---
title: Fair Influence Maximization (FIM)
url: https://www.emergentmind.com/topics/fair-influence-maximization-fim
type: topic
---

# Fair Influence Maximization (FIM)

Searching arXiv for recent and foundational papers on Fair Influence Maximization to ground the article in current literature.
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Fair Influence Maximization (FIM) studies how to choose a seed set in a network so that diffusion is not only large, but also fairly distributed across predefined groups or communities. In the classical influence maximization problem, one selects \(S \subseteq V\) with \(|S|\le k\) to maximize an influence spread function such as \(\sigma(S)\), the expected number of activated nodes under a diffusion model like Independent Cascade; FIM augments this with fairness objectives defined on normalized group reach, minimum group coverage, diversity constraints, social welfare, or related criteria [2005.04074], [1903.00967], [2006.07906], [2311.14288]. The topic now spans graph, hypergraph, cascade-based, mobility-network, probabilistic, and reinforcement-learning formulations, reflecting a broader shift from purely utilitarian spread toward equitable information access, intervention design, and risk mitigation [2306.06820], [2010.03438], [2606.14189], [2403.05564].

## 1. Formal setting and diffusion models

A standard FIM formulation starts from a social network modeled as a graph \(G=(V,E)\), a seed budget \(k\), and a diffusion model. In the classical Independent Cascade model, each activated node gets one chance to activate each inactive neighbor with a given probability, and the expected final number of activated nodes is \(\sigma(S)\). Under IC and Linear Threshold, \(\sigma(S)\) is monotone and submodular, so greedy maximization achieves a \((1-1/e)\)-approximation, although the problem is NP-hard [2005.04074]. FIM retains this seeding-and-diffusion backbone but replaces the group-agnostic objective with criteria that depend on who is reached.

Groups may be specified by sensitive attributes such as race, gender, age, ethnicity, political leaning, or by communities derived from the graph itself. Some formulations assume a partition \(V=V_1\cup\cdots\cup V_m\), while others allow overlapping groups, which is important for intersectional settings [1903.00967]. Group-level utilities are typically normalized influence fractions, such as \(u_c(S)=\sigma_c(S)/|V_c|\) or \(I_{G,R_i}(S)/|R_i|\), so that groups of different sizes can be compared on a per-capita basis [2311.14288].

Several papers generalize the basic graph setting. Time-critical FIM counts only nodes activated before a deadline \(\tau\), using utilities of the form
\[
f_{\tau}(S;Y,\mathcal G)=\mathbb E\Big[\sum_{v\in Y,\; t_v\ge 0}\mathbf 1(t_v\le \tau)\Big],
\]
which makes fairness depend on timely rather than eventual access to information [1905.06618]. Hypergraph FIM replaces pairwise graphs with a hypergraph \(H=(V,E)\) and studies influence under the Susceptible-Infected Contact Process, where an infected node contacts an entire hyperedge rather than a single neighbor; fairness is then defined over hypergraph communities via community activation fractions [2606.14189]. In epidemiological and policy applications, the network itself may be temporal and heterogeneous, as in mobility-based COVID-19 vaccination, where nodes are Census Block Groups and Points of Interest, diffusion follows a calibrated SEIR process, and vaccination acts as an influence intervention on a temporal bipartite network [2403.05564].

## 2. Fairness notions in FIM

One widely used fairness notion is parity of normalized reach across groups. In a two-group case \(A,B\), the requirement is that
\[
\frac{I_A(S)}{|A|}\approx \frac{I_B(S)}{|B|},
\]
or equivalently that the disparity
\[
\left|\frac{I_A(S)}{|A|}-\frac{I_B(S)}{|B|}\right|
\]
be small. This treats fairness as balanced coverage rates rather than balanced raw counts and appears both in graph-embedding work and in later formulations of group influence [2005.04074].

A second major notion is maximin fairness. In one formulation,
\[
MF(S)=\min_i \frac{I_{G,R_i}(S)}{|R_i|},
\]
so the objective is to improve the least well-off group. Closely related is the diversity-constraint view, where each group should receive at least what it could obtain on its own under a proportional seed allocation; the associated Diversity Constraint Violations score is
\[
DCV(S)=\frac{1}{q}\sum_{i=1}^{q}\max\left\{\frac{I_{G[R_i]}(k_i)-I_{G,R_i}(S)}{I_{G[R_i]}(k_i)},0\right\}.
\]
Some methods optimize a composite score such as
\[
F(S)=\lambda\cdot MF(S)-(1-\lambda)\cdot DCV(S),
\]
with \(\lambda\in[0,1]\) [2311.14288].

