Facebook and Instagram are no longer the social media channels they used to be.
Years ago, you could start a page (or a profile) for your shop and within a week, you'd start getting a consistent stream of traffic (and sales) from social channels. It wasn't easy, but you could do it. Today, that's a whole lot harder to achieve.
Every social channel you can think of is overcrowded. As a result, what you post hardly gets seen. If you run a Facebook page, what you post might show up on as little as 6% of your fans. Tweets (which typically have a 24-minute half-life) are even worse. And with the new algorithms on Instagram, businesses are taking a hit.
But some ecommerce brands are killing it, even with all these changes. And there's one reason why: Engagement.
Social Media Engagement is the key to reaching more fans and getting more traffic to your online store.
Every social network prioritises shows their users posts that they are more likely to engage with. Engagement plays a big role in deciding the "reach" of your page, your posts, your photos etc.
Engagement encompasses various forms of interaction, including liking, clicking, sharing, favouriting and commenting. Engagement tells Facebook that this person likes/loves this post (and with it, this page, this type of a post, within this type of a timeframe etc).
Typically, more engagement = more reach.
So the million-dollar question is this: how do we increase social media engagement?
There are many ways to get people to like or comment or share your post (without being unethical about it). Much of it comes out of psychology – the reason we like, comment or share.
In this post, we'll take a look at 15 ways you can implement right now to increase social media engagement, which in turn will lead to a greater reach and more traffic to your store.
There are two things that make Facebook Live work like crazy right now.
Whenever you go live, all your followers, fans and friends get a notification about it. Forget worrying about featuring on someone's timeline. Your fans and followers actually get a "notification".
If you think Facebook Live hasn't got anything to do with an ecommerce store, you might be missing out on a lot of social reach and traffic. While Facebook Live is being used only by news agencies and recipe-makers and marketing gurus, you could be crafting out videos catered to your target market. The best part is the whole canvas is open to you.
You can spot trends if you're active on social media. Twitter makes it easy to find with the trending hashtags. Facebook shows a small trending ticker (that it curates with the help of people and algorithms). And then there's Google Trends.
If you can find an angle for your posts and business that includes the trending thing (use hashtags to get found), you can ride on the wave of trends.
Sure, this is not something that would last long but every trend that you can use multiplies your average engagement and reach.
They get noticed far better than the usual text or emoji replies.
Because they get noticed, they get liked or favourited easily.
Getting a small engagement — even on your replies — helps your profile look highly-engaging to social channels. And this will help you increase your average reach and find more people ready to click-through from your social channels to your store!
This is easily one of the best (and underused) ways to boost social engagement. All it takes is a post dedicated to a customer who reviewed your product.
The reason posts highlighting a customer review work well (and boost engagement) is that they are big ego-boosts. The customer is happy that she was mentioned by your brand (which increases brand loyalty). Your brand comes across as someone who values customers and customer-service highly. And that's a big boost to conversions.
Facebook Groups – once thought to be a dead project from Facebook – have experienced something of a revival in the last couple of years. Today, Ecommerce startups are able to sell tens of thousands worth of merchandise through Facebook Groups alone.
There's one big advantage to having a Facebook group over a page.
The reach on a Facebook Group can be almost 100%.
Every group member sees your post. (This does change depending on the engagement within your group and the way that individual users have their settings, but typically, group posts see a far better reach than pages).
Whilst the latter sounds easy to start with, it's an uphill task to get people to join your group. Becoming a part of existing Facebook groups might sound hard at first but once you start joining groups and contributing (instead of just trying to sell), your engagement will shoot up.
On Instagram or Facebook, have you seen posts that ask you to "Tag a friend"?
Tag their friends — which increases engagement.
Some Ecommerce startups are very good at this. When you are new to the playing field, you should try to partner up with other similar players.
If you have a Facebook page for your sports apparel shop, you could partner up with a variety of pages including gyms, personal trainers, healthy-living startups, food companies etc.
People follow/like brands on social media for different reasons. But one of them stands out as the most popular reason: offers!
People want to know when a brand hosts a giveaway, a contest, a promotion or a sale. This is one of the most important reasons why people follow a brand. Especially if you're an Ecommerce brand.
So, host contests, post about a sale or a discount, and do giveaways to boost engagement by leaps and bounds.
Contests, giveaways or a sale can take up a lot of work but with the right set of tools and planning, you can get it done quicker and with less toil.
Curation of content is the key to making your social profile/page interesting enough for your fans and followers – and therefore a vital step if you're to increase social media engagement.
If you run an online shoe store, there's only so many of your shoes that you can post on your social pages every day. There is a risk that your followers will get bored and stop engaging with your posts.
Pick content from other websites, blogs, YouTube channels, even other Facebook pages / Pinterest profiles etc and share it on yours. This is the hack: when you do it, tag the people/page so as to credit them. This also earns their attention and gives your post an engagement boost.
Social search is so much easier thanks to the widespread use of hashtags. Millions of people use hashtags every day and this gives you an opportunity to find them — and engage with them.
Using the right hashtags and waiting for people to discover your posts is a passive way of doing social media these days.
The active way is to go out there, search for people who use hashtags related to your market and then connect with them.
If you're selling shoes, connect with people using hashtags related to your niche and other verticals (like health, workouts, athletics and sports etc).
Lists of things have some sort of a magnetic power. We are attracted to lists posts most likely because lists are easy to digest, easy to skim-through and get the gist and so on. You've probably skimmed through parts of this article, for example.
Lists posts that are native on social channels (you don't have to click-through to a website to find out what's in the list) are becoming super-effective in earning likes and shares.
The list post gives you a legit excuse to tag a lot of people/brands mentioned. By default, this gives you more exposure whenever the tagged people engage with your post.
Polls and surveys on social networks somehow get more interaction than an average post. If you can incentivise your poll/survey then this is even better.
Twitter and Facebook have built-in polling features that you can try. On other channels, you can run your own polls and ask people to comment their answers.
Despite being a great way to boost social media engagement, polls are generally used very poorly. Get creative and ask questions that will get people's attention and you'll notice your social engagement go up (along with the reach).
A lot of businesses post something concocted on-the-spot during holidays and time-bound events. That's not wrong but it is one the least effective ways of approaching seasonal events and holidays.
If you know what your fans would like to see during these seasonal events and holidays, you can post just that and this will help increase engagement on your posts. To do this, all you need to do is be content-ready.
What this means is to be ready with special promotions and offers. It also means being ready with something creative enough to entertain your fans/followers. Instead of posting what everyone else posts on a special day, post something that meets the expectations (or exceeds them) of your followers. This is one way to boost your engagement — and prevent your posts from disappearing in cyberspace.
The most-overlooked way of boosting social engagement is adding a call-to-action.
You probably see dozens of calls to action every day, even if you don't realise it. "Click share to let your friends know about this!" or "Check it out. Go right here: <somelinkhere.com>"? These are call-to-actions that play on our basic triggers. We are motivated to share or click when we see an explicit CTA.
CTAs are often overrated (because everyone's talking about them). But within the social context, they are super-useful in giving you many spikes on the engagement graph.
Let's say you have a presence on Pinterest, Instagram and Facebook.
The smart social marketer does one thing that the rest of us won't. She promotes her Facebook page on her Instagram profile. She promotes her Pinterest boards on her Facebook page. And so on. You'd be surprised how easy a small cross-promotion is. And while you won't suddenly see a huge spike in followers, they build-up. The trick is to do this every once in awhile (without overdoing it, of course).
Cross-promote all your social channels on other social channels so you can reach your fans/followers on all social media and increase social media engagement without having to win any new fans.
In this article we've highlighted just some of the ideas you can use to boost your social media engagement. Feel free to experiment with newer, better, smarter, more creative ways all the time – and feel free to drop a comment if you've found something that we've missed!


