Original: Individual Fairness in Dynamic Financial Networks
Revised: Individual Fairness in Dynamic Financial Networks

Original: Anonymous Author(s)

Original: AffiliationAddress email

Original: Abstract
Revised: Abstract

Original: In the financial world, a transaction graph is commonly used for modeling theever-changing payee-payor relationships.
Revised: In the financial world, a transaction graph is commonly used for modeling the ever-changing payee-payor relationships.

Original: Every online transaction corresponds toa directed edge from the paying party to the receiving party in this graph.
Revised: Every online transaction corresponds to a directed edge from the paying party to the receiving party in this graph.

Original: Eventhough the superior learning capability of Graph Neural Networks (GNNs) hasled to several successful financial applications like fraud detection and anti-moneylaundering, most of these existing works do not have fairness considerations.
Revised: Even though the superior learning capability of Graph Neural Networks (GNNs) has led to several successful financial applications like fraud detection and anti-money laundering, most of these existing works do not have fairness considerations.

Original: Apparently, the lack of fairness guarantees during the GNN-based decision-makingprocess would cause increasingly serious societal concerns from both buyers andsellers.
Revised: Apparently, the lack of fairness guarantees during the GNN-based decision-making process would cause increasingly serious societal concerns from both buyers and sellers.

Original: Furthermore, the time-varying property of the financial networks makesthe fairness requirements more challenging, since current fairness measures ongraph learning tasks and fairness-aware GNN models are all designed for staticgraphs only.
Revised: Furthermore, the time-varying property of the financial networks makes the fairness requirements more challenging, since current fairness measures on graph learning tasks and fairness-aware GNN models are all designed for static graphs only.

Original: In this work, we present a new generic definition of individual fairnessfor dynamic graphs and propose a regularization-based method to debias the
Revised: In this work, we present a new generic definition of individual fairness for dynamic graphs and propose a regularization-based method to debias the

Original: GNN model in the temporal setting.
Revised: GNN model in the temporal setting.

Original: We perform some preliminary experimentalevaluations on two real-world datasets and demonstrate the potential efficacy ofthe proposed methods.
Revised: We perform some preliminary experimental evaluations on two real-world datasets and demonstrate the potential efficacy of the proposed methods.

Original:  Introduction
Revised:  Introduction

Original: The financial system is one of the most complex systems with massive interacting parties, whichare also constantly updated.
Revised: The financial system is one of the most complex systems with massive interacting parties, which are also constantly updated.

Original: Therefore, graphs are commonly constructed in order to model thedynamic relational data in the financial domain, such as dynamic loan networks [1], Bitcoin trustgraphs [8] and supply chain graphs [14].
Revised: Therefore, graphs are commonly constructed in order to model the dynamic relational data in the financial domain, such as dynamic loan networks [1], Bitcoin trust graphs [8] and supply chain graphs [14].

Original: Almost all of these financial networks can be abstractedinto a dynamic directed graph where the nodes represent financial institutions or clients and theedges represent the directional relations from one node to the other, like the buyer-seller relationship,the borrower-lender relationship and the trustor-trustee relationship.
Revised: Almost all of these financial networks can be abstracted into a dynamic directed graph where the nodes represent financial institutions or clients and the edges represent the directional relations from one node to the other, like the buyer-seller relationship, the borrower-lender relationship and the trustor-trustee relationship.

Original: The dynamic or temporalnature of this financial network makes the downstream learning tasks more challenging, as it iscontinuously evolving with transactions (edges) being created or canceled every second, merchants(nodes) appearing or disappearing every day and clients’ information (node features) updating anytime.
Revised: The dynamic or temporal nature of this financial network makes the downstream learning tasks more challenging, as it is continuously evolving with transactions (edges) being created or canceled every second, merchants (nodes) appearing or disappearing every day, and clients’ information (node features) updating any time.

Original: To gain interesting insights from these graphs, Graph Neural Networks (GNNs) are currentlythe mainstream learning paradigms for various tasks [7].
Revised: To gain interesting insights from these graphs, Graph Neural Networks (GNNs) are currently the mainstream learning paradigms for various tasks [7].

