text:
\section{Appendix}
\subsection{Algorithm for Network of SAGE}
Algorithms \ref{alg:sage} and \ref{alg:nsage}, respectively, define SAGE \cite{sage} and Network of SAGE (N-SAGE).
Algorithm \ref{alg:sage} assumes mean-pool aggregation by \cite{sage}, which performs on-par to their top performer max-pool aggregation. Further, Algorithm \ref{alg:sage} operates in full-batch while \cite{sage} offer a stochastic implementation with edge sampling. Nonetheless, their proposed stochastic implementation should be wrapped  in a network, though we would need a way to approximate (e.g. sample entries) from dense $\hat{A}^k$ as $k$ increases. We leave this as future work.

\begin{figure*}[h]
	\begin{minipage}[t]{2.8in}
		\begin{algorithm}[H]
			\caption{SAGE Model \citep{sage}}
			\label{alg:sage}
			\begin{algorithmic}[1]
				\Require{$\hat{A}$ is a normalization of $A$}
				\Function{SageModel}{$\hat{A}$, $X$, $L$}
				\State {$Z \leftarrow X$}
				\For{$i = 1 $ to $L$}
				\State{$Z \leftarrow \sigma(\left[ \begin{array}{c;{2pt/2pt}c}
					Z & \hat{A} Z
					\end{array} \right] W^{(i)})$ }
				\State{$Z \leftarrow \textsc{L2NormalizeRows}(Z)$ }
				\EndFor
				\State{\Return $Z$}
				\EndFunction
			\end{algorithmic}
		\end{algorithm}
	\end{minipage}
	\hfill
	\begin{minipage}[t]{2.8in}
		\begin{algorithm}[H]
			\caption{N-SAGE}
			\label{alg:nsage}
			\begin{algorithmic}[1]
				\Function{Nsage}{$A$, $X$}
				\State{$D \leftarrow \textbf{diag}(A \mathbf{1}) $}
				\Comment{Sum rows}
				\State{$\hat{A}  \leftarrow D^{-1}A$}
				\State{\Return $\textsc{Network}(\textsc{SageModel}, \hat{A}, X, 2)$}
				\EndFunction
			\end{algorithmic}
		\end{algorithm}
	\end{minipage}
	\hfill
\end{figure*}

Using SAGE with mean-pooling aggregation is very similar to a vanilla GCN model but with three differences. First, the choice of adjacency normalization ($D^{-1}A$ versus $D^{-\frac{1}{2}}AD^{-\frac{1}{2}}$). Second, the skip connections in line 4, which concatenates the features with the adjacency-multiplied (i.e. diffused) features. We believe this is analogous in intuition of incorporating $\hat{A}^0$ in our model, which keeps the original features. Third, the use of node-wise L2 feature normalization at line 5, which is equivalent to applying a layernorm transformation \cite{layernorm}. Nonetheless, it is worth noting \cite{sage}'s formulation of SAGE is flexible to allow different aggregations, such as max-pooling or LSTM, which further deviates SAGE from GCN.

\section{Background}
\label{sec:background}

\subsection{Semi-Supervised Node Classification}
Traditional label propagation algorithms \citep{semiemb, belkin2006} learn a model that transforms node features into node labels and uses the graph to add a regularizer term:
\begin{equation}
\label{eq:labelprop}
    \mathcal{L}_\textrm{label.propagation} =  \mathcal{L}_\textrm{classification}\left(f(X), \mathcal{V}_L\right) + \lambda f(X)^T \Delta f(X),

\end{equation}
The first term $\mathcal{L}_\textrm{classification}$ trains the model $f : \mathbb{R}^{N \times d_0} \rightarrow \mathbb{R}^{N \times C}$ to to predict the known labels $\mathcal{V}_L$. The second term is the graph-based regularizer, ensuring that connected nodes have a similar model output, with  $\Delta$ being the graph Laplacian and $\lambda \in \mathbb{R}$ is the regularization coefficient hyperparameter.

\subsection{Graph Convolutional Networks}
Graph Convolution \citep{bruna} generalizes convolution from Euclidean domains to
graph-structured data.
Convolving a ``filter'' over a signal on graph nodes can be calculated by transforming both the filter and the signal to the Fourier domain, multiplying them, and then transforming the result back into the discrete domain.
The signal transform is achieved by multiplying with the eigenvectors of the graph Laplacian. The transformation requires a quadratic eigendecomposition of the symmetric Laplacian;
however, the low-rank approximation of the eigendecomposition can be calculated using truncated Chebyshev polynomials \citep{chebyshev}. For instance,
\cite{kipf} calculates a rank-1 approximation of the decomposition. They propose a multi-layer Graph Convolutional Networks (GCNs) for semi-supervised graph learning. Every layer computes the transformation:
\begin{equation}
\label{eq:kipflayer}
H^{(l+1)} = \sigma\left(\hat{A} H^{(l)} W^{(l)} \right),
\end{equation}
where $H^{(l)} \in \mathbb{R}^{N \times d_l}$ is the input activation matrix to the $l$-th hidden layer with row $H^{(l)}_i$ containing a $d_l$-dimensional feature vector for vertex $i \in \mathcal{V}$, and $W^{(l)} \in \mathbb{R}^{d_{l} \times d_{l+1}}$ is the layer's trainable weights. The first hidden layer $H^{(0)}$ is set to the input features $X$. A softmax on the last layer is used to classify labels. All layers use the same ``normalized adjacency'' $\hat{A}$, obtained by the ``renormalization trick'' utilized by \cite{kipf}, as $\hat{A}=D^{-\frac12} A D^{-\frac12}$.\footnote{with added self-connections added as $A_{ii} = 1$, similar to \cite{kipf}}

Eq. \eqref{eq:kipflayer} is a first order approximation of convolving filter $W^{(l)}$ over signal $H^{(l)}$ \citep{chebyshev, kipf}. The left-multiplication with $\hat{A}$ averages node features with their direct neighbors; this signal is then passed through a non-linearity function $\sigma(\cdot)$ (e.g, ReLU$(z) = \max(0,z)$). Successive layers effectively \emph{diffuse} signals from nodes to neighbors.

Two-layer GCN model can be defined in terms of vertex features $X$ and normalized adjacency $\hat{A}$ as:
\begin{equation}
\label{eq:kipf2layers}
\textsc{GCN}_\textrm{2-layer}(\hat{A}, X; \theta) = \textrm{softmax}\left( \hat{A} \sigma(\hat{A}X W^{(0)}) W^{(1)}\right),
\end{equation}
where the GCN parameters $\theta = \left\{W^{(0)}, W^{(1)}\right\}$ are trained to minimize the cross-entropy error over labeled examples. The output of the GCN model is a matrix $\mathbb{R}^{N \times C}$, where $N$ is the number of nodes and $C$ is the number of labels. Each row contains the label scores for one node, assuming there are $C$ classes.

\subsection{Graph Embeddings}
Node Embedding methods represent graph nodes in a continuous vector space. They learn a dictionary $Z\in \mathbb{R}^{N \times d}$, with one $d$-dimensional embedding per node. Traditional methods use the adjacency matrix to learn embeddings. For example, Eigenmaps \citep{eigenmaps} performs the following constrained optimization:
\begin{equation}
\label{eq:eigenmaps}
    \sum_{i, j}  ||A_{ij}(Z_i - Z_J)|| \ \ \textrm{s.t.} \ \ Z^T D Z = I,
\end{equation}
where $I$ is identity vector. Skipgram models on text corpora \citep{word2vec} inspired modern graph embedding methods, which simulate random walks to learn node embeddings \citep{deepwalk, node2vec}.
Each random walk generates a sequence of nodes. Sequences are converted to textual paragraphs, and are passed to a word2vec-style embedding learning algorithm \citep{word2vec}.
As shown in \cite{neuralwalk}, this learning-by-simulation is equivalent, in expectation, to the decomposition of a random walk co-occurrence statistics matrix $\mathcal{D}$. The expectation on $\mathcal{D}$ can be written as:
\begin{equation}
\mathbb{E}[\mathcal{D}] \propto \mathbb{E}_{q \sim \mathcal{Q}}\left[\left(\mathcal{T}\right)^q\right] = \mathbb{E}_{q \sim \mathcal{Q}} \left[\left(D^{-1}A\right)^q\right],
\label{eq:exp_D}
\end{equation}
where $\mathcal{T}=D^{-1}A$ is the row-normalized transition matrix (a.k.a right-stochastic adjacency matrix), and $\mathcal{Q}$ is a ``context distribution'' that is determined by random walk hyperparameters, such as the length of the random walk. The expectation therefore weights the importance of one node on another as a function of how well-connected they are, and the distance between them. The main difference between traditional node embedding methods and random walk methods is the optimization criteria: the former minimizes a loss on representing the adjacency matrix $A$ (see Eq. \ref{eq:eigenmaps}), while the latter minimizes a loss on representing random walk co-occurrence statistics $\mathcal{D}$.

