Fair Influence Maximization: A Welfare Optimization Approach

Aida Rahmattalabi1?, Shahin Jabbari2?, Himabindu Lakkaraju2, Phebe Vayanos1Max Izenberg3, Ryan Brown3, Eric Rice1, Milind Tambe2

1University of Southern California2Harvard University

3 RAND Corporation? Equal Contribution. Correspondence to rahmatta@usc.edu and jabbari@seas.harvard.edu

AbstractSeveral behavioral, social, and public health interventions,such as suicide/HIV prevention or community preparednessagainst natural disasters, leverage social network informationto maximize outreach. Algorithmic influence maximizationtechniques have been proposed to aid with the choice of “peerleaders” or “influencers” in such interventions. Yet, traditionalalgorithms for influence maximization have not been designedwith these interventions in mind. As a result, they may dispro-portionately exclude minority communities from the benefitsof the intervention. This has motivated research on fair influ-ence maximization. Existing techniques come with two majordrawbacks. First, they require committing to a single fairnessmeasure. Second, these measures are typically imposed asstrict constraints leading to undesirable properties such aswastage of resources. To address these shortcomings, we pro-vide a principled characterization of the properties that a fairinfluence maximization algorithm should satisfy. In particu-lar, we propose a framework based on social welfare theory,wherein the cardinal utilities derived by each community areaggregated using the isoelastic social welfare functions. Underthis framework, the trade-off between fairness and efficiencycan be controlled by a single inequality aversion design param-eter. We then show under what circumstances our proposedprinciples can be satisfied by a welfare function. The resultingoptimization problem is monotone and submodular and canbe solved efficiently with optimality guarantees. Our frame-work encompasses as special cases leximin and proportionalfairness. Extensive experiments on synthetic and real worlddatasets including a case study on landslide risk managementdemonstrate the efficacy of the proposed framework.

1 IntroductionThe success of many behavioral, social, and public healthinterventions relies heavily on effectively leveraging socialnetworks (Isaac et al. 2009; Valente et al. 2007; Tsang et al.2019). For instance, health interventions such as suicide/HIVprevention (Yonemoto et al. 2019) and community prepared-ness against natural disasters involve finding a small set ofwell-connected individuals who can act as peer-leaders todetect warning signals (suicide prevention) or disseminaterelevant information (HIV prevention or landslide risk man-agement). The influence maximization framework has been

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employed to find such individuals (Wilder et al. 2020). How-ever, such interventions may lead to discriminatory solutionsas individuals from racial minorities or LGBTQ communitiesmay be disproportionately excluded from the benefits of theintervention (Rahmattalabi et al. 2019; Tsang et al. 2019).

Recent work has incorporated fairness directly into influ-ence maximization by proposing various notions of fairnesssuch as maximin fairness (Rahmattalabi et al. 2019) anddiversity constraints (Tsang et al. 2019). Maximin fairnessaims at improving the minimum amount of influence that anycommunity receives. On the other hand, diversity constraints,inspired by the game theory literature, ensure that each com-munity is at least as well-off had they received their shareof resources proportional to their size and allocated theminternally. Each of these notions offers a unique perspectiveon fairness. However, they also come with drawbacks. Forexample maximin fairness can result in significant degrada-tion in total influence due to its stringent requirement to helpthe worst-off group as much as possible, where in reality itmay be hard to spread the influence to some communitiesdue to their sparse connections. On the other hand, while thediversity constraints aim at taking the community’s abilityin spreading influence into account, it does not explicitly ac-count for reducing inequality (i.e., does not exhibit inequalityaversion). Consequently, there is no universal agreement onwhat fairness means and in fact, it is widely known that fair-ness is domain dependent (Narayanan 2018). For example,excluding vulnerable communities from suicide preventionmight have higher negative consequences compared to inter-ventions promoting a healthier lifestyle.

Building on cardinal social welfare theory from the eco-nomics literature and principles of social welfare, we proposea principled characterization of the properties of social in-fluence maximization solutions. In particular, we propose aframework for fair influence maximization based on socialwelfare theory, wherein the cardinal utilities derived by eachcommunity are aggregated using the isoelastic social welfarefunctions (Bergson 1938). Isoelastic functions are in the gen-eral form of uα/α, α < 1, α 6= 0 and log u, α = 0 whereα is a constant and controls the aversion to inequality andu is the utility value. They are used to measure the good-ness or desirability of a utility distribution. However, due tothe structural dependencies induced by the underlying social

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network, i.e., between-community and within-communityedges, social welfare principles cannot be directly applied toour problem. Our contributions are as follows: (i) We extendthe cardinal social welfare principles including the transferprinciple to the influence maximization framework, which isotherwise not applicable. We also propose a new principlewhich we call utility gap reduction. This principle aims toavoid situations where high aversion to inequality leads toeven more utility gap, caused by between-community influ-ence spread; (ii) We generalize the theory regarding theseprinciples and show that for all problem instances, theredoes not exist a welfare function that satisfies all principles.Nevertheless, we show that if all communities are discon-nected from one another (no between-community edges),isoelastic welfare functions satisfy all principles. This resulthighlights the importance of network structure, specificallybetween-community edges; (iii) Under this framework, thetrade-off between fairness and efficiency can be controlledby a single inequality aversion parameter α. This allows adecision-maker to effectively trade-off quantities like utilitygap and total influence by varying this parameter in the wel-fare function. We then incorporate these welfare functions asobjective into an optimization problem to rule out undesirablesolutions. We show that the resulting optimization problem ismonotone and submodular and, hence, can be solved with agreedy algorithm with optimality guarantees; (iv) Finally, wecarry out detailed experimental analysis on synthetic and realsocial networks to study the trade-off between total influencespread and utility gap. In particular, we conduct a case studyon the social network-based landslide risk management inSitka, Alaska. We show that by choosing α appropriately wecan flexibly control utility gap (4%-26%) and the resultinginfluence degradation (36% - 5%).Related Work. Recent work has incorporated fairness di-rectly into the influence maximization framework by rely-ing on either Rawlsian theory (Rawls 2009; Rahmattalabiet al. 2019), game theoretic principles (Tsang et al. 2019) orequality based notions (Stoica, Han, and Chaintreau 2020).Equality based approaches strive for equal outcome acrossdifferent communities. In general, strict equality is hard toachieve and may lead to wastage of resources. This is ampli-fied in influence maximization as different communities havedifferent capacities in being influenced (e.g., marginalizedcommunities are hard to reach). We discuss the other twoapproaches in more details in Sections 3–5. Social welfarefunctions have been used within the economic literature tostudy trade-offs between equality and efficiency (Sen 1997)and have been widely adopted in different decision makingareas including health (Abasolo and Tsuchiya 2004). Re-cently, Heidari et al. (2018) proposed to study similar ideasfor regression problems. The classical social welfare theory,however, does not readily extend to our setting due to de-pendencies induced by the between-community connections.Extending those principles is a contribution of our work. SeeAppendix A for a thorough review of related work.

2 Problem FormulationWe use G = (V, E) to denote a graph (or network) in whichV is the set of N vertices and E is the set of edges. In the

influence maximization problem, a decision-maker choosesa set of at most K vertices to influence (or activate). The se-lected vertices then spread the influence in rounds accordingto the Independent Cascade Model (Kempe, Kleinberg, andTardos 2003).1 Under this model, each newly activated vertexspreads the influence to its neighbors independently and witha fixed probability p ∈ [0, 1]. The process continues until nonew vertices are influenced. We useA to denote the initial setof vertices, also referred to as influencer vertices. The goalof the decision-maker is to select a set A to maximize the ex-pected number of vertices that are influenced at the end of thisprocess. Each vertex of the graph belongs to one of the dis-joint communities (empty intersection) c ∈ C := {1, . . . , C}such that V1 ∪ · · · ∪ VC = V where Vc denotes the set ofvertices that belong to community c. This partitioning canbe induced by, e.g., the intersection of a set of (protected)attributes such as race or gender for which fair treatment isimportant. We use Nc to denote the size of community c,i.e., Nc = |Vc|. Furthermore, communities may be discon-nected, in which case ∀c, c′ ∈ C and ∀v ∈ Vc, v′ ∈ Vc′ , thereis no edge between v and v′ (i.e., (v, v′) /∈ E). We defineA? := {A ⊆ V | |A| ≤ K} as the set of budget-feasible in-fluencers. Finally, for any choice of influencers A ∈ A?, welet uc(A) denote the utility, i.e., the expected fraction of theinfluenced vertices of community c, where the expectation istaken over randomness in the spread of influence. The stan-dard influence maximization problem solves the optimizationproblem

maximizeA∈A?

∑c∈C

Ncuc(A). (1)

When clear from the context, we will drop the dependencyof uc(A) on A to minimize notational overhead.

