1 Introduction
The success of social interventions relies heavily on effectively leveraging social networks [23, 40, 39]. For instance, health interventions such as prevention of suicide, HIV, and substance abuse involve finding a small set of wellconnected individuals who can effectively act as peerleaders to either detect warning signs beforehand and respond appropriately (suicide prevention) or disseminate relevant information (HIV or substance abuse prevention). Such interventions are prone to fairness and ethical concerns as individuals from racial minorities or LGBTQ communities may be disproportionately excluded from the benefits of the intervention [39, 36]. The celebrated influence maximization framework has been employed to find such a set of individuals [24]. However, this framework primarily optimizes for maximizing the influence spread and does not account for fairness.
Recent work has incorporated fairness directly into the influence maximization framework by proposing various notions of fairness such as maximin fairness [36], and diversity constraints [39]
. Maximin fairness aims at improving the minimum amount of influence that any community receives. On the other hand, diversity constraints are inspired by the game theory literature and ensure that each community is at least as welloff had they received their share of resources proportional to their size and allocated them internally. Each of these notions comes with drawbacks for example maximin fairness typically will result in significant degradation in total influence due to its stringent requirement while diversity constraints further exacerbate the influence gap between the communities (see Sections
1.1 and 4 for more details). Moreover, while each notion offers a unique perspective on fairness, there is no universal agreement on what fairness means as fairness is known to be domain specific (see e.g., [31]). Therefore, it is crucial for the policymakers to be able to compare these notions on different applications. Indeed different applications have different fairness requirements e.g., the risk of excluding vulnerable communities from suicide prevention which has life and death impacts might have a higher weight compared to interventions for a healthier life style.Proposed Approach and Contributions
Instead of committing to a single definition of fairness for influence maximization, we build on cardinal welfare theory from the economics literature and provide a family of measures to evaluate fairness in this context. At a high level, the aim of cardinal welfare theory is to design a welfare function to measure the goodness or desirability of an allocation by proposing a set of desiderata that the welfare function should satisfy. However, due to the structural dependencies induced by the underlying social network, these principles are not directly applicable to our problem. We make the following contributions: (i) We extend the cardinal welfare principles including the transfer principle to the influence maximization framework, which is otherwise not applicable. We also propose a new principle which we refer to as utility gap reduction. This principle is particularly relevant in the graphbased problems and aims to avoid situations where high aversion to inequity leads to even more influence gap; (ii) We generalize the theory regarding these principles and describe the circumstances under which there exists a welfare function satisfying the principles. We show that if all community members are disconnected from one another (no underlying social network), our results reduce to the existing theory of social welfare; (iii) We propose a natural family of welfare functions that are governed by a single inequity aversion parameter which allows a decision maker to study taskdependent tradeoffs between fairness and total influence and effectively trade off quantities like influence gap or influence degradation by varying this parameter in the welfare function. We then incorporate these welfare functions as objective into an optimization problem to rule out undesirable allocations. We show that the resulting optimization problem is monotone and submodular and, hence, can be conveniently solved using a greedy algorithm with optimality guarantees; (iv) Finally, we carry out a detailed experimental analysis on synthetic graphs and real social networks of homeless youth participating in substance use interventions to study the tradeoff between total influence and welfare. We show that this tradeoff is rather mild, as high welfare can be achieved without sacrificing the total influence significantly. Interestingly we can trade off influence gap for total influence by varying our inequity aversion parameter and show there exists welfare functions that empirically satisfy all of the principles.
1.1 Related Work
Artificial Intelligence and machine learning algorithms hold great promise in addressing many pressing societal problems. These problems often pose complex ethical and fairness issues which need to be addressed before the algorithms can be deployed in the real world. The nascent field of algorithmic fairness has emerged to address these fairness concerns. In particular, there is a growing body of work focusing on the wellstudied machine learning settings such as classification and regression. To this end, different notions of fairness are defined based on one or more sensitive attributes such as race or gender. These notions mainly aim at equalizing a statistical quantity across different communities or populations [10, 20, 42]. While surveying the entirety of this field is out of our scope (see e.g., [6] for a recent survey), we point out that there is a wide range of fairness notions defined for these settings and it has been shown that the right notion is problem dependent [5, 31] and also different notions of fairness can be incompatible with each other [25]. Thus, care must be taken when we employ these notions of fairness across different applications including resource allocation problems and in particular, influence maximization.