\section{Conclusions and Future Work}
\label{sec:conclusion}
In this paper, we propose a meta-model that can run arbitrary Graph Convolution models, such as GCN \citep{kipf} and SAGE \citep{sage}, on the output of random walks. Traditional Graph Convolution models operate on the normalized adjacency matrix. We make multiple instantiations of such models, feeding each instantiation a power of the adjacency matrix, and then concatenating the output of all instances into a classification sub-network. Each instantiation is therefore operating on different scale of the graph. Our model, Network of GCNs (and similarly, Network of SAGE), is end-to-end trainable, and is able to directly learn information across near or distant neighbors. We inspect the distribution of parameter weights in our classification sub-network, which reveal to us that our model is effectively able to circumvent adversarial perturbations on the input by shifting weights towards model instances consuming higher powers of the adjacency matrix. For future work, we plan to extend our methods to a stochastic implementation and tackle other (larger) graph datasets.

\section{Experiments}
\label{sec:experiments}

\subsection{Datasets}
We experiment on three citation graph datasets: Pubmed, Citeseer, Cora, and a biological graph: Protein-Protein Interactions (PPI).
We choose the aforementioned datasets because they are available online and are used by our baselines. The citation datasets are prepared by \cite{planetoid}, and the PPI dataset is prepared by \cite{sage}. Table \ref{table:dataset-stats} summarizes dataset statistics.

Each node in the citation datasets represents an article published in the corresponding journal. An edge between two nodes represents a citation from one article to another, and a label represents the subject of the article. Each dataset contains a binary Bag-of-Words (BoW) feature vector for each node. The BoW are extracted from the article abstract. Therefore, the task is to predict the subject of articles, given the BoW of their abstract and the citations to other (possibly labeled) articles. Following \cite{planetoid} and \cite{kipf}, we use 20 nodes per class for training, 500 (overall) nodes for validation, and 1000 nodes for evaluation. We note that the validation set is larger than training $|\mathcal{V}_L|$ for these datasets!

The PPI graph, as processed and described by \cite{sage}, consists of 24 disjoint subgraphs, each corresponding to a different human tissue. 20 of those subgraphs are used for training, 2 for validation, and 2 for testing, as partitioned by \cite{sage}.

\subsection{Baseline Methods}

For the citation datasets, we copy baseline numbers from \cite{kipf}. These include label propagation (LP, \cite{lp}); semi-supervised embedding (SemiEmb, \cite{semiemb}); manifold regularization (ManiReg, \cite{manireg}); skip-gram graph embeddings \citep[DeepWalk][]{deepwalk}; Iterative Classification Algorithm \citep[ICA, ][]{ica}; Planetoid \citep{planetoid}; vanilla GCN \citep{kipf}. For PPI, we copy baseline numbers from \citep{sage}, which include GraphSAGE with LSTM aggregation (SAGE-LSTM) and GraphSAGE with pooling aggregation (SAGE). Further, for all datasets, we use our implementation to run baselines DCNN \citep{diffusion-cnn}, GCN \citep{kipf}, and SAGE \citep[with pooling aggregation, ][]{sage}, as these baselines can be recovered as special cases of our algorithm, as explained in Section \ref{sec:nmodel}.

\subsection{Implementation}
We use TensorFlow\citep{tensorflow} to implement our methods, which we use to also measure the performance of baselines GCN, SAGE, and DCNN.
For our methods and baselines, all GCN and SAGE modules that we train are 2 layers\footnote{except as clearly indicated in Table \ref{table:deepgcn}}, where the first outputs 16 dimensions per node and the second outputs the number of classes (dataset-dependent). DCNN baseline has one layer and outputs 16 dimensions per node, and its channels (one per transition matrix power) are concatenated into a fully-connected layer that outputs the number of classes.
We use $50\%$ dropout and L2 regularization of $10^{-5}$ for all of the aforementioned models.

\begin{table*}[t]
\begin{center}
\begin{tabular}{lcccccc}
\textbf{Dataset}
&\textbf{Type}
& \textbf{Nodes}
&\textbf{Edges}
&\textbf{Classes}
& \textbf{Features}
& \textbf{Labeled nodes}
\\
& & $|\mathcal{V}|$ & $|\mathcal{E}|$ & $C$ & $F$ & $|\mathcal{V}_L|$
\\ \hline
Citeseer    & citaction      &3,327 &4,732 &6  (single class)&3,703 & 120 \\
Cora          & citaction   &2,708 &5,429 &7 (single class) &1,433 & 140 \\
Pubmed     & citaction         &19,717 &44,338 &3 (single class)&500 & 60 \\
PPI           & biological   & 56,944&  818,716   & 121 (multi-class) & 50 & 44,906 \\
\end{tabular}
\end{center}
\caption{Dataset used for experiments.
	For citation datasets, 20 training nodes per class are observed, with $|\mathcal{V}_L| = 20 \times C$
}
\label{table:dataset-stats}
\end{table*}

\begin{table*}[t]
\begin{center}
\input{result_summary_table_2.tex}
\end{center}
\caption{
Node classification performance ($\%$ accuracy for the first three, citation datasets, and f1 micro-averaged for multiclass PPI), using data splits of \cite{planetoid, kipf} and \cite{sage}.
We report the test accuracy corresponding to the run with the highest validation accuracy.
Results in rows (a) through (g) are copied from \cite{kipf}, rows (h) and (i) from \citep{sage}, and (j) through (l) are generated using our code since we can recover other algorithms as explained in Section \ref{sec:nmodel}. Rows (m) and (n) are our models.
Entries with ``--'' indicate that authors from whom we copied results did not run on those datasets. Nonetheless, we run all datasets using our implementation of the most-competitive baselines.
}
\label{table:results}
\end{table*}

\begin{figure*}[t]
	\centering
\begin{subfigure}{0.3\linewidth}
	\makebox[\textwidth]{
		\includegraphics[trim={0cm 0 0 0},clip,width=\linewidth]{sweeps/plot2d_citeseer_gcn_fc_1.pdf}
	}
	\caption{N-GCN on Citeseer}
\end{subfigure}
\begin{subfigure}{0.3\linewidth}
	\makebox[\textwidth]{
		\includegraphics[trim={0cm 0 0 0},clip,width=\linewidth]{sweeps/plot2d_cora_gcn_fc_1.pdf}
	}
	\caption{N-GCN on Cora}
\end{subfigure}
	\begin{subfigure}{0.3\linewidth}
	\makebox[\textwidth]{
		\includegraphics[trim={0cm 0 0 0},clip,width=\linewidth]{sweeps/plot2d_pubmed_gcn_fc_1.pdf}
	}
	\caption{N-GCN on Pubmed}
\end{subfigure}

\begin{subfigure}{0.3\linewidth}
	\makebox[\textwidth]{

		\includegraphics[trim={0cm 0 0 0},clip,width=\linewidth]{sweeps/plot2d_citeseer_sage_fc_1.pdf}
	}
	\caption{N-SAGE on Citeseer}
\end{subfigure}
\begin{subfigure}{0.3\linewidth}
	\makebox[\textwidth]{