3 Existing Notions of FairnessProblem (1) solely attempts to maximize the total influencewhich is also known as the utilitarian approach. Existing fairinfluence maximization problems are variants of Problem (1)involving additional constraints. We detail these below. SeeAppendix A for more discussion.Maximin Fairness (MMF). Based on the Rawlsian the-ory (Rawls 2009), MMF (Tsang et al. 2019) aims to maxi-mize the utility of the worst-off community. Precisely, MMFonly allows A ∈ A? that satisfy the following constraint

minc∈C

uc(A) ≥ γ, where A ∈ A?,

where the left term is the utility of the worst-off communityand γ is the highest value for which the constraint is feasible.Diversity Constraints (DC). Inspired by the game theoreticnotion of core, DC requires that every community obtains autility higher than when it receives resources proportional toits size and allocates them internally (Tsang et al. 2019). Thisis illustrated by the following constraint where Uc denotesthe maximum utility that community c can achieve with abudget equal to bKNc/Nc.

uc(A) ≥ Uc, ∀c ∈ C where A ∈ A?. (2)1Our framework is also applicable to other forms of diffusion

such as Linear Threshold Model (Kempe, Kleinberg, and Tardos2003)

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DC sets utility lower bounds for communities based on theirrelative sizes and how well they can spread influence inter-nally. As a result, it does not explicitly account for reducinginequalities and may lead to high influence gap. We showthis both theoretically and empirically in Sections 4.4 and 5.Demographic Parity (DP). Formalizing the legal doctrineof disparate impact (Zafar et al. 2017), DP requires the utilityof all communities to be roughly the same. For any δ ∈ [0, 1),DP implies the constraints (Ali et al. 2019; Stoica, Han, andChaintreau 2020; Aghaei, Azizi, and Vayanos 2019)∣∣uc(A)− uc′(A)

∣∣ ≤ δ, ∀c, c′ ∈ C where A ∈ A?.The degree of tolerated inequality is captured by δ and higherδ values are associated with higher tolerance. We use exactand approximate DP to distinguish between δ = 0 and δ > 0.

4 Fair Influence Maximization4.1 Cardinal Welfare Theory BackgroundFollowing the cardinal welfare theory (Bergson 1938), ouraim is to design welfare functions to measure the goodness ofthe choice of influencers. Cardinal welfare theory proposesa set of principles and welfare functions that are expectedto satisfy these principles. Given two utility vectors, theseprinciples determine if they are indifferent or one of them ispreferred. For ease of exposition, let W denote this welfarefunction defined over the utilities of all individuals in thepopulation (we will formalize W shortly). Then the existingprinciples of social welfare theory can be summarized asfollows. Throughout this section, without loss of generality,we assume all utility vectors belong to [0, 1]N .(1) Monotonicity. If u ≺ u′, then W (u) < W (u′).2 Inother words, if u′ Pareto dominates u, thenW should strictlyprefer u′ to u. This principle also appears as levelling downobjection in political philosophy (Parfit 1997).(2) Symmetry. W (u) = W (P (u)), where P (u) is anyelement-wise permutation of u. According to this principle,W does not depend on the naming or labels of the individuals,but only on their utility levels.(3) Independence of Unconcerned Individuals. Let (u|cb)be a utility vector that is identical to u, except for the util-ity of individual c which is replaced by a new value b. Theproperty requires that for all c, b, b′,u and u′, W (u|cb) <W (u′|cb)⇔ W (u|cb′) < W (u′|cb′). Informally, this prin-ciple states that W should be independent of individualswhose utilities remain the same.(4) Affine Invariance. For any a > 0 and b, W (u) <W (u′) ⇔ W (au + b) < W (au′ + b) i.e., the relativeordering is invariant to a choice of numeraire.(?5) Transfer Principle (Dalton 1920; Pigou 1912). Con-sider individuals i and j in utility vector u such that ui < uj .Let u′ be another utility vector that is identical to u in allelements except i and j where u′i = ui + δ and u′j = uj − δfor some δ ∈ (0, (uj−ui)/2). Then,W (u) < W (u′). Infor-mally, transferring utility from a high-utility to a low-utilityindividual should increase social welfare.

It is well-known that any welfare function W that sat-isfies the first four principles is additive and in the form of

2≺ means uc ≤ u′c for all c ∈ C and uc < u′

c for some c ∈ C.

Wα(u) = ΣNi=1uαi /α for α 6= 0 andWα(u) = ΣNi=1 log(ui)

for α = 0. Further, for α < 1 the last principle is also satis-fied. In this case α can be interpreted as an inequality aversionparameter, where smaller α values exhibit more aversion to-wards inequalities. We empirically investigate the effect of αin Section 5.

4.2 Group Fairness and New PrinciplesApplying the cardinal social welfare framework to influencemaximization problems comes with new challenges. We nexthighlight these challenges and demonstrate our approach.

First, the original framework of cardinal welfare theorydefines the welfare function over individuals. This is equiva-lent to seeking equality in the probability that each individualwill be influenced, similar to the work of Fish et al. (2019). Itis notoriously hard to achieve individual fairness in practice,e.g., Dwork et al. (2012) explore this in the machine learn-ing context. The problem is further exacerbated in influencemaximization because it is not always possible to spread theinfluence to isolated or poorly connected individuals effec-tively. Therefore, we focus on group fairness whereby theutility of each individual is defined as the average utility ofthe members of that community. Let uc denote the averageutility of community c. With this group-wise view, a welfarefunction can be written in terms of the average utilities overcommunities e.g., Wα(u) = ΣNi=1u

αi /α = Σc∈CNcu

αc /α.

Figure 1: The effect of network structure and in particularbetween-community edges on coupling of the utilities ofcommunities. The figure shows two sample networks con-sisting of three communities, differentiated by shape: (a) isthe same as (b) except that between-community edges areremoved. Black fillings show the choice of influencers. Wefurther assume p is small enough such that influence spreaddissipates after one step. Transferring an influencer from cir-cles to squares (top to bottom panel) affects the utility ofdiamonds in (b) but not in (a).

Moreover, while principles 1-4 can be easily extended toour influence maximization problem, this is not the case forthe transfer principle. More precisely, in the influence maxi-mization problem it might not be feasible to directly transferutilities from one community to another without affecting theutilities of other communities. We highlight this effect withan example, see Figure 1. In this figure, each community isrepresented by a distinct shape. The two networks (a) and(b) are identical except that between community edges are

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removed in network (a) (i.e., disconnected communities). Thesolid black vertices determine the choice of influencers. Innetwork (b), if we transfer an influencer vertex from circlesto squares according to Figure 1 (top to bottom panel), itwill indirectly affect diamonds as well. This effect is absentin network (a) as there are no between-community edges toallow the spread of influence across communities. In network(a), the transfer principle prefers the resulting utility vectorafter the transfer. However, this principle cannot be appliedto network (b) as the utilities of more than one communityis modified after the transfer. Additionally, even when directtransfer is possible, it can be the case that there is no symme-try in the amount of utility gained by low-utility communityand the amount of utility lost by high-utility community afterthe transfer. To address both of these shortcomings we intro-duce the influence transfer principle as a generalization ofthe transfer principle for influence maximization problems.

Similar to the original transfer principle, we consider so-lutions in which influencer vertices are transferred from onecommunity to another community. Without loss of generality,we focus on the case where only one influencer vertex istransferred between the two communities. We refer to suchsolutions as neighboring solutions. Clearly, transfer of morethan one influencer vertex can be seen as a sequence of trans-fers between neighboring solutions.(5) Influence Transfer Principle. Let A and A′ ∈ A? betwo neighboring solutions with corresponding utility vectorsu = u(A) and u′ = u(A′). Suppose the elements of u andu′ are sorted in ascending order. We also assume after thetransfer, the ordering of the utilities stays the same across uand u′.

If Σκ∈C:κ≤cNκ(u′κ − uκ) ≥ 0 ∀c ∈ C and u′c > uc forsome c ∈ C, then W (u) < W (u′).

Informally, influence transfer principle states that in a de-sirable transfer of utilities, the magnitude of the improvementin lower-utility communities should be at least as high asthe magnitude of decay in higher-utility communities whileenforcing that at least one low-utility community receivesa higher utility after the transfer. The original transfer is aspecial case of influence transfer principle when communitiesare disconnected and utilities transferred remain the same.

Next, we study whether any of the welfare functions thatsatisfy the first 4 principles satisfy the influence transferprinciple. In Proposition 1, we show any additive and strictlyconcave function satisfies influence transfer principle. Sincefunctions that satisfy the first 4 principles are strictly concavefor α < 1, the influence transfer principle is automaticallysatisfied in this regime. We defer all proofs to Appendix B.

Proposition 1. Any strictly concave and additive functionsatisfies the influence transfer principle.

To measure inequality, notion of utility gap (or analogousnotions such as ratio of utilities) is commonly used (Fishet al. 2019; Stoica, Han, and Chaintreau 2020). Utility gapmeasures the difference between the utilities of a pair of com-munities. In this work, we focus on the maximum utility gap,i.e., the gap between communities with the highest and low-est utilities (utility gap henceforth). For a utility vector u, wedefine ∆(u) = maxc∈C uc−minc∈C uc to denote the utility

gap. Fair interventions are usually motivated by the largeutility gap before the intervention (Commission on SocialDeterminants of Health 2008). Fish et al. (2019) has shownthat in social networks the utility gap can further increaseafter an algorithmic influence maximizing intervention. Weextend this result to the entire class of welfare functions thatwe study in this work and we notice that the utility gap canincrease even if we optimize for these welfare functions. Thisis a surprising result since, unlike the influence maximizationobjective, these welfare functions are designed to incorporatefairness, yet we may observe an increase in the utility gap.We now introduce another principle which aims to addressthis issue. Again we focus on neighboring solutions.(6) Utility Gap Reduction. Let A and A′ ∈ A? be twoneighboring solutions with corresponding utility vectors u =u(A) and u′ = u(A′). If Σc∈CNcuc ≤ Σc∈CNcu

′c. and

∆(u) > ∆(u′) then W (u) < W (u′).The utility gap reduction simply states that the welfare

function should prefer the utility vector whose total utilityis at least as high as the other vector and also has smallerutility gap. We now show that, in general, it is not possible todesign a welfare function that obeys the utility gap reductionprinciple along with the other principles.