Motivated by the importance of fairness when conducting interventions in social initiatives [27], fair influence maximization has received a lot of attention recently [39, 1, 36, 17]. These works have incorporated fairness directly into the influence maximization framework by (1) relying on either Rawlsian theory of justice [37, 36], (2) game theoretic principles [39] or (3) equality based notions [1, 38]. We will discuss the first two approaches in more details in Sections 2.1 and 3, as well as in our experimental section. Equality based approaches do not leverage social network information and imposing strict equality leads to wastage of resources. We will discuss this in more details in Appendix A.
Fish et al. [17]
investigate the notion of information access gap, where they propose maximizing the minimum probability that an individual is being influenced/informed to constrain this gap. As a result they study fairness at an individual level while we study fairness at the group level. Also, their notion of access gap is limited to the gap in a bipartition of the network which is in principle different from utility gap that we study which accommodates arbitrary number of protected groups.
Similar to our work, Ali et al. [1] also study utility gap. They propose an optimization model that directly penalizes utility gap which they solve via a surrogate objective function. Their surrogate functions are in the form of a sum of concave functions of the group utilities which are aggregated with arbitrary weights. Unlike their work, our approach takes an axiomatic approach with strong theoretical justifications and it does not allow for arbitrary concave functions and weights as they violate the welfare principles.
There has also been a long line of work considering fairness in resource allocation problems (see e.g., [7, 26, 9, 8]). More recently, group fairness has been studied in the context of resource allocation problems [12, 15, 4] and specifically in graph covering problems [36]. In resource allocation setting, maximin fairness and proportional fairness are widely adopted fairness notions. Proportional fairness is a notion introduced for bandwidth allocation [9]. An allocation is proportionally fair if the sum of percentagewise changes in the utilities of all groups cannot be improved with another allocation. In classical resource allocation problems, each individual or group has a utility function that is independent of the utilities of others individuals or groups. However, this is not the case in influence maximization due to the underlying social network structure. We note that, while in the bandwidth allocation setting there is also a network structure, the utility of each vertex is still independent of the other vertices and is only a function of the amount of resources that the vertex receives.
Finally, Heidari et al. [22] have recently proposed to study inequity aversion and welfare through cardinal welfare theory in the context of regression problems. Their main contribution is to use this theory to draw attention to other fairness considerations beyond equality. However, we use the welfare theory to propose a tool for policymakers to study the taskspecific tradeoffs between various quantities like influence gap and total influence in different domains. Moreover, the classical social welfare theory does not readily extend to our setting due to dependencies induced by the social network. Indeed, extending those principles is a contribution of our work.
2 Problem Formulation
We use to denote a graph (or network) in which is the set of vertices and is the set of all edges. In the influence maximization problem, a decision maker chooses a set of at most vertices (or seeds) to influence (or activate). The selected vertices then spread the influence in rounds according to the Independent Cascade Model [24]. According to this model, each newly activated vertex will attempt to spread the influence to its neighbors independently and with a fixed probability . The process continues until no new vertex is influenced. We use to denote the initial set of seeds. The goal of the decision maker is to select a seed set to maximize the expected number of vertices that are influenced at the end of this process.
To address the ethical concerns mentioned in Section 1 regarding excluding minority communities, we take a group fairness approach. More concretely, we assume each vertex of the graph belongs to one of the disjoint communities such that where denotes the set of vertices that belong to community . This partitioning can be induced by e.g., the intersection of a set of (protected) attributes such as race or gender for which fair treatment is important. Indeed, our model allows for the case that a vertex belongs to multiple communities by considering the intersections of communities. We use to denote the size of community i.e., . Furthermore, we define as the set of feasible allocations. Finally, for any allocation , we let denote the utility, i.e., the expected fraction of the influenced vertices of community , where the expectation is taken over the spread of influence. The standard influence maximization problem solves the following optimization problem.
When it is clear from the context we drop the dependencies of utilities on .
2.1 Existing Notions of Fairness in Influence Maximization
The optimization problem for the influence maximization solely attempts to maximize the total influence. Many fairnessenforcing interventions can be interpreted as an added constraint to this optimization problem. We discuss a couple of these notions next (see Section 1.1 for more discussion).