Original: Among them, the family of temporal
Revised: Among them, the family of temporal

Original: GNNs [10, 11, 15], a time-dependent variant of GNN models, is the natural fit for the ever-changingfinancial networks with myriad promising applications [13], including fraudster detection at the nodelevel, future transactions at the edge level, and money laundering detection at the sub-graph level.
Revised: GNNs [10, 11, 15], a time-dependent variant of GNN models, is the natural fit for the ever-changing financial networks with myriad promising applications [13], including fraudster detection at the node level, future transactions at the edge level, and money laundering detection at the sub-graph level.

Original: However, high-stakes decisions under strict laws and regulations are often involved in the financialsystem, and thus, fairness concerns beyond the utility of temporal GNNs are in urgent demand.
Revised: However, high-stakes decisions under strict laws and regulations are often involved in the financial system, and thus, fairness concerns beyond the utility of temporal GNNs are in urgent demand.

Original: Forinstance, if a bank makes loan approval predictions based on the transaction record network amongapplicants, the decision must be independent of the sensitive attribute of the applicant, like gender andrace.
Revised: For instance, if a bank makes loan approval predictions based on the transaction record network among applicants, the decision must be independent of the sensitive attribute of the applicant, like gender and race.

Original: Unfortunately, for one thing, current popular temporal GNNs fail to satisfy reasonable notionsof fairness due to the inherent algorithmic bias in GNNs.
Revised: Unfortunately, for one thing, current popular temporal GNNs fail to satisfy reasonable notions of fairness due to the inherent algorithmic bias in GNNs.

Original: For another, most fairness-enhanced static
Revised: For another, most fairness-enhanced static

Original: GNNs cannot be directly generalized in the dynamic setting due to the overwhelming computationalcost and the time-varying fairness-related information loss.
Revised: GNNs cannot be directly generalized in the dynamic setting due to the overwhelming computational cost and the time-varying fairness-related information loss.

Original: Consequently, how to measure fairnessfor temporal graph learning tasks and how to equip dynamic GNNs with fairness considerations arestill open research questions [3].
Revised: Consequently, how to measure fairness for temporal graph learning tasks and how to equip dynamic GNNs with fairness considerations are still open research questions [3].

Original: In this work, we aim to present a principled study of individual fairness for temporal GNNs onfinancial transaction graphs.
Revised: In this work, we aim to present a principled study of individual fairness for temporal GNNs on financial transaction graphs.

Original: Compared with group fairness, individual fairness quantifies potentialbias or discrimination at a finer granularity and follows the general principle that any two similarindividuals should be treated with similar outcomes.
Revised: Compared with group fairness, individual fairness quantifies potential bias or discrimination at a finer granularity and follows the general principle that any two similar individuals should be treated with similar outcomes.

Original: Two fundamental research questions would beraised naturally when we extend individual fairness notions to temporal GNNs.
Revised: Two fundamental research questions would be raised naturally when we extend individual fairness notions to temporal GNNs.

Original: First, how can wequantitatively measure the overall bias of the predictions for temporal graph learning tasks?
Revised: First, how can we quantitatively measure the overall bias of the predictions for temporal graph learning tasks?

Original: Second,how can we alleviate such bias by developing the fairness-aware temporal GNNs model accordingly?
Revised: Second, how can we alleviate such bias by developing the fairness-aware temporal GNNs model accordingly?

Original: For the first question, we introduce a new definition of fairness in retrospect, which requires thepredictions to be individually fair relative to the past at the time of the decision based on the Lipschitzproperty.
Revised: For the first question, we introduce a new definition of fairness in retrospect, which requires the predictions to be individually fair relative to the past at the time of the decision based on the Lipschitz property.

Original: This proposed fairness measure can easily enable us to quantify the overall algorithmicbias in the predicted results for the graph-based learning tasks under the temporal setting.
Revised: This proposed fairness measure can easily enable us to quantify the overall algorithmic bias in the predicted results for the graph-based learning tasks under the temporal setting.