Proposition 2. Let W be a welfare function that obeys prin-ciples 1-5. Then there exists an instance of influence maxi-mization where W does not satisfy the utility gap reduction.

Next, we show on a special class of networks, i.e., networkswith disconnected communities, the utility gap reductionprinciple is satisfied in all influence maximization problems.

Proposition 3. Let W be a welfare function that obeys prin-ciples 1-5. If the communities are disconnected, then W alsosatisfies the utility gap reduction principle.

Propositions 2 and 3 and their proofs establish new chal-lenges in fair influence maximization. These challenges arisedue to the coupling of the utilities as a result of the networkstructure and more precisely the between-community edges.The results in Propositions 2 and 3 leave open the followingquestion: “In what classes of networks, there exists a welfarefunction that satisfies all the 6 principles over all instances ofinfluence maximization problems?” As an attempt to answerthis question, we empirically show that over various real andsynthetic networks including stochastic block models, thereexist welfare functions that obey all of our principles. Weconclude this section by the following three remarks.

Remark 1 (Application to Other Settings). Our welfare-based framework can be theoretically applied to differentgraph-based problems (e.g., facility location) but algorith-mic solution is domain-dependent. The choice of influencemaximization is motivated by evidence about discriminationstudied in previous work (Rahmattalabi et al. 2019; Stoica,Han, and Chaintreau 2020).

Remark 2 (Relationship between Principles and Fairness).Monotonicity ensures there is no wastage of utilities. Sym-metry enforces the decision-maker to not discriminate basedon communities’ names. According to the Independence ofUnconcerned Individuals, between two solutions (choices of

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Notion/Principle Monotonicity Symmetry Ind. of Unconcerned Affine Influence Transfer Gap ReductionExact DP 7 (Prop. 5) X 7 (Prop. 8) X 7 (Prop. 11) X (Prop. 15)

Approximate DP 7 (Prop. 6) X 7 (Prop. 8) 7 7 (Prop. 11) 7 (Prop. 15)DC X (Cor. 1) 7 7 (Prop. 9) 7 7 (Prop. 12) 7 (Prop. 16)

MMF X (Cor. 1) X 7 (Prop. 10) X 7 (Prop. 13) 7 (Prop. 17)Utilitarian X (Cor. 1) X X X 7 (Prop. 14) 7 (Prop. 18)

Welfare Theory X X X X X 7 (Prop. 2)

Table 1: Summary of the properties of different fairness notions through the lens of welfare principles for influence maximization.

influencers) only those individuals/communities whose utili-ties change should impact the decision-maker’s preference.Affine Invariance is a natural requirement that the prefer-ences over different solutions should not change based on thechoice of numeraire. Finally, the Transfer Principle promotessolutions that are more equitable.Remark 3 (Selecting the Inequality Aversion Parameter inPractice). In our approach, α is a user-selected parameterthat the user can vary to tune the trade-off between efficiencyand fairness. Leaving the single parameter α in the handsof the user is a benefit of our approach since the user caninspect the solution as α is varied to select their preferredsolution. Since a single parameter must be tuned, this can bedone without the need for a tailored algorithm. In particular,we recommend that α be either selected by choosing amonga moderate number of values and picking the one with themost desirable behavior for the user or by using the bisectionmethod. Typically, choosing α will reduce to letting the userselect how much utility gap they are willing to tolerate: theywill select the largest possible value of α for which the utilitygap is acceptable.

4.3 Group Fairness and Welfare MaximizationThe welfare principles reflect the preferences of a fairdecision-maker between a pair of solutions. Thus a welfarefunction that satisfies all the principles would always rankthe preferred (in terms of fairness and efficiency) solutionhigher. As a result, we can maximize the welfare function toget the most preferred solution.

We show that the two classes of welfare functionsWα(u) = ΣNi=1u

αi /α for α < 1, α 6= 0 and Wα(u) =

ΣNi=1 log(ui) for α = 0 satisfy 5 of our principles. Hence asa natural notion of fairness we can define a fair solution to bea choice of influencers with the highest welfare as defined inthe following optimization problem.

maximizeA∈A?

Wα(u(A)). (3)

Lemma 1. In the influence maximization problem, any wel-fare function that satisfies principles 1-5 is monotone andsubmodular.

It is well-known that to maximize any monotone submodu-lar function, there exists a greedy algorithm with a (1− 1/e)approximation factor (Nemhauser and Wolsey 1981) whichwe can also use to solve the welfare maximization problem.

Each choice of the inequality aversion parameter α resultsin a different welfare function and hence a fairness notion. A

decision-maker can directly use these welfare functions asobjective of an optimization problem and study the trade-offbetween fairness and total utility by varying α, see Section 5.

4.4 Connection to Existing Notions of FairnessOur framework allows for a spectrum of fairness notions asa function of α. It encompasses as a special case leximinfairness3, a sub-class of MMF, for α → −∞. Proportionalfairness (Bonald and Massoulie 2001), a notion for resourceallocation problems, is also closely connected to the welfarefunction for α = 0. See Appendices A and C.

It is natural to ask which of the fairness principles aresatisfied by the existing notions of fairness for influence max-imization. As we discussed in Section 3, the existing notionsof fairness are imposed by adding constraints to the influencemaximization problem. However, our welfare framework di-rectly incorporates fairness into the objective. In order tofacilitate the comparison, instead of the constrained influencemaximization problems we consider an equivalent reformula-tion in which we bring the constraints into the objective viathe characteristic function of the feasible set. We then have asingle objective function which we can treat as the welfarefunction corresponding to the fairness constrained problem.More formally, given an influence maximization problem andfairness constraints written as a feasible set F

maxA∈A?

∑c∈C

Ncuc(A) s.t. u(A) ∈ F .

We consider the following equivalent optimization problem

maxA∈A?

∑c∈C

Ncuc(A) + IF (u(A)) := maxA∈A?

WF (u(A)),

in which IF (u) is equal to 0 if u ∈ F and −∞ otherwise.Using this new formulation, we can now examine each ofthe existing notions of fairness though the lens of the welfareprinciples. Given the new interpretation, to show that a fair-ness notion does not satisfy a specific principle, it sufficesto show there exist solutions A,A′ ∈ A? and correspondingutility vectors u = u(A) and u′ = u(A′) such that the prin-ciple prefers u over u′ but WF (u) < WF (u′). The resultsare summarized in Table 1 where in addition to comparing

3Leximin is subclass of MMF. According to its definition, amongtwo utility vectors, leximin prefers the one where the worst utility ishigher. If the worst utilities are equal, leximin repeats this processby comparing the second worst utilities and so on.

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with the previous notions introduced in Section 3, we com-pare with the utilitarian notion i.e., Problem (1). We provideformal proofs for each entry of Table 1 in Appendix D.

We observe that none of the previously defined notionsof fairness for influence maximization satisfies all of ourprinciples and each existing notion violates at least 3 out of 6principles. We point out that exact DP is the only notion thatsatisfies the utility gap reduction. However, this comes at acost as enforcing exact DP may result in significant reductionin total utility in the fair solution compared to the optimalunconstrained solution (Corbett-Davies et al. 2017).

5 Computational ResultsWe evaluate our approach in terms of both the total utilityor spread of influence (to account for efficiency) and util-ity gap (to account for fairness). We show by changing theinequality aversion parameter, we can effectively trade-offefficiency with fairness. As baselines, we compare with DCand MMF. To the best our knowledge, there is no prior workthat handles DP constraints over the utilities. We follow theapproach of Tsang et al. (2019) for both problems and viewthese problems as a multi-objective submodular optimiza-tion with utility of each each community being a separateobjective. They propose an algorithm and implementationwith asymptotic (1− 1/e) approximation guarantee whichwe also utilize here. We use Price of Fairness (PoF), definedas the percentage loss in the total influence spread as a mea-sure of efficiency. Precisely, PoF := 1− OPTfair/OPTIM inwhich OPTfair and OPTIM are the the total influence spread,with and without fairness. Hence PoF∈ [0, 1] and smallervalues are more desirable. The normalization in PoF allowsfor a meaningful comparison between networks with differ-ent sizes and budgets as well as between different notionsof fairness. In the PoF calculations, we utilize the genericgreedy algorithm (Kempe, Kleinberg, and Tardos 2003) tocompute OPTIM. To account for fairness, we compare thesolutions in terms of the utility gap. Analogous measures arewidely used in fairness literature (Hardt, Price, and Srebro2016) and more recently in graph-based problems (Fish et al.2019; Stoica, Han, and Chaintreau 2020). We also note thatour framework ranks solutions based on their welfare anddoes not directly optimize utility gap, as such our evaluationmetric of fairness does not favor any particular approach.

We perform experiments on both synthetic and real net-works. We study two applications: community preparednessagainst landslide incidents and suicide prevention amonghomeless youth. We discuss the latter in Appendix E. Inthe synthetic experiments, we use the stochastic block model(SBM) networks, a widely used model for networks with com-munity structure (Fienberg and Wasserman 1981). In SBMnetworks, vertices are partitioned into disjoint communities.Within each community c, an edge between two vertices ispresent independently with probability qc. Between any twovertices in communities c and c′, an edge exists indepen-dently with probability qcc′ and typically qc > qcc′ to capturehomophily (McPherson, Smith-Lovin, and Cook 2001). SBMcaptures the community structure of social networks (Girvanand Newman 2002). We report the average results over 20

random instances and set p = 0.25 in all experiments.Landslide Risk Management in Sitka, Alaska. Sitka,Alaska is subject to frequent landslide incidents. In order toimprove communities’ preparedness, an effective approachis to instruct people on how to protect themselves before andduring landslide incidents. Sitka has a population of morethan 8000 and instructing everyone is not feasible. Our goal isto select a limited set of individuals as peer-leaders to spreadinformation to the rest of the city. The Sitka population is di-verse including different age groups, political views, seasonaland stable residents where each person can belong to multiplegroups. These groups differ in their degree of connectedness.This makes it harder for some groups to receive the intendedinformation and also impacts the cost of imposing fairness.We investigate these trade-offs.