		\includegraphics[trim={0cm 0 0 0},clip,width=\linewidth]{sweeps/plot2d_cora_sage_fc_1.pdf}
	}
	\caption{N-SAGE on Cora}
	\end{subfigure}
	\begin{subfigure}{0.3\linewidth}
		\makebox[\textwidth]{

			\includegraphics[trim={0cm 0 0 0},clip,width=\linewidth]{sweeps/plot2d_pubmed_sage_a_1.pdf}
		}
		\caption{N-SAGE on Pubmed}
\end{subfigure}
\caption{Sensitivity Analysis. Model performance when varying random walk steps $K$ and replication factor $r$. Best viewed with zoom. Overall, model performance increases with larger values of $K$ and $r$. In addition, having random walk steps (larger $K$) boosts performance more than increasing model capacity (larger $r$).}
	\label{fig:sensitivity}
\end{figure*}

\begin{table*}[t]
	\begin{center}

	\input{pubmed_npc.tex}

    \end{center}
\caption{Node classification accuracy (in $\%$) for our largest dataset (Pubmed) as we vary size of training data $\frac{|\mathcal{V}|}{C} \in \{5, 10, 20, 100\}$. We report mean and standard deviations on 10 runs.
We use a different random seed for every run (i.e. selecting different labeled nodes), but the same 10 random seeds across models.
Convolution-based methods (e.g. SAGE) work well with few training examples, but \textit{unmodified} random walk methods (e.g. DCNN) work well with more training data. Our methods combine convolution and random walks, making them work well in both conditions.
}
\label{table:npc}
\end{table*}

\begin{figure*}[t]
	\centering
	\includegraphics[trim={0 0.5cm 0 0},clip,width=\linewidth]{feat_r_plot.pdf}
	\caption{
		Classification accuracy for the Cora dataset with 20 labeled nodes per class $(|\mathcal{V}| = 20\times C)$, but features removed at random, averaging 10 runs.
		We use a different random seed for every run (i.e. removing different features per node), but the same 10 random seeds across models.
	}
	\label{fig:featremoval}
\end{figure*}

\begin{figure*}[t]
	\centering
	\includegraphics[trim={0 0 0 0cm},clip,width=\linewidth]{feat_r_attention_gcn.pdf}

	\caption{
	Attention weights ($m$) for $\textrm{N-GCN}_\textrm{a}$

	when trained with feature removal perturbation on the Cora dataset. Removing features shifts the attention weights to the right, suggesting the model is relying more on long range dependencies.
	}
	\label{fig:attentionfeatremoval}
\end{figure*}

\subsection{Node Classification Accuracy}
Table \ref{table:results} shows node classification accuracy results.
We run 20 different random initializations for every model (baselines and ours), train using Adam optimizer \citep{adam} with learning rate of 0.01 for 600 steps, capturing the model parameters at peak validation accuracy to avoid overfitting.
For our models, we sweep our hyperparameters $r$, $K$, and choice of classification sub-network $\in \{\textrm{fc}, \textrm{a}\}$. For baselines and our models, we choose the model with the highest accuracy on validation set, and use it to record metrics on the test set in Table \ref{table:results}.

Table \ref{table:results} shows that N-GCN outperforms GCN \citep{kipf} and N-SAGE improves on SAGE for all datasets, showing that \textit{unmodified} random walks indeed help in semi-supervised node classification. Finally, our proposed models acheive state-of-the-art on all datasets.

\subsection{Sensitivity Analysis}
We analyze the impact of random walk length $K$ and replication factor $r$ on classification accuracy in Figure \ref{fig:sensitivity}. In general, model performance improves when increasing $K$ and $r$. We note utilizing random walks by setting $K>1$ improves model accuracy due to the additional information, not due to increased model capacity: Contrast $K=1, r>1$ (i.e. mixture of GCNs, no random walks) with $K>1, r=1$ (i.e. N-GCN on random walks) -- in both scenarios, the model has more capacity, but the latter shows better performance. The same holds for SAGE.

\subsection{Tolerance to feature noise}
We test our method under feature noise perturbations by removing node features at random. This is practical, as article authors might forget to include relevant terms in the article abstract,
and more generally not all nodes will have the same amount of detailed information.
Figure \ref{fig:featremoval} shows that when features are removed, methods utilizing unmodified random walks: N-GCN, N-SAGE, and DCNN, outperform convolutional methods including GCN and SAGE. Moreover, the performance gap widens as we remove more features. This suggests that our methods can somewhat recover removed features by \textit{directly} pulling-in features from nearby and distant neighbors.
We visualize in Figure \ref{fig:attentionfeatremoval} the attention weights as a function of \% features removed. With little feature removal, there is some weight on $\hat{A}^{0}$, and the attention weights for $\hat{A}^{1}, \hat{A}^{2}, \dots$ follow some decay function. Maliciously dropping features causes our model to shift its attention weights towards higher powers of $\hat{A}$.

\subsection{Random Walk Steps Versus GCN Depth}

$K$-step random walk will allow every node to accumulate information from its neighbors, up to distance $K$. Similarly, a $K$-layer GCN \citep{kipf} will do the same. The difference between the two was mathematically explained in Section \ref{sec:motivation}. To summarize: the former averages node feature vectors according to the random walk co-visit statistics, whereas the latter creates non-linearities and matrix multiplies at every step. So far, we displayed experiments where our models (N-GCN and N-SAGE) were able to use information from distant nodes (e.g. $K=5$), but for all GCN and SAGE modules, we used 2 GCN layer for baselines and our models.

Even though the authors of GCN \citep{kipf} and SAGE \citep{sage} suggest using two GCN layers, according by holdout validation, for a fair comparison with our models, we run experiments utilizing deeper GCN and SAGE are models so that its ``receptive field'' is comparable to ours.

\begin{table}
	\begin{center}
		\input{deep_gcn_accuracies.tex}
	\end{center}
  \caption{Performance of deeper GCN and SAGE models, both using our implementation. Deeper GCN (or SAGE) does not consistently improve classification accuracy, suggesting that N-GCN and N-SAGE are more performant and are easier to train. They use shallower convolution models that operate on multiple scales of the graph.}
  \label{table:deepgcn}
\end{table}

Table \ref{table:deepgcn} shows test accuracies when training deeper GCN and SAGE models, using our implementation. We notice that, unlike our method which benefits from a wider ``receptive field'', there is no direct correspondence between depth and improved performance.

\section{Introduction}
\label{sec:intro}

Semi-supervised learning on graphs is important in many real-world applications, where the goal is to recover labels for all nodes given only a fraction of labeled ones.
Some applications include social networks, where one wishes to predict user interests, or in health care, where one wishes to predict whether a patient should be screened for cancer. In many such cases, collecting node labels can be prohibitive.
However, edges between nodes can be easier to obtain, either using an explicit graph (e.g. social network) or implicitly by calculating pairwise similarities \citep[e.g. using a patient-patient similarity kernel, ][]{selin}.

Convolutional Neural Networks \citep{lecun98} learn location-invariant hierarchical filters, enabling significant improvements on Computer Vision tasks \citep{alexnet, szegedy, resnet}. This success has motivated researchers \citep{bruna} to extend convolutions from spatial (i.e. regular lattice) domains to graph-structured (i.e. irregular) domains, yielding a class of algorithms known as Graph Convolutional Networks (GCNs).