Maximin Fairness
Maximin fairness [39, 36] is a generalization of Rawlsian theory of justice [37] aiming to maximize the minimum utility that any of the communities obtains. For any allocation , we can compute the minimum utility that any community will receive under . Let denote the maximum of this minimum utility where the maximization is taken over all allocations in . Then the maximin fairness only allows allocations that satisfy the following constraint.
Diversity Constraints
Inspired by the game theoretic notion of core, diversity constraints require that no community should be able to obtain a higher utility with a share of resources proportional to their size and allocating them internally [39]. This is illustrated by the following constraint where denotes the expected utility of the best allocation of seeds to vertices from community .
Diversity constraints set utility lower bounds for the communities only based on their relative sizes. The main drawback of this approach is that it can further exacerbate the influence gap between the communities. We empirically compare our approach with both of these notions in Section 4.
3 Cardinal Welfare and Fairness
Following the cardinal welfare theory, our aim is to design welfare functions to measure the goodness of allocations for the population as a whole [30]. To do so, cardinal welfare theory propose a set of principles and welfare functions are expected to satisfy these principles.
The interpretation of welfare function as a notion of fairness is justified by the concept of veil of ignorance from moral philosophy [37] which is best described with the following thought experiment. Suppose the decision maker knows nothing about the position (e.g., community in our case) that they will be born in. Crucially the decision maker has the full knowledge of the distribution of positions (i.e., the utility of communities and the chance that he will be born into any of these communities). According to the utilitarian doctrine in this hypothetical position a rational decision maker would consider inequity aversion (or risk as they call it) and depending on their level of inequity aversion can aim to guard against the undesirable outcomes (e.g., that they will be born to low utility communities). Note that although this is a purely hypothetical exercise, it can help the decisions maker in making fair decisions by considering the possibility of taking positions of low utility community.
To formally incorporate inequity aversion we follow the constant relative risk aversion utility models and model inequity aversion through a single parameter where models inequity aversion, models inequityneutrality and models inequity seeking behavior [30]. The welfare function then is simply a mapping from the utilities and the inequity aversion parameter to a single number. More formally, let denote the welfare function. The welfare function is defined over the entire population () instead of over the communities. This ensures that the welfare function takes into account the size of the communities in addition to their average utilities. Intuitively, given the utility values for each community and an equity aversion parameter, the welfare function assigns a score to that combination as a measure of goodness in terms of fairness or distributive justice and a higher number indicates a more desirable combination.
To define the utility vector
for the entire population from the utility values for the communities we have two sensible choices. (1) We can replicate the average utility of community by the number of members of that community . This will model “group fairness" where the goal is to improve the utility of the community as a whole. (2) We can let to be the utility of individuals across the population where the utility of an individual is defined as the probability that the individual is influenced by the intervention. This corresponds to “individual fairness". While the cardinal welfare theory can account for both of these situations, we opt for the former. As pointed out by Dwork et al. [14], individual fairness is notoriously hard to achieve in practice. This problem is further exacerbated in influence maximization problems due to scarcity of resources and the network structure. Hence we leave the study of individual fairness to future work. For convenience we write as and omit the dependency on when it is clear from the context.3.1 Principles of Welfare
The welfare economics literature proposes a set of principles (or desiderata) that any welfare function should satisfy [21]. These principles aim to rank different social states, i.e., given any two utility vectors, these principles determine if they are indifferent or one of them is preferred. We revisit these notions and incorporate necessary changes to make them applicable to our influence maximization problem. Throughout, without loss of generality, we assume all utility vectors belong to .
(1) Monotonicity:
If , then . In other words, if Pareto dominates , then should strictly prefer to . This principle also appears as levelling down objection in political philosophy [34]. Equality enforcing notions has been recently proposed for influence maximization problems [1, 38]. Unlike many of the machine learning settings, these notions might not be desirable for influence maximization as they violate monotonicity (see Section 1.1 and Appendix A).
(2) Symmetry:
, where is any elementwise permutation of . According to this desideratum, does not depend on the naming or labels of the communities (or individuals), but only on their utility levels.
(3) Independence of Unconcerned Communities:
Let be a utility vector that is identical to , except for the utility of community which is replaced by a new value . The property requires that for all and , Informally, this desideratum states that should be independent of communities whose utilities remain the same.
(4) Affine Invariance:
For any and , i.e., the relative ordering of two utility vectors is invariant to a choice of numeraire.