A third family uses welfare fairness. One scalable formulation maximizes
\[
F_\alpha(S)=\sum_{c\in C} n_c\,u_c(S)^\alpha,\qquad 0<\alpha<1,
\]
where \(u_c(S)\) is the expected influenced fraction in community \(c\) and \(\alpha\) controls inequality aversion [2306.06820]. A broader welfare treatment uses isoelastic social welfare functions,
\[
W_\alpha(S)=\sum_{c\in\mathcal C}N_c\frac{u_c(S)^\alpha}{\alpha}\quad (\alpha\neq 0), \qquad
W_0(S)=\sum_{c\in\mathcal C}N_c\log u_c(S),
\]
and interprets \(\alpha\) as an inequality-aversion design parameter; this framework encompasses leximin and proportional fairness as special cases [2006.07906].

Recent work also distinguishes seed fairness from outcome fairness. In the many-objective setting, one objective measures whether the seed set itself is evenly distributed across communities, while another measures whether the activated non-seed nodes are evenly distributed across communities. Both are defined through a normalized Jensen–Shannon divergence to the uniform distribution and are maximized as separate objectives [2403.18755].

A different critique targets expectation-only notions. Mutual fairness argues that fairness metrics based only on expected outreach can be misleading because they ignore the stochastic dependence structure of diffusion outcomes. For two groups with realized outreach fractions \((x_1,x_2)\), mutual fairness is
\[
\mathrm{MF}(\gamma)=1-\mathbb E_{\gamma}[|x_1-x_2|],
\]
where \(\gamma\) is the joint distribution of final configurations. A \(\beta\)-weighted extension incorporates both fairness and efficiency through an optimal-transport-inspired cost [2406.17736].

In vaccination-oriented FIM, fairness has also been expressed as equal treatment and equal outcome. Equal treatment compares the demographic composition of vaccinated individuals to the population distribution; equal outcome compares the demographic composition of infections to the population distribution. Both are evaluated with Kullback–Leibler divergence between an observed group distribution \(p(j)\) and a reference \(q(j)=N_j/N\) [2403.05564].

## 3. Algorithmic paradigms

Constraint-based and welfare-based submodular optimization remain a central algorithmic line. Group-fairness under maximin and diversity constraints was formulated as a multi-objective submodular problem, and an algorithmic framework was given that improved the state of the art for general multi-objective submodular maximization; the corresponding fair objectives themselves are not submodular, but the underlying group influence functions are [1903.00967]. Welfare-based FIM keeps the objective monotone and submodular by composing community-level submodular utilities with concave welfare functions, so the standard greedy algorithm yields the usual \((1-1/e)\) guarantee [2006.07906]. Time-critical fair IM similarly uses surrogate objectives based on concave transformations or per-group quotas so that greedy methods still enjoy approximation guarantees under deadline-aware diffusion [1905.06618].

Scalable approximation methods exploit reverse influence sampling and group structure directly. One welfare-fair method introduces an unbiased estimator for the fractional power of the arithmetic mean, adapts RIS, converts the problem to a weighted maximum coverage instance, and obtains a \((1-1/e-\varepsilon)\) approximation with high probability [2306.06820]. For maximin FIM, a two-step decomposition into Inner-group Maximization and Across-group Maximization leverages submodularity of within-group influence. In that framework, AGM-GS achieves a \((1-1/e-\varepsilon)\) approximation when groups are completely disconnected, while AGM-US guarantees a roughly \(\frac{1}{m}(1-1/e-\varepsilon)\) lower bound regardless of group structure [2509.26579].

Randomization provides another route around the inapproximability of deterministic maximin FIM. Two probabilistic formulations were proposed: a node-based problem with independent seeding probabilities \(x\in[0,1]^n\), and a set-based problem with a distribution over seed sets \(p\). Both optimize ex-ante group coverage, \(\min_C \sigma_C(x)\) or \(\min_C \sigma_C(p)\), and both admit approximation algorithms with a constant multiplicative factor of \(1-1/e\) plus an additive arbitrarily small simulation error [2010.03438].

