Fairness-Aware Dense Subgraph Discovery
Authors: Emmanouil Kariotakis
Nicholas D Sidiropoulos
Aritra Konar
Venue: Accepted by TMLR ()

Abstract: Dense subgraph discovery (DSD) is a key graph mining primitive with myriad applications including finding densely connected communities which are diverse in their vertex composition. In such a context, it is desirable to extract a dense subgraph that provides fair representation of the diverse subgroups that constitute the vertex set while incurring a small loss in terms of subgraph density. Existing methods for promoting fairness in DSD have important limitations - the associated formulations are NP-hard in the worst case and they do not provide flexible notions of fairness, making it non-trivial to analyze the inherent trade-off between density and fairness. In this paper, we introduce two tractable formulations for fair DSD, each offering a different notion of fairness. Our methods provide a structured and flexible approach to incorporate fairness, accommodating varying fairness levels. We introduce the fairness-induced relative loss in subgraph density as a price of fairness measure to quantify the associated trade-off. We are the first to study such a notion in the context of detecting fair dense subgraphs. Extensive experiments on real-world datasets demonstrate that our methods not only match but frequently outperform existing solutions, sometimes incurring even less than half the subgraph density loss compared to prior art, while achieving the target fairness levels. Importantly, they excel in scenarios that previous methods fail to adequately handle, i.e., those with extreme subgroup imbalances, highlighting their effectiveness in extracting fair and dense solutions.

Published in Transactions on Machine Learning Research (04/2025)
Bairness-Aware Dense Subgraph Discovery
Emmanouil Kariotakis
ESAT-STADIUS
KU Lewuen
Nicholas D. Sidiropoulos
nukos@@viriinia edu
Department of Electrical Engineering
University of Virginia
Aritra Konar
aritra www.kkknrruuullleee........
ESAT-STADIUS
KU Leuven
Reviewed OpenReview: https: https:://ppenreviau net/ forum? 7d=77g
openreview.
Abstract
Dense subgraph discovery (DSD) is a key graph mining primitive with myriad applications including finding densely connected communities which are diverse in their vertex composition. In such a context, it is desirable extract dense subgraph that provides fair represcntation of the diverse subgroups that constitute the vertex while incurring a small loss in terms of subgraph density. Existing methods for promoting fairness in DSD have important limitations the associated formulations are NP hard in the worst case and they do not provide flexible notions of fairness, making it non-trivial to analyze the inherent trade-off between density and fairness. In this paper, we introduce two tractable formulations for fair DSD, each offering a different notion of fairness. Our methods provide structured and flexible approach to incorporate fairness, accommodating varying fairness levels. We introduce the fairness- induced relative loss in subgraph density as price of fairness measure to quantify the associated trade-off. We the first to study such notion in the context of detecting fair dense subgraphs. Extensive experiments on real-world datasets demonstrate that our hods not only match but frequently out existing solutions, sometimes incurring even less than half the subgraph density loss compared to prior art, while achieving the target fairness levels. Importantly, they excel in scenarios that previous methods fail to adequately handle, i.e, those with rxtreme subgroup imbalances, highlighting their effectiveness in extracting fair and dense solutions.
Without Fairness Constraints
With Faimess Constraints
Figure A toy example of Fair DSG with 2 vertex groups; protected (red) and unprotected (blue). Left: Densest subgraph without fairness constraints; density 2. Right: Densest subgraph with fairness constraints (equal number of red and blue vertices); density 1.875.

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Published in Transactions on Machine Learning Research (04/2025)
1 Introduction
Ensuring that data-driven algorithms do not disproportionately impact different population subgroups is key challenge in human-centered applications of artificial intelligence. The field of algorithmic fairness (Dwork al.] 2012 Feldman et al., 2015 Kleinberg al. 2018 Holstein et al. 2019) focuses on designing lairness-enhancingg mechanisms to mitigate algorithric bias. Fairness considerations have been formulated as optimization problems in both supervised and unsupervised learning (see the surveys Friedler et al. (2019); Chouldechova & Roth (2020) and references therein). In contrast, it remains a nascent area of research in graph mining, where fairness considerations are also important (Dong et al. 2023). To date, only few graph mining problems have been analyzed under the lens of algorithmic fairness.
Dense subgraph discovery (DSD) is one key graph mining primitive that aims to extract subgraphs with high internal connectivity from given graph (sce the survey Lanciano et al. (2024) and references therein). DSD finds widespread applications ranging from mining social media trends (Angel et al. 2012), detecting patterns in gene annotation graphs (Khuller & Saha, 2009), and spotting fraud in e-commerce and financial networks
Hooi ot al. 2016 Li ct al. 2020; Chen & Tsourakakis 2022 A popular formulation used for extracting dense subgraphs maximizes the average induced degree (Goldberg 1984). This is because the problem can
be solved exactly using maximum-flows, or approximated efficiently at scale using greedy vertex-peeling algorithms (Charikar 2000; Boob al. 2020 Chekuri al., 2022), or convex optimization algorithms (Harb)
et al. 2023 Nguyen & Enc 2024
Harb et al. 2022). with guarantees on the sub-optimality of the solution.
In this paper, we study the problem of extracting the densest subgraph from a given graph that meets a target fairness criterion. Such considerations can arise in DSD when the vertices of the graph (e.g,, corresponding to a web or social network) are annotated with sensitive information pertaining to an individual's gender panderr/race/reliiion, /political leaning, etc., which partitions the vertex set into different population subgroups. The fair dense subgraph problem then corresponds to locating a dense subgraph with adequate levels of representation of each subgroup in the extracted subset. Simply applying a pre-existing algorithm for DSD can fail in this regard. Indeed, studies real-world graphs (Anagnostopoulos et al. 2020 2024 have revealed that existing methods return subgraphs which typically exhibit strong homophily, with little to no diversity in the composition of the vertex attributes. Motivating examples of fairness in DSD include selection of diverse teams from a social network of professional contacts (Marcolino et al.) 2013 Rangapuram et al. 2013), and recommending balanced content to social media users that spans the various views of the individuals comprising the network, thus mitigating polarization (Musco et al., 2018).
While DSD is well-studied topic, extracting fair dense subgraphs has received limited attention. A Although the recent works of Anagnostopoulos et al. (2020: 2024); Miyauchi et al. (2023) have initiated progress in this arca, they have some inherent limitations. Notably, their formulations are NP-hard in the worst-case, necessitating approximation algorithms. These methods also struggle with rigid or hard-to-set fairness criteria. Another key issue is the inherent trade-off between subgraph density and fairness, which is difficult to analyze and has not been formally explored due to the NP-hardness of existing approaches (see Section 2 for details).
Contributions: In this paper, we focus on the setting where the vertices of the graph are divided in two subgroups, protected and unprotected. Our objective is to overcome the limitations of prior work by (i) proposing new, tractable (polynomial-time solvable) formulations which are (ii) capable of generating a spectrum of fair dense subgraphs with varying levels of fairness. This approach enables users to analyze the trade-off between subgraph density and fair representation of the protected group, and quantify the price of fairness, i.c., the density loss required to meet a target fairness level. We point out that these features are absent in the formulations of AAnggnostopoulos el al. (2020 2024); Miyauchi et al. (2023). Our key technical idea is to promote fairness in DSD through a pair of judiciously chosen regularizers that create tractable problems and establish regularization paths representing various fairness levels.
To summarize:
We introduce two tractable formulations for fair DSD that are capable of accommodating variable fairness levels. This enables flexible selection across spectrum of target fairness levels, enhancing the applicability of the formulations.
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DDSP
PS
FPS
2-DFSG
Ours
0.5
0.4
0.1
0.0
|
0.52
0.50
0.48
PTBR
RU
Figure 2:
Comparing perfectly balanced fair subgraphs obtained via prior art and FADSG-I (ours, purple) on different TWITCH datasets. (Top): Price of fairness (the lower, the better). (Bottom): Fraction of protected vertices in induced subgraph (set to 50% for perfect balance).
We analyze the inherent trade-off between subgraph density and target fairness for a difficult example using the price of fairness measure. Our results indicate that enhancing fairness can significantly reduce density, regardless of the algorithm used.
Through extensive experiments on diverse datasets, we demonstrate superior performance (see Figure and practical utility of our formulations over existing approaches.
Preliminarics and Notation: Consider a simple, undirected graph g (V,E) E) on == DV vertices and m :- [8] edges. Let w E - 1 denole a set of positive weights defined on the edges of G; i.e., each edge {i.j} E is assigned a positive weight W(is))- If each edge has unit weight, the graph 9 is said to be unweighted. Let denote the n x n symmetric adjacency matrix of g with entries ai = W(i.s), V {i,j] E & and e be the n-dimensional vector of all-ones. Given vertex subset S C V, let Gs (S,Es) denote the subgraph induced by S. The (weighted) sum of edges in 9s is denoted e(S) subgraph density of 9s its average (induced) degree, which expressod as p(S) == 2c(S)/[S). Given a vertex subset S _ V, let x E 10, denote its corresponding binary indicator vector. In terms of x, the subgraph density can be equivalently expressed as p(x) (x Ax)/(e x). Tn addition, each vertex v V possesses binary group label that designates it as belonging to the protected the vertices of G can be partitioned into a protected Sp C V, with np == [Spl. and an unprotected subset Sp = :>> Spp A real-valued set function f= 2V - IR defined on a ground set V is supermodular if and only if (iff) f(A) + f(B) & f(AUB)+ f(AnB) ((ANB) for all A, 1,5 B V. Equivalently, f is supermodular iff f(ulA) I f(u(B), for all A B V\ [w}, where [((]S) = (SS (u)) - J(S) denotes the marginal value of v with respect to submodular iff ff supermodular, A function is modular if it is both supermodular and submodular. Relevant to this paper, the function e(S) is monotone supermodular, whereas |S| is modular.
Enwrees (((s))- The
unprotected group. Thus,
S. f(0) = 0, f is said be normalized. A function
2 Related Work
Densest subgraph and it variants: Given a graph G, the classic DENSEST SUBGRAPII (DSG) problem Goldberg 1984) aims to extract dense vertex subset S C V that maximizes the average (induced) degree. DSG can be solved exactly in polynomial time via maximum flows, but this is computationally demanding for large graphs. In practice, scalable greedy algorithm that itcratively peels vertices is used, providing a 1/2-factor approximation (Asahiro el al. 2000 Charikar 2000). The algorithm was generalized in the recent work of Boob et al. (2000), which proposed a multi-stage greedy algorithm GREEDY++ that exhibits fast convergence a ncar-optimal solution of DSG without relying on flows. The follow-up work of Chekuri et al. (2022) provided performance analysis of GREEDY++, and demonstrated that can attain (1 e)- approximate solutions of DSG in O(1/e2) iterations. Building this approach, general peeling-based
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framework SUPER-GREEDY'S was introduced by Chekuri et al. (2022) for obtaining (1 -)-aprroximaee solutions for the broader problem class of computing densest supermodular subsets, which can be expressed as (aX)cv(P(S)/[S], where F(S) is a monotone, normalized, supermodular function. Note that DSG is a special case, with F(S) = e(S). More generally, it was shown by Harb et al. (2023) that SREER- exhibits convergence to an optimal dense decomposition vector of J, which corresponds to a partition of of V into a finite collection of subsets St,...,ou, Sk. such that each for each i E "......,,, S, is the unique maximal subset that maximizes A (F(S, -I - P) over V Mi,i,,, where X;- =- Uj The value of λ corresponds to density of the densest supermodular subset, whereas subsequent values monotonically decrease, A > .......... > AR- For DSG, we refer to such a decomposition a dense decomposition on a graph G.
A separate line of research (Danisch et al. 2017 Harb et al. 2022 Nguyen & Ene 2024) for developing fast approximation algorithms for DSG has blossomed around applying suvex-optimmzzaion algorithms for solving a tight linear programming (LP) relaxation of DSG originally proposed by Charikar (2000). The work of Danisch et al. 2017) considered quadratic programming problem derived from the dual of the LP relaxation of Charikar [2000]; and applied the Frank- Wolfe algorithm (Frank el. al. 1956) for solving it. Recently, the authors of Harb et al. (2022) developed faster algorithms for DSG and dense decompositions
graphs by solving the dual LP relaxation via accelerated proximal gradient (Beck & Teboulle, 2009). This was later improved upon by Nguyen & Ene (2024) using tools from area convexity |Sherman (017)) and accelerated
random coordinate descent Enee & Nguyen (0055) to develop faster approximation algorithms for DSG and dense decompositions on graphs respectively.