Formally, we are interested in semi-supervised learning where we are given a graph $\mathcal{G}=(\mathcal{V}, \mathcal{E})$ with $N=|\mathcal{V}|$ nodes;  adjacency matrix $A$; and matrix $X \in \mathbb{R}^{N \times F}$ of node features. Labels for only a subset of nodes $\mathcal{V}_L \subset \mathcal{V}$ are observed. In general, $|\mathcal{V}_L| \ll |\mathcal{V}|$. Our goal is to recover labels for all unlabeled nodes $\mathcal{V}_U = \mathcal{V} - \mathcal{V}_L$, using the feature matrix $X$, the known labels for nodes in $\mathcal{V}_L$, and the graph $G$.
In this setting, one treats the graph as the ``unsupervised'' and labels of $\mathcal{V}_L$ as the ``supervised'' portions of the data.

Depicted in Figure \ref{fig:model}, our model for semi-supervised node classification builds on the GCN module proposed by Kipf and Welling \cite{kipf}, which operates on the normalized adjacency matrix $\hat{A}$, as in $\textrm{GCN}(\hat{A})$, where $\hat{A} = D^{-\frac12} A D^{-\frac12}$, and $D$ is diagonal matrix of node degrees. Our proposed extension of GCNs is inspired by the recent advancements in random walk based graph embeddings \citep[e.g.][]{deepwalk, node2vec, neuralwalk}.
We make a Network of GCN modules (N-GCN), feeding each module a different power of $\hat{A}$,
as in $\{\textrm{GCN}(\hat{A}^0), \textrm{GCN}(\hat{A}^1), \textrm{GCN}(\hat{A}^2), \dots\}$.
The $k$-th power contains statistics from the $k$-th step of a random walk on the graph.
Therefore, our N-GCN model is able to combine information from various step-sizes (i.e. graph scales). We then combine the output of all GCN modules into a classification sub-network, and we jointly train all GCN modules and the classification sub-network on the upstream objective for semi-supervised node classification.
Weights of the classification sub-network give us insight on how the N-GCN model works. For instance, in the presence of input perturbations, we observe that the classification sub-network weights shift towards GCN modules utilizing higher powers of the adjacency matrix, effectively widening the ``receptive field'' of the (spectral) convolutional filters.
We achieve state-of-the-art on several semi-supervised graph learning tasks, showing that explicit random walks enhance the representational power of vanilla GCN's.

The rest of this paper is organized as follows. Section \ref{sec:background} reviews background work that provides the foundation for this paper. In Section \ref{sec:method}, we describe our proposed method, followed by experimental evaluation in Section \ref{sec:experiments}. We compare our work with recent closely-related methods in Section \ref{sec:related}. Finally, we conclude with our contributions and future work in Section \ref{sec:conclusion}.

\begin{figure*}[t]
  \centering
  \begin{subfigure}{0.47\linewidth}
	\makebox[\textwidth]{
		\scalebox{0.6}{
			\input{fig1tikz.tex}
		}
	}
  	\caption{N-GCN Architecture}
  \end{subfigure}
  \begin{subfigure}{0.47\linewidth}
  	\makebox[\textwidth]{

	  	\includegraphics[width=\linewidth]{tsne_fc.pdf}
	  }
  	\caption{t-SNE visualization of fully-connected (fc) hidden layer of NGCN when trained over Cora graph.}
  \end{subfigure}
  \caption{
  	Left: Model architecture, where $\hat{A}$ is the normalized normalized adjacency matrix, $I$ is the identity matrix, $X$ is node features matrix, and  $\times$ is matrix-matrix multiply operator. We calculate $K$ powers of the  $\hat{A}$, feeding each power into $r$ GCNs, along with $X$. The output of all $K \times r$ GCNs can be concatenated along the column dimension, then fed into fully-connected layers, outputting
  	$C$ channels per node, where $C$ is size of label space. We calculate cross entropy error, between rows \textit{prediction} $N \times C$ with known labels, and use them to update parameters of classification sub-network and all GCNs. Right: pre-relu activations after the first fully-connected layer of a 2-layer classification sub-network. Activations are PCA-ed to 50 dimensions then visualized using t-SNE.
  }
  \label{fig:model}
\end{figure*}

\section{Our Method}
\label{sec:method}

\subsection{Motivation}
\label{sec:motivation}
Graph Convolutional Networks and random walk graph embeddings are individually powerful.
\cite{kipf} uses GCNs for semi-supervised node classification. Instead of following traditional methods that use the graph for regularization (e.g. Eq. \ref{eq:eigenmaps}), \cite{kipf} use the adjacency matrix for training and inference, effectively diffusing information across edges at all GCN layers (see Eq. \ref{eq:kipf2layers}).
Separately,
recent work has showed that random walk statistics can be very powerful for learning an unsupervised representation of nodes that can preserve the structure of the graph \citep{deepwalk, node2vec, neuralwalk}.

Under special conditions, it is possible for the GCN model to learn random walks. In particular, consider a two-layer GCN defined in Eq. \ref{eq:kipf2layers} with the assumption that first-layer activation is identity as $\sigma(z) = z$, and weight $W^{(0)}$ is an identity matrix (either explicitly set or learned to satisfy  the upstream objective). Under these two identity conditions, the model reduces to:
\begin{align*}
\textsc{GCN}_\textrm{2-layer-special}(\hat{A}, X) &= \textrm{softmax}\left( \hat{A} \hat{A}X W^{(1)}\right) \\
 &= \textrm{softmax}\left( \hat{A}^2 X W^{(1)}\right)
\end{align*}
where $\hat{A}^2$ can be expanded as:
\begin{align}
\nonumber
\hat{A}^2 &= \left(D^{-\frac12} A D^{-\frac12}\right) \left(D^{-\frac12} A D^{-\frac12}\right) = D^{-\frac12} A \left[D^{-1} A\right] D^{-\frac12}  \\
  &= D^{-\frac12} A \mathcal{T} D^{-\frac12}
\label{eq:a_hat_2}
\end{align}
By multiplying the adjacency $A$ with the transition matrix $\mathcal{T}$ before, the $\textsc{GCN}_\textrm{2-layer-special}$ is effectively doing a one-step random walk i.e. diffusing signals from nodes to neighbors, without non-linearities, then applying a non-linear Graph Conv layer.

\subsection{Explicit Random Walks}

The special conditions described above are not true in practice. Although stacking hidden GCN layers allows information to flow through graph edges, this flow is \emph{indirect} as the information goes through feature reduction (matrix multiplication) and a  non-linearity (activation function $\sigma(\cdot)$).
Therefore, the vanilla GCN cannot directly learn high powers of $\hat{A}$, and could struggle with modeling information across distant nodes.
We hypothesize that making the GCN directly operate on random walk statistics will allow the network to better utilize information across distant nodes, in the same way that node embedding methods (e.g. DeepWalk, \cite{deepwalk}) operating on $\mathcal{D}$ are superior to traditional embedding methods operating on the adjacency matrix (e.g. Eigenmaps, \cite{eigenmaps}).
Therefore, in addition to feeding only $\hat{A}$ to the GCN model as proposed by \cite{kipf} (see Eq. \ref{eq:kipf2layers}), we propose to feed a $K$-degree polynomial of $\hat{A}$ to $K$ instantiations of GCN. Generalizing Eq.~\eqref{eq:a_hat_2} to arbitrary power $k$ gives:
\begin{equation}
\hat{A}^k = D^{-\frac12} A \mathcal{T}^{k-1} D^{-\frac12}.
\end{equation}
We also define $\hat{A}^0$  to be the identity matrix. Similar to \cite{kipf}, we add self-connections and convert directed graphs to undirected ones, making $\hat{A}$ and hence $\hat{A}^k$ symmetric matrices. The eigendecomposition of symmetric matrices is real. Therefore, the low-rank approximation of the eigendecomposition \cite{chebyshev} is still valid, and a one layer of \cite{kipf} utilizing $\hat{A}^k$ should still approximate multiplication in the Fourier domain.