Since collecting the social network data for the entire cityis cumbersome, we assume a SBM network and use in-personsemi-structured interview data from 2018-2020 with mem-bers of Sitka to estimate the SBM parameters. Using theinterview responses in conjunction with the voter lists, weidentified 5940 individuals belonging to 16 distinct commu-nities based on the intersection of age groups, political views,arrival time to Sitka (to distinguish between stable and tran-sient individuals). The size of the communities range from112 (stable, democrat and 65+ years of age) to 693 (republi-can, transient fishing community, age 30-65). See Appendix Efor details on the estimation of network parameters.

Figure 2 summarizes results across different budget valuesK ranging from 2% to 10% of the network size N for ourframework (different α values) as well as the baselines. In theleft panel, we observe that as α decreases, our welfare-basedframework further reduces the utility gap, achieving lowergap than DC and competitive gap as MMF. As we notedin Section 4.4, our framework recovers leximin (which hasstronger guarantees than MMF) as α → −∞, though weshow experimentally that this is achieved with moderate val-ues of α. Overall, utility gap shows an increasing trend withbudget, however the sensitivity to budget decreases whenmore strict fairness requirements are in place, e.g. in MMFand α = −9.0. From the right panel, PoF varies significantlyacross different approaches and budget values surpassing40% for MMF. This is due to the stringent requirement ofMMF to raise the utility of the worst-off as much as possible.Same holds true for lower values of α as they exhibit higheraversion to inequality. The results also indicate that PoF de-creases as K grows which captures the intuition that fairnessbecomes less costly when resources are in greater supply.Resource scarcity is true in many practical applications, in-cluding the landslide risk management domain which makesit crucial for decision-makers to be able to study differentfairness-efficiency trade-offs to come up with the most effec-tive plan. Figure 3 depicts such trade-off curves where eachline corresponds to a different budget value across the rangeof α. Previous work only allows a decision-maker to chooseamong a very limited set of fairness notions regardless of theapplication requirements. Here, we show that our frameworkallows one to choose α to meaningfully study the PoF-utilitygap trade-offs. For example, given a fixed budget and a toler-ance on utility gap, one can choose an α with the lowest PoF.

6

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) Budget2%4%6%8%10%

Figure 2: Left and right panels: utility gap and PoF for different K and α values for our framework and baselines.

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Figure 3: PoF vs. utility gap trade-off curves. Each line cor-responds to a different budget K across different α values.

We now investigate the effect of relative connectedness. Westudy the effect of relative community size in Appendix E.2.

0

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q300.010.020.030.040.050.06

Figure 4: Utility gap and PoF for various levels q3. All resultsare compared across different values of α and the baselines.

Relative Connectedness. We sample SBM networks con-sisting of 3 communities each of size 100 where com-munities differ in their degree of connectedness. We setq1 = 0.06, q2 = 0.03, q3 = 0.0 to obtain three commu-

nities with high, mid and low relative connectedness. Wechoose these values to reflect asymmetry in the structureof different communities which mirrors real world scenar-ios since not every community is equally connected. We setbetween-community edge probabilities qcc′ to 0.005 for all cand c′ and K = 0.1N . We gradually increase q3 from 0.0 to0.06. Results are summarized in Figure 4, where each groupof bars correspond to a different approach. We observe asq3 increases utility gap and PoF decrease until they reach aminimum around at around q3 = 0.03. From this point, thetrend reverses. This U-shaped behavior is due to structuralchanges in network. More precisely, for q3 < 0.03 we are inthe high-mid-low connectedness regime for the three groups,where the third community receives the minimum utility. Asa result, as q3 increases it becomes more favorable to choosemore influencer vertices in this community which in turnreduces the utility gap. For q3 > 0.03, the second communitywill be become the new worst-off community due its lowestconnectedness. Hence, further increase in q3 causes moreseparation in connectedness and we see previous behavior inreverse. Thus, by further increasing q3, communities 1 and3 receive more and more influencer vertices. This behaviortranslates to PoF as the relative connectedness of communi-ties impacts how hard it is to achieve a desired level of utilitygap. Finally, we see that the U-shaped behavior is skewed,i.e., we observe higher gap and PoF in lower range of q3which is due to higher gap in connectedness of communities.

We can also compare the effect of relative connectednessand community size (see Appendix E.2. ) We observe thatconnectedness has a more significant impact on PoF (up to25%) compared to community size (less than 4%). In otherwords, when communities are structurally different it is morecostly to impose fairness. This is an insightful result giventhat in different applications we may encounter differentpopulations with structurally different networks. Utility gapon the other hand is affected by both size and connectedness.

Finally while our theory indicates that in the network set-ting, no welfare function can satisfy all principles includingutility gap reduction over all instances of the influence max-imization, we observe that our class of welfare functionssatisfies all of the desiderata on the class of networks that weempirically study. Our theoretical results showed this for aspecial case of networks with disconnected communities. Inparticular, we see higher PoF is accompanied by lower utilitygap which complies with utility gap reduction principle.

7

Ethics Statement and Broader ImpactAs the empirical evidence highlighting ethical side effectsof algorithmic decision-making is growing (Angwin et al.2016; Miller 2015), the nascent field of algorithmic fairnesshas also witnessed a significant growth. It is well-establishedby this point that there is no universally agreed-upon notionof fairness, as fairness concerns vary from one domain toanother (Narayanan 2018; Berk et al. 2017). The need fordifferent fairness notions can also be explained by theoret-ical studies that show that different fairness definitions areoften in conflict with each other (Kleinberg, Mullainathan,and Raghavan 2017; Chouldechova 2017; Friedler, Scheideg-ger, and Venkatasubramanian 2016). To this end, most ofthe literature on algorithmic fairness proposes different fair-ness notions motivated by different ethical concerns. A majordrawback of this approach is the difficulty of comparing thesemethods against each other in a systematic manner to choosean appropriate notion for the domain of interest. Instead offollowing this trend, we propose a unifying framework con-trolled by a single parameter that can be used by a decision-maker to systematically compare different fairness measureswhich typically result in different (and possibly also problem-dependent) trade-offs. Our framework also accounts for thesocial network structure while designing fairness notions –a consideration that is mainly overlooked in the past. Giventhese two contributions, it is perceivable that our approachcan be used in many of the public health interventions suchas suicide, HIV or Tuberculous prevention that rely on so-cial networks. This way, the decisions-makers can compare amenu of fairness-utility trade-offs proposed by our approachand decide which one of these trade-offs are more desirablewithout a need to understand the underlying mathematicaldetails that are used in deriving these trade-offs.

There are crucial considerations when deploying our sys-tem in practice. First, cardinal welfare is one particular wayof formalizing fairness considerations. This by no means im-plies that other approaches for fairness e.g. equality enforcinginterventions should be completely ignored. Second, we haveassumed that the decision-maker has the full knowledge ofthe network as well as possibly protected attributes of theindividuals which can be used to define communities. Third,while our experimental evaluation is based on utilizing agreedy algorithm, it is conceivable that this greedy approx-imation can create complications by imposing undesirablebiases that we have not accounted for. Intuitively (and aswe have seen in our experiments) the extreme of inequalityaversion (α→ −∞) can be used as a proxy for pure equality.However, the last two concerns require more care and weleave the study of such questions as future work.

Acknowledgement We would like to thank David GrayGrant, Matthew Joseph, Andrew Perrault and Bryan Wilderfor helpful discussions about this work. This work is sup-ported in part by the Smart & Connected Communities pro-gram of the National Science Foundation under NSF awardNo. 1831770, the US Army Research Office under grantnumber W911NF1710445 and Center for Research on Com-putation and Society (CRCS) at the Harvard John A. PaulsonSchool of Engineering and Applied Sciences.

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9

A Additional Related WorkArtificial Intelligence and machine learning algorithms holdgreat promise in addressing many pressing societal problems.These problems often pose complex ethical and fairness is-sues which need to be addressed before the algorithms can bedeployed in the real world. The nascent field of algorithmicfairness has emerged to address these fairness concerns. Tothis end, different notions of fairness are defined based onone or more sensitive attributes such as age, race or gender.For example, in the classification and regression setting, thesenotions mainly aim at equalizing a statistical quantity acrossdifferent communities or populations (Hardt, Price, and Sre-bro 2016; Zafar et al. 2017). While surveying the entirety ofthis field is out of our scope (see e.g., (Berk et al. 2018) fora recent survey), we point out that there is a wide range offairness notions defined across different settings and it hasbeen shown that the right notion is problem dependent (Berket al. 2017; Narayanan 2018) and also different notions offairness can be incompatible with each other (Kleinberg,Mullainathan, and Raghavan 2017). Thus, care must be takenwhen we employ these notions of fairness across differentapplications.