An important variant of DSG is finding the k-core decomposition of a graph Seidman [1883] A k-core is the maximal subgraph (by size) where every vertex has an induced degree of at least k. The largest value of l for which such a subgraph exists is known as the degeneracy, and the corresponding subgraph is known as the max-core. Extracting the max-core is tantamount to finding the vertex subset which maximizes the minimam induced degree. A slight modification of the greedy peeling algorithm for DSG exactly solves this problem.
Recently, it has been shown (Veldt et al., 2021 that the solution of DSG and the k-core both correspond to special cases of a more general framework known as the GENERALIZED MEAN DENSEST SUBGRAPH (GMDSG) problem. This framework is parameterized by a single parameter p. based on computing generalized means of degroe sequences of subgraph, with DSG and k-core corresponding to the choices of p and p == -o0 respectively. Hence, by varying p, one can extract a family of dense subgraphs which obey different nolions of density. For p 2 1, GMDSG can be solved optimally in polynomial-tine via maximum flows, or approximated via a generalized greedy peeling algorithm. Additionally, it was shown by Chandrasekaran et al. (2024) that for < 1, GMDSG is NP-hard for p in the range p E (-1/8, 0) U 104), and that the solutions of DSG and max-core provide a 1 1/2aapprxximatio for GMDSG when p < 1.
Other variants of DSG consider imposing size restrictions on the extracted subgraph. The DENSEST-K- SUBGRAPH (D&S) problem (Feige et al., (001) seeks to determine the edge densest subgraph of size k which is NP-hard and also notoriously difficult to approximate (Bhaskara et al. 2012 Manurangsi 2017). Notwithstanding this fact, practical algorithms which work well for this problem on real graphs include Papailiopoulos et al. (2014); Konar & Sidiropoulos (2021). Another variant, DamkS (Andersen & Chellapilla 2009), maximizes average degree subject to the size of the subgraph being at most k and is NP hard. Mcanwhile, the DalkS problem (Andersen & Chellapilla| (2009) maximizes average degree subject to the subgraph size being al least k. DalkS is also NP hard (Khuller & Saha, 2009), with greedy and linear programming approximation algorithms known.
Fairness in DSD: Subgraphs extracted via DSG or its variants are not guaranteed to fulfill a pro-specified fairness criterion (regardless of the algorithmic approach employed), as the respective formulations do not explicitly take into account any protected attributes that the vertices may possess. This issue was recently brought up by Anaguostoppolls et al. (2020) 2024); ; Miyauchi et al. (2023), which considered the task of extracting dense subgraphs that satisfy a group-fairnes fairness criterion. In Anagnostopoulos et al. (2020) 2024), , each vertex is member of either a protected or an unprotected group, and the goal is to extract the densest. subgraph with an equal number of verticos from both groups (perfect balance). The follow-up work of Miyauchi let al. (2023) proposed a pair of fairness promoting formulations which can handle multiple vertex groups. The first (Myyauchi et al. 2023. Problem 1) corresponds to an extension of Anagnostopoulos et al. (2020)
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2024) that aims to find the densest subgraph subject to the constraint that the number of vertices from each group in the solution does not exceed a fixed fraction of the solution size. The second formulation (Miyauchi et al. 2023 Problem 2) concerns finding the densest subgraph while ensuring that at least a pre-specified number of vertices from each group are present in the solution. A separate work (Oettershagen et al. 2024 considered a variation of the fair DSG problem where the edge relationships (as opposed to vertices) in a graph are modcled as belonging to different groups. The authors of (ettershagen et al. (2024) proposed formulations for extracting fair dense subgraphs containing exactly, at most, and at least certain number of edges from each edge type.
While serving as reasonable means of promoting fairness in DSD, the aforementioned approaches have important limitations. (i): Fairness is enforced using hard combinatorial constraints, which renders the problem formulations of Amaguostoopouls et al. (220 )024); Miyauchi et al. (2023) NP-hart in the worst case. Mcanwhile, the decision versions of the problems considered in Oettershagen et al. (2024) are demonstrated to be No-coppee Since optimal solution cannot be guaranteed in polynomial-time for every problem instance, it is a non-trivial lask to analyze the trade-off between density and fairness made by these formulations on real-world datasets. (ii): The approximation algorithms developed in Anagnostopoulos et al. (2020) 2024);
Miyauchi et al. (2023) do not tackle the problem directly, but adopt a two-pronged approach. In the first part, which corresponds to a relaxation step, the fairness constraints are decoupled from the problem formulation and a dense subgraph is extracted. This is followed by a "rounding" step that takes the dense subgraph as imput and aims to make the solution feasible with respect to the fairness constraints. However, such a strategy, which employed in Anagnostopoolls et al. (2020) 2024) and Miyauchi et al. (2023) Problem 1), can cause large drop in practical performance. Although the spectral relaxation algorithms proposed in Anagnostopoulos ct al. [200 2024 provide sub-optimality guarantees, these apply under the assumption that the degree distribution of the underlying graph is near uniform; a condition that rarely holds in real-world graphs which typically exhibit a skewed-degree distribution (Newman 2003). In practice a heuristic procedure adjusts the output of DSG by ad-hoc addition of vertices from the least represented group until perfect balance is reached. Meanwhile, Miyauchi et al. (2023) compute an approximation to DalkS to obtain an initial solution for Problem without imposing fairness, which is then refined via post-processing step that adds or removes vertices until the target fairness constraints are met. Regarding the second formulation of Miyauchi et al.] [2023 Problem 2), linear programming relaxation with quality guarantees is developed, a albeit the algorithm exhibits high complexity. A low complexity greedy algorithm is proposed as an alternative, which offers quality guarantees under very specific conditions on the optimal solution of the problem, which are difficult to verify a priori. On the other hand, Oettershagen et al. (2024) provides a constant-factor approximation algorithm for their fairness problem, but these guarantees apply to graphs which are everywhere sparse restrictive assumption. (iii): The fairness criteria employed in Anagnostopoulos et al. (2020) 2024); MMyauchi et al. 2023) can be restrictive, or difficult to properly tune in order to achieve a target representation level. The case of perfect balance is the sole consideration of [faannossooooslss et al. (2020) 2024), whereas Miyauchi et all 20233 Problem 1) only guarantee that the representation level of each subgroup is no more than a fixed fraction of the total vertices in the solution. However, it is not possible to specify a priori which subgroup attains maximum representation or whether this upper bound is attained at all. This implies that even in the simplest setting of two subgroups (i.e., protected and umprotected), the method may not be effective in guaranteeing a target level of representation of the protected group. The second formulation Miyauchi et al. (2023 Problem 2) ensures only that at least a certain number of vertices from the protected group are included in the solution, and cannol guarantee allaining a specific proportion of protected vertices, which is the focus of our work. Meanwhile, (Dettershagen et al. (024)) quantifies fairness w.r.t. edges via edge-type constraints, which is not directly comparable with our work as we mcasure fairness w...t. vertices.
3 Problem Statement
Recall the classic DENSEST SUBGRAPH (DSG) problem (Goldberg) 1984). Given an undirected graph G, DSG aims to extract a dense vertex subset S Ç V that maximizes the average (induced) degree. The problem can be expressed as
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Definition 1 (DSG).
x argmax p(x): xe[0,1]
where p(x) := (x7 (xTAx) Ax) denotes the subgraph density of a candidate subgraph x E (0, 1]" in terms of average induced-degree.
In contrast to prior work on fairness in DSD where fairness requirements are explicitly enforced as constraints, adopt regularization based approach, where group-fairness in the extracted subgraph is promoted by an appropriately chosen regularizer. In this section, we introduce our formulations.
Given a graph & with designated protected and unprotected subgroups of vertices, let e, E 10,1 denote the binary indicator vector of the protected subset S,. Define the regularizer r (x() = x)/x)//² x and consider the first FAIRNESS-AWAR DENSEST SUBGRAPH (FADSG-I) problem. Definition 2 (FADSG-I).
{ ==xx,))==(x)) p(x) x +x))))))))
(A) argmax
(2)
xE(0,1)
Here, 2 the regularization paramcter. The objective function g(x, A) consists of two terms the first term p(x) corresponds subgraph density, whereas for À 0, (x) promotes the inclusion of vertices from the protected subset Sp. This can be seen by equivalently expressing r-)-) the set function
() |8 S [S]
(3)
It is evident. that 0 S (8) S 1, with the upper bound being attained for any subset S C Sp. Meanwhile, (i( 0 iff S does not contain any protected vertices. Thus, larger values of r-(-) correspond to vertex subsets containing a greater fraction of protected vertices. Note that (S = 1/2 corresponds to the case of perfect balance (Anagnostopoulos et al. 2020 2024), with equal number of protected and unprotected vertices in S. Moreover, the choice of ri (S() = =np/n implies that the number vertices in the solution is proportional to the number of vertices in the population, a property known as demographic pority (Dwork et al. 2012). By varying \; we allow r-(-) to span a range of values, which affords flexibility in extracting dense subgraphs exhibiting varying fairness levels.
When À 0, FADSG-1 reduces to DSG, the solution of which is not guaranteed to contain any protected vertices. As À is increased, greater emphasis is placed on maximizing r1 (x(),, enhancing the inclusion of protected vertices. For À + 00, rı (xx)))) 1, and the solution of problem 22) corresponds to the densest subset of Sp. Clearly, by varying À in the interval [0, oo), the solution of 2 spans a spectrum which lies between these two extremes. By appropriately selecting X, desired representation levels of the protected subset can be achieved. Hence, in contrast to Anagnostoooolos al. (2020) 2024);; Miyauchi al. (2023), group-fairness formulation (2) affords much greater flexibility in trading-off subgraph density in exchange for protected subset representation.
It is worth pointing out that in certain scenarios, the solution of FADSG-1 may not provide satisfactory levels of representation of the protected subset. Although the regularizer r.(+) promotes the inclusion of protected vertices in the solution of (2). it does not explicitly take into account the fraction of such vertices that are represented. Even for (infinitely) large values of X (2), the maximal level of representation attainable corresponds the densest subset of Sp. However, this does not imply that the fraction of the vertices of Sp that will be included in the solution of FADSG-I will be large (please see Table 7 Appendix H.3 for experimental results indicating this). These considerations motivale more focused formulation for incorporating protected vertices in the solution, which we designate as the FADSG-II problem.
Definition 3 (FADSG-1I).
x' x²()) argmax h(x, p(x) A 2-12(x)
(4)
x0(0,1)
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In this case, (x(x) == (|xx - e x is regularizer and À 2 0 is a regularization parameter. The above problem differs from FADSG-T in the choice of the regularizer. Note that the numerator of r2(+) can be viewed as the Hamming distance between the sets S and Sp, since ||x - e 1 -iev<xx(x) - ei)) corresponds to the number of disagrcements between S (with indicator vector x) and Sp (with indicator vector Op). Hence, for given value of À > O, the objective function of 4 aims to maximize subgraph density while penalizing the distance between the extracted and the protected subgraphs. As À -> 20, r2(x" - and the solution x (A) coincides with the entire protected subset itself. Building on this observation, consider the case where the target fairness requirement stipulates that 50% of the vertices in the protected subset be included in S.
We have the following result. (see Appendix A.1 for proof).
Lemma 4. (2( = - + sns,I Sp1
= 1
2'
More generally, although problems (1) and (3) promote different notions of fairness, it is interesting to note that the two formulations are not completely unrelated. In particular, we have shown that the following relationship holds (see Appendix A.2 for the proof).
Lemma 5. For the regularizers Ti (S) and r2(S), the following inequalities hold: ri( + r2(S) 2 1 and 25)(5) + (2) & 1+ Rp
2
The implication of the above result is that given the value of one regularizer, we can attain bounds on the other (see Figure g Appendix A.2 for a visualization). Since providing explicit representation of the protected subset is absent in the fairness formulations of Anagnostopoolls et al. (2020 FADSG-II goes beyond what is attainable using the prior art.