\subsection{Network of GCNs}
Consider $K$ instantiations of $\{ \textsc{GCN}(\hat{A}^0, X)$,  $\textsc{GCN}(\hat{A}^1, X)$, $\dots$,
$\textsc{GCN}(\hat{A}^{K-1}, X)  \}$. Each GCN outputs a matrix $\mathbb{R}^{N \times C_k}$, where the $v$-th row describes a latent representation of that particular GCN for node $v \in \mathcal{V}$, and $C_k$ is the latent dimensionality. Though $C_k$ can be different for each GCN, we set all $C_k$ to be the same for simplicity. We then combine the output of all $K$ GCN and feed them into a classification sub-network, allowing us to jointly train all GCNs and the classification sub-network via backpropagation. This should allow the classification sub-network to choose features from the various GCNs, effectively allowing the overall model to learn a combination of features using the raw (normalized) adjacency, different steps of random walks (i.e. graph scales), and the input features $X$ (as they are multiplied by identity $\hat{A}^0$).

\subsubsection{Fully-Connected Classification Network}
From a deep learning prospective, it is intuitive to represent the classification network as a fully-connected layer.  We can concatenate the output of the $K$ GCNs along the column dimension, i.e. concatenating all $\textrm{GCN}(X, \hat{A}^k)$, each $\in \mathbb{R}^{N \times C_k}$ into matrix $\in \mathbb{R}^{N \times C_K}$ where $C_K = \sum_k C_k$. We add a fully-connected layer $f_\textrm{fc} : \mathbb{R}^{N \times C_K} \rightarrow  \mathbb{R}^{N \times C}$, with trainable parameter matrix $W_\textrm{fc} \in \mathbb{R}^{C_K \times C}$, written as:
\begin{align}
\label{eq:ngcnfc}
&\textrm{N-GCN}_\textrm{fc}(\hat{A}, A; W_\textrm{fc}, \theta) = \textrm{softmax}\bigg( \\
&
\nonumber
\ \ \ \ \ \ \ \ \ \left[
\begin{array}{c;{2pt/2pt}c;{2pt/2pt}c}
\textrm{GCN}(\hat{A}^0, X; \theta^{(0)}) & \textrm{GCN}(\hat{A}^1, X; \theta^{(1)}) & \dots
\end{array}
\right] W_\textrm{fc} \bigg).
\end{align}
The classifier parameters $W_\textrm{fc}$ are jointly trained with GCN parameters $\theta=\{\theta^{(0)}, \theta^{(1)}, \dots \}$. We use subscript fc on N-GCN to indicate the classification network is a fully-connected layer.

\subsubsection{Attention Classification Network}
We also propose a classification network based on ``softmax attention'', which learns a convex combination of the GCN instantiations. Our attention model ($\textrm{N-GCN}_\textrm{a}$) is parametrized by vector $\widetilde{m} \in \mathbb{R}^K$, one scalar for each GCN. It can be written as:
\begin{align}
\textrm{N-GCN}_\textrm{a}( \hat{A}, X; m, \theta) &= \sum_{k} m_k \textrm{GCN}(\hat{A}^k, X; \theta^{(k)})
\end{align}
where $m$ is output of a softmax: $m = \textrm{softmax}(\widetilde{m})$.

This softmax attention is similar to ``Mixture of Experts'' model, especially if we set the number of output channels for all GCNs equal to the number of classes, as in $C_0 = C_1 = \dots = C$.
This allows us to add cross entropy loss terms on all GCN outputs in addition to the loss applied at the output NGCN, forcing all GCN's to be independently useful. It is possible to set the $m \in \mathbb{R}^K$ parameter vector ``by hand''  using the validation split, especially for reasonable $K$ such as $K \le 6$. One possible choice might be setting $m_0$ to some small value and remaining $m_1, \dots, m_{K-1}$ to the harmonic series $\frac{1}{k}$; another choice may be linear decay $\frac{K - k}{K-1}$. These are respectively similar to the context distributions of GloVe \citep{glove} and word2vec \citep{word2vec, levy}. We note that if on average a node's information is captured by its direct or nearby neighbors, then the output of GCNs consuming lower powers of $\hat{A}$ should be weighted highly.

\subsection{Training}
We minimize the cross entropy between our model output and the known training labels $Y$ as:
\begin{equation}
\min \textrm{diag}(\mathcal{V}_L)  \left[Y \circ \log \textrm{N-GCN}(X, \hat{A})\right],
\end{equation}
where $\circ$ is Hadamard product, and $\textrm{diag}(\mathcal{V}_L)$ denotes a diagonal matrix, with entry at $(i, i)$ set to 1 if $i \in \mathcal{V}_L$ and 0 otherwise. In addition, we can apply intermediate supervision for the $\textrm{NGCN}_\textrm{a}$ to attempt make all GCN become independently useful, yielding minimization objective:
\begin{align}
\nonumber
\min_{m, \theta} \textrm{diag}(\mathcal{V}_L)  \bigg[ & Y \circ \log \textrm{N-GCN}_\textrm{a}(\hat{A}, X; m, \theta) \\
& + \sum_k Y \circ \log \textrm{GCN}(\hat{A}^k, X; \theta^{(k)}) \bigg].
\end{align}

\subsection{GCN Replication Factor $r$}

To simplify notation, our N-GCN derivations (e.g. Eq. \ref{eq:ngcnfc}) assume that there is one GCN per $\hat{A}$ power. However, our implementation feeds every $\hat{A}$ to $r$ GCN modules, as shown in Fig. \ref{fig:model}.

\subsection{Generalization to other Graph Models}
\label{sec:nmodel}
In addition to vanilla GCNs \citep[e.g. ][]{kipf}, our derivation also applies to other graph models including GraphSAGE \citep[SAGE,][]{sage}.
Algorithm \ref{alg:nmodel} shows a generalization that allows us to make a network of arbitrary graph models (e.g. GCN, SAGE, or others). Algorithms \ref{alg:gcn} and \ref{alg:sage}, respectively, show pseudo-code for the vanilla GCN \citep{kipf} and GraphSAGE\footnote{Our implementation assumes mean-pool aggregation by \cite{sage}, which performs on-par to their top performer max-pool aggregation. In addition, our Algorithm \ref{alg:sage} lists a full-batch implementation whereas \citep{sage} offer a mini-batch implementation.} \citep{sage}. Finally, Algorithm \ref{alg:ngcn} defines our full Network of GCN model (N-GCN) by plugging Algorithm \ref{alg:gcn} into Algorithm \ref{alg:nmodel}. Similarly, Algorithm \ref{alg:nsage} defines our N-SAGE model by plugging Algorithm \ref{alg:sage} in Algorithm \ref{alg:nmodel}.

\begin{figure*}[h]
\begin{minipage}[t]{5.52in}
	\begin{algorithm}[H]
		\caption{General Implementation: Network of Graph Models}
		\label{alg:nmodel}
		\begin{algorithmic}[1]
			\Require{$\hat{A}$ is a normalization of $A$}
			\Function{Network}{$\textsc{GraphModelFn}$, $\hat{A}$, $X$, $L$, $r=4$, $K=6$, \textsc{ClassifierFn}=\textsc{FcLayer}}
			\State{$P \leftarrow I$}
			\State{GraphModels $ \leftarrow $ []}
			\For{$k = 1 $ to $K$}
			\For{$i = 1 $ to $r$}
			\State{GraphModels.append($\textsc{GraphModelFn}(P, X, L)$)}
			\EndFor
			\State{$P \leftarrow \hat{A} P$}
			\EndFor

			\State{\Return \textsc{ClassifierFn}(GraphModels)}
			\EndFunction
		\end{algorithmic}
	\end{algorithm}
\end{minipage}
\hfill
\begin{minipage}[t]{2.5in}
	\begin{algorithm}[H]
		\caption{GCN Model \citep{kipf}}
		\label{alg:gcn}
		\begin{algorithmic}[1]
			\Require{$\hat{A}$ is a normalization of $A$}
			\Function{GcnModel}{$\hat{A}$, $X$, $L$}
			  \State {$Z \leftarrow X$}
			  \For{$i = 1 $ to $L$}
			  	\State{$Z \leftarrow \sigma(\hat{A} Z W^{(i)})$ }
			  \EndFor