Motivated by the importance of fairness when conductinginterventions in social initiatives (Kube, Das, and Fowler2019), fair influence maximization has received a lot of atten-tion recently (Tsang et al. 2019; Ali et al. 2019; Rahmattalabiet al. 2019; Fish et al. 2019). These works have incorporatedfairness directly into the influence maximization frameworkby (1) relying on either Rawlsian theory of justice (Rawls2009; Rahmattalabi et al. 2019), (2) game theoretic princi-ples (Tsang et al. 2019) or (3) equality based notions (Ali et al.2019; Stoica, Han, and Chaintreau 2020). We will discussthe first two approaches in more details in Sections 3 and 4,as well as in our experimental section. Equality based ap-proaches strive for equal outcomes across different commu-nities. In general, strict equality is hard to achieve and maylead to wastage of resources. This is amplified in influencemaximization as different communities have different capaci-ties in being influenced (e.g., marginalized communities arehard to reach).

Fish et al. (2019) investigate the notion of informationaccess gap, where they propose maximizing the minimumprobability that an individual is being influenced/informedto constrain this gap. As a result they study fairness at anindividual level while we study fairness at the group level.Also, their notion of access gap is limited to the gap in abipartition of the network which is in principle differentfrom utility gap that we study which accommodates arbitrarynumber of protected groups.

Similar to our work, Ali et al. (2019) also study utility gap.They propose an optimization model that directly penalizesutility gap which they solve via a surrogate objective function.Their surrogate functions are in the form of a sum of concavefunctions of the group utilities which are aggregated witharbitrary weights. Unlike their work, our approach takes anaxiomatic approach with strong theoretical justifications andit does not allow for arbitrary concave functions and weightsas they violate the welfare principles.

There has also been a long line of work considering fair-

ness in resource allocation problems (see e.g., (Bertsimas,Farias, and Trichakis 2011; Kleinberg, Rabani, and Tardos1999; Bonald and Massoulie 2001; Biswas and Barman2018)). More recently, group fairness has been studied inthe context of resource allocation problems (Conitzer et al.2019; Elzayn et al. 2019; Benabbou et al. 2018) and specifi-cally in graph covering problems (Rahmattalabi et al. 2019).In resource allocation setting, maximin fairness and propor-tional fairness are widely adopted fairness notions. Propor-tional fairness is a notion introduced for bandwidth allo-cation (Bonald and Massoulie 2001). An allocation is pro-portionally fair if the sum of percentage-wise changes inthe utilities of all groups cannot be improved with anotherallocation. In classical resource allocation problems, each in-dividual or group has a utility function that is independent ofthe utilities of others individuals or groups. However, this isnot the case in influence maximization due to the underlyingsocial network structure i.e., the between-community edgeswhich makes our problem distinct from the classical resourceallocation problems. We note that, while in the bandwidthallocation setting there is also a network structure, the utilityof each vertex is still independent of the other vertices andis only a function of the amount of resources that the vertexreceives.

Finally, Heidari et al. (2018) have recently proposed tostudy inequality aversion and welfare through cardinal wel-fare theory in the context of regression problems. Their maincontribution is to use this theory to draw attention to other fair-ness considerations beyond equality. However, the classicalsocial welfare theory, does not readily extend to our settingdue to dependencies induced by the between-community con-nections. Indeed, extending those principles is a contributionof our work.

B Omitted Proofs from Section 4.2Proof of Proposition 1. Let F : RN → R be an additivefunction in the form F (u) =

∑Ni=1 f(ui) where f : R→ R

is a monotonically increasing and strictly concave function.We are focusing on group fairness where the utility of eachindividual is given by the average utility of their commu-nity. Hence, we can rewrite F (u) =

∑c∈C Ncf(uc). Let

u = u(A) and u′ = u(A′) denote the utility vectors cor-responding to neighboring solutions A and A′, respectively.Suppose u and u′ are sorted in ascending order and for allc ∈ C, index c in both vectors corresponds to the same com-munity, i.e., after the transfer the ordering of the utilities hasnot changed.

Furthermore, assume Σκ∈C:κ≤cNκ(uκ− u′κ) ≥ 0, ∀c ∈ Cand uc > u′c for some c ∈ C. Clearly u and u′ sat-isfy the assumptions of the influence transfer principle.We need to show that Σc∈CNcf(uc) > Σc∈CNcf(u′c) orΣc∈CNc (f(uc)− f(u′c)) > 0.

The proof is by induction. We iteratively sweep the vectorsu and u′ from the smallest index to the largest and show thatfor any κ ∈ C, Σc≤κNc (f(uc)− f(u′c)) ≥ 0 with inequalitybecoming strict for at least one κ. To do so we repeatedly usea property of strictly concave functions known as decreasingmarginal returns. According to this property f(x + δx) −

10

f(x) > f(y + δy)− f(y) for x < y and δx ≥ δy > 0.More specifically, in our inductive step, we keep track of a

“decrement budget” which we denote by ∆. Intuitively if wecan show that Σc≤κNc (f(uc)− f(u′c)) > 0 with budget ∆for some κ, we can then use the decreasing marginal returnproperty along with the assumption that u′ is sorted to showthat as long as Nκ+1

(u′κ+1 − uκ+1

)≤ ∆ it is the case

that Σc≤κ+1Nc (f(uc)− f(u′c)) > 0. After each round weupdate the ∆ and move on to the next element in the utilityvectors.

Formally, let ∆ = 0 to start at the begining of this induc-tive process. After visiting the cth community, we simplyupdate ∆ by ∆← ∆ +Nc (uc − u′c). By the assumption ofthe transfer principle ∆ is non-negative at all points of thisiterative process and is strictly positive at some point duringthe process. Observe that f(u1) ≥ f(u′1) since u1 ≥ u′1 bythe assumption of the transfer principle and monotonicity off . We can use this as the base case. Since u and u′ are sorted,given that ∆ is non-negative, the fact that f is strictly concave(so that the decreasing marginal return property can be used)immediately implies that Σc≤κNc (f(uc)− f(u′c)) ≥ 0 atany iteration κ of the process. The inequality becomes strictfor some κ given the assumption of the transfer principle.This proves the claim.

Proof of Proposition 2. Figure 5 is an illustration of thegraph that is used in the proof to witness the statement.We set p = 1 (deterministic spread) and number of ini-tial seeds K = 4. Consider two choices of influencer ver-tices A and A′. Let A denote the choice that consists ofthe center of all-star structures that consist of a single com-munity. Let A′ denote the solution that is identical to Awith the sole difference that only the center of one of theall-square structures is chosen and the last seed is selectedto be the center of the star structure that is the mix ofcircle and diamond communities. Clearly these two solu-tions are neighboring. The average utilities for these so-lutions are (diamond = 0.3, square = 0.7, circle = 0.8) in uand (diamond = 0.34, square = 0.6, circle = 0.86) in u′, re-spectively. Both solutions correspond to a total utility of 180but the utility gap is ∆(u) = 0.5 for u as opposed to theutility gap of ∆(u′) = 0.52 for u′. So a welfare functionthat obeys the utility gap reduction should prefer u over u′.

We now show that no welfare function that satisfies the first5 principles will prefer u over u′. Recall that such welfarefunctions are in the form Wα(u) = Σc∈CNcu

αc /α for α < 1

and α 6= 0, Wα(u) = Σc∈CNc log(uc) for α = 0. Weverify this claim numerically. In particular Figure 6 plots thedifference ofWα(u)−Wα(u′) forWα(u) = Σc∈CNcu

αc /α

when α ∈ (0, 1) (top panel) and α < 0 (bottom panel).This difference is always negative so u′ is preferred by thesewelfare functions. ForWα(u) = Σc∈CNc log(uc),Wα(u)−Wα(u′) ≈ −4.3.

We point out that the instance used in the proof (graphstructure, probability of spread and the number of seeds) isdesigned with the sole purpose of simplifying the calculationsof the utilities. It is possible to modify this instance to morecomplicated and realistic instances.

Proof of Proposition 3. Let A and A′ denote two neighbor-ing solutions with corresponding utility vectors u = u(A)and u′ = u(A′). Let u denote any of the two utility vectorssuch that Σc∈CNcuc ≥ Σc∈CNcu

′c. Without loss of general-

ity, we assume u′ is sorted in ascending order of the utilitiesand u is permuted so that index c ∈ C in both u and u′ cor-responds to the same community. This is because we assumethat W satisfies the symmetry principle due to which by per-muting a utility vector the value of the welfare function doesnot change. Let ν and κ ∈ C denote the communities whoseutilities are changed between u and u′, i.e., we assume ν andκ are the two communities where taking influencer verticesfrom ν and giving them to κ will transfer u′ into u.

To satisfy the condition of the utility gap reduction prin-ciple, it should be the case that u′ν ≥ u′κ (i.e., we transferinfluencer vertices from the group with higher utility to agroup with lower utility), otherwise after the transfer from u′

to u the utility gap could not get smaller (i.e., ∆(u) ≥ ∆(u′)in which case the utility gap reduction is not applicable).

Assuming u′ν ≥ u′κ, if ∆(u) ≥ ∆(u′), again the assump-tion of the utility gap reduction principle is not satisfied,hence the principle is not applicable and there is no need tostudy this case. Therefore, we further assume ∆(u) < ∆(u′).We would like to show in this case a welfare function W thatsatisfies all the 5 other principles witnesses W (u) > W (u′).

By assumption Σc∈CNcuc ≥ Σc∈CNcu′c. From this, it

follows that: ∑c∈C

Nc (uc − u′c) ≥ 0 (4)

⇔Nν (uν − u′ν) +Nκ (uκ − u′κ) ≥ 0 (5)

⇔∑

y∈C:y≤x

Ny(uy − u′y) ≥ 0, ∀x ∈ C, (6)

where both inequalities (5) and (6) follow directly from thefact that the utilities of all the other communities are the samein both u and u′. Finally, since uκ > u′κ (we are transferringinfluencer vertices to the community κ), we can apply theinfluence transfer principle to show that W (u) > W (u′) asclaimed.