2024); Miyauchi et al. (0023
Miyauchi et al. 2023 is that they are Na-hard in the worst-casc. Since optimality cannot be guaranteed, this implics that the solution obtained by their approximation algorithms may not be satisfactory in terms of the loss in density incurred for a target fairness level. In contrast, it turns out that problems (2) and (4) can always be solved exactly in polynomial-tine irrespective of X (see Appendix B for details). This implies that by varying A, we can trace the optemal boundary between the two components comprising the objective function (i.e., density and fairness) of both problems 2 and 4 This enables us to reveal the underlying trade-off between density and fairness on real-world datasets. As mentioned before, the intrinsic hardness of the formulations
Tractability: A limitation of the formulations of Anagnostopouuls et al. 2020 2024):
in Anagnostopoulos et al. 22020 2024
Miyauchi et al. (2023) does not allow for such feature.
Our formulations are explicitly designed to study the density-fairness trade-off in a disciplined way for the foundational setting of two vertex groups. As we will see later, they can outperform existing two-group (protected, umprotected) approaches by significant margin. The extension of our approaches to the case of multiple vertex groups is deferred to future work.
4 Selecting the Regularization Parameter A
A key component of problems (2 and (4h is the choice of the À parameter. Clearly, increasing the value of À promotes the inclusion of protected vertices in the subgraph extracted by FADSG-I and II, albeit at the expense of subgraph density. This section examines how À can be selected in our formulations to guarantce that the solutions obtained correspond to the densest subgraph that attains a target fairness level.
The first step is to determine the start and end points of the regularization paths of problems 2 and 4.. which determine the intervals characterizing all possible solutions generated by each formulation. While == O corresponds to the start of the regularization path for both problems (note that this is tantamount to solving DSG in both cases), determining the end is non-trivial task in general. This stems from the fact that both problems are combinatorial in nature. In the context of FADSG-I, such that for all A 2 Amax: the solution x' (A) is the densest subset of the protected subset S,- Let S S S, denote the densest protected subset and p(S}) denote its density. While it is difficult to pin down the exact analytical form of Amax; the following result can be derived (see Appendix C.1 for proof).
wish to determine Amax
Proposition 6. For X 2 Amax [maxsss, (p((S) p(8)))/(1 -)))), the solution of (2) is the densest protected subsel S7.
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The above result shows that there exists a finite value of À that corresponds to the end of the regularization path of FADSG-I. However, it is difficult to further simplify it. Meanwhile, for FADSG-IT, the analogous definition of Amax corresponds to the value such that for all À 2 Amax: the solution x x*) is the protected subsct Sp. For the end of its regularization path, a finite À can be shown to exist, similarly to FADSG-I (sec Appendix C.2 J. In practice, is computationally easier to find a suitable Amax (which we know exists owing to Propositions G and 10 Appendix C via trial and error (sce Appendix [ Table @ for values used). This enables us to bound the interval of the regularization path for a given dataset, which in turn facilitates the selection of the regularization parameter that corresponds to target fairness level, as explained next.
4.1 Selection of Target Fairness Level
FADSG-I: For x = 0, let (0) denote the solution of problem (2), ie., DSG. We also denote 4 == ri rx 0 as the fraction of protected vertices present in the solution of DSG. This corresponds to the level of fairness (as measured by ri))) that is attainable by the solution of the non-fair DSG problem, and is not guaranteed to be non-zero in general. Meanwhile, we know that x*(Amax) corresponds to the densest protected subset for which == ri (x' (Amax))= =1 Let a E [1,,"]] denote the desired fairness level; i.e., we wish to select. À such that the solution of xA), is the densest subgraph that satisfies rı (xxx (A)) = a. To this end, we will exploit the fact that the function ri ri(xx()) (A)) is non-decreasing in A. Consequently, we define the function
4 (A) = rx((x(A)) a.
we can employ bisection on À to solve the equation v2(A) = 0, in order to meet the target fairness level a E u1). The lower and upper limits of the bisection interval correspond to 0 and Xmax respectively, for which (0) = 4 -a a S 0 and ti (Amax) = 1 - a 2 0. Moreover, note that each choice of À E [0, Amax] corresponds to solving an instance of (2), which, as pointed out before, is tractable.
FADSG-II: A similar technique can be applied to select X in problem [ as well. Tet x" x*()) denote the
solution of problem 4 which again corresponds to DSG, and define U2 == r2(x* (0)) > 0. In this case, x" xa(maa) ) corresponds to the entire protected subset, for which l2 == r2 (Amax) ) = 0. Since r2(x*(X) is
non non-increasing in X, given a desired value of r2(-) defined as å E [12, u2], we can solve the equation
va2 = r==r2xx (A)) - =0
(6)
via bisection on X. The initial bisection interval is [0, Amax), with (0) = 122 5 2 0 and the (Amax) -5 < 0. For cach X E [0, Amax], an instance of problem has to be solved, which is also tractable.
Unlike FADSG-I, in this case it is more difficult to set 8 to a pre-determined value as it does not admit a trraight-forward interpretation of corresponding to a particular target fairness level. An important exception is the case where = 1, which is equivalent to the requirement that 50% of the protected subset is represented in the solution (see Lemma 4. Thus, the bisection-search strategy can be applied to determine the value of X that will extract the densest subgraph that contains 50% of the protected vertices.
A subtle point is that for the bisection framework to provably terminate, the functions e'z and (A) should be continuous for every target fairness level, and for every graph. Although we cannot show this at present, we observed in our experiments that depending on the nature of the underlying Pareto-optimal frontier, some target fairness levels may be more difficult to attain than others. However, in general, the overall procedure works very well on real datasets.
5 Quantifying the Price of Fairness
FADSG-I and II trade off subgraph density in order to promote the inclusion of protected vertices. Con- sequently, the density of the fair subgraphs obtained via (2) and { is no larger than the density of their non-fair counterpart DSG; i.c., a loss in subgraph density the price paid for ensuring fairness. The loss incurred in attaining a target fairness level o, either a for FADSG-I or 5 for FADSG-II, can be quantified using the price of fairness.
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For given graph 9, the price of fairness (PoF) for fixed value of Φ is
POF Φ) == PG PG(() = PG(0)
Pg
Pg
where PG(6) is the density of the densest subgraph in G attaining target fairness level é, and P is the density of the solution of DSG. Simply stated, PoF Φ) is the ratio of the loss in subgraph density suffered in meeting the fairness requirement & and the maximum density Pg in the absence of fairness considerations. Let do denote the fairness level of the solution of DSG (obtained with explicitly imposing fairness), then P (00) = P3. and thus PoF do) = As & diverges from do, the resulting density (G(() decreases, and consequently PoF o) approaches 1. For a desired value of Φ, the closer PoF Φ) to 1, the greater is the price paid (in terms of density) for extracting the desired fair dense subgraph.
Similar definitions have been considered for analyzing fairness in influence maximization (Taang et al. 2019 and graph covering (Rahmattalabi et al.) 2019)) To the best of our knowledge, however, we are the first to study such a notion in the context of delecting fair dense subgraphs. Analyzing the price of fairness is inherently complex due to multiple factors like the type of graph considered, the protected subgraph, and the desired level fairness.
The "Lollipop" Graph: We base our study on a difficult example which exemplifies the underlying tension between density and fairness. Consider the "Jollipop" graph (demoted as Ln). In this construction, the unprotected vertices constitute a clique of size /n (C (Cym), while the protected vertices form path graph on n - vn vertices (Pn-va) with one end connected to Cym. (so that the overall graph is connected). We point out that the "lollipop" graph mimics the properties of real-world graphs in the following manner: (i) the graph Cn has O(n) edges, and is hence sparse, (ii) the majority of vertices (n vn to be precise) have low degree (S 2), while the remaining vertices have high degroe (at least vn-1) - 1). Intuitively, we expect the solution of DSG for the "lollipop" graph to be the clique Cym, as it corresponds to region of high density. Since this solution does not contain any protected vertices, from a fairness perspective, it is the worst possible solution. In addition, as the protected subset Pn-ya is minimally connected, and thus has low density, we expect that improving the representation of protected vertices will cost large decrease in density, reflected by a large price of fairness.
Figure 3: The "lollipop" graph for 16. The un- protected set (blue) forms clique on 4 vertices whereas the protected set (red) forms a path graph on 12 vertices.
Let analyse how the PoF is affected by altering the balance between the majority and minority vertices in the desired solution. Consider the case where the desired proportion of unprotected and protected vertices in the extracted subgraph is 1 y, with y E 21,2.... n vn), and the desired balance is a, == (1 + +). Note that the case of y = 1 corresponds to the perfect balance scenario considered in Anaguostopoulos et al. 2020 2024). Then, the PoF can be expressed as follows (see Appendix D.1 for proof).
FADSG-I:
Proposition 7.
0gy
vn-1
Cry
E
2: ... 1
vm
an)
(8)
ag E {
[va)
vn-I
Vn+i
12 Vn+l
Since the larget fairness level a, increases with 7, the above result reveals that the PoF increases with 7, as expected. Perhaps surprisingly, even when ay is large but (strictly) lesser than 1, it is still possible that the PoF is at least 1 - (n), which can again be made arbitrarily close to 1. For smaller values of ay, we have that lim PoF a.) at.. In this case, the PoF is at most the desired fairness level aty.
FADSG-II: Regarding FADSG-II, we can show that for the lollipop graph, there exists a one-to-one correspondence between the target fairness parameters FADSG-II and a of FADSG-I such that solutions of the two problems coincide (refer Appendix D.2 for proof). Hence the analysis remains the same as that of FADSG-I.
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6 Algorithmic Approach
Although FADSG-I and 1I can be optimally solved using maximum flows, we refrain from using such an approach. The main reason is that for target fairness level, we have to solve multiple instances of cach problem within binary search framework to determine the appropriate regularization parameter A. Since solving each maximum flow problem incurs Q(n²) complexity (even for sparse graphs), the overall procedure for computing an exact solution can be computationally prohibitive. Thus, we adopt an approximation approach that seeks to quickly compute high quality sub-optimal solutions for a given instance of FADSG-I and II (i.e., for a fixed 2). The key observation that both of these problers correspond to special cases of a general problem, the DENSEST SUPERMODULAR SUBSET (DSS) problem (Chekuri et al., 2022). Given supermodular function F 2² - IR, the problem seeks subset that maximizes the ratio F(S) / For this problem class, several algorithmic strategies are available. One option is to employ SUPER- GREEDY+
[Chekuri et al. 2022), a multi-stage greedy algorithm which generalizes the prior-art of [Asahiro et al.
20000;
Boob et al. 2 Veldt et al. 2 If F()) is normalized, non-negative, and monotone, for a given e E (0, 1), the algorithm outputs an (1 (1-cl e)-approximation of the optimal objective value of DSS in O (1/e² iterations. The peeling process dominates the computational complexity of each iteration and it can be implemented to run in O(m + n log n.), by handling the vertices of using a Fibonacci heap (Fredman & Tarjan 1987) with key values being &, + FoosS - v), similar to Miyauchi et al. (2023). Hence, SUPER-GREEDY ++ can run in O(mm + m)ggn)/e performance (Boob et al. 2020 Nguyen & Enc. 2024). Additionally, it is straightforward to implement it. using the objective functions of FADSG-I and II (for more details, see [Algorithm
Its main appeal is that it can be efficiently implemented, and exhibits strong empirical
Appendix
An alternative is to consider the iterative convex-optimizaate
algorithms developed in Danisch et al. 2017 ;
Harb et al. (2022); Nguyen & Enc (2024). These methods are centered around solving the dual LP relaxation of DSG (Charikar 2000). However, modifying and implementing these algorithms (which are tailored for DSG) for our problems poses a greater challenge relative to SUPER-GREEDY+ + FADSG-T and IT have to be reformulated as constrained convex optimization problem via extension of the dual linear programming (LP) relaxation of DSG proposed in Charikar (2000). Moreover, since the algorithms of Danisch et al. 2017)) Harb et al.] [2 Nguyen & Ene (2024) are tailored to exploit the structure of the dual LP relaxation based on DSG, it is not clear to what extent they would need to be modified in order to be applicable for our problems, and whether the requisite changes would still result in low-complexity updates. Auother practical drawback of these continuous methods (even when applied to plain DSG) is that a subroutine known as fractional peeling (Harb al. 2022) has to be applied on each of the generated iterates if one desires to extract estimate of the densest supermodular subset (which corresponds to the densest subgraph for DSG) at cach iteration. However, performing a single round of fractional peeling incurs a complexity of O(m + n log n), which has the same complexity order as a single iteration of SUPER- GREEDY++. Consequently, determining the point at which the estimate of the DSS has converged from the continuous itcrates can prove to be computationally expensive, unless the fractional peeling subroutine is sparingly used. Even then, this can come at the cost of executing the convex optimization algorithms for a larger number of iterations than required for the sequence of DSS estimates to converge. We point out that this is non-issue in CUPER-GREEDY+++; as its combinatorial nature makes it easy to keep track of the best estimate of the DSS at each iteration. Owing to these reasons, we elect to choose SUPER- + as the algorithmic primitive for FADSG-T and IT.