			  \State{\Return $Z$}
			\EndFunction
		\end{algorithmic}
		\end{algorithm}
	\end{minipage}
\hfill
\begin{minipage}[t]{2.5in}
	\begin{algorithm}[H]
		\caption{SAGE Model \citep{sage}}
		\label{alg:sage}
		\begin{algorithmic}[1]
			\Require{$\hat{A}$ is a normalization of $A$}
			\Function{SageModel}{$\hat{A}$, $X$, $L$}
			\State {$Z \leftarrow X$}
			\For{$i = 1 $ to $L$}
			\State{$Z \leftarrow \sigma(\left[ \begin{array}{c;{2pt/2pt}c}
				Z & \hat{A} Z
				\end{array} \right] W^{(i)})$ }
			\State{$Z \leftarrow \textsc{L2NormalizeRows}(Z)$ }
			\EndFor
			\State{\Return $Z$}
			\EndFunction
		\end{algorithmic}
	\end{algorithm}
\end{minipage}
	\hfill
	\begin{minipage}[t]{2.5in}
		\begin{algorithm}[H]
			\caption{N-GCN}
			\label{alg:ngcn}
			\begin{algorithmic}[1]
			\Function{Ngcn}{$A$, $X$, $L=2$}
				\State{$D \leftarrow \textbf{diag}(A \mathbf{1}) $}
				\Comment{Sum rows}
			    \State{$\hat{A}  \leftarrow D^{-1/2}AD^{-1/2}$}
				\State{\Return $\textsc{Network}(\textsc{GcnModel}, \hat{A}, X, L)  $}
			\EndFunction
			\end{algorithmic}
		\end{algorithm}
	\end{minipage}
	\hfill
		\begin{minipage}[t]{2.5in}
		\begin{algorithm}[H]
			\caption{N-SAGE}
			\label{alg:nsage}
			\begin{algorithmic}[1]
				\Function{Nsage}{$A$, $X$}
				\State{$D \leftarrow \textbf{diag}(A \mathbf{1}) $}
				\Comment{Sum rows}
				\State{$\hat{A}  \leftarrow D^{-1}A$}
				\State{\Return $\textsc{Network}(\textsc{SageModel}, \hat{A}, X, 2)$}
				\EndFunction
			\end{algorithmic}
		\end{algorithm}
	\end{minipage}
	\hfill
\end{figure*}

We can recover the original algorithms GCN \citep{kipf} and SAGE \citep{sage}, respectively, by using Algorithms \ref{alg:ngcn} (N-GCN) and \ref{alg:nsage} (N-SAGE) with $r=1$, $K=1$, identity \textsc{ClassifierFn}, and modifying line 2 in Algorithm \ref{alg:nmodel} to $P \leftarrow \hat{A}$. Moreover, we can recover original DCNN \citep{diffusion-cnn} by calling Algorithm \ref{alg:ngcn} with $L=1$,  $r=1$, modifying line 3 to $\hat{A} \leftarrow D^{-1}A$, and keeping $K>1$ as their proposed model operates on the power series of the transition matrix i.e. \textit{unmodified} random walks, like ours.

\section{Related Work}
\label{sec:related}
The field of graph learning algorithms is quickly evolving. We review work
most similar to ours.

Defferrard et al \cite{fast-spectrals} define graph convolutions as a $K$-degree polynomial of the Laplacian, where the polynomial coefficients are learned. In their setup, the $K$-th degree Laplacian is a sparse square matrix where entry at $(i, j)$ will be zero if nodes $i$ and $j$ are more than $K$ hops apart. Their sparsity analysis also applies here. A minor difference is the adjacency normalization. We use $\hat{A}$ whereas they use the Laplacian defined as $I-\hat{A}$. Raising $\hat{A}$ to power $K$ will produce a square matrix with entry $(i, j)$ being the probability of random walker ending at node $i$ after $K$ steps from node $j$. The major difference is the order of random walk versus non-linearity. In particular, their model calculates learns a linear combination of $K$-degree polynomial and pass through classifier function $g$, as in $g(\sum_k q_k \widetilde{A}^k)$, while our (e.g. $\textrm{N-GCN}$) model calculates $\sum_k q_k g(\widetilde{A}^k)$, where $\widetilde{A}$ is $\hat{A}$ in our model and $I-\hat{A}$ in theirs, and our $g$ can be a GCN module. In fact, \cite{fast-spectrals} is also similar to work by \cite{neuralwalk}, as they both learn polynomial coefficients to some normalized adjacency matrix.

Atwood and Towsley \cite{diffusion-cnn} propose DCNN, which calculates powers of the transition matrix and keeps each power in a separate channel until the classification sub-network at the end. Their model is therefore similar to our work in that it also falls under $\sum_k q_k g(\widetilde{A}^k)$. However, where their model multiplies features with each power $\widetilde{A}^k$ once, our model makes use of GCN's \citep{kipf} that multiply by $\widetilde{A}^k$ at every GCN layer (see Eq. \ref{eq:kipflayer}). Thus, DCNN model \citep{diffusion-cnn} is a special case of ours, when GCN module contains only one layer, as explained in Section \ref{sec:nmodel}.

redpajama_set_name: RedPajamaArXiv


Death Of Chris Msando, the IEBC's tech guru who was brutally murdered in unclear circumstances continues to linger on many Kenyans minds an year later. One of the most outstanding memories is his wife's testimony and strength shown during those trying moments.
A widow left with orphans and without a job since the IT guru was the family's breadwinner, there was immediate need for her to find a job or s business to sustain her young family.
Struggling with life, Eva at one point had reached out to the public for any job opportunity. Her prayers seemingly, have been heard, in a gazette notice Peter Munya the CS Industry, Trade and Cooperatives, Eva Buyu Msando has been appointed to be member of the Export Processing Zones Authority for three years effective 20th September 2018.
The good news could be attributed to the handshake between President Uhuru and PM Raila Odinga. Msando who ails from Odinga's community was suspected to have been assassinated by hit Squad allied to the state owing to his stiff grounds that would've limited rigging of the elections. The late Msando is on record saying to have developed a hack free electoral system. His death was widely read as a sabotage to the electoral system.
Many questions on the murder of Chris remains unresolved with the state reluctant to follow the trail.


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The course comprises eight compulsory assessed modules, a group project and an individual research project. The modules include lectures, practical sessions and tutorials.
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Introduction to the circular economy: opportunities for businesses; opportunities for consumers.
Case study: trade-offs, synergies, and opportunities to enhance well-being and ecosystem service provision in terms of energy, food, feed and wood for a case study area.
Critique associated terms such as "human well-being", "sustainability", and "biodiversity".
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Water governance and stewardship: Abstraction licencing and regulation; Water storage & reservoirs; Working together and abstractor groups; Water trading.
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Critically appraise the role of water in future challenges to food sustainability.
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Innovations for agricultural production (FFS).
Explain the principal responses of plants to the climatic environment.
Describe the functional role of soil biology in soil systems.
Create a soil assessment in the context of land restoration (LRR).
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Plant breeding strategies including conventional selection and marker assisted selection.
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Seed production and seed treatment technologies.
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Biocontrol agents – principles and case studies.
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SC collaboration approaches and types of partnerships.
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An overview of the reasons why organisations engage in foresight research.
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Candidates must normally possess, or be expected to achieve, a first or second class UK Honours degree in a relevant science or social science-based discipline; or the international equivalent of these UK qualifications. Other relevant qualifications together with industrial experience may be considered.