Proof of Lemma 1. As we have shown earlier welfare func-tions Wα(u) = Σc∈CNc log(uc) for α = 0 and Wα(u) =Σc∈CNcu

αc /α for α < 1, α 6= 0 satisfy all the first 5 princi-

ples. Lin and Bilmes (2011) show that the composition of anon-decreasing concave function (in our case log(x), α = 0or xα/α for α < 1, α 6= 0) and a non-decreasing submodularfunction (in our case uc(A)) is submodular. Since the sum ofsubmodular functions is submodular, our proposed class ofwelfare functions is submodular. Our welfare functions alsosatisfy monotonicity. This is because uc(A) is monotonicallynon-decreasing so its composition with another monotoni-cally non-decreasing function (log(x) for α = 0 or xα/αfor α < 1, α 6= 0) will be monotonically non-decreasing.Since our welfare functions are the sum of monotonicallynon-decreasing function they are also monotone.

C Leximin Fairness and Social WelfareIn this section, we show that leximin fairness can be capturedby our welfare maximizing framework. See (Heidari et al.

11

Figure 5: An illustration for the graph used in the proof of Proposition 2 without the correct scaling. There are three communities(circle, square and diamond) and they all have size 100. The circle community consists of an “all-circle” star structure with80 vertices, 14 isolated vertices and a mixed star structure (shared with the diamond community) with 6 circle vertices. Thesquare community consists of two “all-square” star structures with sizes 60 and 10 plus a set of 30 isolated vertices. The diamondcommunity consists of an “all-diamond” star structure with 30 vertices, 66 isolated vertices and a mixed star structure (sharedwith the circle community) with 4 diamond vertices.

Wα(u

)−W

α(u′�)

−4−3−2−1

0

0.00 0.25 0.50 0.75 1.00α value

1

10−2

4 (Wα(u

)−W

α(u′�) )

−2

−1

0

−50 −40 −30 −20 −10 0α value

Figure 6: The difference of Wα(u)−Wα(u′) on the verticalaxis versus α on the horizontal axis for different welfarefunctions (this difference is scaled by a factor of 10−24 onthe bottom panel). Top panel: Wα(u) = Σc∈CNcu

αc /α for

α ∈ (0, 1); bottom panel: Wα(u) = Σc∈CNcuαc /α for α <

0.

2018) for more details.

Proposition 4. Welfare optimization is equivalent to the lex-imin fairness, i.e., there exists a constant α0, such for α < α0,an optimal solution to the welfare maximization satisfies lex-imin fairness and vice versa.

Proof. Let u = (u1, . . . , uN ) < u(A) ∀A ∈ A?, where“<” is the lexicographic ordering sign and it indicates thatu is a leximin fair solution (w.l.o.g. and with a slight abuseof notation, we assume that both u and u(A) are sorted inincreasing order). We aim to show that ∃α0 < 0 such that forany α ≤ α0, ΣNi=1u

αi /α ≥ ΣNi=1u

αi (A)/α, ∀A ∈ A?. For

simplicity we multiply both sides of the inequality by −1/αand since α < 0 the direction of the inequality sign does notchange.

We now prove this inequality by contradiction. Suppose∀α1 < 0, ∃α < α1 such that ΣNi=1 − uαi < ΣNi=1 −uαi (A),∃A ∈ A?. Since u is a leximin solution then bydefinition u1 ≥ u1(A). We consider two cases. First suppose

u1 > u1(A).

N∑i=1

−uαi <N∑i=1

−uαi (A)⇔

∑Ni=1−u

αi

min(u1, u1(A))α<

∑Ni=1−uαi (A)

min(u1, u1(A))α=∑N

i=1−uαi

u1(A)α<

∑Ni=1−uαi (A)

u1(A)α⇒

limα→−∞

∑Ni=1−u

αi

u1(A)α≤ limα→−∞

∑Ni=1−uαi (A)

u1(A)α⇒

0 ≤ −N1.

This is a contradiction since N1 > 0. Now, suppose u1 =u1(A). In this case, we can eliminate the first terms thatinvolve uα1 and uα1 from the two sides of inequality and redothe above steps iteratively starting from the second biggestelement in uα.

Next, we prove the other direction. Let us assume u is autility vector such that ∃α0 < 0,∀α ≤ α0,

∑Ni=1−u

αi ≥∑N

i=1−uαi (A), ∀A ∈ A?. W.l.o.g, we can assume thatu1 6= u1 otherwise we can remove those terms that are equaland the proof still holds. However, we assume this for easeof exposition. It follows that∑N

i=1−uαi

min(u1, u1)α≥∑Ni=1−uαi (A)

min(u1, u1)α, ∀A ∈ A?.

If min(u1, u1) = u1 meaning that u1 > u1 we have−C − ε(α) ≥ −δ(α,A), ∀A ∈ A? where C > 0 is aconstant (equal to the number of entities in u that are equalto u1) and both ε ≥ 0 and δ ≥ 0 are functions of α and canbe made arbitrarily small by decreasing α. This is a contra-diction which means that min(u1, u1) = u1, i.e., u1 ≥ u1.By continuing this procedure, we can establish that u < u.This completes the proof.

D Omitted Proofs from Table 1In this section we provide detailed description of the entriesof Table 1 and their derivations.

D.1 MonotonicityProposition 5. Exact DP does not satisfy monotonicity.

12

Figure 7: Companion figure to Proposition 5. The networkconsists of two communities circle and square, each of sizeN .

Proof. Let K = 2 and p ∈ (0, 1). Consider a graph G asshown in Figure 7 consisting of two communities, squareand circle, each of size N (for large enough N ). The circlecommunity consists of a star network of size N . The squarecommunity contains a star network of size 2 + p(N − 2) and(N − 2)(1 − p) singletons. Consider two solutions A andA′. A will select a seed from the periphery of the star for thecircle community and allocate the other seed to the centerof the star for the square community. A′ on the other handallocates each of the seeds to the center of the stars. Let u =u(A) and u′ = u(A′) denote the corresponding allocationsof A and A′. The utility vectors for these allocations areu = ((1 + p + p2(N − 2))/N, (1 + p + p2(N − 2))/N)and u′ = ((1 + p(N − 1))/N, (1 + p + p2(N − 2))/N),respectively. Clearly, u < u′. So by monotonicity u′ ispreferred to u. However, only u satisfies the exact DP. Hence,DP does not satisfy monotonicity.

Proposition 6. Approximate DP does not satisfy monotonic-ity.

Proof. Consider a graph G as shown in Figure 8 consistingof two communities, square and circle, each of size N . Wechoose an arbitrary δ ∈ (0, 1), to reflect the arbitrary strict-ness of a decision maker. Let δ < p <

√δ, K = 2 and

N > max(3p/(p− δ), 1/(δ − p2)

). The optimal solution

A of the influence maximization problem chooses the centerof the star and any disconnected square vertex. In A, the util-ity of circle and square communities are (1 + (N − 1)p) /Nand (1 + 2p)/N , respectively and the utility gap exceeds δ(so this solution does not satisfy the DP constraints). By im-posing DP, any fair solution is to choose one vertex from theperiphery of the circle community and one from the isolatedsquare vertices. For a fair solution A′, the utilities of circleand square are

(1 + p+ p2(N − 2)

)/N and (1 + 2p2)/N ,

respectively. Given the range of N , the utility gap is less thanδ so approximate DP is satisfied. Since the utility of bothcommunities have degraded, any monotone welfare functionwill prefer A′ (and its corresponding utility vector) over A.However, only A′ is DP fair and hence it is preferred overA by DP. We point out that the graph used in the proof isdirected. This is for ease of exposition. It is possible to createa more complex example with an undirected graph.

Proposition 7. Consider a general fairness notion as a setof constraints F =

{u ∈ [0, 1]C : uc ≥ lc, ∀c ∈ C

}where

lc ∀c ∈ C are arbitrary lower-bound values. The consideredfairness notion satisfies the monotonicity principle.

Figure 8: Companion figure to Proposition 6. The networkconsists of two communities circle and square, each of sizeN . All edges except the two shown by arrows are undirectedmeaning that influence can spread both ways.

Proof. Let A and A′ ∈ A? denote two solutions whosecorresponding utility vectors u = u(A) and u′ = u(A)are feasible (u,u′ ∈ F) such that u < u′. Given theobjective function of the influence maximization is equiv-alent to Σc∈CNcuc(A) and that all Nc values are positivethe objective values of u′ is strictly better than u. Hence,W (u) < W (u′) and the monotonicity is satisfied.

As we have shown in Section 3, both maximin and DCcan be written as constraints that are compatible with thefairness definition in Proposition 7. The utilitarian solutioncorresponds to setting all the lower bounds to 0. Hence all ofthem satisfy monotonicity.Corollary 1. DC, MMF and utilitarian satisfy monotonicity.

D.2 SymmetryIt is straightforward to show that DP (exact and approximate),maximin and utilitarian fairness satisfy the symmetry prin-ciple. DC, however, does not satisfy the symmetry principle.Based on its definition, DC can place different lower-boundson the utility of different communities. Hence, by permutinga utility vector we may no longer be able to satisfy the DCconstraints (see Definition 2).

D.3 Independence of Unconcerned IndividualsProposition 8. Exact and approximate DP do not satisfy theindependence of unconcerned individuals.