It is wellknown that any welfare function that satisfies the above desiderata is additive and can be written as for , for and for [13], where with slight abuse of notation we used to denote the th element of the utility vector . To highlight this notational difference we use the subscript to denote the utility of the th element (or individual) in and subscript to denote the denote the (average) utility for community . With this in mind, the welfare function can also be written based on the utility of communities e.g., for the case of .
These desiderata are general in the sense that they do not depend on the influence maximization problem and can be applied to any resource allocation problem (graphbased or not). We next study another two desiderata which focus on the transfer of resources between two communities and unlike the previous desiderata crucially depend on the network structure.
(*5) Transfer Principle [19, 35]
: We start by describing the original transfer principle and then describe how we can modify this principle to account for network structure and communities. Consider two individuals and in utility vector such that . Let be another utility vector that is identical to in all elements except entries and where and for some . Then by the transfer principle . Informally, this states that transferring utility from a highutility to a lowutility individual should increase social welfare.
We are concerned with communities rather than individuals so instead we focus on transfer of utilities from one community to another. In the presence of network structure it might not always be feasible to directly transfer utilities from one community to another without modifying the utilities of other communities. Furthermore, even if the direct transfer is possible, it can be the case that there is no symmetry in the amount of utility gained by the lowutility community and the amount of utility lost by the highutility community. The generalized transfer principle addresses both of these shortcomings. In the spirit of the original transfer principle we only consider allocations in which some units of resources are transferred from individuals in one community to individuals in another community. We refer to such allocations as neighboring allocations.
(5) Generalized Transfer Principle:
Suppose and are two neighboring allocations. Let and denote the utility vectors from these allocations. Suppose is sorted in ascending order and is permuted so that, for all , index in both vectors corresponds to the same community. If , and for some then .
Informally, the generalized transfer principle states that in a desirable transfer of utilities the magnitude of the improvement in lowerutility communities should be at least as high as the magnitude of decay in higherutility communities while enforcing that at least one lowutility community is receiving a higher utility after the transfer. In the absence of the network structure, the generalized transfer principle becomes equivalent to the original transfer principle.
We next study whether any of the welfare functions that satisfy the first four desiderata can also satisfy the generalized transfer principle. In particular in Proposition 1 we show that any additive and strictly concave welfare function satisfies the generalized transfer principle. Since the functions that satisfy the first three desiderata are strictly concave for (the inequity neutral and inequity averse behaviors), the transfer principle can also be simultaneously satisfied in this regime.
Proposition 1.
Any strictly concave and additive function satisfies the generalized transfer principle.
Proof are deferred to Appendix A. A key issue in fairness considerations in resource allocation problems is the gap in the utility of communities and in particular the gap between the communities with the highest and lowest utilities (which we simply refer to as utility gap henceforth). For a utility vector , we define to denote the utility gap. Imposing fairness interventions in allocation problems are usually motivated by the large utility gap before the intervention [33]. In the context of influence maximization, Fish et al. [17] has shown that in social networks the utility gap can further increase after an influence maximizing intervention. Thus, we introduce another desideratum to address this issue and again focus on neighboring allocations.
(6) Utility Gap Reduction:
Suppose and are two neighboring allocations. Let and denote the utility vectors from these allocations such that . If then . The utility gap reduction simply states that the welfare function should prefer the utility vector with the smaller utility gap if the total utility of the vector with the smaller gap is at least as large as the total utility of the other vector.
We next show that, in general, there is no hope to design a welfare function that obeys the utility gap reduction along with the other desiderata. However, it is possible to satisfy all the desiderata if the communities are disjoint (i.e., there is no edge between individuals from different communities).
Proposition 2.
Let be a welfare function that obeys desiderata 15. If the communities are disjoint then also obeys the utility gap reduction.
However, this result does not hold for general networks as we show next.
Proposition 3.
Let be a welfare function that satisfies desiderata 15. Then there exists an instance of the influence maximization problem and two neighboring allocations and with corresponding utility vectors and such that and but .
We showed that the network structure make it impossible to design a welfare function that satisfies all of the principles in the general case. This is perhaps consistent with the emerging theory demonstrating the fundamental inconsistency of different fairness desiderata for classification [25, 11, 18]. In Section 4 we empirically show that there exists welfare functions that obey all of these principles.
3.2 Fair Influence Maximization Through Welfare Maximization
A welfare function measures the desirability of an allocation. Hence as a natural notion of fairness we can require an allocation with the highest welfare as stated in the following optimization problem.