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Within this fast changing realm of competitors and development, it has been a undeniable fact that people barely get enough time to spend in their homes, looking after themselves. It's mandatory for all of us that people spend at least half of the times time in office, performing our responsibility. When the office lacks in correct tools and facilities for that employees, the employees get severe health issues. Likewise, the decorum and also the tools of the workplaces are provided enormous importance. These days, the majority of the tools are selected inside a careful manner in order to ensure the workers comfort. One such essential product at work may be the ergonomic workplace seats. As most of the offices have turned out to be wise, meaning they depend more about machine than hard physical work, ergonomic desk office chairs are specifically created for this kind of offices. They are the seats used for the use of the workplaces having computer work stations. These have been developed specifically seats which help within the reduction of demands on the a number of nerves from the spinal wires, thereby limiting the risk of muscle and nerve illnesses. Also they assist in maintaining the right physique position. It aligns the forearms, arms, lower back, throat and also the mind in the correct jobs, therefore decreasing lower back pain, function related musculoskeletal disorders along with other such illnesses. And when you just looking for a chair to relax, then consider among the reclining chair from our list. In the following paragraphs we will discuss about the very best fifteen best ergonomic seats to be able to provide the visitors using the accurate understanding to ensure that once they purchase this kind of chairs, it might be simple for them to select the correct one.
If you are looking for some severe rest, this loveseats strong pillow back and folded hands will deliverbut at a higher cost. Every seat cushioning consists of 30 person pocketed circles for spring and assistance, and high strength chair froth for an extra coating of plushness and sturdiness. Plus, the froth cushions are topped with super soft, blended lower down. For your additional comfort, there are three regular positions: sitting, a slight recline by having an elevated footrest, along with a full lie down by having an elevated foot rest. It does not harm this lying loveseat looks lux, possibly. Its all-leather-based upholstery and decorative nail head cut provide a vintage look that can function with lots of house dcor styles, from traditional to nation to arts and crafts. Plus, leather holds up well against everyday use, children and pet locks and fingernails. Just remember to condition the leather each and every six to twelve months.
The Lawson could be known as perfect American couch style. It's comfortable and simple: The boxy form is large and characteristically has three back again soft cushions in addition to three chair soft cushions. The traditional Lawson also offers A taller back and box-formed cushions that have welts in the edges, as do the rear cushions. Ideal for cuddling and sleeping, this couch design was created for Thomas Lawson, an American copper tycoon at the change from the century. He wanted a sofa which was quite different from the picky Victorian designs which were typical at that time. Present versions from the Lawson may also include metal or wood included in the arms. Most significantly, this sofa design is the one that will change its feel using the textile you choose for the upholstery. The shape is really versatile that it can be attractive, magnificent, industrial, informal or formal based on your material choice.
With almost 100 5-star evaluations, this reclining loveseat is recommended for its comfort, sturdiness, look and ease of set up. In addition to gentle, but supportive cushions, it features a gliding mechanism, so you can place your ft up and rock (although, not simultaneously.) Padded with extremely-durable bonded leather, this reclining loveseat will avoid tears, releases and stainswhich is a good factor, since it also features a storage space compartment system (where you can conceal your treats, obviously) and a two-mug owner. Whether you are settling set for a Netflix binge or viewing football together with your team, this lying loveseat offers the durability and comfort with out emptying your wallet.
With almost 100 5-star reviews, this reclining loveseat is recommended because of its comfort, sturdiness, look and ease of set up. Along with gentle, but supportive cushions, it features a gliding mechanism, so that you can place your ft up and rock and roll (though, not at the same time.) Padded with ultra-long lasting glued leather, this reclining loveseat will resist tears, rips and stainswhich is a good thing, since it also features a storage compartment system (where one can conceal your treats, of course) along with a two-mug owner. Whether youre settling in for a Blockbuster online binge or viewing football with your team, this reclining loveseat provides the comfort and durability with out breaking the bank.
The Bridgewater is really a couch style that's mainly informal and definitely comfortable. This design may be the beloved of many a designer because you can use it to produce a casual, friendly space. Bridgewater couches are versatile, depending on the upholstery you choose. Completed in an unbiased fabric, this kind of couch doesnt compete with other elements within the room that may be more dramatic, for example artwork or other big functions. More official material will produce a more grand type of sofa. Numerous qualities differentiate Bridgewater couches: Typically they've low arms and a high back again, which contributes to the sporadic appear. Most also have a tailored dress that conceals the legs and loose soft cushions for that chair and back again. This specific design was created to support slipcovers, that also give a casual air to the couch.
This established is padded inside a push and comfortable textured padded velvet. This gives it the amazing comfort that it is loved for. Once again it supports chaise style seats for adequate comfort and ease. Furthermore, the nose spring foundation increases the comfort and durability. Additionally, the sofa has hard wood structures that increase its toughness and sturdiness. The sofa has cushions on both the chair and the back again. Yet still, the couch can change from a sitting position to a lying down placement effortlessly. That said, simply to repeat, the sofa is extremely gentle and gives exceptional comfort and ease.
This set is padded inside a push and comfortable textured cushioned velvet. This provides it the amazing comfort and ease that it is loved for. Again it supports chaise style seating for sufficient comfort. Moreover, the sinus spring base increases the comfort and durability. Additionally, the couch has hard wood structures that increase its toughness and sturdiness. The sofa has cushions on both the seat and also the back again. Yet still, the couch can move from a sitting placement to some lying down position effortlessly. Having said that, just to replicate, the couch is extremely soft and gives outstanding comfort and ease.
Just like its namesake, the camelback sofa includes a hump or maybe two that highlights the main outline of the sofa design. This is a traditional style that was prominent in the 1700s by Chippendale and households who wanted aristocratic, formal furniture often chosen this style. Today, this couch design lends a far more official air to any room, particularly when padded in a formal material. A far more informal fabric choice would make it appropriate for a family space that doesnt need a lot of formality. In either case, the camelback couch has some factors that characterize the style. The legs are usually uncovered, the couch doesn't have back cushions and it usually has square or folded hands.
The Bridgewater is a couch style that's primarily casual and certainly comfortable. This design is the darling of numerous an artist simply because you can use it to produce a casual, pleasant space. Bridgewater couches are flexible, with respect to the furniture you select. Done in an unbiased material, this style of sofa doesnt contend with additional factors in the room which may be more dramatic, such as artwork or any other big features. Much more official fabric will yield a far more stately type of couch. Numerous qualities distinguish Bridgewater couches: Usually they have low hands and a high back, which contributes to the casual appear. Most also have a tailored dress that conceals the legs and loose soft cushions for that seat and back. This specific style was designed to accommodate slipcovers, that also lend a casual atmosphere to the couch.
This sofa is praised for unique style and luxury. Its impressive pazazz-arm couch with seats offers you maximum room for stretching out and using your room. So usually the modern sofa chaise adds to the great thing about this set. The established also includes cushion top hands cushioned back and seats for maximum comfort. The couch includes a large part-obstructed wooden frame that provides the sofa sufficient sturdiness. The sofa steps 89 inches broad by 62 in . in depth by 4 in . tall which makes it ideal for medium size rooms. You may want to do small putting together for example of the thighs.