Our football team has progressed to the next round of the cup by beating Quarry Hill Primary 2-0 in their latest game.
The game was watched by a large number of parents and staff.
From everyone at Chadwell St Mary Primary, well done boys and good luck in the next round!


text:
Richard Leo Strauss
Obituary of Richard Leo Strauss
Richard Leo Strauss, 76, passed away on July 30, 2022 in Mill Creek, WA, in the arms of his beloved family.
He was born in Yuba City, CA, on June 22, 1946 to Ada Lavene Young and Nathan A. Strauss.
Richard was a veteran of the United States Marine Corps, serving from 1963-1967, and a recipient of the Navy Unit Commendation Medal for actions during the Vietnam War.
He was a gifted mechanic who practiced his trade in Davis, CA, for over four decades, winning recognition for his skill and dedication to his customers. Richard was passionate about his home and his garden, always working and building to make them a more comfortable and beautiful refuge for himself and his family. He was self-taught in all of his pursuits and loved to read, absorbing anything from the westerns of Louis L'Amour to the obtuse novels of Umberto Eco. He loved the laughter of children, the twang of a country guitar, the passion of an operatic tenor, and the crash of waves on an ocean beach.
He is survived by his wife of 54 years, Heidi; his son Richard Nicholas; and his sister Merry Lavene Morrissey. A Celebration of Life event will be held in Mill Creek, WA.
Memorial contributions may be made to The United States Marine Corps' Toys for Tots program or to the St. Jude Children's Hospital.
If you wish to contact Richard's family, please email dickstraussmemorial@gmail.com.
"I was by no means a scholar, simply an interested reader
with nothing to do but live and learn."

redpajama_set_name: RedPajamaCommonCrawl


text:
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text:
\section{Introduction}
\noindent Graphs, such as social networks, biomedical networks, and traffic networks, are commonly observed in many real-world applications. A lot of graph-based machine learning methods have been proposed in the past decades, and they have shown promising performance in tasks like node similarity measurement, node classification, graph regression, and community detection. In recent years, graph neural networks (GNNs) have been actively studied~\citep{scarselli2008graph, wu2020comprehensive, jiang2019censnet, jiang2020co, zhu2021automated, zhu2021transferable, zhu2021cross, hua2020less,chu2021graph}, which can model graphs with high-dimensional attributes in the non-Euclidean space and have achieved great success in many areas such as recommender systems~\citep{sheu2021knowledge}.
However, it has been observed that many graphs are biased, and thus GNNs trained on the biased graphs may be unfair with respect to certain sensitive attributes such as demographic groups. For example, in a social network, if the users with the same gender have more active connections, the GNNs tend to pay more attention to such gender information and lead to gender bias by recommending more friends to a user with the same gender identity while ignoring other attributes like interests. And from the data privacy perspective, it is possible to infer one's sensitive information from the results given by GNNs~\citep{sun2018adversarial}. In a time when GNNs are widely deployed in the real world, this severe unfairness is unacceptable. Thus, fairness in graph learning emerges and becomes notable very recently.

Existing work on fair graph learning mainly focuses on the pre-processing, in-processing, and post-processing steps in the graph learning pipeline in order to mitigate the unfairness issues. The pre-processing approaches modify the original data to conceal sensitive attributes. Fairwalk~\citep{Rahman2019fairwalk} is a representative pre-processing method, which enforces each group of neighboring nodes an equal chance to be chosen in the sampling process. In many in-processing methods, the most popular way is to add a sensitive discriminator as a constraint, in order to filter out sensitive information from original data. For example, FairGNN~\citep{dai2021say} adopts a sensitive classifier to filter node embeddings. CFC~\citep{bose2019compositional} directly adds a filter layer to deal with unfairness issues. The post-processing methods directly force the final prediction to satisfy fairness constraints, such as \citep{hardt2016equality}.

When the graphs have complete node attributes, existing fair graph learning methods could obtain promising performance on both fairness and accuracy. However, in practice, graphs may contain nodes whose attributes are entirely missing due to various reasons (e.g., newly added nodes, and data privacy concerns). Taking social networks as an example, a newly registered user may have incomplete profiles. Given such incomplete graphs, existing fair graph learning methods would fail, as they assume all the nodes have attributes for model training. Although FairGNN~\citep{dai2021say} also involves the missing attribute problem, it only assumes that a part of the sensitive attributes are missing. To the best of our knowledge, addressing the unfairness issue on graphs with some nodes whose attributes are entirely missing has not been investigated before. Another relevant topic is graph attribute completion~\citep{jin2021heterogeneous, chen2020learning}. It mainly focuses on completing a precise graph but ignores the unfairness issues. In this work, we aim to jointly complete a graph with missing attributes and mitigate unfairness at both feature and topology levels.

In this paper, we study the new problem of learning fair embeddings for graphs with missing attributes. Specifically, we aim to address two major challenges: (1) how to obtain meaningful node embeddings for graphs with missing attributes, and (2) how to enhance fairness of node embeddings with respect to sensitive attributes. To address these two challenges, we propose a Fair Attribute Completion (FairAC) framework. For the first challenge, we adopt an autoencoder to obtain feature embeddings for nodes with attributes and meanwhile we adopt an attention mechanism to aggregate feature information of nodes with missing attributes from their direct neighbors. Then, we address the second challenge by mitigating two types of unfairness, i.e., feature unfairness and topological unfairness. We adopt a sensitive discriminator to regulate embeddings and create a bias-free graph.

The main contributions of this paper are as follows:
(1) We present a new problem of achieving fairness on a graph with missing attributes. Different from the existing work, we assume that the attributes of some nodes are entirely missing.