Lin and Bilmes [28] show that the composition of a nondecreasing concave function (in our case , for or for ) and a nondecreasing submodular function (in our case is submodular. Hence our proposed class of welfare functions is submodular. Our welfare functions also satisfy monotonicity. It is wellknown that for maximizing any monotone submodular function, there exists a greedy algorithm with a approximation factor [32] which we can also use to solve the welfare maximization problem.
Each choice of the inequity aversion parameter will result in a different welfare function and hence a fairness notion. For any such notion, the decision maker can easily impose a minimum threshold on the welfare function and study the tradeoff between welfare (fairness) and total utility by varying this minimum threshold. As we show in our experimental evaluations in Section 4, high welfare is usually achievable without sacrificing the total influence significantly in our realworld networks.
3.3 Connection to Existing Notions of Fairness
The welfare maximization framework allows for a spectrum of fairness notions as a function of . While maximin fairness cannot be cast in this framework, the specific case of leximin fairness^{1}^{1}1Among two utility vectors, leximin prefers the one where the worst utility is higher. If the worst utilities are equal, leximin repeats this process by comparing the second worst utilities and so on. corresponds to the welfare function for [22]. Proportional fairness [9], a notion for resource allocation problems, is also closely connected to welfare function for (see Section 1.1).
4 Computational Results
We investigate the effect of the inequity aversion parameter on both the utility gap and total influence and show that by varying we can effectively trade off utility gap with total influence. Furthermore, we compare our approach against proposed notions of fairness for influence maximization. Finally, we explore the effectiveness of the greedy algorithm in solving the welfare maximization problem.
Baselines
As baselines, we compare against diversity constraints (DC) and Maximin fairness. We follow the approach of Tsang et al. [39] for both problems and view these problem as a multiobjective submodular optimization with utility of each each community being a separate objective. They propose an algorithm and implementation with asymptotic approximation guarantee which we also utilize here.^{2}^{2}2https://github.com/bwilder0/fair_influmax_code_release We also compare our approach against the solution of the influence maximization problem with no fairness considerations (IM). Particularly, we compare solutions in terms of utility gap, total influence spread and Price of Fairness (PoF) defined as the percentage loss of the total influence due to fairness. Precisely, in which and are the the total influence in influence maximization problems with and without fairness. Hence PoF, with smaller values being more desirable. As we can not the solve influence maximization problem optimally, in the PoF calculations we utilize the standard greedy algorithm [24]. We use the same greedy algorithm for welfare maximization. We set across all experiments.
Synthetic Experiments
We start by demonstrating our setup. The synthetic networks are random instances of the Stochastic Block Model (SBM) network, a widely used model for networks with community structure [16]. In SBM networks, vertices are partitioned into disjoint communities. Within each community , an edge between two vertices is present independently with probability . Between any two vertices in communities and , an edge exists independently with probability and typically to capture homophily. SBM is suitable as it captures the community structure.
The first result is on SBM networks where we consider 3 communities each of size 60 where the communities differ in the their degree of connectedness. In particular, we set . We choose these values to reflect asymmetry in the structure of different communities with at least one community having sparse connections. This reflects real world scenarios since not every community is equally connected. Later, we explore the effect of different connectivity parameters. We also keep the betweencommunity edge probabilities the same and equal to 0.01. All synthetic results are averaged over 25 different realization of the SBM networks.
In Figure 1, we investigate the effects of the inequity aversion parameter on influence and utility gap. More specifically in the top left and right panels of Figure 1, for five budget values of 5, 10, 15, 20 and 25, we depict the utility gap and PoF for different levels of ranging from 5 to 0.9 and our baselines. According to the top left panel of Figure 1, has a relatively small utility gap and as we increase (lower levels of inequity aversion) the utility gap increases. At , which is close to being inequityneutral (), the utility gap is almost as high as DC. Finally, Maximin has the lowest utility gap which is explained by an extreme level of inequity aversion. Indeed, decreasing the utility gap comes at a cost. As shown in the top right panel, Maximin exhibits the highest PoF values. Also increasing generally results in a decrease in PoF. For example for , PoF is very close to zero. We note that the negative values are due to the fact that we used the solution to the greedy as a proxy for . This inherent tradeoff between utility gap and PoF is depicted in the top right panel in which each color corresponds to a different fairness approach plotted for budget values of 5 to 25 (lines connecting the points). According to this figure, for any given budget and a PoF tolerance level, one can choose an approach with minimum utility gap. In particular, if lower PoF is expected, the decision maker can choose, among different values of , one that minimizes the gap. Moreover, the bottom panel shows that the PoF decreases as grows. This captures the intuition that fairness becomes a less costly constraint when resources are in greater supply. We also observe that DC is dominated by the other methods both in terms of utility gap and PoF. When deciding the lowerbound utility that every community should receive, DC does not take the betweencommunity edges into account and this will lead to lower quality solutions. Also, by definition DC provides more resources to communities that “will do better with the resources” so it does not show an aversion to inequity. Hence, it does not satisfy the utility gap reduction.