Note that the objective function of FADSG-1 can be expressed in the form of the DSS problem, with a non-negative, monotone supermodular numerator. Hence, the SUPER- GREEDY algorithm can be applied directly to problem (2) to obtain a high-quality approximate solution. For FADSG-II, its objective function can be expressod as well as the ratio F(S) where the numerator is supermodular, making problem (4) special case of DSS. However, since the numerator may not always be non-negative for every 0, SUPER- GREEDY++ does not guarantee an (1 appoooxxittooo in all cases (see Huang et al. (2024) for a counter-oxample. Nonetheless, SUPER-GREEDY can still be applied on problem (4) and our experiments indicate that it can serve as an effective heuristic. For details regarding implementation, refer to Appendix L
It is worth pointing out that although we employ approximation algorithms for our problem formulations, there is a difference compared to that of the prior art Anagnostopoolos et al. (2024) 2020); Miyanchi et al. (203).. More specifically, we employ approximation as a means of improving scalability, as opposed to dealing with intractability.
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Table 1: Summary of dataset statistics: number of vertices (n); number of edges (m); and size of protected subset (mp). The datasets contain multiple graphs and the statistics represent mean values with standard deviations. The full list of datasets can be found in Appendix F
DATASET
7
Up
AMAZON (10)
3699 + 2764
22859 + 18052
1927 = 2537
LASTFM (19)
7624
27806
424 1 454
TWITCH (6)
5686 1 2393
71519 1 45827
2523 1 1787
7 Experimental Evaluation
summarrziing their statistics (for more information, see Appendix The implementation of our approach is publicly available on Githul
Experimental Sctup: We consider various real-world datasets, with Table 1
The following baselines were selected as performance benchmarks:
2-DFSG (Anagnostopoulos et al. 2020: 2024).
The algorithm starts from the solution of DSG and subsequently introduces verlices from the underrepresented group in an arbilvary fashion, until perfect balance is reached.
2024). Paired Sweep (PS) and Fair Paired Sweep (FPS) are spectral algorithms. They categorize the elements based on their group and sort them in a manner reminiscent of spoctral clustcring.
PS, FPS Anagnostopoulos et al. 2020
DDSP (Miyauchi et al. 2023). The algorithm initially computes a constant-factor approximate solution to DalkS with k= [1/a], with a being the maximum desired fraction of monochromatic vertices. Then, the algorithm makes the solution feasible by adding arbilrary vertex from the least represented group, similar to 2-DFSG.
7.1 Evaluation of FADSG-I
Comparison with prior art: In order to compare FADSG-I with the baselines, we set the target fairness level a = 0.5. Therefore, our formulation aims to extract dense subgraph with perfect balance. We point out that we exclude datasets for which the protected vertices in the DSG solution are more than the unprotected, as they already meet and excoed the target a = 0.5. Figures 2 and g iilustrate a comparative analysis with prior art. The top sections showcase the price of fairness (PoF) for each algorithm across various dalasets, while the bottom sections depict the fraction of protected vertices in the solutions. As the results indicate, our proposed formulation consistently yields the lowest PoF values across a broad spectrum of scenarios ?
It can be observed from Figures 2 and that for certain dalasets, FADSG-I does not find a perfectly balanced subgraph (see Figure 4 AMAZON HPC & OP). To obtain a better understanding, we plot the regularization path of ri (S) for these datasets in Figure 5 A noteworthy observation is the emergence of "fairness plateaus" where the fairness level remains constant across intervals of A. Since FADSG-I is tractable, these figures correspond to the actual Pareto frontier which characterizes the trade-off between fairness and density. This phenomenon suggests that the graph's inherent structure leads certain fairness levels that arise naturally. For instances where the data's underlying structure does not. inherently support. perfect division (i.e., AMAZON HPC & OP), FADSG-I does not enforce it. Figure X depicts the outcomes in datasets where there exists an extreme imbalance in the composition of the protected and unprotected subgroups, i.c., we have np < n (see Table g Appendix D.. In such cases, DDSP consistently fails to produce balanced/ fair solutions. In fact, we observe an extreme gap to the desired fairness level of 0.5, as well as a PoF that is always worse than that of FADSG-I. The other prior methods always achieve perfect balance, but with a significantly higher PoF compared our algorithm. This is an important feature of FADSG-I, as it can provide a near optimal solution, when the nature of the graph allows it, while almost always paying the least in terms of PoF.
httpp:// htttps://githbb... http:///ggthuuuueeerrieeeeeeeeee wwtp///aaarttttii//taaeeeeeeeeee tm12
Note that, in experiments, order compute Pg. we employ the exact max flow algorithm
Goldberg 1984
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DDSP
FPS
2-DFSG
Ours
0.5
0.3
0.1
0.0
n Spl
0.5
0.3
B
HPC
OP
ACS
5O
TMI
Figure 4: Comparing induced perfectly balanced fair subsets of prior art and FADSG-1 (Ours), on different AMAZON datasets. Top: PoF. Bottom: Fraction of protected vertices in induced subsets, (S).
1,00
1,00
1.00
Ss Spl
0.75
ISA Spl
0.75
Is Syl
0.75
0.50
0.50
55 0.50
0.25
0.25
0.25
0.00
0.00
0.00
20
40
50
100
A
Figure 5: The fraction of protected vertices in the induced subset, ri (S), as function of X, for FADSG-I. Left: Middle: AMAZON OP Right: TwITCH PTBR.
AMAZON HPC
1.00
DDSP
PS
FPS
2-DFSG
Ours
0.75
PoF
0.50
0.00
1.00
ss Spl
0.75
0.50
0.25
0.00
5
10
11
12
13
100
Figure 6: Comparing induced perfectly balanced fair subsets of prior art and FADSG-I (Ours), on different. LASTEM datasets. Top: PoF. Bottom: Fraction of protected vertices in induced subsets, rS)
Various target fairness levels: Next, we investigate the ability of FADSG-I to extract subgraphs corresponding lo various fairness levels, contrary to PS, FPS, or 2-DFSG. Although DDSP allows adjusting group representation levels, it can't guarantee priori which group (i.e., protected or unprotected) will meet them. Returning to Figure it can be observed that the fairness levels are non-dccrcasing with À. Notably, this value can only surpass or equal that of the classic DSG solution (A = 0), aligning with our objective of enhancing fairness. The achievable fairness levels contingent. upon the underlying nature of the dataset,
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Twitch
Twitch
DE
ENGB
ES
PTBR
RU
0.9
0.9
I Spl
0.7
0.5
0.5
0.1
0.1
0.5
0.91.0
0.3
0.5
0.7
1.0
Target Fairness Level
Target Fairness Level
Figure 7: Fraction of protected vertices each induced subgraph, rı (S), of FADSG-I, for various values of target fairness level a, for the TWITCH datasots. (Dotted lines: fairness value in the solution of classic DSG that want to enhance.)
DSG
Ours
0.3
PoF
0.2
0.1
0.0
0.5
ss Sal
0.3
0.1
DE
ENGB
ES
FR
PTBR
RU
Figure 8: TWITCH datasets. Top: PoF of FADSG-II (Ours) for = Bottom: Comparing Ours (5 = 1)
with DSG. Fraction of the protected subset that belongs the solutions.
with datascts like TWITCH PTBR exhibiting smoother curves (Figure 5 degrees of fairness. In contrast, certain datasets, such AMAZON IPC & OP (Figure 5), display plateaus that correspond to specific fairness levels induced in the final solution. Figure 7 demonstrates the flexibility of FADSG-I in extracting fair dense subgraphs that attain varying target fairness levels a, which correspond to the fraction of protected vertices in the induced subgraph. Starting from the fairness lovel attained by the DSG solution, we can introduce more fairness to our solution by increasing the desired value of a.
Right), resulting in naturally varying
7.2 Evaluation of FADSG-II
Direct comparison with prior art not feasible here, since FADSG-IT utilizes a different. fairness criterion, as detailed in Section In Figure S. we compare this formulation with DSG. The top section depicts the PoF of the solution obtained from FADSG-II, and the bottom one illustrates the fraction of all of the protected vertices in the solutions of DSG and FADSG-IL. For the TWITCH datasets, DSG's solution includes only about 1/5 of the protected vertices. This is clcarly due to the fact that DSG docs not take into account the protected altributes of the vertices. On the other hand, FADSG-II for = 1, yields solutions with nearly half of the protected vertices, while achieving a relatively low PoF. Similar to FADSG-I, there are instances where the solution does not prociscly align with the desired fairness level. This can be attributed to the structure of each graph, which may not inherently exhibit the specific fraction we are aiming for (see Appendix H for more details).
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8 Conclusions
We addressed the fairness-aware densest subgraph problem by introducing two tractable formulations, each incorporating a different regularizer in the objective function, to provide two different notions of fairness. The inclusion of fairness introduces the inherent barness-density trade-off. In order to quantify this trade-off we proposed a mcasure of the price of fairness, and we analyzed based on an illuminating example- which revealed the underlying tension between density and fairness. To assess the effectiveness of the proposed solutions, we employed peeling-based approximation algorithm, and compared their performance against the prior art using real-world datasets. Our findings revealed that FADSG-I generally incurs lower price of fairness (PoF) in achieving perfect balanced subgraphs, sometimes even less than half of that of prior art. FADSG-I can also induce perfectly balanced subgraph in some extreme scenarios where existing approaches fall short, and it is capable of providing various fairness levels, other than perfect balance. Furthermore, FADSG-II promotes an alternative notion of inclusion of protected vertices, often achieving this with a relatively low PoF. Together, these formulations offer flexible solutions for balancing fairness and density in subgraph discovery. Extending our approach to handle multiple vertex groups using regularizers which are designed to ensure that no protected group is inadequately represented constitutes a direction of future work.
Broader Impact
The authors posit this work on the assumption that practitioners will appropriately select the protected subset and publicly disclose it along with the results of their analysis.
Acknowledgments
Supported in part by the National Science Foundation, USA under grant IIS- and the KU Leuven Special Research Fund BOF/STG-22-040.
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Jianmo NI, Jiacheng Li, and Julian McAuley. Justifying recommendations using distantly-labeled reviews and fine-grained aspects. In Proceedings of the 2019 conference on empirical methods in natural language processing and the 9th international joint conference on nalural language processing (EMNLP-IJCNLP) pp. 188-197, 2019.
Lutz Oettershagen, Honglian Wang, and Aristides Gionis. Finding densest subgraphs with edge-color con- straints. In Proceedings of the ACM Web Conference 2024, 936 947. ACM, 2024. ISBN 9798400701719.
James Orlin. Max flows in (nm) time, or better. In Proceedings of the forty-fitth annual ACM symposium Theory of computing, pp. 765-774, 2013.
Dimitris Papailiopoulos, loannis Mitliagkas, Alexandros Dimakis, and Constantine Caramanis. Finding dense subgraphs via low-rank bilinear optimization. International Conference on Machine Learning, pp- 1890-1898. PMLR, 2014.