(2) We propose a new framework, FairAC, for fair graph attribute completion, which jointly addresses unfairness issues from the feature and topology perspectives.
(3) FairAC is a generic approach to complete fair graph attributes, and thus can be used in many graph-based downstream tasks.
(4) Extensive experiments on benchmark datasets demonstrate the effectiveness of FairAC in eliminating unfairness and maintaining comparable accuracy.

\section{Related Work}

\subsection{Fairness in Graph Learning}
Recent work promotes fairness in graph-based machine learning~\citep{bose2019compositional,Rahman2019fairwalk,dai2021say,wang2022unbiased}. They can be roughly divided into three categories, i.e., the pre-processing methods, in-processing methods, and post-processing methods.

The pre-processing methods are applied before training downstream tasks by modifying training data. For instance, Fairwalk~\citep{Rahman2019fairwalk} improves the sampling procedure of node2vec~\citep{grover2016node2vec}.
Our FairAC framework can be viewed as a pre-processing method, as it seeks to complete node attributes and use them as input of graph neural networks. However,
our problem is much harder than existing problems, because the attributes of some nodes in the graph are entirely missing, including both the sensitive ones and non-sensitive ones.
Given an input graph with missing attributes, FairAC generates fair and complete feature embeddings and thus can be applied to many downstream tasks, such as node classification, link prediction~\citep{liben2007link, taskar2003link}, PageRank~\citep{haveliwala2003topic}, etc. Graph learning models trained on the refined feature embeddings would make fair predictions in downstream tasks.

There are plenty of fair graph learning methods as in-processing solutions.
Some work focus on dealing with unfairness issues on graphs with complete features. For example, GEAR~\citep{ma2022learning} mitigates graph unfairness by counterfactual graph augmentation and an adversarial learning method to learn sensitive-invariant embeddings. However, in order to generate counterfactual subgraphs, they need precise and entire features for every node. In other words, it cannot work well if it encounters a graph with full missing nodes since it cannot generate counterfactual subgraph based on a blank node. But we can deal with the situation.
The most related work is FairGNN~\citep{dai2021say}. Different from the majority of problem settings on graph fairness.
It learns fair GNNs for node classification in a graph where only a limited number of nodes are provided with sensitive attributes.  FairGNN adopts a sensitive classifier to predict the missing sensitive labels. After that, it employs a classic adversarial model to mitigate unfairness.Specifically, a sensitive discriminator aims to predict the known or estimated sensitive attributes, while a GNN model tries to fool the sensitive discriminator and meanwhile predicts node labels. However, it cannot predict sensitive information if a node misses all features in the first place and thus will fail to achieve its final goal. Our FairAC can get rid of the problem because we recover the node embeddings from their neighbors. FairAC learns attention between neighbors according to existing full attribute nodes, so we can recover the node embeddings for missing nodes from their neighbors by aggregating the embeddings of neighbors. With the help of the adversarial learning method, it can also remove sensitive information.
In addition to attribute completion, we have also designed novel de-biasing strategies to mitigate feature unfairness and topological unfairness.

\subsection{Attribution Completion on Graphs}
The problem of missing attributes is ubiquitous in reality. Several methods~\citep{liao2016slr, you2020handling, chen2020learning, he2022analyzing, jin2021heterogeneous,Jin2022Amer,Tu2022Initializing,taguchi2021graph} have been proposed to address this problem. GRAPE~\citep{you2020handling} tackles the problem of missing attributes in tabular data using a graph-based approach. SAT~\citep{chen2020learning} assumes that the topology representation and attributes share a common latent space, and thus the missing attributes can be recovered by aligning the paired latent space. \cite{he2022analyzing} and \cite{jin2021heterogeneous} extend such problem settings to heterogeneous graphs. HGNN-AC~\citep{jin2021heterogeneous} is an end-to-end model, which does not recover the original attributes but generates attribute representations that have sufficient information for the final prediction task. It is worth noting that existing methods on graph attribute completion only focus on the attribute completion accuracy or performance of downstream tasks, but none of them takes fairness into consideration. Instead, our work pays attention to the unfairness issue in graph learning, and we aim to generate fair feature embeddings for each node by attribute completion, which contain the majority of information inherited from original attributes but disentangle the sensitive information.

\begin{figure*}[t]
      \centering
      \includegraphics[width=0.85\linewidth]{FairAC.png}
      \vspace{-2mm}
      \caption{Overview of our FairAC framework. FairAC is composed of three major modules, i.e., an autoencoder for embedding nodes, an attributes completion module, and sensitive classifiers for mitigating feature unfairness and topological unfairness. The solid circles indicate nodes with full attributes, while the empty circles indicate nodes without any attributes.}
      \vspace{-4mm}
      \label{fig:framework}
\end{figure*}

\section{Methodology}

\subsection{Problem Definition}
\label{3.1}
Let $\mathcal{G=(V,E,X)}$ denote an undirected graph, where $\mathcal{V} = \{v_1, v_2, ..., v_N\}$ is the set of $N$ nodes, $\mathcal{E} \subseteq \mathcal{V} \times \mathcal{V}$ is the set of undirected edges in the graph, $\mathcal{X} \in \mathbb{R}^{N\times D}$ is the node attribute matrix, and $D$ is the dimension of attributes. $\mathcal{A} \in \mathbb{R}^{N\times N}$ is the adjacency matrix of the graph $\mathcal{G}$, where $\mathcal{A}_{ij} = 1$ if nodes $v_i$ and $v_j$ are connected; otherwise, $\mathcal{A}_{ij} = 0$. In addition, $\mathcal{S}=\{s_1, s_2, ..., s_N\}$ denotes a set of sensitive attributes (e.g., age or gender) of $N$ nodes, and $\mathcal{Y}=\{y_1, y_2, ..., y_N\}$ denotes the node labels. The goal of fair graph learning is to make fair predictions of node labels with respect to the sensitive attribute, which is usually measured by certain fairness notations like statistical parity~\citep{dwork2012fairness} and equal opportunity~\citep{hardt2016equality}.
Statistical Parity and Equal Opportunity are two group fairness definitions. Their detailed formulations are presented below. The label $y$ denotes the ground-truth node label, and the sensitive attribute $s$ indicates one's sensitive group. For example, for binary node classification task, y only has two labels. Here we consider two sensitive groups, i.e. $s\in\{0,1\}$.
\begin{itemize}
    \item Statistical Parity~\citep{dwork2012fairness}. It refers to the equal acceptance rate, which can be formulated as:
    \begin{equation}
    \small
    \label{equ:Parity}
	P(\hat{y}|s=0) = P(\hat{y}|s=1),
    \end{equation}
    where $P(\cdot)$ denotes the probability that $\cdot$ occurs.