Original: For thesecond question, we propose a novel individual fairness regularization term that can be integratedinto almost all temporal GNNs without much effort.
Revised: For the second question, we propose a novel individual fairness regularization term that can be integrated into almost all temporal GNNs without much effort.

Original: As far as we are concerned, we are the first toinvestigate the individual fairness measures and methods for dynamic GNNs.
Revised: As far as we are concerned, we are the first to investigate the individual fairness measures and methods for dynamic GNNs.

Original: We also present somepreliminary experiments to demonstrate the potential applications in real-world financial networks.
Revised: We also present some preliminary experiments to demonstrate the potential applications in real-world financial networks.

Original:  Methodology2.1 Problem Formulation
Revised:  Methodology 2.1 Problem Formulation

Original: A static graph can be denoted as G =
Revised: A static graph can be denoted as G =

Original: ( V, E ) with the node set V = { v 1 , . .
Revised: ( V, E ) with the node set V = { v 1 , . .

Original: . , v n } and the edgeset E ⊆ V × V .
Revised: . , v n } and the edge set E ⊆ V × V .

Original: The nodes could be associated with features X = { x v | v ∈ V } and the edgescould also have features F = { f uv | ( u, v ) ∈ E } .
Revised: The nodes could be associated with features X = { x v | v ∈ V } and the edges could also have features F = { f uv | ( u, v ) ∈ E } .

Original: The dynamic or temporal graph under the discretesetting can be viewed as a sequence of static graphs G = { G t } Tt =1 , where each graph has a uniquetimestamp t .
Revised: The dynamic or temporal graph under the discrete setting can be viewed as a sequence of static graphs G = { G t } Tt =1 , where each graph has a unique timestamp t .

Original: In each snapshot G t = ( V t , E t ) , the note set V t ( | V t | = n t ), the edge set E t and theircharacteristics could vary with t .
Revised: In each snapshot G t = ( V t , E t ) , the note set V t ( | V t | = n t ), the edge set E t and their characteristics could vary with t .

Original: This way of modeling can handle the dynamics of financial networksin the real world.
Revised: This way of modeling can handle the dynamics of financial networks in the real world.

Original: Current dynamic graph representation learning aims to learn latent representationsh tv ∈ R d for each node at time step t ∈ { 1 , . .
Revised: Current dynamic graph representation learning aims to learn latent representations h tv ∈ R d for each node at time step t ∈ { 1 , . .

Original: , T } , such that h tv can encode both the neighborhoodinformation centered at v and its temporal evolutionary behaviors of the graph [11].
Revised: , T } , such that h tv can encode both the neighborhood information centered at v and its temporal evolutionary behaviors of the graph [11].

Original: In this work, we push this goal further and restrict the representation h tv to be individually fair withrespect to the ground-truth results.
Revised: In this work, we push this goal further and restrict the representation h tv to be individually fair with respect to the ground-truth results.

Original: For example, if the learning task is the node classification, wewant to predict the label of a node u at time t based on the previous snapshots { G τ } t − 1 τ =1 .
Revised: For example, if the learning task is the node classification, we want to predict the label of a node u at time t based on the previous snapshots { G τ } t − 1 τ =1 .

Original: We use
Revised: We use

Original: Y t ∈ { 0 , 1 } n t × c and ˆ Y t ∈ R n t × c to denote the ground truth and the predicted class membershipmatrix, respectively.
Revised: Y t ∈ { 0 , 1 } n t × c and ˆ Y t ∈ R n t × c to denote the ground truth and the predicted class membership matrix, respectively.

Original: Then we hope that ˆ Y t [ u ] ∈ R c (the u -th row of ˆ Y t ) and ˆ Y t [ v ] are similar aslong as node u and node v are similar on the temporal graph.
Revised: Then we hope that ˆ Y t [ u ] ∈ R c (the u -th row of ˆ Y t ) and ˆ Y t [ v ] are similar as long as node u and node v are similar on the temporal graph.