Representation learning methods encode fairness into latent space rather than imposing it directly on the combinatorial objective. Adversarial graph embeddings co-train an auto-encoder on adjacency-based node features and a discriminator on sensitive attributes, yielding embeddings whose group distributions are aligned; a clustering-based fair selection procedure then chooses seeds from these embeddings [2005.04074]. At a much larger scale, diffusion-cascade methods learn influence and susceptibility directly from cascades rather than from social connectivity. Fairness-based participant sampling penalizes unfair influencers by down-sampling their cascade participants during training, whereas fairness as context uses a two-headed neural architecture with a shared embedding layer so that fairness and spread are modeled jointly; both methods feed a fairness-aware greedy seed selector and are designed for arbitrary sensitive attributes at scale [2306.01587].

Search-based and learning-based approaches broaden the design space beyond submodularity. CEA-FIM combines Louvain communities, an attribute-urgency score, PageRank-like node scores, and an evolutionary algorithm driven by a composite fairness objective \(F(S)=\lambda MF(S)-(1-\lambda)DCV(S)\) [2311.14288]. DQ4FairIM formulates FIM as a finite-horizon MDP, uses maximin fairness as the fairness signal, and learns a policy with deep Q-learning and Structure2Vec embeddings; its reward combines marginal outreach and marginal fairness through a weight \(\phi\) [2512.00545]. MOEIM treats influence, seed-set size, outcome fairness, seed fairness, budget, and time as simultaneous objectives in an NSGA-II framework with graph-aware mutations [2403.18755]. In hypergraphs, FIMH uses approximate contact probabilities and a parameter-free utopia-distance criterion to balance influence and community-level disparity under SICP diffusion [2606.14189].

## 4. Evaluation practice and empirical regularities

A consistent empirical pattern is that fairness-agnostic influence maximization often favors larger, denser, or more central groups. Greedy IM, normal embeddings without adversarial debiasing, and similar baselines repeatedly concentrate seeds in advantaged regions of the network and yield much lower normalized reach for smaller or structurally marginalized groups [2005.04074]. In the original group-fairness formulation, standard greedy can heavily neglect smaller groups, and the gap can worsen further when groups overlap [1903.00967]. Time-critical diffusion intensifies this pattern because short deadlines privilege well-connected regions that are reachable in a few hops; the paper on time-critical IM explicitly shows that deadlines can exacerbate disparity across groups [1905.06618].

At the same time, the literature repeatedly reports that fairness need not eliminate efficiency. Welfare-based optimization produces clear PoF–gap tradeoff curves and often achieves substantial gap reduction at moderate price of fairness [2006.07906]. Adversarial graph embeddings dramatically reduce disparity while remaining competitive with state-of-the-art influence maximization on both synthetic and real networks [2005.04074]. CEA-FIM attains lower diversity-constraint violations and higher maximin fairness than several baselines, and is reported to be up to two orders of magnitude faster than Tsang et al.’s fairness-constrained approach while maintaining low PoF [2311.14288]. DQ4FairIM achieves the highest maximin fairness across the evaluated synthetic and real datasets while remaining competitive in outreach and generalizing across graph sizes and budgets without retraining [2512.00545].

Distributional critiques have also altered how empirical fairness is interpreted. Mutual fairness demonstrates that two strategies can have identical expected group reach yet very different stochastic behavior; outcomes of the form “in 50% of the cases, no one in group 1 gets the information, while everyone in group 2 does, and in the other 50%, it is the opposite” are classified as fair by several expectation-based metrics but are treated as unfair by the optimal-transport-based criterion [2406.17736]. Randomization-based FIM reinforces this point: probabilistic strategies substantially improve ex-ante fairness values relative to deterministic approaches, and the reported ex-post fairness values also dominate previous methods on most tested instances [2010.03438].

Scalability has become an empirical theme in its own right. Cascade-based FPS and FAC are evaluated on Sina Weibo and Digg and are reported to outperform state-of-the-art solutions in spread, fairness, and scalability, precisely because they learn from historical cascades rather than simulating diffusion over the full network [2306.01587]. Hypergraph FIMH, evaluated on seven real-world hypergraphs, preserves influence close to the best IM baselines while reducing disparity; compared to MIE, it reduces disparity by about \(24.3\%\) on average [2606.14189]. Across welfare experiments, community connectedness rather than community size appears to have the larger effect on the price of fairness [2006.07906].