    \item Equal Opportunity~\citep{hardt2016equality}. It means the probability of a node in a positive class being classified as a positive outcome should be equal for both sensitive group nodes. It mathematically requires an equal true positive rate for each subgroup.
    \begin{equation}
    \small
    \label{equ:Equality}
	P(\hat{y}=1|y=1,s=0) = P(\hat{y}=1|y=1,s=1).
    \end{equation}
\end{itemize}

In this work, we mainly focus on addressing unfairness issues on graphs with missing attributes, i.e., attributes of some nodes are totally missing. Let $\mathcal{V}^+$ denote the set of nodes whose attributes are available, and $\mathcal{V}^-$ denote the set of nodes whose attributes are missing, $\mathcal{V}=\{\mathcal{V}^+, \mathcal{V}^- \}$. If $v_i \in \mathcal{V}^-$, both $\mathcal{X}_i$ and $s_i$ are unavailable during model training.

With the notations given below, the fair attribute completion problem is formally defined as:

\textit{
\textbf{Problem 1}. Given a graph $\mathcal{G=(V,E,X)}$, where node set $\mathcal{V}^+\in \mathcal{V}$ with the corresponding attributes available and the corresponding sensitive attributes in $\mathcal{S}$, learn a fair attribute completion model to generate fair feature embeddings $\mathcal{H}$ for each node in $\mathcal{V}$, i.e.,
\begin{equation}
    \small
    \label{equ:Problem definition}
	f(\mathcal{G}, S) \rightarrow \mathcal{H},
\end{equation}
where $f$ is the function we aim to learn. $\mathcal{H}$ should exclude any sensitive information while preserve non-sensitive information.
}

\subsection{Fair Attribute Completion (FairAC) Framework}

\begin{algorithm}[tb]
\caption{FairAC framework algorithm}
\label{alg:algorithm}
\textbf{Input}: $\mathcal{G=(V,E,X)}$, $\mathcal{S}$\\
\textbf{Output}: Autoencoder $f_{AE}$, Sensitive classifier $C_s$, Attribute completion $f_{AC}$
\begin{algorithmic}[1]
\STATE Obtain topological embedding $\mathcal{T}$ with DeepWalk
\REPEAT
\STATE Obtain the feature embeddings H with $f_{AE}$
\STATE Optimize the $C_s$ by Equation \ref{loss:Cs}
\STATE Optimize $f_{AE}$ to mitigate feature unfairness by loss $\mathcal{L}_F$

\STATE Divide $\mathcal{V}^+$ into $\mathcal{V}_{keep}$ and $\mathcal{V}_{drop}$ based on $\alpha$
\STATE Obtain the feature embeddings of nodes with missing attributes $\mathcal{V}_{drop}$ by $f_{AC}$
\STATE Optimize $f_{AC}$ to achieve attribute completion by loss $\mathcal{L}_C$
\STATE Optimize $f_{AC}$ to mitigate topological unfairness by loss $\mathcal{L}_T$
\UNTIL{convergence}
\STATE \textbf{return} $f_{AE}$, $C_s$, $f_{AC}$
\end{algorithmic}
\end{algorithm}

We propose a fair attribute completion (FairAC) framework to address Problem 1. Existing fair graph learning methods tackle unfairness issues by training fair graph neural networks in an end-to-end fashion, but they cannot effectively handle graphs that are severely biased due to missing attributes. Our FairAC framework, as a data-centric approach, deals with the unfairness issue from a new perspective, by explicitly debiasing the graph with feature unfairness mitigation and fairness-aware attribute completion. Eventually, FairAC generates fair embeddings for all nodes including the ones without any attributes. The training algorithms are shown in Algorithm \ref{alg:algorithm}.

To train the graph attribute completion model, we follow the setting in~\citep{jin2021heterogeneous} and divide the nodes with attributes (i.e., $\mathcal{V}^+$) into two sets: $\mathcal{V}_{keep}$ and $\mathcal{V}_{drop}$. For nodes in $\mathcal{V}_{keep}$, we keep their attributes, while for nodes in $\mathcal{V}_{drop}$, we temporally drop their attributes and try to recover them using our attribute completion model. Although the nodes are randomly assigned to $\mathcal{V}_{keep}$ and $\mathcal{V}_{drop}$, the proportion of $\mathcal{V}_{drop}$ is consistent with the attribute missing rate $\alpha$ of graph $\mathcal{G}$, i.e., $\alpha = \frac{|\mathcal{V}^-|}{|\mathcal{V}|} = \frac{|\mathcal{V}_{drop}|}{|\mathcal{V}^+|}$.

Different from existing work on fair graph learning, we consider unfairness from two sources. The first one is from node features. For example, we can roughly infer one's sensitive information, like gender, from some non-sensitive attributes like hobbies. It means that non-sensitive attributes may imply sensitive attributes and thus lead to unfairness in model prediction. We adopt a sensitive discriminator to mitigate feature unfairness. The other source is topological unfairness introduced by graph topological embeddings and node attribute completion. To deal with the topological unfairness, we force the estimated feature embeddings to fool the sensitive discriminator, by updating attention parameters during the attribute completion process.

As illustrated in Figure \ref{fig:framework}, our FairAC framework first mitigates feature unfairness for nodes with attributes (i.e., $\mathcal{V}_{keep}$) by removing sensitive information implicitly contained in non-sensitive attributes with an auto-encoder and sensitive classifier (Section~\ref{sec3.1}). For nodes without features (i.e., $\mathcal{V}_{drop}$), FairAC performs attribute completion with an attention mechanism (Section~\ref{sec3.2}) and meanwhile mitigates the topological unfairness (Section~\ref{sec3.3}). Finally, the FairAC model trained on $\mathcal{V}_{keep}$ and $\mathcal{V}_{drop}$ can be used to infer fair embeddings for nodes in $\mathcal{V}^-$.

The overall loss function of FairAC is formulated as:
\begin{equation}
\small
\label{loss:final}
	\mathcal{L} = \mathcal{L}_{F} + \mathcal{L}_{C} + \beta\mathcal{L}_{T},
\end{equation}
where $\mathcal{L}_{F}$ represents the loss for mitigating feature unfairness, $\mathcal{L}_{C}$ is the loss for attribute completion, and $\mathcal{L}_{T}$ is the loss for mitigating topological unfairness. $\beta$ is a trade-off hyperparameter.