Original: 2.2 Individual Fairness Measures for Dynamic Graphs
Revised: 2.2 Individual Fairness Measures for Dynamic Graphs

Original: In this section, we propose a novel metric to quantitatively measure the level of overall bias in thepredicted results on the dynamic graphs.
Revised: In this section, we propose a novel metric to quantitatively measure the level of overall bias in the predicted results on the dynamic graphs.

Original: This new metric can motivate the design of our proposedmethod, which potentially enhances the individual fairness of dynamic GNNs.
Revised: This new metric can motivate the design of our proposed method, which potentially enhances the individual fairness of dynamic GNNs.

Original: First, let us revisit the definition of the individual fairness measure for static graphs, following the
Revised: First, let us revisit the definition of the individual fairness measure for static graphs, following the

Original: Lipschitz condition [4].
Revised: Lipschitz condition [4].

Original: Definition 2.1 ( ( D 1 , D 2 ) -Lipschitz property) .
Revised: Definition 2.1 ( ( D 1 , D 2 ) -Lipschitz property) .

Original: Given a learning model f , denote f ( v ) as the predictedoutcome of node v on a static graph G .
Revised: Given a learning model f , denote f ( v ) as the predicted outcome of node v on a static graph G .

Original: Then f satisfies ( D 1 , D 2 ) -Lipschitz property if
Revised: Then f satisfies ( D 1 , D 2 ) -Lipschitz property if

Original: [Equation]
Revised: [Equation]

Original: It is also worth noting that we assume there is an oracle similarity matrix S ∈ R n × n to measurethe pairwise node similarity according to domain knowledge or human judgment [12].
Revised: It is also worth noting that we assume there is an oracle similarity matrix S ∈ R n × n to measure the pairwise node similarity according to domain knowledge or human judgment [12].

Original: Existingworks promote individual fairness on static graphs through the defined ( D 1 , D 2 ) -Lipschitz property.
Revised: Existing works promote individual fairness on static graphs through the defined ( D 1 , D 2 ) -Lipschitz property.

Original: The basic intuition is that individual nodes with higher similarity should be constrained to have ashorter distance in the embedding space.
Revised: The basic intuition is that individual nodes with higher similarity should be constrained to have a shorter distance in the embedding space.

Original: Therefore, if we set D 1 ( f ( u ) , f ( v )) =
Revised: Therefore, if we set D 1 ( f ( u ) , f ( v )) =

Original: ∥ ˆ Y [ u ] − ˆ Y [ v ] ∥ 2 ,
Revised: ∥ ˆ Y [ u ] − ˆ Y [ v ] ∥ 2 ,

Original: D 2 ( u, v ) =
Revised: D 2 ( u, v ) =

Original: 1 S [ u,v ] and L = ϵ > 0 (as a fairness tolerance constant), we immediately obtain thefollowing condition Eq.
Revised: 1 S [ u,v ] and L = ϵ > 0 (as a fairness tolerance constant), we immediately obtain the following condition Eq.

Original: [Equation]
Revised: [Equation]

Original: The difference between the predicted results of a node pair ( u, v ) is upper bounded by the RHS of
Revised: The difference between the predicted results of a node pair ( u, v ) is upper bounded by the RHS of

Original: Eq.
Revised: Eq.

Original: The more similar the node u and the node v are, the tighter the upper bound in Eq.
Revised: The more similar the node u and the node v are, the tighter the upper bound in Eq.

Original: (2) is,and the smaller the difference between ˆ Y [ u ] and ˆ Y [ v ] will be, achieving the individual fairness goalfrom the perspective of Lipschitz constraint in Definition 2.1.
Revised: (2) is, and the smaller the difference between ˆ Y [ u ] and ˆ Y [ v ] will be, achieving the individual fairness goal from the perspective of Lipschitz constraint in Definition 2.1.

Original: By summing the LHS of Eq.
Revised: By summing the LHS of Eq.

Original: (1) overall node pairs, we can get the overall bias for the static graph in terms of individual fairness (IF-S) as
Revised: (1) over all node pairs, we can get the overall bias for the static graph in terms of individual fairness (IF-S) as

Original: Definition 2.2 that is widely employed in the current literature [6, 2, 12].
Revised: Definition 2.2 that is widely employed in the current literature [6, 2, 12].