Proof. Consider two utility vectors: u = ((1 + 3δ)/8, (1−δ)/8) and u′ = ((1 + δ)/4, (1− δ)/8) for δ ∈ [0, 1). Bothexact and approximate DP strictly prefer u over u′. Let ussubstitute the second component of both vectors by (1+δ)/4.Therefore, we obtain v = u|2(1 + δ)/4 = ((1 + 3δ)/8, (1 +δ)/4) and v′ = u′|2(1 + δ)/4 = ((1 + δ)/4, (1 + δ)/4). Incontrast to the previous case, both approximate and exactDP prefer v′ over v. Note that while the construction doesdoes not involve an instance of the influence maximizationproblem, it is possible to provide such an instance to witnessthe claim as follows.

Figure 9 demonstrates the instance witnessing the claim.We consider an influence maximization problem with twocommunities: circle (first component of the utility vector)and square (second component of the utility vector), eachof size N (for N large enough). We assume p = 1 andK = 2. The circle community consists of three components:two star components of size N(1 + δ)/4 (small) and N(1 +3δ)/8 (large) and 5N(1− δ)/8 isolated vertices. The squarecommunity consists of three components as well: two star

13

Figure 9: Companion figure to Proposition 8. The networkconsists of two communities circle and square each of sizeN .

components of size N(1 − δ)/8 (small) and N(1 + δ)/4(large) and N(5 − δ)/8 isolated vertices. Solution u (u′)corresponds to selecting a seed vertex from the large (small)star component of the circle community and a vertex from thesmall star component of the square community. Allocationv (v′) corresponds to selecting a seed vertex from the large(small) star component of the circle community a vertex fromthe large star component of the square community. Note thatthe choice of p = 1 is merely for the ease of expositionand the example network can be modified to accommodatep < 1.

Henceforth, we only discuss utility vectors when appropri-ate. In all such cases, there exist instances of the influencemaximization problem which witness these utility vectors.We have demonstrated one such instance in the proof ofProposition 8, but we omit the details from the remainingproofs for simplicity.

Proposition 9. DC does not satisfy the independence of un-concerned individuals.

Proof. Consider an instance of the influence maximizationproblem with 2 communities were the lower bound setby DC for both communities is 0.4. Also consider 2 solu-tions with corresponding utility vectors u = (0.5, 0.5) andu′ = (0.5, 0.3). Therefore, only u satisfies DC and henceDC prefers u over u′. Let us substitute the first componentwith 0.35. Therefore, we obtain v = u|10.35 = (0.35, 0.5)and v′ = u′|10.35 = (0.35, 0.3). In contrast to the previ-ous case, both solutions are infeasible with respect to theDC (hence -∞ welfare, see Section 4.4). Therefore, whileW (u) > W (u′), it does not hold that W (v) > W (v′).

Proposition 10. MMF does not satisfy the independence ofunconcerned individuals.

Proof. Consider an instance of the influence maximizationproblem with 3 communities of equal size. Also consider 2 so-lutions with corresponding utility vectors u = (0.3, 0.6, 0.4)and u′ = (0.3, 0.2, 0.8). Maximin fairness strictly prefersu over u′. Let us substitute the first component with 0.1.Therefore, we obtain v = u|10.1 = (0.1, 0.6, 0.4) andv′ = u′|10.1 = (0.1, 0.2, 0.8). Maximin fairness is indiffer-ent between v and v′ (both have the same worst-case utilityand total utility) which shows that maximin fairness does notsatisfy the independence of unconcerned individuals.

Note that the utilitarian satisfies the independence of un-concerned individuals because if W (u) = Σc∈CNcuc >

Σc∈CNcu′c = W (u′) then W (u|cb′) < W (u′|cb′) since Nc,

uc, u′c and b′ are all non-negative.

D.4 Affine InvarianceExact DP satisfies affine invariance principle because a lin-ear transformation over a uniform vector will remain uni-form. However, for approximate DP this is not the case.More particularly, for any utility vector u that is δ-DPfor δ ∈ (0, 1) and an affine transformation of the formu′ = αu + β, u′ satisfies αδ-DP. Therefore, for α > 1, u′does not satisfy δ-DP. Similarly, DC does not satisfy this prin-ciple either. This is because after the transformation the con-straints may not be satisfied (e.g., when α < 1/minc∈C Uc).It is known that MMF satisfies this principle (Bertsimas,Farias, and Trichakis 2011). The same holds for the utili-tarian objective because if Σc∈CNcuc > Σc∈CNcu

′c, then

Σc∈CNcαuc + β > Σc∈CNcαu′c + β since α > 0.

D.5 Influence Transfer Principle.Proposition 11. Exact and approximate DP do not satisfythe influence transfer principle.

Proof. Let δ ∈ [0, 1) denote the parameter of DP. Considerutility vectors u = ((1 + δ)/2, 0) and u′ = (δ, 0). Thesizes of the all the communities are the same. Based on theinfluence transfer principle, W (u) > W (u′), however, DPstrictly prefers u′ over u.

Proposition 12. DC does not satisfy the influence transferprinciple.

Proof. Let u = (0.5, 0.5) and u′ = (0.3, 0.6) denote theutility vectors of two allocations where sizes of the all thecommunities are the same. Suppose the lower bounds setby DC are 0.25 and 0.55, respectively. This means that onlyu′ satisfies DC. Based on the transfer principle, W (u) >W (u′), however, u does not satisfy DC and DC strictlyprefers u′ over u.

Proposition 13. MMF does not satisfy the influence transferprinciple.

Proof. Consider two utility vectors u = (0.2, 0.4, 0.6) andu′ = (0.2, 0.2, 0.8). The sizes of the all the communities arethe same. A fairness notion satisfying the influence transferprinciple strictly prefers u over u′. However, Maximin isindifferent between u and u′ as they both obtain the sameworst-case utility.

Proposition 14. Utilitarian does not satisfy the influencetransfer principle.

Proof. Consider two utility vectors u = (0.5, 0.5) and u′ =(0.3, 0.7). The sizes of the all the communities are the same.Based on the transfer principle, W (u) > W (u′), however,the utilitarian approach is indifferent between u and u′ asboth solutions lead to the same total utility.

14

D.6 Utility Gap Reduction.Proposition 15. DP satisfies the utility gap reduction if andonly if δ = 0.

Proof. It is easy to show that if δ = 0, DP satisfies the utilitygap reduction principle. We can prove this by contradiction.Suppose that δ = 0 and DP does not satisfy the utility gapreduction principle. From this, it follows that given two utilityvectors u,u′ such that

∑c∈C Ncuc ≥

∑c∈C Ncu

′c if ∆u <

∆u′, DP can strictly prefer u′. If u′ is preferred, then u′ isfeasible and it must be that ∆u′ = 0 and ∆u < 0 which isnot possible. Next, we show that if δ 6= 0, DP does not satisfythis principle over all instances of the influence maximizationproblem.

For the proof, we use the example used in the proof ofProposition 2. In that setting there are two solutions withutility gap 0.52 and 0.5 with the same total utility. First, letus assume δ > 0.02. In this case, since both solutions satisfyDP constraints, they are both feasible and the total utilityof both solutions is equal (= 180), however, DP does notstrictly prefer the solution with smaller gap. In fact, bothsolutions are feasible with the same objective value and DPdoes not favor one solution to the other. For δ ≤ 0.02, we canuse the same example graph (see Figure 5) and add enoughisolated vertices to each community until the gap between thesolutions becomes small enough to pass the δ threshold.

Figure 10: Companion figure to Proposition 16 of a graphwith two communities: N black vertices and N/3 white ver-tices for N = 9. We choose K = 4 and arbitrary p < 1. Alledges are undirected, meaning that influence can spread bothways.

Proposition 16. DC does not satisfy utility gap reductionprinciple.

Proof. Consider the network G as in Figure 10 consistingof two communities white and black with size N/3 and N ,respectively. SupposeK = 4 and p < 1. Without DC, an opti-mal solution places one seed vertex at the center of the blackgroup and allocates the remaining 3 vertices to the whitegroup. We let u denote this solution. Thus, the utility of theblack and white groups will be equal to (1 + (N − 1)p) /Nand 9/N . Since there is no edge between the black and whitecommunities, DC (See Definition 2) reduces to how to opti-mally choose one seed vertex from white and the remaining3 from the black group. After imposing DC, the utility of theblack and white groups will be equal to (3 + (N − 3)p)/Nand 3/N . We let w denote one such solution that satisfies DC.While u has a higher total utility and a smaller utility gap,DC strictly prefers w with higher utility gap and lower totalutility. Therefore, DC does not satisfy utility gap reductionprinciple.

Proposition 17. MMF does not satisfy the utility gap reduc-tion

Proof. We prove the statement via the example in Figure 11which depicts a network with three groups: blue, black andwhite. We fix K = 1 and p > 3/4. The graph corresponds tothe case where p = 1 but the example will hold for arbitrary pby setting the number of isolated green vertices to be d21/pe.

Figure 11: Companion figure to Proposition 17 for the caseof p = 1. The network consists of three groups: white, blueand black. The edges are undirected so the influence canspread both ways. For arbitrary p, the number of isolatedblack vertices should scale to d21/pe.

Consider one solution that targets the center of the biggerstar component. Thus, the utilities of blue, black and whitewill be (1 + 4p)/11, 4p/11 and 0, respectively. This resultsin utility gap equal to (1 + 4p)/11. By imposing leximin,the optimal fair solution selects the center of the smaller starcomponent and the optimal fair utilities of blue, black andwhite will be 6p/11, p/11 and 1/(1 + d21/pe), respectivelyand we observe a utility gap 6p/11 − 1/(1 + d21/pe) ≥6p/11 − 1/22 > (1 + 4p)/11, where we used p > 3/4 inthe last inequality. In conclusion, while the first solution hasa higher total utility (= 9) and lower utility gap comparedto the second solution (total utility = 8), leximin still strictlyprefers the second solution. This concludes the proof.