\subsubsection{Mitigating feature unfairness}
\label{sec3.1}

The nodes in $\mathcal{V}_{keep}$ have full attributes $\mathcal{X}$, while some attributes may implicitly encode information about sensitive attributes $\mathcal{S}$ and thus lead to unfair predictions. To address this issue, FairAC aims to encode the attributes $\mathcal{X_i}$ of node $i$ into a fair feature embedding $\mathcal{H}_i$. Specifically, we use a simple autoencoder framework together with a sensitive classifier. The autoencoder maps $\mathcal{X}_i$ into embedding $\mathcal{H}_i$, and meanwhile the sensitive classifier $C_s$ is trained in an adversarial way, such that the embeddings are invariant to sensitive attributes.

\paragraph{Autoencoder.}
The autoencoder contains an encoder $f_E$ and a decoder $f_D$. $f_E$ encodes the original attributes $\mathcal{X}_i$ to feature embeddings $\mathcal{H}_{i}$, i.e., $\mathcal{H}_{i} = f_E(\mathcal{X}_i)$, and $f_D$ reconstructs attributes from the latent embeddings, i.e., $\hat{\mathcal{X}_i} = f_D(\mathcal{H}_i)$,
where the reconstructed attributes $\hat{\mathcal{X}}$ should be close to $\mathcal{X}_i$ as possible.
The loss function of the autoencoder is written as:
\begin{equation}
\small
\label{loss:reconstruction}
	\mathcal{L}_{ae} = \frac{1}{|\mathcal{V}_{keep}|}\sum_{i\in \mathcal{V}_{keep}|}\sqrt{(\hat{\mathcal{X}_i} - \mathcal{X}_i)^2}.
\end{equation}

\paragraph{Sensitive classifier}
The sensitive classifier $C_s$ is a simple multilayer perceptron (MLP) model. It takes the feature embedding $\mathcal{H}_i$ as input and predicts the sensitive attribute $\hat{s}_i$, i.e., $\hat{s}_i = C_s(\mathcal{H}_i)$. When the sensitive attributes are binary, we can use the binary cross entropy loss to optimize $C_s$:
\begin{equation}
\small
\label{loss:Cs}
	\mathcal{L}_{C_s} = -\frac{1}{|\mathcal{V}_{keep}|}\sum_{i\in \mathcal{V}_{keep}}s_i\log{\hat{s_i}} + (1-s_i)\log{(1-\hat{s_i})}.
\end{equation}
With the sensitive classifier $C_s$, we could leverage it to adversarially train the autoencoder, such that $f_E$ is able to generate fair feature embeddings that can fool $C_s$.
The loss $\mathcal{L}_F$ is written as: $\mathcal{L}_F = \mathcal{L}_{ae} -\beta \mathcal{L}_{C_s}$.

\subsubsection{Completing node embeddings via attention mechanism}
\label{sec3.2}
For nodes without attributes ($\mathcal{V}_{drop}$), FairAC makes use of topological embeddings and completes the node embeddings $\mathcal{H}_{drop}$ with an attention mechanism.

\paragraph{Topological embeddings.} Recent studies reveal that the topology of graphs has similar semantic information as the attributes~\citep{chen2020learning,mcpherson2001birds,pei2020geom,zhu2020beyond}. Inspired by this observation, we assume that the nodes' topological information can reflect the relationship between nodes' attributes and the attributes of their neighbors. There are a lot of off-the-shelf node topological embedding methods, such as DeepWalk~\citep{perozzi2014deepwalk} and node2vec~\citep{grover2016node2vec}. For simplicity, we adopt the DeepWalk method to extract topological embeddings for nodes in $\mathcal{V}$.

\paragraph{Attention mechanism.} For graphs with missing attributes, a commonly used strategy is to use average attributes of the one-hop neighbors. This strategy works in some cases, however, simply averaging information from neighbors might be biased, as the results might be dominated by some high-degree nodes. In fact,
different neighbors should have varying contributions to the aggregation process in the context of fairness. To this end, FairAC adopts an attention mechanism~\citep{vaswani2017attention} to learn the influence of different neighbors or edges with the awareness of fairness, and then aggregates attributes information for nodes in $\mathcal{V}_{drop}$.

Given a pair of nodes $(u,v)$ which are neighbors, the contribution of node $v$ is the attention $att_{u,v}$, which is defined as: $att_{u,v} = Attention(T_u, T_v)$, where $T_u, T_v$ are the topological embeddings of nodes $u$ and $v$, respectively. Specifically, we only focus on the neighbor pairs and ignore those node pairs that are not directly connected. $Attention(\cdot,\cdot)$ denotes the attention between two topological embeddings, i.e., $Attention(T_u, T_v) = \sigma(T_u^TWT_v)$, where $W$ is the learnable parametric matrix, and $\sigma$ is an activation function. After we get all the attention scores between one node and its neighbors, we can get the coefficient of each pair by applying the softmax function:
\begin{equation}
\small
\label{equ:coefficient}
    c_{u,v} = \text{softmax}(att_{u,v}) = \frac{\exp(att_{u,v})}{\sum_{s\in N_u} \exp(att_{u,s})},
\end{equation}
where $c_{u,v}$ is the coefficient of node pair $(u,v)$, and $N_u$ is the set of neighbors of node $u$.
For node $u$, FairAC calculates its feature embedding $\hat{\mathcal{H}}_u$ by the weighted aggregation with multi-head attention:
\begin{equation}
\small
\label{equ:multi_aggregation}
    \hat{\mathcal{H}}_u = \frac{1}{K}\sum_{k=1}^K\sum_{s\in N_u} c_{u,s}\mathcal{H}_s,
\end{equation}
where $K$ is the number of attention heads. The loss for attribute completion with topological embedding and attention mechanism is formulated as:
\begin{equation}
\small
\label{loss:completion}
    \mathcal{L}_{C} = \frac{1}{|\mathcal{V}_{drop}|}\sum_{i\in \mathcal{V}_{drop}|}\sqrt{(\hat{\mathcal{H}}_i - \mathcal{H}_i)^2}.
\end{equation}

\subsubsection{Mitigating topological unfairness}
\label{sec3.3}
The attribute completion procedure may introduce topological unfairness since we assume that topology information is similar to attributes relation. It is possible that the completed feature embeddings of $\mathcal{V}_{drop}$ would be unfair with respect to sensitive attributes $\mathcal{S}$. To address this issue, FairAC leverages sensitive classifier $C_s$ to help mitigate topological unfairness by further updating the attention parameter matrix $W$ and thus obtaining fair feature embeddings $\mathcal{H}$.
Inspired by \citep{gong2020jointly}, we expect that the feature embeddings can fool the sensitive classifier $C_s$ to predict the probability distribution close to the uniform distribution over the sensitive category, by minimizing the loss:
\begin{equation}
\small
\label{loss:Cadv2-1}
	\mathcal{L}_{T} = -\frac{1}{|\mathcal{V}_{drop}|}\sum_{i\in \mathcal{V}_{drop}}s_i\log{\hat{s_i}} + (1-s_i)\log{(1-\hat{s_i})}.
\end{equation}

\subsection{FairAC for Node Classification}
The proposed FairAC framework could be viewed as a generic data debiasing approach, which achieves fairness-aware attribute completion and node embedding for graphs with missing attributes. It can be easily integrated with many existing graph neural networks (e.g., GCN~\citep{kipf2016semi}, GAT~\citep{velickovic2018graph}, and GraphSAGE~\citep{hamilton2017inductive}) for tasks like node classification. In this work, we choose the basic GCN model for node classification and assess how FairAC enhances model performance in terms of accuracy and fairness.