Original: The smaller the IF-S, themore individually fair the predicted results.
Revised: The smaller the IF-S, the more individually fair the predicted results.

Original: Definition 2.2 (Individual fairness on static graphs) .
Revised: Definition 2.2 (Individual fairness on static graphs) .

Original: Given a static graph G =
Revised: Given a static graph G =

Original: ( V, E ) with | V | = n ,the predicted membership matrix ˆ Y ∈ R n × c and the oracle pairwise similarity matrix S ∈ R n × n ,
Revised: ( V, E ) with | V | = n , the predicted membership matrix ˆ Y ∈ R n × c and the oracle pairwise similarity matrix S ∈ R n × n ,

Original: Eq.
Revised: Eq.

Original: (2) measures the overall bias regarding the individual fairness.
Revised: (2) measures the overall bias regarding the individual fairness.

Original: [Equation]
Revised: [Equation]

Original: Here, L S is the Laplacian matrix of S .
Revised: Here, L S is the Laplacian matrix of S .

Original: For dynamic graphs, the most straightforward extension of IF-S is to sum all IF-S over all thetimestamps as (cid:80)
Revised: For dynamic graphs, the most straightforward extension of IF-S is to sum all IF-S over all the timestamps as (cid:80)

Original: Tt =1 IF-S t , where each IF-S t follows Definition 2.2 at t -th snapshot.
Revised: Tt =1 IF-S t , where each IF-S t follows Definition 2.2 at t -th snapshot.

Original: However,this metric views each snapshot independently, and thus fails to take the variation of the fairnessconstraint across time into account.
Revised: However, this metric views each snapshot independently, and thus fails to take the variation of the fairness constraint across time into account.

Original: Nodes with different timestamps but with similar contextsshould also share similar outcomes.
Revised: Nodes with different timestamps but with similar contexts should also share similar outcomes.

Original: Therefore, in order to incorporate the temporal property into thefairness measurement, we first present a novel Lipschitz condition under the dynamic setting andthen construct a new individual fairness metric on dynamic graphs accordingly.
Revised: Therefore, in order to incorporate the temporal property into the fairness measurement, we first present a novel Lipschitz condition under the dynamic setting and then construct a new individual fairness metric on dynamic graphs accordingly.

Original: Definition 2.3 ( ( K, D 1 , D 2 ) -Lipschitz in retrospect property) .
Revised: Definition 2.3 ( ( K, D 1 , D 2 ) -Lipschitz in retrospect property) .

Original: Given a learning model f , denotef ( v t ) as the predicted outcome of node v on a dynamic graph at time step t .
Revised: Given a learning model f , denote f ( v t ) as the predicted outcome of node v on a dynamic graph at time step t .

Original: Then f satisfies( K, D 1 , D 2 ) -Lipschitz in retrospect property with respect to a monotonically non-decreasing function
Revised: Then f satisfies ( K, D 1 , D 2 ) -Lipschitz in retrospect property with respect to a monotonically non-decreasing function

Original: L ( · ): R + → R + and a non-negative integer K ∈ Z ∗ if
Revised: L ( · ): R + → R + and a non-negative integer K ∈ Z ∗ if

Original: [Equation]
Revised: [Equation]

Original: Note that in Definition 2.3, by setting L ( · ) to be a monotonically non-decreasing function, we canmodel the scenario where the past predictions have a diminishing impact on the future predictionsdepending on the amount of time passed.
Revised: Note that in Definition 2.3, by setting L ( · ) to be a monotonically non-decreasing function, we can model the scenario where the past predictions have a diminishing impact on the future predictions depending on the amount of time passed.

Original: This insight is motivated by the assumption that nodeswith similar features should achieve similar predictions compared with their counterparts in recenthistory, but could obtain dissimilar treatments compared with those in the distant past.
Revised: This insight is motivated by the assumption that nodes with similar features should achieve similar predictions compared with their counterparts in recent history, but could obtain dissimilar treatments compared with those in the distant past.