## 5. Application domains and modeling regimes

FIM is motivated by domains in which under-serving a minority or hard-to-reach community is substantively problematic rather than merely statistically uneven. Early and continuing examples include HIV prevention, suicide prevention, substance abuse interventions, public-health messaging, and disaster preparedness [1903.00967], [2006.07906]. In these settings, influence corresponds not to product adoption but to access to information, training, or protective behaviors, so fairness is linked to public policy and social protection rather than only to marketing efficiency.

The literature also broadens the structural notion of group. Some work defines groups by sensitive attributes such as race, age, gender, ethnicity, or political leaning [2311.14288]. Other work replaces demographic attributes with empirical network communities, arguing that community structure directly shapes diffusion and may be preferable when node attributes are unavailable, unreliable, or poorly aligned with how information actually spreads [2305.08791]. That community-centric approach uses a fitted SBM or DCSBM and defines fairness through maximum entropy of community coverage, which also makes it suitable for partially observed and noisy social networks [2305.08791].

Data regimes vary sharply across papers. In some formulations, the network is a conventional graph and fairness is assessed over predefined groups. In others, the graph itself is not the primary object: the large-scale cascade-based approach learns influence and susceptibility embeddings directly from observed diffusion traces on Weibo and Digg, allowing arbitrary sensitive attributes and their combinations to be handled without repeated graph-based diffusion simulations [2306.01587]. The COVID vaccination case study moves even further from classical IM by using temporal mobility networks and a calibrated SEIR process over Census Block Groups and Points of Interest, with race, income, and age entering as protected attributes or risk weights [2403.05564]. Hypergraph FIM extends the field to group interactions such as co-authorship, simultaneous proximity, or joint reviews, where clique-expanded pairwise graphs can obscure fairness-relevant structure [2606.14189].

## 6. Limitations and open directions

A persistent limitation is that FIM does not have a single accepted fairness definition. Maximin fairness, diversity constraints, welfare fairness, demographic-parity-style reach parity, equal treatment, equal outcome, maximum entropy, mutual fairness, seed fairness, and outcome fairness all target different objects. This plurality is not merely terminological: one paper shows that expectation-based group metrics can label as fair stochastic outcomes that are unfair in virtually every realization, which implies that metric choice changes the substantive meaning of fairness [2406.17736]. A plausible implication is that FIM should be understood as a family of optimization problems rather than a single canonical objective.

Theoretical gaps also remain. Maximin FIM is non-submodular, which is why efficient algorithms with strong guarantees are difficult; even the recent IGM–AGM framework only attains the full \((1-1/e-\varepsilon)\) guarantee when groups are completely disconnected [2509.26579]. Several influential methods are heuristic and explicitly lack theoretical guarantees on fairness or approximation, including adversarial embeddings, community-based evolutionary search, deep RL, and hypergraph utopia-distance seeding [2005.04074], [2311.14288], [2512.00545], [2606.14189]. Even when guarantees exist, they can depend on a small number of communities or on specific diffusion assumptions [2306.06820].

Fairness supervision itself is often a bottleneck. Representation-learning approaches require sensitive-attribute labels and only debias with respect to known attributes, leaving hidden or unmeasured biases untouched [2005.04074]. Community-based approaches reduce dependence on protected labels, but then fairness is defined over structural groups rather than protected groups, which may or may not align with policy goals [2305.08791]. Multi-attribute and intersectional settings remain difficult: some methods support multiple discriminators or combined attributes, but richer intersectional formulations are repeatedly identified as open problems [2311.14288], [2306.01587], [2403.05564].

Scalability remains uneven across modeling regimes. Monte Carlo evaluation is still a dominant cost in many RL, evolutionary, and epidemic-simulation approaches [2512.00545], [2403.05564]. Welfare-based RIS methods scale well when the number of communities is small, but their time and space costs grow with \(|C|\) [2306.06820]. Time-critical, dynamic, and partially observed networks, heterogeneous diffusion parameters, competition between campaigns, and adaptive multi-round interventions are all named as future work across different papers [1905.06618], [2311.14288], [2403.05564]. Taken together, these limitations suggest that future FIM research will continue to revolve around three unresolved tensions: fairness definition versus tractability, ex-ante guarantees versus ex-post outcomes, and structural realism versus algorithmic scalability.

Source: https://www.emergentmind.com/topics/fair-influence-maximization-fim