\section{Experiments}
\label{others}

\begin{table*}[t]
\centering
\caption{Comparisons of our FairAC method and baselines on three graphs. M refers to missing or not. M is true means that some nodes' attributes are entirely missing and the ratio is controlled by $\alpha$. Otherwise, full attributes are provided. The attribute missing rate $\alpha$ is set to 0.3. GCN and FairGNN are trained on averaging attribute completed graphs. Bold fonts denote the best results.
}
\vspace{-2mm}
\resizebox{\columnwidth}{!}{%
\begin{tabular}{c|c|c||c|c|c|c|c}
		\toprule\toprule
		Dataset &Method & M & Acc $\uparrow$ & AUC $\uparrow$ & $\Delta SP$ $\downarrow$ & $\Delta EO$ $\downarrow$ & $\Delta SP$+$\Delta EO \downarrow$\\
		\hline
			        & GCN   & \checkmark & \textbf{70.66$\pm$0.24} & 74.23$\pm$0.63 & 3.43$\pm$2.44 & 2.74$\pm$0.67 & 6.16$\pm$3.1 \\
	    	        & ALFR  & $\times$ & 64.3$\pm$1.3 & 71.5$\pm$0.3 & 2.3$\pm$0.9 & 3.2$\pm$1.5 & 5.5$\pm$2.4\\
	                & ALFR-e  & $\times$ & 66.0$\pm$0.4 & 72.9$\pm$1.0 & 4.7$\pm$1.8 & 4.7$\pm$1.7 & 9.4$\pm$3.4    \\
	    NBA         & Debias & $\times$ & 63.1$\pm$1.1 & 71.3$\pm$0.7 & 2.5$\pm$1.5 & 3.1$\pm$1.9 & 5.6$\pm$3.4\\
	                & Debias-e & $\times$ & 65.6$\pm$2.4 & 72.9$\pm$1.2 & 5.3$\pm$0.9 & 3.1$\pm$1.3 & 8.4$\pm$2.2\\
	                & FCGE  & $\times$  & 66.0$\pm$1.5 & 73.6$\pm$1.5 & 2.9$\pm$1.0 & 3.0$\pm$1.2 & 5.9$\pm$2.2\\
	                & FairGNN & \checkmark & 70.73$\pm$0.44 & \textbf{76.77$\pm$0.1} & 0.95$\pm$0.7 & 1.63$\pm$0.67 & 2.58$\pm$1.37 \\
	                & FairAC (Ours) & \checkmark & \textbf{70.66$\pm$0.73} & 74.44$\pm$0.67 & \textbf{0.28$\pm$0.25} & \textbf{0.63$\pm$0.34} & \textbf{0.91$\pm$0.59}\\

	    \midrule\midrule
		            & GCN & \checkmark & 67.4$\pm$0.88 & 72.04$\pm$1.78 & 1.96$\pm$0.64 & 4.17$\pm$0.54 & 6.12$\pm$1.18\\
                    & ALFR  & $\times$ & 65.4$\pm$0.3 & 71.3$\pm$0.3 & 2.8$\pm$0.5 & 1.1$\pm$0.4 & 3.9$\pm$0.9\\
	                & ALFR-e  & $\times$ & \textbf{68.0$\pm$0.6} & \textbf{74.0$\pm$0.7} & 5.8$\pm$0.4 & 2.8$\pm$0.8 & 8.6$\pm$1.2\\
	    Pokec-z     & Debias & $\times$ & 65.2$\pm$0.7 & 71.4$\pm$0.6 & 1.9$\pm$0.6 & 1.9$\pm$0.4 & 3.8$\pm$1.0\\
	                & Debias-e & $\times$ & 67.5$\pm$0.7 & 74.2$\pm$0.7 & 4.7$\pm$1.0 & 3.0$\pm$1.4 & 7.7$\pm$2.4\\
	                & FCGE & $\times$ & 65.9$\pm$0.2 & 71.0$\pm$0.2 & 3.1$\pm$0.5 & 1.7$\pm$0.6 & 4.8$\pm$1.1\\
	                & FairGNN & \checkmark & 66.54$\pm$0.45 & 70.10$\pm$0.07 & 0.95$\pm$0.70 & 1.63$\pm$0.67 & 2.58$\pm$0.57 \\
	                & FairAC (Ours) & \checkmark & 66.94$\pm$0.14 & 72.87$\pm$0.13 & \textbf{0.19$\pm$0.07} & \textbf{0.12$\pm$0.07} & \textbf{0.31$\pm$0.14}\\
	    \midrule\midrule
	                & GCN & \checkmark & 66.12$\pm$0.88 & 71.5$\pm$0.1 & 0.46$\pm$0.1 & 1.41$\pm$0.14 & 1.87$\pm$0.24 \\
	                & ALFR  & $\times$ & 63.1$\pm$0.6 & 67.7$\pm$0.5 & 3.05$\pm$0.5 & 3.9$\pm$0.6 & 3.95$\pm$1.1\\
	                & ALFR-e & $\times$  & 66.2$\pm$0.4 & 71.9$\pm$1.0 & 4.1$\pm$1.8 & 4.6$\pm$1.7 & 8.7$\pm$3.5    \\
	    Pokec-n     & Debias  & $\times$ & 62.6$\pm$1.1 & 67.9$\pm$0.7 & 2.4$\pm$1.5 & 2.6$\pm$1.9 & 5.0$\pm$3.4\\
	                & Debias-e & $\times$ & 65.6$\pm$2.4 & 71.7$\pm$1.2 & 3.6$\pm$0.9 & 4.4$\pm$1.3 & 8.0$\pm$2.2\\
	                & FCGE  & $\times$ & 64.8$\pm$1.5 & 69.5$\pm$1.5 & 4.1$\pm$1.0 & 5.5$\pm$1.2 & 9.6$\pm$2.2\\
	                & FairGNN & \checkmark & \textbf{68.54$\pm$0.45} & 70.10$\pm$0.07 & 0.76$\pm$0.15 & 0.48$\pm$0.09 & 1.24$\pm$0.24 \\
	                & FairAC (Ours) & \checkmark & 66.35$\pm$0.24 & \textbf{72.32$\pm$0.08} & \textbf{0.27$\pm$0.12} & \textbf{0.14$\pm$0.11} & \textbf{0.41$\pm$0.23}\\
		\bottomrule \bottomrule
	\end{tabular}%
}
\label{tab:experiment1}
\vspace{-2mm}
\end{table*}

In this section, we evaluate the performance of the proposed FairAC framework on three benchmark datasets in terms of node classification accuracy and fairness w.r.t.~sensitive attributes. We compare FairAC with other baseline methods in settings with various sensitive attributes or different attribute missing rates. Ablation studies are also provided and discussed.