Original: For example,in the transaction network for load approval, both Alice and Bob earn almost the same amount ofmoney annually, but Alice applies for the loan several years before Bob does, appearing in the earliersnapshots of the network.
Revised: For example, in the transaction network for load approval, both Alice and Bob earn almost the same amount of money annually, but Alice applies for the loan several years before Bob does, appearing in the earlier snapshots of the network.

Original: It would still be considered individually fair if the bank approves Alice’sapplications but rejects Bob’s since the average personal income could grow over the years.
Revised: It would still be considered individually fair if the bank approves Alice’s applications but rejects Bob’s since the average personal income could grow over the years.

Original: Anotherinterpretation of Definition 2.3 is that it can be considered as a generalized version of Definition 2.1.
Revised: Another interpretation of Definition 2.3 is that it can be considered as a generalized version of Definition 2.1.

Original: If we set K = 0 ( t ′ =
Revised: If we set K = 0 ( t ′ =

Original: t ) in Definition 2.3, then the degenerate definition becomes the same as
Revised: t ) in Definition 2.3, then the degenerate definition becomes the same as

Original: Definition 2.1 with L =
Revised: Definition 2.1 with L =

Original: L (0) .
Revised: L (0) .

Original: Therefore, Definition 2.3 is a more strict constraint by restricting thecurrent treatment to be individually fair relative to the recent past within the interval of K snapshots.
Revised: Therefore, Definition 2.3 is a more strict constraint by restricting the current treatment to be individually fair relative to the recent past within the interval of K snapshots.

Original: Similar to Eq.
Revised: Similar to Eq.

Original: (1), if we still set D 1 ( f ( u ) , f ( v ))
Revised: (1), if we still set D 1 ( f ( u ) , f ( v ))

Original: =
Revised: =

Original: ∥ ˆ Y t ′ [ u ] − ˆ Y t [ v ] ∥ 22 , D 2 ( u, v ) = 1 S t ′ ,t [ u,v ] and set
Revised: ∥ ˆ Y t ′ [ u ] − ˆ Y t [ v ] ∥ 22 , D 2 ( u, v ) = 1 S t ′ ,t [ u,v ] and set

Original: L ( t − t ′ )
Revised: L ( t − t ′ )

Original: =
Revised: =

Original: ϵ exp( t − t ′ ) with ϵ > 0 , we can get the individual fairness requirement for the temporalgraphs as Eq.
Revised: ϵ exp( t − t ′ ) with ϵ > 0 , we can get the individual fairness requirement for the temporal graphs as Eq.

Original: [Equation]
Revised: [Equation]

Original: Intuitively, Eq.
Revised: Intuitively, Eq.

Original: (3) leads to a tighter upper bound on the inconsistency of the treatments for a pairof nodes if their similarity S t ′ ,t [ u, v ] is greater and their existence falls into a shorter time interval( t − t ′ ) .
Revised: (3) leads to a tighter upper bound on the inconsistency of the treatments for a pair of nodes if their similarity S t ′ ,t [ u, v ] is greater and their existence falls into a shorter time interval ( t − t ′ ) .

Original: We then propose a new I ndividual F airness metric for D ynamic graphs (IF-D) accordinglyby summing all node pairs as Definition 2.4.
Revised: We then propose a new I ndividual F airness metric for D ynamic graphs (IF-D) accordingly by summing all node pairs as Definition 2.4.

Original: [Table]
Revised: [Table]

Original: Definition 2.4 (Individual fairness on dynamic graphs) .
Revised: Definition 2.4 (Individual fairness on dynamic graphs) .

Original: Given a dynamic graph G = { G t } Tt =1 , where
Revised: Given a dynamic graph G = { G t } Tt =1 , where

Original: G t =
Revised: G t =

Original: ( V t , E t ) with | V t | = n t , the predicted membership matrix ˆ Y t ∈ R n t × c at time step t andthe oracle pairwise similarity matrices S t ′ ,t ∈ R n t ′ × n t denoting the similarity distance between thenodes at the t -th snapshot with nodes from any previous K snapshots with max(0 , t − K ) ≤ t ′ ≤ t .
Revised: ( V t , E t ) with | V t | = n t , the predicted membership matrix ˆ Y t ∈ R n t × c at time step t and the oracle pairwise similarity matrices S t ′ ,t ∈ R n t ′ × n t denoting the similarity distance between the nodes at the t -th snapshot with nodes from any previous K snapshots with max(0 , t − K ) ≤ t ′ ≤ t .