Proposition 18. Utilitarian does not satisfy the utility gapreduction

Proof. Consider an instance of the influence maximizationproblem where all the communities are of size 10. Letu = (0.5, 0.5, 0.5) and u′ = (0.2, 0.5, 0.8). Both u andu′ achieve the same total utility (=15). Thus, the utilitarianapproach is indifferent between u and u′. According to theutility gap reduction u is strictly preferred. We note that inthis special instance, both influence transfer principle andutility gap reduction apply and according to both principlesu is strictly preferred to u′.

E Omitted Details from Section 5E.1 Estimating the SBM Parameters for

Landslide Risk Management ApplicationIn order to qualitatively and formatively describe the networkstructure, the research team conducted several in-person semi-structured interviews in Sitka, Alaska from 2018-2020. Theseinterviews were conducted with individuals who were identi-fied as “community leaders” or “popular community mem-bers” through word-of-mouth, and then subsequently throughrespondent-driven sampling, a broader range of community

15

members were interviewed (n=14). In these semi-structuredinterviews, respondents were asked to 1) sort and describecommunity groups and 2) identify “cliques” and “isolates” asthey relate to an early landslide warning system. The formerresulted in developing, to the extent possible, discrete a prioricommunity groups. The latter helped to inform the relation-ships between and within these groups. The interviewer tooknotes which listed the responses and through a tallying andpile sorting exercise, attempted to seek consensus in defini-tions of community groups. The formative research resultedin cliques based on occupation, political affiliation, age, andlocal recreational activities. Many cliques were overlappingwith shared attributes (e.g. people from two different occupa-tions share a political affiliation and frequent the same localpub), however for the purposes of this formative exercise,these community groups were qualitatively coerced into dis-crete classifications. These resulted in 16 community groupsthat include political affiliation, time spent in Sitka (e.g. newarrival or tourist vs. long-term resident), occupation, andwhether or not a parent of a child in the public-school system.The community size estimates were developed based on a2018 Sitka Economic Development Survey, particularly forthe occupation-based community groups, as well as publiclyavailable voter records for political affiliations. Several at-tributes, namely age, specific occupation, time spent in Sitka,and parental status were unavailable in existing datasets, andtherefore required the use of proxies and assumptions for es-timating community group sizes. Once the community groupsizes were estimated, based on the formative research noteson social cohesion, cliques, and isolates, we further devel-oped assumptions on within and between-community con-nectedness. For example, if a respondent suggested that theremay be very close relationships between two cliques, we as-sumed a higher relative p(b) than between two cliques whichhad less similar attributes. For simplicity, we limited theabsolute probabilities for withing-community and between-community probabilities between 0.00 and 0.10. We thensense-checked these absolute probabilities with several ofthe initial formative research respondents. These absoluteprobabilities were then organized into a 16× 16 adjacencymatrix to facilitate simulations for influence maximization.

E.2 Relative Community SizesIn this section we study the effect of relative community sizeon both utility gap and efficiency. We consider synthetic sam-ples of SBM network consisting of two communities eachof size 100 and we gradually increase the size of one com-munity from 100 to 900 in order to study its effect on theutility values of each community. We set qc = 0.005 andqcc′ = 0.001. Results are summarized in Figure 12. Thisresult indicates that the utility gap increases with the relativecommunity size, suggesting that minorities can be adverselyaffected without appropriate fairness measures in place. Wealso note that the strength of our approach is in its flexibilityto trade-off fairness with efficiency. We may encounter sce-narios where the fairness-efficiency trade-offs are mild (asin the particular setting of Figure 12), but this does not un-dermine our approach as there are many practical situations(as discussed in real-world applications in the paper) where

0

10

20

30

−9 −7 −5 −2 0 0.5 DC MMFMethod

Util

ity G

ap (

%)

Ratio13579

0

1

2

3

4

−9 −7 −5 −2 0 0.5 DC MMFMethod

PoF

(%

) Ratio13579

Figure 12: Utility gap and PoF for various relative communitysizes where the ratio changes from 1 to 9.

clearly this is not the case and our approach can handle allthose cases effectively. DC exhibits relatively high utility gap.This is because by definition DC allocates more resourcesto communities that “will do better with the resources” andit does not always show an aversion to inequality, a resultwhich we show theoretically in Section 4.4.

E.3 Suicide Prevention ApplicationInfluence maximization has been previously implemented forhealth promoting interventions among homeless youth (Ya-dav et al. 2016; Wilder et al. 2020). In this section, weconsider the problem of training a set of individuals whocan share information to help prevent suicide (e.g., how toidentify warning signs of suicide among others). We presentsimulation results over six different social networks of home-less youth from a major city in US as described in Barman-Adhikari et al. (2016). We provide aggregate summaries ofthese networks (e.g., size, edge density and community statis-tics) in Table 2. The data set consists of six different socialnetworks of homeless youth from a major city in US asdescribed in detail in Barman-Adhikari et al. (2016). Eachsocial network consists of 7 racial groups, namely, Blackor African American, Latino or Hispanic, White, AmericanIndian or Alaska Native, Asian, Native Hawaiian or OtherPacific Islander and Mixed race. Each individual belongs toa single racial group. We use these partitioning by race todefine our communities. However, to avoid misinterpretationof the results, we combine racial groups with a population< 10% of the network size N under the “Other” category.After this pre-processing step, each dataset will contain 3to 5 communities. Results are summarized in Table 2. Weremark that the absent of a racial category in a given net-work is due to their small sizes and hence being merged intothe “Other” category after pre-processing (e.g., Hispanic innetwork W2SPY.)

We compare our welfare-based framework for different

16

Network Name # of Vertices # of Edges White Black Hispanic Mixed Race OtherW1MFP 219 217 16.4 41.5 20.5 16.4 5.0W2MFP 243 214 16.8 36.6 21.8 22.2 2.4W3MFP 296 326 22.6 34.4 15.2 22.9 4.7W2SPY 133 225 55.6 10.5 – 22.5 11.3W3SPY 144 227 63.0 – – 16.0 20.0W4SPY 124 111 54.0 16.1 – 14.5 15.3

Table 2: Racial composition (%) after pre-processing as well as the number of vertices and edges of the social networks (Barman-Adhikari et al. 2016).

Measure (%)Fairness Approach

K α = −5 α = −2 α = 0 α = 0.5 α = 0.9 DC Maximin IM

utility gap

5 4.8 7.5 8.5 9.7 11.4 8.2 3.5 12.510 4.6 6.6 7.3 9.5 12.9 6.9 2.0 11.715 3.6 5.2 5.9 8.9 13.5 7.6 2.4 15.320 3.6 4.4 5.9 7.3 14.0 5.8 2.3 17.225 2.6 3.5 4.6 5.8 13.2 7.0 2.0 16.630 2.4 3.2 4.3 6.4 8.6 8.2 2.0 15.7

PoF

5 6.9 5.1 3.4 1.5 0.7 14.4 16.6 0.010 6.6 3.8 2.8 0.7 0.1 14.3 13.1 0.015 3.8 2.5 1.6 1.1 0.1 14.6 10.5 0.020 4.6 3.8 2.9 2.0 1.0 13.9 10.5 0.025 4.0 3.2 2.5 1.9 1.0 13.4 9.9 0.030 3.9 3.4 2.9 2.3 1.8 12.7 10.8 0.0

Table 3: Summary of the utility gap and PoF results averaged over 6 different real world social networks for various budget,fairness approaches and baselines. Numbers in bold highlight the best values in each setting (row) across different approaches.

values of α against DC, MMF and influence maximizationwithout fairness considerations (IM). Table 3 provides a sum-mary of the results averaged over all network instances wherethe numbers in bold highlight the best values (minimum util-ity gap and PoF) for each budget and across different fairnessapproaches. As seen, IM typically has a large utility gap (upto 17.2% for K = 20 which is significant because the totalinfluence is only 28.40%). By imposing fairness we can re-duce this gap. In fact, we observe that across different valuesof α ranging from -5 to 0.5, there is a decreasing trend inutility gap, where for K = 20 and with α = −5, we areable to decrease the utility gap by 3.6%. Consistent withprevious results on SBM networks, both MMF and α = −5exhibit very low utility gaps, however, MMF results in higherPoF. Furthermore, across the range of α we observe a mildtrade-off between fairness and utility. This shows that in thesenetworks enforcing fairness comes at a low cost, though aswe see in the landslide setting, this is not always the case.

Figure 13 shows the results for each network separately(X axis) for a fixed budget K = 30. Figure 13 shows that thetrade-offs can also be very network-dependent (compare e.g.W2SPY and W3MFP). This highlights the crucial need fora flexible framework that can be easily adjusted to meaning-fully compare these trade-offs.

0

5

10

15

W1MFP W2MFP W3MFP W2SPY W3SPY W4SPY

Util

ity G

ap (

%)

Maximinα = −5α = −2α = 0α = 0.5α = 0.9DC

0

5

10

15

20

W1MFP W2MFP W3MFP W2SPY W3SPY W4SPY

PoF

(%

)

Maximinα = −5α = −2α = 0α = 0.5α = 0.9DC

Figure 13: Top and bottom panels: utility gap and PoF foreach real world network instances (K = 30).

17