Aida Rahmattalabi, Phebe Vayanos, Anthony Fulginiti, Eric Rice, Bryan Wilder, Amulya Yadav, and Milind Tambe. Exploring algorithmic fairness in robust graph covering problems. Advances in Neural Information Processing Systems, 32, 2019.
Syama Sundar Rangapuram, Thomas Bühler, and Matthias Hein. Towards realistic team formation in social networks based on densest subgraphs. In Proceedings of the 22nd international conference on World Wide Web, pp. 1077-1088, 2013.
Ryan Rossi and Nesreen Ahmed. The network data repository with interactive graph analytics and visualization. In Proceedings of the AAAT conference artificial intelligence, volume 29, 2015.
Benedek Rozemberezki and Rik Sarkar. Characteristic functions on graphs: Birds of a feather, from statistical descriptors parametric models. In Proceedings of the 29th ACM international conference on information g knowledge management, pp. 1325-1334, 2020.
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Stephen B Seidman. Network structure and minimum degree. Social networks, 5(3):269 267,27,7777 1983.
Jonah Sherman. Area-convexity, fxr regularization, and undirected multicommodity flow. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pp. 452 460, 2017.
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A Lemmas
A.1 Proof of Lemma 4
Proof. We express r2()) in set notation as
r2(S) [|] + Ss 2 n Spp
(9)
[S]
The claim then follows by noting that r2(S) = 1 + [Sp] - - n Spl 0.
A.2 Proof and Intuition of Lemma 5
Proof. Note that r2(+) can be expressed in terms of rı(+) as
follows
S2(S)
(2 1+ +
nhp
2rl
(10)
"p +2
(S) =1
For the first inequality: Since S 2 = ri ()) implies that
Syl [S] 2 (8)). Hence, obtain
R
(S 2
- 11-1ri (S).
(11)
2r) + (5))==
For the second inequality: We have that [s] S The solution has at least one edge. Hence, we obtain
since each
2
(8)
(8) + (r) - np 2
(12)
(1(S) (8)
Figure 9: The joint fairness region of the regularizers ri(S) and r2)),, for a graph with np protected vertices.
Furthermore, we define the joint faimesss region characterized by the above inequalities as the fol- lowing polytope:
R: R (m:((((()))225)) :0 1 r2(S) S1+ "p (S) + r2() 2 1,2r +r( < 11+ up (13)
2
(rr = 0A 1
Note that the point (0, is excluded from R, since if (S) 0 |Sn = and r2( = e s, = H1 = [Sp] = O, which is a contradiction since we assume that there always exists at least one protected vertex the graph. We can approximate this point when Un < n. In this case, when ri - 0, it can still hold that half of the protected vertices belong in the solution, i.e. r2(S) =1
Let consider the three remaining corner-points of R.
(Bottom right:) We can attain (1, when r2(S) = 0, meaning that the solution the protected set Sp. hence Tı (8( ssns,
1.
(Top left:) We can attain (0, =x=+ only when the solution contains just one edge that connects two unprotected vertices. This holds because when (S) 0 +> S, = 0 + r2(S) equal to the desired value only for |S| 2.
= ne + 1, which is 2
(Top right:) We can attain (1, ~د only when the solution contains just one odge that connects two protected vertices. This holds because when (S) + s =|S| = rz(S) "p 1, which is equal to the desired value only for [S]= = + |Sn Sp 2.
2
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B
Tractability details
First, consider problem 2 Define the functions Pi (x,, A) := x Ax + A e x, and == x. Thus, the objective function can be expressed as g A) = (xx monotone supermodular function and a non-negative modular function, the result is also non-negative, monotone supermodular (for every X 2 0). This implics that FADSG-I corresponds to maximizing the ratio of monolone, non-negative supermodular function and a modular function. The solution such a problem can be determined in polynomial-tima by solving finite number of supermodular maximization (SFM) sub-problems within a binary-search framework (Fujishige 2005 Bai et al. 2016). Since SFM incurs polynomial complexity (Grötschel et al.) 2012), the claim follows. Furthermore, the quadratic form of (ı(:,)) can be exploited to show that, each sub-problem is equivalent to solving a minimum-cut problem GGoldber 1984 Kolmogorov & Zabih 2002), which can be accomplished via maximum flow, as we show in Appendix B.1.1 Each maximum-flow problem incurs complexity (nm) (Orlin 2013) and an upper-bound the maximum number of binary-search steps required is Ollog(n + A) + log(n)) (see Appendix B.2
A Since (ı,,)) A) is the sum of a non-negative,
Next, consider problem and define the function pa A) == x Ax À (e- 2ep)) x- X Tp. Then, the objective function of FADSG-II can be expressed as the ratio h(x, A) = p2 (x)())(x)). While p2(-, A) is a supermodular function (being the sum of a supermodular and modular function), it is not guaranteed to be non-negative for every choice of A >> O. Additionally, it is not normalized either, since (2 = -X ny.
Nevertheless, we can still show that 4 can be solved via a sequence of maximum-flow problems using a small modification of the binary-search strategy, as we explain in Appendix B.1.2] Thus, FADSG-II can also be solved exactly in polynomial-time.
B.1 Exact solutions via max-flow
B.1.1 Applying max-flow on FADSG-I
Based on the interpretation of Lanciano et al. (2024 Algorithm 1) for Goldberg's max-flow based algorithm (Goldberg. 1984). we only need to make a small modification in order to apply it to FADSG-I.
The algorithm maintains estimate of the unknown optimal value by tracking upper and lower bounds, which it progressively tightens through binary search. To update them, the algorithm employs a max-flow (equiva- lently min-cut) computation. Let 3 2 0 be the midpoint of the bounds kept in the current iteration of the max-flow based algorithm. For = (V,E), Sp the protected set and A the regularization parameter of FADSG-I (con- stant), the algorithm constructs the following edge-weighted directed graph (U, wh) (Figure 10), with U ( (), A = A, AE UA Ay, where A. = ((s, v)le As {(u, w), (v, (u, v) E), AB (lu t)lu 8p}
degful
A,6 # (S))
dlgglu
27 3 (
and Ap- ((u, l)(c S and w/ A + R such that
Figure 10: An edge-weighted directed graph (4, wa) con- (14) structed from G, B and X.
deg(v)
e = (8, E As
1
EE A
(e)(e)
3 3-X X
E Ap
E Ae
An L cut of A,, ws) is partition of (X,3) of U, such that X and L y. Then define the cost of s cut ()), as cost(X )) := E(.N))AAEE vey wp(u,)).
Lemma 8. Let (X be an s cut of the edge-weighted directed graph (U, A, w/), and S X (s}. Then it holds that cost(X, y) 2m + B|S (S, X), where pi (S 2e(S) X/S n S.l.
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Proof. Any edge from X' to y is contained in exactly one of the following sets: {(s, E Aslo E VVSS
uu, E Aelu E S, E V \ 8), ((v11) E Aflv E S), ((v, 1) E A E SJ. Therefore the cost of (X, }) is
cost(22))) = Σ w)((,0)+ +
u)((, + (.. (a:))AeeESS
Σ
Σ w + Σ w)((,1)
(V.))EA
Σ deg(v) + ( (3- 3(S) s
(15)
VEVIS
2m 2e(5) + 5|5) s = 2m + +|S| 20(S) - N!l
= 2m + 3|5| - A))
Let (X)) be the minimum s t cut (or max-flow) of (U, A, w/). If X + {s), then 8 = [8} is vertex 18 2 3, and we can update the lower bound by 3. If X = {s), then there no S V that satisfies ( > B, and the upper bound can be replaced by B. Therefore, we can apply the same procedure as that of Lanciano et al. 2024 Algorithm 1), with a small change at the initial upper bound 3000 m.
subset that satisfies 8| Pi (S A) s 0 +> p1(S.A)
[s]
B.1.2 Applying max-flow on FADSG-I1
We follow the same procedure as we did in Appendix B.1.1 but with a slightly different edge-weighted graph.
For 9 = (V, Sp the protected set and À the regularization parameter of FADSG-II (constant), the algorithm constructs the following alge-weighten directed graph (44 AA, wzf",, with 4 = [u(s, A = A, A.
where A, ((s,))]o V), A {(u, v), (v, u)) fu, w) E). A {uu, E Sp} and A, ((u,t) E Sp)
and w !! AA- -> IR, such that
deg(v)
== (s, v) E
1
e E A
w/ w (e)
(16)
B-
e E Af
B+1
e Af
Lemma 9. Let (X,Y) be an s - cul of the edge-weightee directed graph (U, A, w and S = X [s}. Then
it holds that ))) = (2m - A n,) + 3|5| - p2(S, A), where A2(S, = 2e(S) - A(S + |S, - 2|5
Proof. Any edge from X' to y is contained in exactly one of the following sets: ((s, E A.lo E VV ((),
{(u,e) E Aelu E S E V \ S), {(e,t) E A e S), ((v, t) E ABlv E S}. Therefore the cost of (X' y) is
c)st(X()))=
Σ
w (s +
Σ (W
ub'(u,e) +
Σ (u,
(f,tt, Σ (v,
Σ deg(v) + [[(U,,)) U) EEI 8) + (8 -
(3 ( - -18 n Spl) (17)
"
2m + 3|S| - 2e(S) + (Slk 2A18 nS = 2m + S| (p2(S, A) + ((S)))
(2m - X- 12) + 5| - AS)
Similarly to FADSG-I, let (2)3) be the minimum - t cut (or max-flow) of (U, w/¹). If X + (s), then S 8=xx- X { is a vertex subset that satisfies B|S| p2 A) < < Pa(S.A) bound by 8. X' = (s), then there is no S C that satisfies (²(
> B, and we can update the lower [S] > 8, and the upper bound can be replaced by B. Therefore, we can again apply the same procedure as that of Lanciano et al. (2024 Algorithm 1), with small change in the initial upper bound ((0) (m X hnp)/2 since X -p is constant.
[
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B.2 Upper bound on the number of binary search iterations for FADSG-I and II
Applying binary search lo solve problems (2) and (4p requires specifying an initial interval [Z. U], where UL
denote lower and upper bounds on the optimal value of (2) and ( respectively, with L < U. The number U-L with A being the difference of
of binary search itcrations needed to solve problems (2) and (4) is log A
any two distinct values of the objective function of FADSG-I and II. It is straight-fore forward to prove that
A 2 O(1/n²), similarly to Fazzone et. al. (2022 Lemma 1).
B.2.1 For FADSG-1
For FADSG-I, we have the following bounds:
min g(.xx)
min p(x) + A min ri(x) + L1 =0
(18)
(c(0.1)
xc(0.1) xE(O.1)"
and
(1,
p(x) +
rx(x) = U1 (n- 1) X.
(19)
XE {0.1)
x (0.1)
xE(0.1)
Thus, the number of iterations is
U1 La
1+ + 1/12
log
< log
log(n + A) + n² O(log(n A) + llggn).
(20)
A
B.2.2 For FADSG-II
For FADSG-II, we have the following bounds:
min h(),x) x)
min p(x) - max r2(x) L2 - xE(0,13" x=(0,1)
(21)
xE(0,1)"
and
max x(1,x)
max p(x) - min r2(x) = U2 n 1. xE(0.1)
(22)
XE (0.1))
xc (0.1)*
Thus, the number of iterations is
U2 L2 < log
n 1- log((1+ A)- n 1) logn? = O(log(1 + A) + logn).
log
(23)
1/n²
C Identifying the end of the regularization path
In order to determine the choice of À that corresponds to the end of the regularization path of FADSG-I and II, we can consider analyzing the optimality conditions of problems ( and (4) Our reasoning behind this step is that for continuous, convex regularization problems, studying the Barush-KKuhnTTuckee (KKT) conditions of the problem (which are necessary and sufficient for optimality) can facilitate the selection the regularization parameter (see Bach et al. (2012)). However, while our formulations are tractable, they are combinatorial in nature, for which there is no analogue of the KKT conditions in general. We note that for the special case where the objective function being maximized is supermodular, the following characterivation of the optimality conditions is known.
Consider the problem maxAgy F(A), where F(A) is a supermodular function. Then, a set A is optimal if and only if
Fact 1: (Fujishige 2005)
and V VB2 A".