Next, we study how the relative degree of connectedness of communities affects these two measures. We again consider 3 communities of size 60 and fix to obtain three communities with low, mid and high relative connectivity. Betweencommunity probabilities are set to and to 25. We gradually increase from 0.0 to 0.1. Results are summarized in Figure 2 where each group of bars correspond to a different approach for different values of . The left panel shows that, for all approaches expect Maximin, by increasing the utility gap gradually decreases until it reaches its minimum level at around , after which it shows an increasing trend. This is due to structural changes in the network. More precisely, for we are in the lowmidhigh connectivity regime, where the third community receives the minimum utility. As a result, as increases it becomes more favorable to allocate more resources to the this community which in turn reduces the utility gap. For , the second community will be become the new worseoff community due its lowest connectivity. Hence, further increase in causes more separation in connectedness and we see the previous behavior in reverse. While the utility gap for Maximin remain unaffected due its extreme inequity aversion, it exhibits a similar shaped behavior in PoF (right panel). Unlike Maximin, across different values PoF shows a decreasing trend with .
Real World Experiments
The data set consists of six different social networks of homeless youth from a major city in US as described in detail in BarmanAdhikari et al. [3]. Each social network consists of 7 racial groups, namely, Black or African American, Latino or Hispanic, White, American Indian or Alaska Native, Asian, Native Hawaiian or Other Pacific Islander and Mixed race. Each individual belongs to a single racial group. We use these partitioning by race to define our communities. However, to avoid misinterpretation of the results, we combine racial groups with a population of the network size under the “Other” category. After this preprocessing step, each dataset will contain 3 to 5 communities. The size of these communities (in percentage), as well as the sizes of the networks and their edge densities are summarized in Table 1. We remark that the absent of a racial category in a given network is due to their small sizes and hence being merged into the “Other" category after preprocessing (e.g., Hispanic in network W2SPY.)
Network Name  of Vertices  of Edges  White  Black  Hispanic  Mixed Race  Other 

W1MFP  219  217  16.4  41.5  20.5  16.4  5.0 
W2MFP  243  214  16.8  36.6  21.8  22.2  2.4 
W3MFP  296  326  22.6  34.4  15.2  22.9  4.7 
W2SPY  133  225  55.6  10.5  –  22.5  11.3 
W3SPY  144  227  63.0  –  –  16.0  20.0 
W4SPY  124  111  54.0  16.1  –  14.5  15.3 
Table 2 provides a summary of the results averaged over all network instances and the numbers in bold highlight the best values (minimum utility gap and PoF) for each budget value and across different fairness approaches. IM typically has a large utility gap (up to 17.2% for which is significant because the total influence is only 28.40%). By imposing fairness we can reduce this gap. In fact, we observe that across different values of ranging from 5 to 0.5, there is a decreasing trend in utility gap, where for and with , we are able to decrease the utility gap by . Consistent with previous results on SBM networks, both Maximin and exhibit very low utility gaps, however, Maximin results in higher PoF. Furthermore, across the range of we observe a mild tradeoff between fairness and utility.