Original: We define the bias regarding the individual fairness on dynamic graphs at the t -th snapshot IF-D t as
Revised: We define the bias regarding the individual fairness on dynamic graphs at the t -th snapshot IF-D t as

Original: [-IF-D t = 
 [Equation] (cid:88)-]
Revised: [-IF-D t = 
 [Equation] (cid:88)-]IF

Original: [Equation]
Revised: [Equation]

Original: 2.3 Individual Fairness Regularization for Dynamic Graphs
Revised: 2.3 Individual Fairness Regularization for Dynamic Graphs

Original: To achieve the individual fairness goal on dynamic graphs, we naturally propose a simple yet effectiveregularization-based method that could be plugged into almost all current state-of-the-art dynamic
Revised: To achieve the individual fairness goal on dynamic graphs, we naturally propose a simple yet effective regularization-based method that could be plugged into almost all current state-of-the-art dynamic

Original: GNN models [5, 11, 10].
Revised: GNN models [5, 11, 10].

Original: Based on the proposed Definition 2.4, we can make the assumption that| V t ′ | = | V t | ( n t ′ = n t ) when K is small.
Revised: Based on the proposed Definition 2.4, we can make the assumption that | V t ′ | = | V t | ( n t ′ = n t ) when K is small.

Original: This corresponds to the fact that the number of clients ina transaction network will barely change in a sufficiently short time interval.
Revised: This corresponds to the fact that the number of clients in a transaction network will barely change in a sufficiently short time interval.

Original: Even if the numberof nodes changes, we can still add some “pseudo-nodes" with random features and links during thetraining process to make up for the difference in terms of the number of nodes.
Revised: Even if the number of nodes changes, we can still add some “pseudo-nodes" with random features and links during the training process to make up for the difference in terms of the number of nodes.

Original: Thus, we can constructa simplified individual fairness regularization loss for dynamic graphs based on Definition 2.4 as
Revised: Thus, we can construct a simplified individual fairness regularization loss for dynamic graphs based on Definition 2.4 as

Original: Eq.
Revised: Eq.

Original: [Equation]
Revised: [Equation]

Original: Note that now S t ′ ,t ∈ R n t × n t and L S t ′ ,t ∈ R n t × n t (the Laplacian matrix of S t ′ ,t ) are squarematrices.
Revised: Note that now S t ′ ,t ∈ R n t × n t and L S t ′ ,t ∈ R n t × n t (the Laplacian matrix of S t ′ ,t ) are square matrices.

Original: For the sake of training efficiency, we can set K = 3 .
Revised: For the sake of training efficiency, we can set K = 3 .

Original: Then, suppose that the utilityloss in the backbone dynamic GNN model is L util , then the total training loss can be obtained as
Revised: Then, suppose that the utility loss in the backbone dynamic GNN model is L util , then the total training loss can be obtained as

Original: L =
Revised: L =

Original: L util + α (cid:80) kt
Revised: L util + α (cid:80) kt

Original: =1 L IF-D t .
Revised: =1 L IF-D t .

Original: Here, k is the total number of snapshots for training, and α is thebalancing factor.
Revised: Here, k is the total number of snapshots for training, and α is the balancing factor.

Original:  Preliminary Experimentsand Discussions
Revised:  Preliminary Experiments

Original: In this section, we present some preliminary experiments on two real-world datasets from the financialdomain to validate the efficacy of the proposed method.
Revised: In this section, we present some preliminary experiments on two real-world datasets from the financial domain to validate the efficacy of the proposed method.

Revised: This is ongoing work, and we are still conducting more experiments on more temporal graph datasets with more baseline models as well.

Original:  The datasets we use are Bitcoin-OTC and Bitcoin-Alpha datasets.
Revised: Datasets The datasets we use are Bitcoin-OTC and Bitcoin-Alpha datasets.