(24)
Hence, in order to certify that a candidate solution optimal, it suffices to check only all its subsets and supersets, instead of examining all subsets of V. The above property of supermodular functions can be viewed as statement to the effect that local optimality implies global optimality, akin convex functions in the continuous domain. Going forward, this property will prove useful.
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C.1 FADSG-1
Unfortunately, problem (2) is not supermodular, and hence (24) does not directly apply. To circumvent this limitation, we make the key observation that (2) is equivalent to soloing supermodular maximization problem of the following form.
Fact 2: Let Ux be the optimal value of a given instance of Then mazimizer of 2 also a mazimizer of the supermodular problem
{ F(())= Pi(; v² v
(25)
Furthermore, the optimal value of the above problem 0.
Consequently, the optimality condition (24 be applied for the above problem, which in turn will allow
to determine the smallest value of the regularization parameler À for which the solution of (2) is the densest. protected subgraph of Sp. Let S. S Sp denote the densest protected subset and p(S;) denote its density,
When the solution (2) equals Sy. the optimal value is ox = P(S)) + X, since r1())) =. We are now ready to prove Proposition (restated below for convenience).
Proposition 6 For X 2 Amax (max))s,((p(S) -0(0))) protected subsel Sp.
2g )1,
the solution of is the densest
Proof. Applying the optimality conditions of 24 problem
25 we obtain the condition
F(B) S ((S))
v B2 S.
(26)
Since F(S))= = 0, the condition simplifies to
Pi (B, A) 03 (BB < 8, S
+ 2B( |B S, ox |B] S0
(27)
2e(BB 13 s
S p(S)) X. [
B
Define p(B) BB to be the density of the subgraph induced by B. Then, we obtain
[ [1-
[B
p(B) ((S)) S A.
A 1BB2
(28)
B
We now distinguish between two categories of supersets Fither we have (i): Sp 2 B 2 Sp or (ii):
B Sp 2 Sp. If the first case is true, condition 28 reduces to
(B < (())), v Sp 2 2
(29)
which is always guaranteed to hold (irrespective of X) since St is the densest protected subset. This implics that we only need consider the second calegory of supersets. Then, 28 becomes
( ((S))
>
AB Sp-
(30)
Bns_
B
Since [Bn = =|S, = npp v BB Sp, we finally obtain
((B)
Ri p(S))
A max BDS,
(31)
B
Note that the optimality conditions 24 also require us to verify that the above choice of Amax satisfies
F(B) F(S,),
(32)
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Note that for B 0, the above condition is guaranteed to hold irrespective of A. For subsets B satisfying 0 C B C Sp, have 18 n Spl = |8|. Consequently,
28 becomes
p(B) p( S 20,00 B Sp.
(33)
which is again always satisfied irrespective of X, since p(S}) is the densest protected subset. The claim then follows.
C.2 FADSG-I1
We adopt the same line of reasoning as the previous subsection. Our starting point is the fact that given an instance of (4) with optimal value vx a maximizer is also a maximizer the following supermodular maximization problem
F( p2(, vx 9(5)
(34)
SEV
with optimal value equal to 0. This allows us to use the optimality conditions of the above problem (which are characterized by (24)) to determine the smallest value of X required so that the solution of (4) corresponds to the protected subset Sp. Note that in such a case we have o = p(Sp). We are now ready to prove Proposition 4.2.
Proposition 10. Define the quantities
(( p(Sp)
p(B) p(Sp)
max A
A2
max
B3SP
1 [8]
BCBCS,
B
Then, for X 2 max(An )2), the solution of
4
is the protected subset Sp.
Proof. Since F()) is supermodular, the optimality conditions 24 assert that
F(B) F(S))
(35)
Since F(S,) = 0, the condition becomes
P2 (B, A) 1x (B)) S 0, B S
2c(B) A Ar) & p(Sp),
(36)
[
* P(B) p(S S Ar2(B)
For superset B 3 Sp, we have TB( = 1 (n./ BB)) Thus we obtain the condition
p(B) - p(Sp)
A x max =
(37)
B3S,
"2
we also have the condition From 24h,
F(B) S ((S))
(38)
which can simplified as
2e(B) X [[8] + |Sp1 2 s S p(S,) BB Sp.
(39)
A little calculation reveals that when B 0. the above condition holds irrespective of 2 Hence, we restrict our attention to subsets of the form 0
Sp- In this case (39 becomes
((B) p(S,) < S
(40)
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Since r2( (np/1B)) 1, we finally obtain
p(B) p(Sp)
A M A2
max BCB
(41)
B
Combining the inequalities and
we obtain
À 2 max max (21;22)
(42)
from which the claim follows.
D Analyzing the Price of Fairness
Recall that the lollipop graph on n vertices (denoted as Ln) comprises an unprotected clique of size vn (denoted as C,m), while the protected vertices form a path graph on n v vertices (denoted as Pa Pn-va), with one end connected to Cya (so that the overall graph is connected).
In order to analyze the price of fairness (PoF) on Lu the following facts will be useful. Let Cs denote a clique h vertices with density p(Cx) = 1. Additionally, let Pk denote path graph on k vertices. A little calculation reveals that the density of the path graph is p(Px) == 2 (1 10--1p 1/k). Given two distinct vertex subsets X, Y with no overlap, let e(X')) V) denote the number of edges with one endpoint in X' and the other in y. Then, the number of edges induced by the combined vertex subset X' U y can be expressed as
c(XU)) = e(x) e()) + e(X:)).
(43)
Proposition 11. For 9, the densest subgraph of L. is the umprotected clique C.z-
Proof. Consider candidate subgraph comprising r S v umprotected vertices and s S n vn protected vertices. The solution can be expressed S C, UP,, with density ((S) given by
p(8) 2e(C, UPd)
2(e(Cr) + e(Pa) + e(C,;; 2/e(C,,) + t eP +1] r r+
T+8
r r
(44)
S max (2((C,) ((e(P.) +1]
max m (p(C,,, = max(r - 1,2).
The first inequality stems from the fact that P.C S 1, whereas the second inequality is due to the following fact: given positive numbers a, b, c, we have (a + c)/(b dd N max(a/b, e/d).. For r > 3, the maximum is attained by the first term (irrespective of s) note that a necessary condition is n > 9. Hence we obtain
p(S) 15 -n-
(45)
for any candidate subgraph S. Both inequalities are satisfied with equality when S = Cr with vn, which establishes the desired claim.
]
We now turn towards formalizing this intuition. From Proposition 11 it follows that Pč, vn 1, and
4 = 0. Hence, the admissible range of a is the unit interval [0, 1] (which is the largest possible range). First, consider the case where the target fairness level induced by the protected vertices F This case corresponds to the opposile extreme of DSG (where no fairness is imposed). It turns out that the PoF can be expressed as follows.
which corresponds to extracting the subgraph Pa-va
Proposition 12. PoF(C., > 12/(Vn- 1)].
Proof. For n > 9, have established that Pcc - 1. Since p(Pa-va) 2, it follows from the definition that PoF (Cn, 1>>1>> > 22(2/ 1)]
Note that by construction, the densest protected subset coincides with the entire protected subset in this case.
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The above result is pessimistic as it demonstrates that in the worst-case, the price paid in extracting the fair component can be made arbitrarily close to 1, via a suitable choice of n. Additionally, it demonstrates that the lollipop graph is indeed a suitable candidate for analyzing the worst-case trade-off between density and fairness. Next, we present the proof of Proposition 7 in order to understand how the PoF is affected by altering the balance between the majority and minority vertices in the desired solution.
D.1 Proof of Proposition 7
Recall that the desired proportion of umprotected and protected vertices in the extracted subgraph is 1 1 where 7 E 21, ... n vn) is a positive integer. In this case, the target fairness level can be expressod as the ratio Cty := 7/(1 + 7). We aim to show that
aa
aty
E
2 ,1
1 v
POF(G,
(46)
2
1
1
+
Oy vn +
(vn)
vn
n - v +
for which a, lies in the range 1 Given a value y lying in the above range, the optimal fair densest subgraph comprises vn unprotected vertices of C.z and Tyn protected vertices of the path graph Pn-va- The optimal density is
Proof. Consider the range y E (L....VV--- - 1},
Pcn (an) 2e(C vn UP PPy
(vn-1+27
(47)
(1+7)/n
(1+?)
Hence, the PoF is
PoF 0n)) =1 Pcn (an)
2
1
(48)
Pc.
Vn
Next, we consider in the range {vT. .... n vn} for which ay ranges from {1 va+1
v
In this range, the optimal fair densest subgraph comprises the n v vertices of the protected subgraph Pa-va and r < vn unprotected vertices from C,m such that r satisfies = (n - vn)/7- The density of the solution is given by
Pč. (a,) = 2c(C, UPn-va)
(r (r-1 1+ (n - 7(++7)
vn(Vū - + (1+7)
27
(49)
(1r
(1
+1+
Hence, the PoF in this case can be expressed as
PoF = -
Vn + (1 (1+7)(v-1))
2--1 -
>1
n1 72
+
2
v
+
2
+1+)
-1
Vn
[vn)2
vn - (50)
= =1-0(1/Vm
D.2 Equivalence of 8 and
As we have already explained in Section 5 the solution of DSG without any fairness considerations is the clique C.m that contains the umprotected vertices. For this solution, it holds that r2(S) attains its maximum value, which is r2(S)= = Vn. As we decrease the target fairness value ó of FADSG-II from this upper bound 6, = vñ, the solution will gradually inchude more and more protected vertices from the protected subgraph Pa- Pa-vas until we reach the point that we have the whole graph 9, for which point it holds that r2(S) = / Up to this point, it always holds that we have a certain fraction (say 1/x < 1) of the protected vertices in the solution, i.c., [SNS,] [Sp]
<11 e SSS ne n-va This implics that
r (S) = s
v
Vn-1
Vn + a-va + rr = vn+ (x 1)²
a
(51)
[S]
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Moreover, it follows that
r2(S) = 1 + "p
25)(S) =1 -14 Up - 2 Sp
= 1 + pp 2
_ 2)
=1
[
|S|
|||
[S]
vn + x-1 Up
vn + --1 (n vn)
+ z-1 (v -1) 1)
(x 1) n2 x +1
Vn + x
vn + (n (1)
+ // n2 -
x + v n1
(52)
(x 1l
+ (2( =
Vn + (x 1)
(1 + (2( (1-)) Vn)
=
r2(S) vn
which implies that
(S) (n + - (19))) +1yn
(53)
71 1
If we further decrease the target fairness value 6, the solution will contain all of the protected verticos and it is going to start removing the unprotected ones. Hence, it will always hold that 18nS,1 This results in
1 + |S S = Rp.
[.
r2(S)= = + Up
|s Spl |8| |S|
|5|= "p -2/8
nhp
(54)
6-1
1 (S)
and
Sp
(S)
+ (8) -2rr
(55)
[S]
ut I-)a(8)
Finally, we get the following mapping of 5 to a:
Ny if 8 > 1 with SSS = <<1,
then
a (x+
-パ-
(8--)_
(corresponds to the 1st branch of Eq. 46
n-1
if <
Vn
with ShS Rp
1 <> Sp
=1p
then
a 6.
[corresponds to the 2nd branch of Eq. 46
E +
is a efficient multi-stage greedy algorithm designed to retrieve high-quality solutions for problems referred to as DENSEST SUPERMODULAR SUBSET (DSS), i.e., maximization of F(S)/(S), for F:2V- TR. This algorithm can extract. - )>aapproximate solutions, for e (0, 1) under some conditions O(L/e)) iterations, the following result of Chekuri al. (2022) indicates.
Theorem 13. Let F 2 R>o be normalized, non-negative, and monotone supermodular function over the ground set V, with DV. Let e E (0. Let A = max,ev F(u(V) and OPT the marimam of FSS)/88) over all S C V. For T 2 O A-na SUPER-GREEDYY (PP?>))
oulpuls a (1 -)-pproorimaae solution to DSS.