Measure (%)  Fairness Approach  

DC  Maximin  IM  
utility gap  5  4.8  7.5  8.5  9.7  11.4  8.2  3.5  12.5 
10  4.6  6.6  7.3  9.5  12.9  6.9  2.0  11.7  
15  3.6  5.2  5.9  8.9  13.5  7.6  2.4  15.3  
20  3.6  4.4  5.9  7.3  14.0  5.8  2.3  17.2  
25  2.6  3.5  4.6  5.8  13.2  7.0  2.0  16.6  
30  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  0.0 
10  6.6  3.8  2.8  0.7  0.1  14.3  13.1  0.0  
15  3.8  2.5  1.6  1.1  0.1  14.6  10.5  0.0  
20  4.6  3.8  2.9  2.0  1.0  13.9  10.5  0.0  
25  4.0  3.2  2.5  1.9  1.0  13.4  9.9  0.0  
30  3.9  3.4  2.9  2.3  1.8  12.7  10.8  0.0 
The results for each network separately (X axis) is shown in Figure 3 for a fixed budget . Figure 3 shows that that the tradeoffs can also be very networkdependent (compare e.g. W2SPY and W3MFP). This highlights the crucial need for a flexible framework that can be easily adjusted to meaningfully compare these tradeoffs.
Finally, while theoretical results indicate that in the network setting, no welfare function can satisfy all principles including utility gap reduction, we observe empirically, both in synthetic and real world settings, that we can systematically control the gap by choosing appropriately.
5 Discussion and Future Work
As the empirical evidence highlighting ethical side effects of algorithmic decisionmaking is growing in many different domains [2, 29, 41], the nascent field of algorithmic fairness has also seen witnessed a significant growth. It is wellestablished by this point that there is no universally agreedupon notion of fairness, as fairness concerns can vary from one domain to another [31, 5]. This need for different fairness notions can also be explained by theoretical studies that show that different fairness definitions are often in conflict with each other [25, 11, 18]. To this end, most of the literature on algorithmic fairness proposes different fairness notions motivated by different ethical concerns. A major drawback of this approach is the difficulty of comparing these methods against each other in a systematic manner. Instead of following this trend, we propose a unifying framework controlled by a single parameter that can be used by a decision maker to systematically compare different fairness measures which typically result in different (and possibly also problemdependent) tradeoffs (e.g., between utility and fairness violation). Our framework also accounts for the social network structure while designing fairness notions – a consideration that is mainly overlooked in the past. Given these two contributions, it is perceivable that our approach can be used in many of the public health interventions such as HIV or Tuberculous prevention that rely on social networks. This way, the decisions makers, such as social workers, can compare a menu of fairnessutility tradeoffs proposed by our approach and decide which one of these tradeoffs are more desirable for their domain without a need to understand the underlying mathematical details that are used in deriving these tradeoffs.
There are crucial considerations when deploying our system in practice. First, cardinal welfare is one particular way of formalizing fairness considerations. This by no means implies that other approaches for fairness e.g. equality enforcing interventions should be completely ignored specially in domains that pure equality is more desirable. Second, we have assumed that the decision maker has the full knowledge of the network structure as well as possibly protected attributes of the individuals which can be used to define communities. Third, while our experimental evaluation is based on utilizing a greedy algorithm to solve our optimization problem, it is conceivable that this greedy approximation can create complications by imposing undesirable biases that we have not accounted for. Intuitively (and as we have seen in our experiments) the extreme of inequity aversion () can be used as a proxy for pure equality. However, the second and third concerns require more care and we leave the study of such questions as interesting future work directions.
Acknowledgement
We would like to thank David Gray Grant, Matthew Joseph, Andrew Perrault and Bryan Wilder for helpful discussions about this work. This work was supported in part by the Smart & Connected Communities program of the National Science Foundation under NSF award No. 1831770 and by the US Army Research Office under grant number W911NF1710445.
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Appendix A Omitted Details from Section 3
Proof of Proposition 1.
Let be a strictly concave function. Let and denote the utility vectors from allocations and . Suppose we sort and permute entries of so that, for all , index in both vectors corresponds to the same community. Furthermore, assume , and for some . Clearly and satisfy the assumptions of the transfer principle. We need to show that or .
The proof is by induction. We iteratively sweep the vectors and from the smallest index to the largest and show that for any value , with inequality becoming strict for at least one . To do so we repeatedly use a property of strictly concave functions known as decreasing marginal returns. According to this property for and
More specifically, in our inductive step, we keep track of a “decrement budget” which we denote by . Intuitively if we can show that with budget for some , we can then use the decreasing marginal return property along with the assumption that is sorted to show that as long as it is the case that . After each round we update the and move on to the next element in the utility vectors.
