning to approximate the expected profit through supervised (cascade-based) regression. The profit is predicted differentiably as a function of soft seed masks, enabling direct optimization in continuous latent space (Sharma et al., 1 Feb 2026). Empirically, such methods achieve up to 60% more profit than classical heuristics in medium-scale networks, with seed sets 20–30% smaller.

4. Key Extensions and Variants

4.1 Pricing and Inequity Aversion

Variants of PMCSN introduce per-node pricing, user valuations, inequity aversion constraints, and multiple price tiers (Lu et al., 2012, Amanatidis et al., 2016). These models often require joint optimization of seed selection and price assignment, subject to revenue, valuation, and fairness constraints. For fixed price vectors, profit remains submodular in the seeds. For general pricing, efficient exact and approximate algorithms exist for small numbers of price tiers; APX-hardness is encountered as pricing complexity increases (Amanatidis et al., 2016).

4.2 Team Grouping and Task Allocation

TeamGrouping considers formation of socially compatible teams to maximize multi-task profit (e.g., group-based promotions/assignments), formalized as a covering integer program with skill, compatibility, and load constraints. Approximation algorithms combine LP/ellipsoid relaxations with rounding and support practical deployment in closed network organizations (Tang et al., 2016).

4.3 Dynamic and Competitive Settings

Extensions to competitive scenarios (multi-firm viral campaigns), robust profit maximization under graph/diffusion uncertainty, open-loop equilibrium in dynamic budget allocation, and multi-stage decision problems have been addressed with concave-programming (single player) and regret minimization (multi-player) (Etesami, 2021). These admit polynomial algorithms when stage utility is concave in resource allocations.

5. Empirical Findings and Practical Guidelines

Experimental evaluations on real social networks (Email-Eu-core, Facebook, Wiki-Vote, Epinions, Slashdot) confirm the superiority of profit-maximization-aware algorithms over baselines such as High-Degree or naive heuristics. Key empirical observations include:

• Marginal-gain-based heuristics and double-greedy/binary search methods achieve 20–50% higher profit than degree-based baselines, scaling to networks with $n \sim 10^4-10^5$.
• Two-phase/greedy methods yield an additional 5–40% gain compared to single-phase, especially when observation and re-optimization is possible (Sharma et al., 22 Jan 2026, Sharma et al., 2022).
• Iterative pruning and reverse sampling dramatically reduce candidate set size and computational effort, maintaining empirical ratios near the theoretical guarantees (Tang et al., 2017, Guo et al., 2020).
• Learning-based approaches (DeepPM) achieve further profit increases, especially in heterogenous and dense networks, and produce compact seed sets (Sharma et al., 1 Feb 2026).

Recommendations for practice:

• Use sampling-based or DR-submodular double greedy methods when theoretical guarantees are important and compute resources are available.
• Marginal gain heuristics provide the best tradeoff for large graphs requiring fast solutions.
• For two-phase or adaptive settings, empirical tuning of budget splits can yield substantial gains.
• Learning-based surrogates are suited for networks where standard diffusion models are misaligned with actual propagation.
• For strict closure settings (limited $\ell$), ensure diffusion graphs are constructed accordingly prior to seed selection.

6. Future Directions and Open Challenges

• Development of scalable, theoretically sound algorithms for highly constrained, non-monotone, or non-submodular PMCSN variants.
• Integration of partially observed, dynamic, or privacy-limited social graph scenarios, combining robust optimization and online learning (Tang et al., 2017, Guo et al., 2020).
• Multi-phase and multi-objective formulations accounting for user retention, competitive campaigns, or heterogeneous product features.
• Extending deep learning frameworks—currently limited to offline and per-network training—to enable transfer learning and hierarchical scaling.
• Algorithmic research on joint pricing, fairness (inequity aversion), and profit maximization remains open beyond small price-set cases (Amanatidis et al., 2016, Lu et al., 2012).
• Investigation of privacy-preserving protocols and partial-feedback models in closed or private social networks is ongoing (Sharma et al., 22 Jan 2026, Tang et al., 2017).

7. Summary Table: Major Algorithmic Methods

Method	Theoretical Guarantee	Time Complexity
Sampling-based (SBA)	(1-1/e-ε)-approx, w.h.p.	O(x·n(m+n))
Marginal gain heuristic	None (empirically near-opt.)	O(n^2(m+n)·ln(1/ε))
Double greedy (binary)	1/2-approx (DR-submodular)	O(n·log B)
Iterative pruning + RIS	1/2-ε approx (exp. speedup)	O((m+n)n²·log(1/δ)/ε²)
Two-phase greedy	None, strong empirics	O(n(m+n))
DeepPM (learning-based)	No formal ratio	O((T·(

These methods are selected and adapted according to the specific PMCSN variant, network size, budget, closure constraints, and practical deployment requirements (Sharma et al., 1 Feb 2026, Sharma et al., 1 Feb 2026, Gu et al., 2022, Sharma et al., 22 Jan 2026, Guo et al., 2020).