Original: They contain who-trusts-whom transaction networks of people who trade onthe OTC and Alpha platforms [8, 9].
Revised: They contain whotrusts-whom transaction networks of people who trade on the OTC and Alpha platforms [8, 9].

Revised: The statistics of the datasets are summarized in Table 1.

Original:  We evaluate the proposed method over the future link predictiontask (edge classification).
Revised: Task We evaluate the proposed method over the future link prediction task (edge classification).

Original: We concatenate the embeddings of two linked nodes from the last layer andapply an MLP to obtain the class membership probability matrix.
Revised: We concatenate the embeddings of two linked nodes from the last layer and apply an MLP to obtain the class membership probability matrix.

Original: At each time step t , we can utilizethe history information accumulated up time t to predict the edge in the snapshot t + 1 .
Revised: At each time step t , we can utilize the history information accumulated up time t to predict the edge in the snapshot t +1 .

Revised: For other tasks like node classification or graph classification, we plan to add them in the future to further verify the capability of the model.

Original:  We choosethe mean reciprocal rank (MRR) to assess the models in terms of utility and choose the proposed IF-D to evaluate the models in terms of individual fairness.
Revised: Evaluation metrics We choose the mean reciprocal rank (MRR) to assess the models in terms of utility and choose the proposed IF-D to evaluate the models in terms of individual fairness.

Revised: MRR score is the mean of reciprocal ranks over all nodes u .

Original: We use the last 10% of snapshots fortesting.
Revised: We use the last 10% of snapshots for testing.

Revised: For each node u with positive edge ( u, v ) at t + 1 , we randomly sample 1,000 negative edges starting from node u and identify the rank of edge ( u, v ) ’s prediction score among all other negative edges.

Original:  For the backbone dynamic GNN models, we choose the most popular EvolveGCN [10].
Revised: [Table] Baselines For the backbone dynamic GNN models, we choose the most popular EvolveGCN [10].

Original: Both EvolveGCN-H and EvolveGCN-O use an RNN to dynamically update the training parameters of 
Revised: Both EvolveGCN-H and EvolveGCN-O use an RNN to dynamically update the training parameters of GNNs.

Original: GNNs.
Revised: The former can be implemented by using a standard GRU, while the latter requires a straightforward extension of the standard LSTM from the vector version to the matrix version.

Original: From Table 1, we can conclude that our proposed method can greatly improve individualfairness for temporal financial networks without sacrificing the utility of the backbone model toomuch.
Revised: From Table 2, we can conclude that our proposed method can greatly improve individual fairness for temporal financial networks without sacrificing the utility of the backbone model too much.

Revised: Surprisingly, in some datasets, our method can even improve the utility when combined with some backbone models.

Original: We can further conduct more experiments to verify the potential of our method in terms offairness awareness on more temporal graph datasets.
Revised: We can further conduct more experiments to verify the potential of our method in terms of fairness awareness on more temporal graph datasets.

Revised:  Conclusion

Revised: In this work, we extend the investigation of individual fairness from static graphs to temporal graphs.

Revised: We first revisit the current individual fairness metric for static graphs based on the ( D 1 , D 2 )

Revised: -Lipschitz property and then generalize it under the temporal setting by taking the time-aware fairness constraint into account.

Revised: A new fairness requirement termed ( K, D 1 , D 2 ) -Lipschitz is presented accordingly based on the principle of individual fairness in retrospect.

Revised: We also propose a novel bias measure with regard to individual fairness on dynamic graphs, together with a regularization-based fairnessaware method for dynamic GNNs.

Revised: Some preliminary experimental results on two financial networks are shown to validate the promising efficacy of the proposed method.

Revised: As an ongoing work, we will try to deal with the challenge of the varying number of nodes in the different snapshots, the training efficiency for large-scale real-world datasets, and the theoretical guarantee of the proposed method.

Revised: We will also conduct more experiments on more temporal datasets, try other designs of the monotonically non-decreasing function L ( · ) , and evaluate performance in the live-update training paradigm [15].