Formally, let to start at the begining of this inductive process. After visiting the th community, we simply update by . By the assumption of the transfer principle is nonnegative at all points of this iterative process and is strictly positive at some point during the process. Observe that since by the assumption of the transfer principle. We can use this as the base case. Since is sorted, given that is nonnegative, the fact that is strictly concave (so that the decreasing marginal return property can be used) immediately implies that at any iteration of the process. The inequality becomes strict for some given the assumption of the transfer principle. This proves the claim. ∎
Proof of Proposition 2.
Let and denote two neighboring allocations with corresponding utility vectors and . Let denote any of the two utility vectors such that . Without loss of generality, we assume is sorted in ascending order of the utilities and is permuted so that index in both and corresponds to the same group. This is because we assume that satisfies the symmetry principle due to which by permuting a utility vector the value of the welfare function does not change. Let and denote the communities whose utilities are changed between and , i.e., we assume and are the two communities where taking resources from and giving them to will transfer into .
To satisfy the condition of the utility gap reduction principle, it should be the case that (i.e., we transfer resources from the group with higher utility to a group with lower utility), otherwise after the transfer from to the utility gap could not get smaller (i.e., in which case the utility gap reduction is not applicable).
Assuming , if , again the assumption of the utility gap reduction principle is not satisfied, hence the principle is not applicable and there is no need to study this case. Therefore, we further assume . We would like to show in this case a welfare function that satisfies all the 5 other desiderata witnesses .
By assumption . From this, it follows that:
(1)  
(2) 
Proof of Proposition 3.
Figure 4 is an illustration of the graph that is used in the proof to witness the statement. We set (deterministic spread) and number of initial seeds . Consider two allocations and . Let denote the allocation that consists of the center of all star structures that consist of a single community. Let denote the allocation that is identical to with the sole difference that only the center of one of the all square structures is chosen and the last seed is selected to be the center of the star structure that is the mix of circle and diamond communities. Clearly these two allocations are neighboring. The average utilities for these allocations are in and in , respectively. Both allocations correspond to a total utility of 180 but the utility gap is for as opposed to the utility gap of for . So a welfare function that obeys the utility gap reduction should prefer over .
We now show that no welfare function that satisfies the first 5 desiderata will prefer over . Recall that such welfare functions include for , for and for . We verify this claim numerically. In particular Figure 5 plots the difference of for when (left panel) and when (right panel). This difference is always negative so is preferred by these welfare functions. For .
We point out that the instance used in the proof (graph structure, probability of spread and the number of seeds) used in this proof is designed with the sole purpose of simplifying the calculations of the utilities. It is possible to modify this instance to more complicated and realistic instances.
∎
Next, we study equalitybased notions of fairness for influence maximization [38, 1]. In particular, we show through an example that the fair allocation violates the monotonicity principle (a condition known as levelling down in political philosophy [34]).
Equality Enforcing Influence Maximization and Levelling Down
We start by introducing the most commonly used equalitybased notion of fairness known as demographic parity or statistical parity.
Demographic Parity
Demographic parity requires that the utility of each community to be roughly the same. This notion formalizes the legal doctrine of disparate impact [42]. Let . Then the demographic parity implies the following constraint on the influence maximization problem.
We use to denote the above set of constraints.
The parameter in the definition captures the amount of relaxation that is permitted by the demographic parity from strict equality and larger values correspond to more relaxation (equivalently, less strictness). We next show that demographic parity does not satisfy monotonicity.
Proposition 4.
For all , there exists an instance of the influence maximization problem such that for any fair allocation satisfying demographic parity, there exists an allocation that Pareto dominates . Mathematically, this can be written as:
Proof.
Consider a graph as shown in Figure 6 consisting of two communities, square and circle, each of size . We choose an arbitrary to reflect the arbitrary strictness of a decision maker. Let , and be large enough, i.e., . The optimal solution of the influence maximization problem chooses the center of the star and any disconnected square vertex. In the optimal solution, the utility of circle and square communities are and , respectively and the utility gap exceeds (so this solution does not satisfy demographic parity constraints). By imposing demographic parity, any fair solution is to choose one vertex from the periphery of the circle community and one from the isolated square vertices. The utilities of circle and square are and , respectively. Given the range of , the utility gap is less than . It is easy to verify the utility of both communities have degraded so the fair solution does not satisfy the monotonicity.
We point out that the graph used in the proof is directed. This is for ease of exposition. It is possible to create a more complex example with an undirected graph. ∎