Across-group Maximization (AGM)

• Across-group Maximization (AGM) is a fairness-oriented formulation for influence maximization that balances influence spread among groups by enforcing diversity constraints and coordinated seed selection.
• It employs two main strategies—using diversity constraints directly and a two-step coordination (IGM then AGM)—to address fairness while maximizing total influence.
• Empirical studies show that AGM effectively reduces group disparity with only a marginal sacrifice in total influence, achieving a price of fairness typically between 1.05 and 1.15.

Across-group Maximization (AGM) is a fairness-oriented formulation for influence maximization in which seed selection is evaluated by its consequences across predefined groups rather than only by total expected spread. In the influence-maximization literature summarized here, the term appears in two closely related senses. In "Group-Fairness in Influence Maximization" (Tsang et al., 2019), AGM is the diversity-constraint formulation requiring each group to receive at least the influence it could achieve alone under a proportional seed allotment. In "Efficient Approximation Algorithms for Fair Influence Maximization under Maximin Constraint" (Rui et al., 30 Sep 2025), AGM is the second step of a two-step optimization framework that coordinates seed selection across groups after group-specific candidate lists have been constructed. Both formulations address the same central issue: standard influence maximization techniques oftentimes neglect smaller groups which contribute less to overall utility, resulting in disparity across groups.

1. Problem setting and formal objectives

Influence maximization selects up to $k$ seed nodes in a graph $G$ in order to maximize expected spread of influence. In the group-fairness setting, the population is partitioned into groups $\mathcal{C} = \{C_1,\ldots,C_m\}$, where each $C_i \subseteq V$, and the groups are not necessarily disjoint, which allows intersectionality. Influence is measured globally by $I_G(A)$, the expected number of nodes activated when seeding $A \subseteq V$, and groupwise by $I_{G,C_i}(A)$, the expected number of activated nodes in group $C_i$ when seeding $A$ (Tsang et al., 2019).

A central maximin objective is to maximize the minimum proportional influence across groups: $U^{Maximin}(A)=\min_i \frac{I_{G,C_i}(A)}{|C_i|}.$ The 2025 maximin-constraint formulation writes the same idea as

$G$0

where $G$1 is the expected number of influenced nodes in group $G$2. The utility of a solution is therefore the utility of the worst-off group. This objective is direct and intuitive, but the global maximin objective does not satisfy submodularity, which is the main obstacle to importing classical greedy influence-maximization guarantees (Rui et al., 30 Sep 2025).

2. AGM as diversity constraints

In the 2019 formulation, AGM is the requirement that each group should get at least as much influence as it could achieve alone if given its fair share of seeds. The seed allocation for group $G$3 is

$G$4

For that group, one computes the influence in the subgraph $G$5, denoted $G$6. A seed set $G$7 satisfies the AGM constraints if

$G$8

The associated objective maximizes total expected influence subject to all group constraints: $G$9 The AGM solution is written as $\mathcal{C} = \{C_1,\ldots,C_m\}$0, with

$\mathcal{C} = \{C_1,\ldots,C_m\}$1

This formulation is called a diversity-constraint or group-rational baseline: it does not directly maximize the minimum proportional influence, but instead enforces a lower bound for every group and then optimizes total spread subject to those lower bounds (Tsang et al., 2019).

The same work defines a price of fairness for both the diversity-constraint and maximin formulations as the reduction in unconstrained influence caused by imposing fairness constraints. The reported empirical price of fairness is small, typically between 1.05 and 1.15. This suggests that, at least on the studied networks, substantial disparity reduction can be obtained with little loss in total influence (Tsang et al., 2019).

3. AGM as the across-group coordination stage in a two-step framework

The 2025 work introduces a two-step optimization framework consisting of Inner-group Maximization (IGM) and Across-group Maximization (AGM). The key structural observation is that the influence spread function within each group, $\mathcal{C} = \{C_1,\ldots,C_m\}$2, remains submodular: $\mathcal{C} = \{C_1,\ldots,C_m\}$3 for all $\mathcal{C} = \{C_1,\ldots,C_m\}$4, $\mathcal{C} = \{C_1,\ldots,C_m\}$5. This enables effective optimization within groups even though the global maximin objective is not submodular (Rui et al., 30 Sep 2025).

IGM uses a greedy-style algorithm, specifically IMM, to generate for each group $\mathcal{C} = \{C_1,\ldots,C_m\}$6 a ranked list of $\mathcal{C} = \{C_1,\ldots,C_m\}$7 candidate seeds maximizing $\mathcal{C} = \{C_1,\ldots,C_m\}$8. AGM is then the coordination stage: it chooses $\mathcal{C} = \{C_1,\ldots,C_m\}$9 seeds from the union of the IGM-generated seed lists for each group so that $C_i \subseteq V$0 is maximized. The decomposition is motivated by the fact that, even after high-quality group-specific candidates have been identified, the final seed set must still balance groups so as to lift the worst-off group.

A plausible implication is that AGM is best understood not as a standalone influence objective, but as a coordination layer that exploits submodularity where it survives—inside each group—while managing the non-submodular interaction across groups at the final selection stage.

4. AGM-US and AGM-GS

The AGM stage provides two strategies: Uniform Selection (AGM-US) and Greedy Selection (AGM-GS). Both take as input the graph $C_i \subseteq V$1, group structuring $C_i \subseteq V$2, the IMM output lists $C_i \subseteq V$3 for each group $C_i \subseteq V$4, the budget $C_i \subseteq V$5, and optionally precomputed RR-sets for efficient utility evaluation (Rui et al., 30 Sep 2025).