\subsection{Datasets and Settings}

\noindent \textbf{Datasets.} In the experiments, we use three public graph datasets, \textbf{NBA}, \textbf{Pokec-z}, and \textbf{Pokec-n}. A detailed description is shown in supplementary materials.

\textbf{Baselines.} We compare our FairAC method with the following baseline methods: GCN~\citep{kipf2016semi}, ALFR~\citep{ALFR}, ALFR-e, Debias~\citep{zhang2018mitigating}, Debias-e, FCGE~\citep{bose2019compositional}, and FairGNN~\citep{dai2021say}.
ALFR-e concatenates the feature embeddings produced by ALFR with topological embeddings learned by DeepWalk~\citep{perozzi2014deepwalk}.  Debias-e also concatenates the topological embeddings learned by DeepWalk with feature embeddings learned by Debias.
FairGNN is an end-to-end debias method which aims to mitigate unfairness in label prediction task. GCN and FairGNN uses the average attribute completion method, while other baselines use original complete attributes.

\textbf{Evaluation Metrics.} We evaluate the proposed framework with respect to two aspects: classification performance and fairness performance. For classification, we use accuracy and AUC scores
As for fairness, we adopt $\Delta SP$ and $\Delta EO$ as evaluation metrics, which can be defined as:
\begin{equation}
    \small
    \label{equ:DeltaParity}
	\Delta SP = P(\hat{y}|s=0) - P(\hat{y}|s=1),
    \end{equation}
\begin{equation}
    \small
    \label{equ:DeltaEquality}
	\Delta EO = P(\hat{y}=1|y=1,s=0) - P(\hat{y}=1|y=1,s=1).
    \end{equation}
The smaller $\Delta SP$ and $\Delta EO$ are, the more fair the model is. In addition, we use $\Delta SP$+$\Delta EO$ as an overall indicator of a model's performance on fairness.

\begin{table*}[t]
\centering
\caption{Comparisons of our method with the baselines on pokec-z dataset with four levels of attribute missing rates $\alpha$. FairAC generates fair and complete node features, and then GCN is trained for node classification. BaseAC is a simplified version of FairAC, which only has the attention-based attribute completion module, but does not contain the module for mitigating feature unfairness and topological unfairness. Bold fonts denote the best results.}
\begin{tabular}{c|c||c|c|c|c|c}
		\toprule\toprule
		$\alpha$ &Method & Acc (\%)$\uparrow$ & AUC (\%)$\uparrow$ & $\Delta SP$ (\%)$\downarrow$ & $\Delta EO$ (\%)$\downarrow$ & $\Delta SP$+$\Delta EO \downarrow$\\
		\hline
	                & GCN           & 66.10 & 69.14 & 0.88 & 0.20 & 1.08 \\
	    0.1         & FairGNN       & \textbf{69.37} & \textbf{76.93} & 0.17 & \textbf{0.29} & 0.46 \\
	                & BaseAC (Ours)  & 66.37 & 69.34 & \textbf{0.07} & 1.46 & 1.53 \\
	                & FairAC (Ours)  & 66.33 & 69.35 & 0.14 & 0.37 & \textbf{0.51}\\
	    \midrule\midrule
		            & GCN           & 67.40 & 72.04 & 1.96 & 4.17 & 6.12\\
	    0.3         & FairGNN       & 66.54 & 70.10 & 0.95 & 1.63 & 2.58 \\
	                & BaseAC (Ours)  & 66.10 & 69.67 & 0.78 & 1.57 & 2.35\\
	                & FairAC (Ours)  & \textbf{66.94} & \textbf{72.87} & \textbf{0.19} & \textbf{0.12} & \textbf{0.31}\\
	    \midrule\midrule
		            & GCN           & 66.41 & 70.14 & 0.07 & 2.11 & 2.18\\
	    0.5         & FairGNN       & 66.25 & 70.13 & 0.47 & 1.71 & 2.18 \\
	                & BaseAC (Ours)  & 66.13 & 69.93 & 0.38 & 1.60 & 1.98 \\
	                & FairAC (Ours)  & \textbf{66.45} & \textbf{72.95} & \textbf{0.14} & \textbf{1.06} & \textbf{1.20}\\
	    \midrule\midrule
		            & GCN           & \textbf{66.99} & \textbf{73.13} & 0.57 & 1.74 & 2.31 \\
	    0.8         & FairGNN       & 66.60 & 71.35 & 0.94 & \textbf{0.09} & 1.03\\
	                & BaseAC (Ours)  & 66.06 & 71.21 & 0.08 & 2.20 & 2.28 \\
	                & FairAC (Ours)  & 66.10 & 71.66 & \textbf{0.01} & 0.69 & \textbf{0.70}\\
		\bottomrule \bottomrule
	\end{tabular}

\label{tab:experiment2}
\end{table*}
\subsection{Results and Analysis}

\subsubsection{Unfairness issues in Graph Neural Networks}
According to the results showed in Table \ref{tab:experiment1}, they reveal several unfairness issues in Graph Neural Networks. We divided them into two categories.
\begin{itemize}
    \item \textbf{Feature unfairness} Feature unfairness is that some non-sensitive attributes could infer sensitive information. Hence, some Graph Neural Networks may learn this relation and make unfair prediction. In most cases, ALFR and Debias and FCGE have better fairness performance than GCN method. It is as expected because the non-sensitive features may contain proxy variables of sensitive attributes which would lead to biased prediction. Thus, ALFR and Debias methods that try to break up these connections are able to mitigate feature unfairness and obtain better fairness performance. These results further prove the existence of feature unfairness.
    \item \textbf{Topological unfairness} Topological unfairness is sourced from graph structure. In other words, edges in graph, i.e. the misrepresentation due to the connection\citep{mehrabi2021survey} can bring topological unfairness. From the experiments, ALFR-e and Debias-e have worse fairness performance than ALFR and Debias, respectively. It shows that although graph structure can improve the classification performance, it will bring topological unfairness consequently. The worse performance on fairness verifies that topological unfairness exists in GNNs and graph topological information could magnify the discrimination.
\end{itemize}