Pseudo-code of the algorithm is provided in (Algorithm 1 In order to implement this algorithm for FADSG-T and 11, one has to derive the expressions of F(v|S - v), which are provided in Section E.1 For programming simplicity, our implementation at this point does not include heaps. Instead, we adopt a direct approach in which each time we recompute the values of F(w|S - v) and extract the minimum, while updating vertex degrees after peeling. Therefore, the computational complexity per iteration of our implementation is O(n²), resulting in an overall complexity of O
E.1 Analytical Expressions of F(u - v).
As defined in Section D f(](S)= == ( Thus, F(v|S- v) F(S) F(S u), where {s-u) o} is the set S without the vertex Furthermore, with degs(v), we denote the degree of v in the induced set S.
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Algorithm 1 SUPER- ++
Input: F:2²= + R2o: T 2+
Output: Stensees
Sdenseet f- V
for all E V, set 6 =0
for i: 1 + T do
St, +
for j + do
E argmin,
11-1) F(v - v)
(( - +f(e | - v²)
+ F
St,41 + (Si, -Sij- -2"
if F(St,)) > P(Sttmenes) then
18,,
Sdensest + St,5
end if
end for end for
FADSG-L: In FADSG-I, the objective function is g(x, A) (x Ax X ee,,, (e (T)).
x). For given X, can
set F(x) = x Ax + A ee, x = pi (x,,)) and express it terms of S C V, to get F(S) = 2c(S) + A/S n Spl-
We also have that:
((8-1)= 2e(S) 2dogs (v) + (v)+
X|SnS, -1), f
(56)
2e(SS) 2degs (v) + 1pS
if sns,
Thus:
F(y -) -) F(S -b)=
2degs (v) + A, EveSnn, 2degs(U),
(57)
if $ SnS,
FADSG-II: In FADSG-II, the objective function is h(x, A) (x Ax A/Ix (p x). If we expand the norm of the numerator we get
|x 1xc1 = ||x|2 + 1lepp12 20, x = |S| + S Spl
(58)
For given X, can set F(x) x Ax A||x ep1 and express it in terms of S V, to get. F(S)
2e(S) A([S) |spl 2 Spl). We also have that:
2e(S) 2degs (v) X([S) 1+ 111| - 8 - 1)), if f
F(S - v)
20(S) 2degs (v) - X([S) - -1 -51-22Snn 2(5
if fugSns,
(59)
2c(S) 2degs (V) A([S) + Spp -2 - fue
20(S) 2degs (v) X(SS) + Spl - 2/Sn +\, i
Thus:
F(u S - v) F(S) - F(S - v) = 2degs(v)+ (w)+ iuussns, 2degs(u) - X, fugSns,
(60)
F Dataset details.
A summary of the statistics of all the datasets used in experiments be found in Table 2
Political Books Dataset (PolBooks): The vertices of the network are books on US politics included in the Amazon catalog, with an edge comnecting two books if they are frequently co-purchased by the same buyer Each book is categorized based on its political stance, with possible labels including liberal, neutral, and conservative. In our experiments, we focused solely on the subgraph formed by liberal and conservative
httpss//vebsites.. unich. edu/ jmjja//ttaaaa/
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Table Full summary of dataset statistics: number of vertices (n); number of edges (m), and size of protected subsct (Mp).
DATASET
7
m
PP
PoLBooKs
92
374
43
POLBLOGS
1222
16717
586
AMAZON (10)
3699 + 2764
22859 + 18052
1927 1 2537
DEEZER
28281
92752
12538
LASTFM (19)
7624
27806
424 1: 454
GITHUR
37700
289003
9739
TWITCH (6)
5686 + 2393
71519 + 45827
2523 + 1787
TWITTER
18470
48053
11355
books, resulting in 92 vertices (43 associated with conservative and 49 with liberal worldview) connected
by a total of 374 edges, curated in Anagnostopoulos et al. (2020 2024
Political Blogs Dataset (PolBlogs) [Adamic & Glance 2005 The vertices of the network are weblogs on US politics, with edges representing hyperlinks. Each blog is categorized by its political stance, left or right, with the left ones composing the protected set.
Amazon Products Metadata (Amazon) (Ni et al. 2019: The vertices represent Amazon products, and the edges denote frequent co-purchasing product pairs. The attribute associated with each product is their category. In our experiments used 10 different such datasets (see Table 3 assembled in Anagnostopoulos et al. 22200 2024). We name them based on the initials of the categories.
Table 3: Summary of Amazon Product Metadata and Twicch Users Social Networks datasets statistics: the number of vertices (nt), the number of edges (m), and size of protected subset (np).
DATASET
m
Thp
AMAZON
3699 1 2764
22859 1 18052
1927 1: 2537
AMAZON B
230
592
83
AMAZON
1809
14099
230
AMAZON IS
2009
8341
1495
AMAZON HPC
3011
10004
1352
AMAZON OP
2281
16542
162
AMAZON AH
10378
53679
8163
AMAZON ACS
5056
33827
987
AMAZON
6435
56553
5376
AMAZON SO
3216
19331
604
AMAZON TMI
2565
15619
520
TWITCH
5686 + 2393
71519 + 45827
2523 + 1787
TWITCH DE
9498
153138
5742
TWITCH ENGB
7126
35324
3888
TWITCH ES
4648
59382
1360
TWITCH
65449
112666
2415
TWITCH PTBR
1912
31299
661
TWITCH
4385
37304
1075
The vertices represent users of LastEM from Asian countries, and the edges are mutual follower connections among them. The attribute associated with each user their location.
LastFM Asia Social Network (LastFM) (Rozemberczki & Sarkar 2020):
For the LastFM Asia Social Network datasct, we choose each time different group be the protected one, and denote the resulting dataset. LASTFM-I, with being the index of the set that is considered
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protected. The number of vertices of each protected set can be found in Table 4 We also consider the LASTFM-100 werw for which we combine the vertices of all locations with less than 100 vertices and consider them as the protected set.
Table 4: Number of protected vertices Rp for each of the LASTF M-1 datasets.
6
7
8
9
10
11
12
13
14
15
16
17
100
Pp
1098
54
73
515
16
391
655
82
468
58
1303
138
57
63
570
257
254
1572
387
The vertices represent users of Deezer from European countries, and the edges are mutual follower connections among them. The attribute associated with each user their gender.
Deezer Europe Social Network (Deezer) (Rozemberezki Sarkar 2020
The vertices represent developers in GitHub who have starred at least 10 repositories, and the edges are mutual follower relationships between them. The attribute associated with each user is whether they are web or machine learning developers.
GitHub Developers (GitHub) (Rozemberezki al. 2019
Twitch Users Social Networks (Twitch) (Rozemberezki et al. 2019): The vertices represent users of Twitch that stream in a certain language, and the edges are mutual friendships between them. The attribute associated with each user if the user is using mature language.
Twitter Retweet Political Network (Twitter) (Rossi & Ahmed 2015 Conover et al 2011 The vertices are Twitter users, and the edges represent whether the users have retweeted each other. The attribute associated with each user is their political orientation (Tsiuutiioulkkis et al. 2021
G Details of Experiments
In order to find the appropriate À value for the desired target fairness level that we had for each experiment, we performed bisection on À values. The lower bound was always 0 and the upper was Amax- The value of Amax was computed experimentally for each dataset. We specifically chose the value of Xmax based on our observation that led to the densest protected subgraph for FADSG-I and the entire protected subgraph for FADSG-II. Moreover, increasing Amax beyond this point did not alter the desired outcomes. The specific values employed for each dataset and formulation are detailed in Table 6 Furthermore, this bisection was X, e 10-9. Thus, the number of iterations for each bisection experiment was bounded by [log2 أmma} ]. Furthermore, in recent work of Fazzone et al. (2022), a stopping criterion for the number of iterations in the SUPER-GREDDY ++ algorithm is being introduced, ensuring a certain level of solution's quality. However, we observed that this criterion tends to be conservative for our purposes. Through experimentation with various datasets, we determined that a value of = 5 iterations provided satisfactory results across all our datasets in our implementation of SUPER-GREEDY Table shows an example of the running time of cach algorithm, on AMAZON TMI datasct.
performed with tolerance
single machine, with Intel 17-13700K CPU @ 3.4GHz and 128GB of main memory rumning Python 3.12.0. The code of the baselines were obtained via personal communication with the respective authors.
All experiments were performed
Table Comparison of running time of cach algorithm for AMAZON TMI datasct. Using heaps will drastically reduce the complexity of our algorithm by O(n).
DDSP
PS
FPS
2-DFSG FADSG-I (Ouns)
TIME (SEC) 0.135372 0.371171 0.167832 10.079753
918
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Table 6: Amax values for each dataset and formulation.
DATASET
FADSG-I FADSG-II
DATASET
FADSG-I FADSG-II
AMAZON
20
PoLBooKs
1.2
AMAZON PS
18
50
POLBLOGS
100
100
AMAZON 18
35
150
DEEZER
40
125
AMAZON HPC
35
70
LASTFM (ALL)
40
50
AMAZON OP
50
50
GITHUB
100
200
AMAZON
40
200
TWITCH (ALL)
100
200
AMAZON AGS
40
50
TWITTER
150
200
AMAZON
40
140
AMAZON SO
40
50
AMAZON TMM
H Additional Experimental Results
In this section, we offer further insights by presenting additional results obtained from various datascts, aiming to reinforce the validity of findings.
H.1 FADSG-I:
In Figure 1 comparison with prior art illustrated, for the remaining datasels. Once more, we exclusively showcase datasets where the DSG solution exhibits fairness level lower than the desired one of 0.5.
DDSP
FPS
2-DFSG
0.8
0.6
0.4
0.2
0.0
0.55
ssn Spl
0.50
0.45
PolBooks
PolBlogs
Deezer
GitHub
Figure 11: Comparing induced perfectly balanced fair subsets of prior art and FADSG-I (Ours), on PoLBooks, BLOGS, DEEZER and GITHUB datasots. Top: PoF. Bottom:
Fraction of protected vertices in induced subscts.
Figures 12 and 14 illustrate the ficxibility of our definition of fairness level a. They reveal that certain desired fairness levels may be unattainable. As explained in Section 7.1 this limitation arises from the inherent structure of the graph, which does not naturally accommodate certain fairness levels. This can be clearly observed in Figure 13 where the right subfigure exhibits notable discontinuity just before the curve reaches the value of 0.6, jumping to a value greater than 0.8. This is translated in the left subfigure, to an attained fairness level slightly greater than 0.8, despite the desired level being 0.6.
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amazon_
Js
amazon_
amozon
amazon_g g
amazon
amazon amazon
emazon_5o
1.0
0.9
0.9
0.7
0.7
| Sp)
Syl
0.5
0.5
0.3
0.0
0.0
0.0
0.1
0.3
0.5
0.9
1.0
0.0
0.1
0.3
0.7 0.9
Figure 12: Fraction of protected vertices in each induced subgraph of FADSG-I, for various values target fairness level a, for the AMAZON datasets. The dotted lines represent the fairness value in the solution of classic DSG, which want to enhance.
Piice of Falmess (PoF)
0.8
0.8
0.6
1.0
Target Faimess Level
Target Faimess Level a
Figure 13: POIBBLOGS dataset FADSG-L. Left: Fraction of protected vertices in each induced subgraph, for various values of target fairness level Middle: PoF for various values of target fairness level a. The dotted lines represent the fairness value in the solution DSG. Right: Fraction of protected vertices each induced subgraph, for various values of the regularization parameter
0.3
1s s,1
of
0.1
0.9
0.5
0.7
Target Fairness Level
Target Fairness Level a
Figure 14: TWITTER dataset. FADSG-L. Left: Fraction of protected vertices in each induced subgraph, for various values of target fairness level The dotted lines represent the fairness value in the solution of DSG. Right: PoF for various values of target fairness level a.
1.00
1.00
0.50
0.50
0.25
0.25
20
Figure 15: Fraction of protected vertices in the induced subset, function of X, for FADSG-I. (Left: LASTFM-12 Right: LASTFM-13).