AGM-US performs column-wise coordination. It repeatedly selects the first seed from each group’s list, then the second, and so on, uniformly. When the last column would exceed the budget, it greedily chooses from that column to fill the remaining slots by evaluating

$C_i \subseteq V$6

for each candidate $C_i \subseteq V$7 in that column and adding the node that maximizes $C_i \subseteq V$8. Its defining property is even spread across groups, which the paper describes as fair representation.

AGM-GS is utility-driven. At each iteration, for each group $C_i \subseteq V$9, it considers the current prefix, namely the next available node in $I_G(A)$0 not already included. From that candidate set it selects the node maximizing the current minimum utility $I_G(A)$1 if added. After selection, all group prefixes are updated if the chosen node appears in multiple group lists, thereby avoiding duplicates in the seed set. Its defining property is that it greedily targets the largest increase in the current min utility, regardless of which group provides the candidate.

The contrast between the two strategies is therefore explicit. AGM-US enforces an even allocation pattern across groups; AGM-GS greedily pursues the worst-group objective itself.

5. Approximation guarantees and structural assumptions

AGM-US has a general lower bound: $I_G(A)$2 where $I_G(A)$3 is the number of groups and $I_G(A)$4 is the error parameter from IGM/IMM. The abstract summarizes this guarantee as a roughly $I_G(A)$5 lower bound regardless of the group structure. The significance of the result is that it is structure-agnostic: overlapping, connected, or highly interlaced group structures do not invalidate the guarantee (Rui et al., 30 Sep 2025).

AGM-GS has the stronger approximation guarantee

$I_G(A)$6

when groups are completely disconnected, meaning there are no edges between groups and each group’s influence is determined only by its own seeds. In that setting, AGM-GS recovers the classic greedy submodular approximation for the maximin objective. The paper is equally explicit about the limitation: there is no general guarantee when groups are interconnected, because cross-group influence breaks submodularity of the global min-utility function.

The practical role of group connectivity is emphasized through the parameter $I_G(A)$7. Low $I_G(A)$8 means groups are almost disconnected, in which case AGM-GS is preferable; high $I_G(A)$9 means many cross-group edges, in which case the theoretical bound for AGM-GS disappears and AGM-US provides the robust bound. The empirical claim that AGM-GS usually outperforms AGM-US on connected networks is presented as an observed tendency rather than a theorem.

6. Algorithmic context and empirical behavior

For the diversity-constraint form of AGM, the 2019 paper reduces the problem to multi-objective submodular maximization. For each group $A \subseteq V$0, it defines the submodular function $A \subseteq V$1 and the target $A \subseteq V$2, then adds the total objective $A \subseteq V$3 and performs binary search for the highest possible total utility subject to $A \subseteq V$4 for all groups. The proposed optimization pipeline consists of threshold inclusion, continuous relaxation and optimization via a Frank-Wolfe type method called MultiFW, a stochastic saddle-point mirror descent subroutine called S-SP-MD, and swap rounding to produce a discrete seed set. For $A \subseteq V$5, the approximation ratio approaches $A \subseteq V$6 as $A \subseteq V$7 increases, matching the best-known ratio for unconstrained submodular influence maximization (Tsang et al., 2019).

Empirically, the paper evaluates Greedy, AGM/DC, and Maximin on four real-world social networks of homeless youth and on 20 simulated networks for obesity prevention. The reported findings are that standard Greedy substantially neglects small groups, with large diversity-constraint violations and near-zero minimum proportional influence for some sensitive attributes; AGM/DC reduces diversity-constraint violations by 55–65% versus Greedy; AGM/DC performs almost as well as, or better than, Maximin with respect to minimum influence and much better with respect to constraint violations; and the price of fairness is typically between 1.05 and 1.15. In overlapping or intersectional groups, the algorithm still reduces unfairness at modest cost. These results situate AGM within a broader empirical claim: fairness constraints can materially reduce disparity without requiring large sacrifices in total spread.

7. Terminological boundaries and related concepts

Across-group Maximization in influence maximization should be distinguished from several unrelated arXiv uses of the acronym “AGM.” In database theory, “AGM” denotes the AGM bound, a worst-case upper bound on join output size used in runtime guarantees such as $A \subseteq V$8 (Kim et al., 2023). In belief change, “AGM” denotes the Alchourrón-Gärdenfors-Makinson framework for expansion, revision, and contraction, extended in one recent work to the Desirability-Indifference framework (Coussement et al., 10 Feb 2025). In analysis and number theory, “AGM” denotes the arithmetic-geometric mean, including generalizations of the mean and finite-field dynamical studies [(Farhi, 2010); (Vargas, 25 Mar 2025)].

Within fairness research, AGM in influence maximization is also related to, but not identical with, minimax group fairness in supervised learning. The latter minimizes the maximum loss across all groups rather than equalizing group losses, and supports regression and classification settings with oracle-efficient reductions to weighted empirical risk minimization (Diana et al., 2020). This suggests a conceptual family resemblance—both prioritize worst-case group outcomes—but the mathematical objects, constraints, and algorithms are different. In the influence-maximization setting, the core difficulty is the interaction between diffusion dynamics, submodularity, and groupwise fairness constraints.

A common misconception is that group fairness in influence maximization is simply a matter of adding protected-group constraints to a standard greedy algorithm. The cited work shows instead that both the diversity-constraint objective and the maximin objective are not generally submodular, and that provable algorithms therefore require either a multi-objective continuous-relaxation framework or a two-step decomposition into inner-group optimization and across-group coordination.