\subsubsection{Effectiveness of FairAC on mitigating feature and topological unfairness}
The results of our FairAC method and baselines in terms of the node classification accuracy and fairness metrics on three datasets are shown in Table \ref{tab:experiment1}. The best results are shown in bold. Generally speaking, we have the following observations. (1). The proposed method FairAC shows comparable classification performance with these baselines, GCN and FairGNN. This suggests that our attribute completion method is able to preserve useful information contained in the original attributes. (2). FairAC outperforms all baselines regarding fairness metrics, especially in $\Delta SP$+$\Delta EO$. FairAC outperform  baselines that focus on mitigate feature fairness, like ALFR, which proves that FairAC also mitigate topological unfairness. Besides, it is better than those who take topological fairness into consideration, like FCGE, which also validates the effectiveness of FairAC. FairGNN also has good performance on fairness, because it adopts a discriminator to deal with the unfairness issue. Our method performs better than FairGNN in most cases. For example, our FairAC method can significantly improve the performance in terms of the fairness metric $\Delta SP + \Delta EO$, i.e., 65$\%$, 87$\%$, and 67$\%$ improvement over FairGNN on the NBA, pokec-z, pokec-n datasets, respectively. Overall, the results in Table \ref{tab:experiment1} validate the effectiveness of FairAC in mitigating unfairness issues.

\subsection{Ablation Studies}

\paragraph{Attribute missing rate} In our proposed framework, the attribute missing rate indicates the integrity of node attribute matrix, which has a great impact on model performance. Here we investigate the performance of our FairAC method and baselines on dealing with graphs with varying degrees of missing attributes. In particular, we set the attribute missing rate to 0.1, 0.3, 0.5 and 0.8, and evaluate FairAC and baselines on the pokec-z dataset. The detailed results are presented in Table \ref{tab:experiment2}.
From the table, we have the following observation that with varying values of $\alpha$, FairAC is able to maintain its high fairness performance. Especially when $\alpha$ reaches 0.8, FairAC can greatly outperform other methods. It proves that FairAC is effective even if the attributes are largely missing.

\paragraph{The effectiveness of adversarial learning} A key module in FairAC is adversarial learning, which is used to mitigate feature unfairness and topological unfairness. To investigate the contribution of adversarial learning in FairAC, we implement a BaseAC model, which only has the attention-based attribute completion module, but does not contain the adversarial learning loss terms.  Comparing BaseAC with FairAC in Table \ref{tab:experiment2}, we can find that the fairness performance drops desperately when the adversarial training loss is removed. Since BaseAC does not have an adversarial discriminator to regulate feature encoder as well as attribute completion parameters, it is unable to mitigate unfairness. Overall, the results confirm the effectiveness of the adversarial learning module.

\begin{wrapfigure}[14]{r}{0.45\textwidth}
    \centering
    \vspace{-4mm}
    \includegraphics[width=\linewidth]{beta.png}
    \caption{Accuracy and $\Delta SP$+$\Delta EO$ of FairAC when varying $\beta$ on Pokec-z dataset with $\alpha = 0.3$.}
    \label{fig:beta}
\end{wrapfigure}
\paragraph{Parameter analysis}
We investigate how the hyperparameters affect the performance of FairAC. The most important hyperparameter in FairAC is $\beta$, which adjusts the trade-off between fairness and attribute completion. We report the results with different hyperparameter values.
We set $\beta$ to 0.2, 0.4, 0.7, 0.8 and 0 that is equivalent to the BaseAC. We also fix other hyperparameters by setting $\alpha$ to 0.3. As shown in Figure \ref{fig:beta}, we can find that, as $\beta$ increases, the fairness performance improves while the accuracy of node classification slightly declined. Therefore, it validates our assumption that there is a trade-off between fairness and attribute completion, and our FairAC is able to enhance fairness without compromising too much on accuracy.

\section{Conclusions}
In this paper, we presented a novel problem, i.e., fair attribute completion on graphs with missing attributes. To address this problem, we proposed the FairAC framework, which jointly completes the missing features and mitigates unfairness. FairAC leverages the attention mechanism to complete missing attributes and adopts a sensitive classifier to mitigate implicit feature unfairness as well as topological unfairness on graphs.
Experimental results on three real-world datasets demonstrate the superiority  of the proposed FairAC framework over baselines in terms of both node classification performance and fairness performance. As a generic fair graph attributes completion approach, FairAC can also be used in other graph-based downstream tasks, such as link prediction, graph regression, pagerank, and clustering.

\subsubsection*{Acknowledgement}
This research is supported by the Cisco Faculty Award and Adobe Data Science Research Award.

redpajama_set_name: RedPajamaArXiv


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To ensure safety and patient comfort, Royal Victoria Regional Health Centre (RVH) recently completed a major upgrade to its cooling system.
RVH installed a new 27,000 pound air-cooled chiller which will provide an additional 350 tons of cooling to the health centre. As the health centre continues to grow, this chiller was required for additional cooling capacity.
This chiller will service the air handling units for RVH's operating rooms and the Medical Device Reprocessing Department (MDRD) ensuring these areas maintain consistent temperature and humidity levels which is critically important as the health centre gradually begins to ramp up surgical procedures.
"Just like medical equipment, the equipment that keeps our building safely running needs to be upgraded or replaced," says Darrell Sewell, vice president, facilities and chief human resources officer. "With the warmer weather coming, and with resumption of some surgeries, this new chiller means the key areas supporting surgeries – the operating room and MDRD, which does all the equipment sterilization, will not be impacted by fluctuations in temperature. This also adds to the overall safety and comfort of our patients and staff."
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This peak is #12 in Prominence List for Washington .Whitehorse Mountain rises with great bulk above Darrington, WA (over 6000 feet in 3 miles). The mountain has low brushy flanks, and is usually approached from the Lone Tree Pass trail.
Name Notes: The name comes from an event when a white horse ran away from the homestead of pioneer Fred Olds. Someone remarked that the horse-shaped snowpatch on the mountain must be the missing horse.
There was mining activity in the 1890s to high on the mountain.
A made-for-tv movie called "High Ice" was filmed on the mountain in the 1980s. There were exciting scenes of drama on the peak. The Darrington high school gymnasium was used to construct a set of snow and cliff.
Finally, in the late 1990s, a Bulgarian hermit lived in hovels dug into the mountainside. He broke into cabins for food. Residents tolerated him for years, but finally got a professional tracker to look for him when he began vandalizing.