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H.2 FADSG-I1:
In Figures 16 and 18
compare FADSG-II with DSG. The top sections show what fraction of all of the protected nodes belongs in the solutions, and the bottom ones depict the PoF for FADSG-IL, Again, for some datasets the solution not exactly the desired onc, which is something that can be attributed to the structure of each graph, as can be seen in Figure
DSG
Ours
0.4
0.2
0.0
1.0
is
SS
0.5
0.0
Figure 16: LASTFM datasets. Bottom: PoF of FADSG-I1 for 8 1. Top: Comparing FADSG-1I (Ours) for with DSG. Fraction of the protected subset that belongs in induced subsets.
DSG
Ours
0.50
PoF
0.25
0.00
0.0
HPC
15
OP
PS
AH
ACS
G
SO
TMI
Figure 17: AMAZON datasets. Bottom: PoF of FADSG-II for Top: Comparing FADSG-11 (Ours) for with DSG. Fraction of the protected subset that belongs induced subsets.
DSG
Ours
0.4
0.0
0.50
n Spl
0.00
PolBooks
PoiBlogs
Deezer
Github
Figure 18: Rost of the datascts. Bottom: PoF of FADSG-II for 5 z= 1. Top: Comparing FADSG-II (Ours)
for = with DSG. Fraction of the protected subset that belongs in induced subsets.
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1.00
1.00
0.75
|So Sql
[sn.s S.
0.25
0.25
50
100
20
40
Figure 19: The fraction of all of the protected vertices included in induced subsets as function of for FADSG-II. (Left: AMAZON Right: AMAZON OP).
1.00
1.00
SS Syl
SSn.
0.50
0.25
0.25
20
Figure 20: Fraction of all of the protected verticos that belong the induced subsct, as function of X, for FADSG-II. (Left: A-STEM-4 Right: LASTFM-14)..
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H.3 Resulting proportions of protected vertices in FADSG-I solution
Table 7: Resulting proportions of protected vertices in FADSG-1 solution S. ny the total number of protected
vertices, |S n Spl the number of protected vertices in S, |Sns,] the proportion of protected vertices in S, and [S|
[ the proportion among all of the protected vertices that belong in S. [
LASTFM DATASET
DATASET
Tpp
|
SSnS,
1 [S]
ISnSpl [Sp
PROTECTED SET T4
Spl
SS
Jsnspl [S]
ISnS [8p]
PoLBooKs
42
26
1.0
0.6046
1098
62
1.0
0.0565
POLBLOGS
586
117
1.0
0.1997
54
10
1.0
0.1852
AMAZON
83
10
1.0
0.1205
73
26
1.0
0.3562
AMAZON PS
230
17
0.0739 1.0
3
515
43
0.0835 1.0
AMAZON
1495
134
0.0896 1.0
16
1.0000 1.0
AMAZON HPC
1352
25
0.0185 1.0
391
44
0.1125 1.0
AMAZON OP
462
1.0
0.0173
655
0.0702
AMAZON AH
8163
62
0.0076 1.0
7
82
14
0,1707 1.0
AMAZON ACS
987
35
0.0355 1.0
35
1.0
0.0748
AMAZON C
5376
65
1.0
0.0121
58
13
0.2241 1.0
AMAZON SO
604
16
1.0
0.0265
10
1303
69
0.0530 1.0
AMAZON TMI
520
17
1.0
0.0327
11
138
40
0.2899 1.0
TWTTCH DE
5742
382
1.0
0.0665
12
57
11
0.1930 1.0
TWITCH ENGB
3888
34
1.0
0.0874
13
63
25
0.3968 1.0
TWITCH ES
1360
160
1.0
0.1176
14
570
59
0.1035 1.0
TWITCH
2415
287
1.0
0.1188
15
257
52
1.0
0.2023
TWITCH PTER
661
107
1.0
0.1619
16
254
24
1.0
0.0945
TWITCH RU
1075
85
1.0
0.0791
1572
79
1.0
0.0503
DEEZER
12538
32
1.0
0.0026
100
387
51
1.0
0.1318
CITHUB
9739
186
1.0
0.0191
TWITTER 11355
156
1.0
0.0137
In the experiments presented in Table we set the target fairness level |Sns,] in FADSG-I to extract [S|
the densest protected subset, and examined what fraction of the entire protected set it constitutes; i.., if S is the densest protected subset obtained using FADSG-I, we evaluate the value of |8ns,] |s,l As we can see from these tables, there are many cases where the solution comprises small proportion of protected vertices (it can be even less than 5% see GITHUB). Note that this stems from the structure of the protected set, since the solution that we get always the densest subset of the protected set. Hence, FADSG-I may not be able to extract a dense subgraph that contains even 50% of the protected set motivating the need, some cases, for FADSG-11.
34


Reviews:
[{"review_id": "GI5bsnt64v", "reviewer": "Reviewer_SQso", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Strengths: \n\n-The paper generalizes the prior work and introduces two meaningful problems that have a compromise between DSD and fairness using a parameter  $\\lambda$ \n\n\n-The paper includes experiments and comparison to baselines which show an advantage. \n\n\n## Weaknesses: \n\n-The paper is only using two groups which is a limitation. \n\n-It seems that the paper claims that it straightforwardly follows that the problem is solvable in polynomial time for any $\\lambda$ using max flow but where is this proof/explanation?\n\n-From Section 6, the problem is solved using binary search and a fast approximate algorithm. But this makes the paper loose the claimed advantage over prior work since they involve an NP-hard problem that cannot be solved exactly. \n\n-It would be better if there was also an upper bound on $r_2$ using $r_1$.\n\n-The mathematical derivations do not seem very informative/substantial,e.g., Lemma 1, Proposition 3. \n\nMinor point: using $\\boldsymbol{1}$ as a vector of ones is more clear than using $\\boldsymbol{e}$.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper is concerned with the densest subgraph (DSD) problem. Unlike prior work it introduces two objectives that tradeoff the value of DSD objective and fairness using a regularization parameter. Further, unlike prior work the optimization is tractable. Finally, the paper shows better performance over a collection of baselines experimentally.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1-In Section 5 for PoF, I don’t see the point behind lollipop example? How is it that useful since it is a very specific example? Also, a figure would improve presentation here. \n\n\n2-Please add an explanation on how is this setting significantly different from for example Miyauchi? Specifically, what if the algorithm from Miyauchi is run at different fraction values using binary search? \n\n3-As mentioned under weaknesses explain the tractability advantage of your setting given that you are using approximation algorithms", "broader_impact_concerns": "I don't think that there are broader impact concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "dr3xQjionD", "reviewer": "Reviewer_RYxZ", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**  \nThe paper's primary contribution lies in its novel approach to incorporating fairness considerations into dense subgraph discovery through regularization. The authors provided rigorous theoretical analysis in presenting their formulations and provide valuable insights regarding the effects of regularization parameters. In general, the manuscript is well-structured and systematic in its presentation. I also appreciate their comprehensive theoretical analysis of fairness-density trade-offs and provide the price of fairness bounds.\n\n**Weakness**\nI have concerns regarding the theoretical soundness of the proposed formulations in Equations (1) and (3).  In Equation (1), the regularizer $r_1(x)$ is designed to encourage inclusion of vertices from the protected subset $S_p$. However, this formulation may have an unintended consequence: at any given density level, selecting vertices exclusively from $S_p$ could potentially yield the optimal solution, as there are no constraints preventing this outcome. Similarly, in Equation (3), minimizing the penalized term could be achieved simply by selecting vertices predominantly from $S_p$, potentially undermining the intended balance between protected and unprotected groups. How could these formulations still promote fairness that the numbers of vertices from protected set and unprotected set are balanced?", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper makes three significant contributions to algorithmic fairness in graph mining:\n- First, it introduces two novel, polynomial-time solvable formulations for fair dense subgraph discovery (DSD), addressing scenarios where graph vertices contain sensitive attributes like gender or race. Unlike previous NP-hard approaches, these formulations are computationally tractable.\n- Second, through carefully designed regularizers, the methods enable flexible control over fairness levels, allowing the balance between graph density and fair representation of protected groups. \n- Third, the authors analyze the trade-off between subgraph density and fairness through their \"price of fairness\" metric, quantifying the density sacrifice needed for achieving desired fairness levels. They validate their approach through comprehensive empirical evaluation on multiple datasets.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- Please put all the problem formulations in a dedicated problem formulation section. It was a pain when I was confused about the definition of densest subgraph problem, but ended up finding it in the related work section. Also, please highlight the definitions of Equation (1) and Equation (3).\n- I would suggest the authors to further clarify the connections between the two fairness-aware densest subgraph problems and the conventional fairness measures like demographic parity and equality of opportunity.", "broader_impact_concerns": "This work provided valuable insights on the responsible algorithms for dense graph detection. I do not foresee any potential ethical concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "sYhBmBL1CO", "reviewer": "Reviewer_hAip", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths\n1.Both FADSG-I and FADSG-II can be computed in polynomial time, overcoming the NP-hardness drawback of previous methods in the worst-case scenario.\n2.FADSG-I and FADSG-II exhibit monotonicity, enabling the use of binary search to determine the strength of regularization.\n3.The experimental results seem to validate the effectiveness of the two regularization methods.\n\nWeaknesses\n1.From the experimental results, FADSG-I appears capable of achieving a similar proportion of protected nodes being selected. What, then, is the advantage of FADSG-II over FADSG-I? The paper mentions that in certain scenarios, FADSG-I cannot adequately represent protected nodes. What are these scenarios, and how frequently do they occur in real-world applications?\n2.The paper focuses on relatively simple fairness scenarios, considering only binary sensitive attributes. However, in real-world scenarios, sensitive attributes may involve multiple groups. Can the proposed methods be easily extended to handle multi-valued sensitive attributes?\n3.In the experiments, why is FADSG-II not compared against FADSG-I and existing baseline methods? Although the paper mentions that FADSG-II uses a different fairness criterion, the metrics presented in the figures (POF and the proportion of protected nodes) are the same as those used for FADSG-I.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper focuses on the problem of fair densest subgraph discovery and addresses the high computational complexity of previous methods by proposing two regularization terms and proving their computability. It introduces the POF metric to measure the trade-off for incorporating fairness. The optimization problems corresponding to the two regularization terms are formulated as the Densest Supermodular Subset (DSS) problem, which is approximately solved using the Super-Greedy++ algorithm, significantly reducing time complexity compared to solving via the maximum flow algorithm. Experiments on real-world datasets demonstrate the effectiveness of the proposed regularization terms in achieving fairness.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. Clarify the Scenarios for FADSG-I and FADSG-II:  \n   Provide a detailed explanation of the scenarios where FADSG-I fails to adequately represent protected nodes. Include examples or case studies from real-world datasets to illustrate these scenarios and discuss their prevalence.\n\n2. Extend Fairness to Multi-Valued Sensitive Attributes:\n   Discuss whether the proposed methods can be extended to handle multi-valued sensitive attributes. If possible, include a brief theoretical or experimental analysis to demonstrate the feasibility of such an extension.\n\n3. Experimental Comparisons for FADSG-II:  \n   Include direct comparisons of FADSG-II with FADSG-I and baseline methods using the same metrics (e.g., POF and the proportion of protected nodes). If FADSG-II uses a different fairness criterion, explain why this criterion is necessary and how it relates to the metrics used for FADSG-I.\n\n4. Highlight the Advantages of FADSG-I:  \n   Clearly articulate the advantages of FADSG-II over FADSG-I in terms of fairness, performance, or other criteria. Provide experimental evidence to support these claims.", "broader_impact_concerns": "The proposed method carries the risk of misuse in sensitive applications such as social network analysis, hiring networks, or political influence mapping. For instance, it could be exploited to justify biased decisions under the pretense of fairness by selectively defining fairness criteria or subgroup representations to suit specific agendas.\n\nWhile the paper introduces multiple notions of fairness, these concepts are inherently context-dependent and may not universally align with ethical or widely accepted definitions of fairness. If the fairness criteria are poorly chosen or intentionally manipulated, the resulting dense subgraph may inadvertently reinforce existing biases or inequalities instead of addressing them.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The current version of this paper is self-standing, with clear motivations, theoretical analyses, and experimental verification. With the excellent review team, I mainly agree with their comments but have a minor issues on the figure visualization. I suggest the authors improving the figures in terms of figure size (some figures are too small) and removing the necessary background color.
Decision: Accept (poster)