Surveying the remote reefs of Australia's North-west shelf aboard Eviota.
My wife's first response after I was caught up in Cyclone Debbie two years earlier was entirely predictable. "December through January is peak cyclone season up there, I think you're mad".
The lure of diving the reefs off the North West Shelf was too strong however, and I found myself with Ben, Nestor and Graham provisioning Eviota in Broome midway through December, hoping we had enough to keep us well fed until we arrived in Darwin on the 15th January.
We planned to re-survey reefs visited by Reef Dragon 6 years ago, plus we had a few more we now had permission to survey, so it was a pretty ambitious task Graham had set for us.
We were lucky throughout that we had winds that generally helped us get where we needed to be, and although the distances were quite long on some legs, sailing overnight where possible meant we were able to dive every day for over 23 days, and keeping in mind each dive usually meant two transects from each buddy pair, there was plenty of activity around the computers each night.
Squalls were pretty much a daily occurrence, and the preliminary cyclone warning we received in the first week of our trip only heightened our awareness of the prevailing weather conditions. Eviota has been equipped with a state-of-the-art drogue designed to keep the stern of the vessel facing into the sea, thus preventing broaching or pitch poling, so it was deemed wise to hold a practice deployment, just so the crew would know what was involved in the event it was needed. A rather lengthy process that was quite physically demanding, especially the retrieval afterwards, but very educational and everyone felt much more comfortable having now learned the steps required, and in some cases, what not to do.
Thankfully however the weather warning for the cyclone turned out to be a false alarm, and after a few more days the drogue was stowed and we got on with the daily routine of diving, data entry, cooking, compressor duty and sailing.
We found that typically the condition of corals on most of the reefs was poor. Turbidity around inshore islands and reefs usually meant the corals were in better condition, and there were larger fish present. Those reefs further afield are showing signs of past damage, few live corals but masses of dead corals and coral rubble. Subsequently we recorded fewer large fishes and sharks, and the phenomenon of the disappearance of sea snakes from Ashmore Reef in recent years still holds, in all of the surveys over 5 or 6 days, no snakes were sighted.
We visited Cartier Island, part of the Ashmore Island group, after RLS had received a permit to conduct surveys there. Off limits to everyone for decades, due to it being a practice bombing target during the war and the subsequent risk of unexploded ordinance, we at last got a glimpse of what it could look like elsewhere if the reefs were afforded the same protection. Abundant corals, larger fish and more diversity – if proof was needed that protection works, Cartier Island is the evidence.
During what would normally be the monsoon season it was expected we would not see many other vessels. In fact, apart from the Border Force plane that flew over every day, and the Border Force crew aboard the patrol vessel Cape York, moored at Ashmore Reef, we saw no-one for the full four weeks. The Border Force crew were very helpful, guiding us to a safer mooring up the channel at Ashmore, and advising us of where we could legally land on the island. They have a presence there 365 days of the year, and while it is quiet for them now, during the winter months they are paid regular visits by fishermen from Indonesia. Their presence helps to protect the reefs from illegal fishing methods and subsequent damage.
On the way to Darwin after the surveys were completed, we called into the King George River Gorge, for an overnight stay and some welcome sight seeing. Graham and Nestor climbed to the top of the gorge at the Eastern Arm and Twin Falls, neither of which had water flowing, and returned with some great photos. A spectacular part of the country, it is easy to see why government tourism boards are promoting the Kimberley Region as they do.
We were back in Darwin 2 days later; the monsoon still hadn't started, and Darwin was hot and sticky. The chores of cleaning Eviota ready for the next crew were begun and the realisation our trip was over finally hit. All up we had completed transects at 78 sites, we all did at least 40 dives each, and we four were still talking to each other! I think you would have to say it was a very successful trip.


text:
Alisha Saxena to represent Central Puget Sound at 2014 Intel International Science & Engineering Fair
Alisha Saxena, a senior from Interlake High School, was named the First Runner-Up in the 2014 Central Sound Regional Science & Engineering Fair. Alisha's project, "Analyzing and Preventing Quick Response Code Based Malware and Phishing Attacks for Smartphones," was conducted in UW CSE's Security and Privacy Research Laboratory, where she has been mentored by Ph.D. student Franzi Roesner and professor Yoshi Kohno.
In addition to being named First Runner-Up, Alisha also won first place in the Computer Science category and the Intel Excellence in Computer Science award. We reported in December on Alisha's recognition as a Winner in the 2014 National NCWIT Award for Aspirations in Computing.
Alisha and the Grand Champion – Meera Srinivasan, a junior at Interlake High School – will receive an all-expense-paid trip to represent the Central Puget Sound region at the 2014 Intel International Science & Engineering Fair in Los Angeles this May.
Congratulations, Alisha!

redpajama_set_name: RedPajamaCommonCrawl


CNS News: House Financial Services Chairwoman Maxine Waters said on CNN on Sunday that congressional Democrats "believe in border security." Waters made the point while explaining he concerns that President Donald Trump will not engage in fruitful negotiations on the issue.
"It's hard to believe that he is prepared to seriously negotiate when he's making his demands outside of the negotiation already," said Waters.
"First of all, he's talking about he's got to have a certain amount of money," she said. "He's also talking about how terrible it is with the rapists and the drug dealers and on and on and on. That's not how you enter a negotiation.
"We want to make sure that he understands that it's not simply about a wall," she said. "I want him to think about, and we want him to think about, technology. We want him to think about drones. We want him to think about all of those things that go into having a secure border.
She then cited air traffic controllers as an example. "So, I hope that he would have sense enough or his advisers would have sense enough to send him into a negotiation in a legitimate way," she said.
They believe in it so much, that's why they don't want voter ID laws established.
This loud mouth nasty woman should be arrested for her remarks about the people that voted for the President. She has caused all of the attacks on good people, like the Jewish men that have been attacked, or the death threats on all of the Trump people. She is a sick clown of a person.
Maxine is proof, more nut wards needed.
Anyone with half a brain knows a wall without technology and staff will fail. Trump wants all of these. Any other news is fake news.


text:
Madden 21: Still flawed
Madden 21 is a game with positives, but still has the flaws of previous iterations. Photo / ClutchPoints
Sports video games franchises are among some of the most money-hungry corporations in the modern economy.
Many sports video game fans are fed up with the current cycle that companies follow: update rosters, change the design a tiny bit and sell millions.
EA Sports released Madden 21 as the National Football League (NFL) season kicks off this fall, ushering in a new iteration of the storied football video game franchise.
This year, EA Sports' big selling point for Madden is "The Yard," along with the typical game modes like "Franchise," "Career," "Ultimate Team" and "Superstar KO."
The Yard is a six vs. six "schoolyard" style of football game, where players play both sides of the ball and can use customized characters to play with friends or alone. This game mode provides a fresh, new style of football that fans can play, and the addition of co-op mode is a definite plus for those who are looking to enjoy time with their friends remotely.
What EA Sports has not changed in their annual game is the focus on "Ultimate Team." "Ultimate Team" brings them money through the purchasing of packs to get rare cards in their card-collecting game mode that has dominated the sports gaming industry.
"Franchise" mode and "Career" mode are virtually the same, except for "Career" mode having a small storyline tweak. But what will be different every year is "Ultimate Team."
New cards, new programs and new packs to make even more money, "Ultimate Team." is the focal point of Madden year-after-year. That did not change with Madden 21.
Despite the success of "The Yard," Madden once again fails to provide a fresh, new product. The problem with releasing a game yearly is the quick turnaround that they must make. Making big strides in the mechanics, game modes and appearance of the game is made difficult with a quick deadline.
That doesn't excuse the explicit money-grab that Madden has become though. One good new game mode does not excuse negligence to others.
The strength of previous Madden games was the "Franchise" mode. There were many features within the game mode that made the consumer feel like a real general manager or coach of an NFL team. Now, the game mode is a shell of itself, with very barebones features that provide an enjoyable playthrough on only the first play.
After the first franchise file, each subsequent playthrough becomes less and less exciting. The consumer knows how to play the game the way it wants to be played and can manipulate the algorithms with ease. The simulation realism is skewed, with role players and below-average players winning "Most Valuable Player" or leading the league in a statistical category.
The "Franchise" mode has clear, distinct flaws, but "Franchise" doesn't have downloadable content. It doesn't encourage kids to pick up their mom's credit card and buy a pack, so they spend their time improving those money-making game modes. On one hand, it's hard to blame EA for this. But on the other hand, it's deplorable to continue to annually release a half-complete game for more money.
With the newest installment of Madden, EA Sports had some positives. The user interface and presentation are great, as always, and "The Yard" is a fresh, fun and new game mode. But the negatives are still there. The same negatives that have been there for the past five or six years.
To make complete, quality games, EA either needs to stop the yearly release schedule or focus on the aspects of their game that they have neglected for half of a decade. For half of a complete game, this addition to the franchise gets a half score.
Rating: 2.5/5 stars
Elex Martinez • Sep 17, 2020 at 1:19 PM
For the past 8 years, Madden has singlehandedly been the most downward spiraling franchise I've ever seen & the gameplay is beyond horrendous. For a game that copies something that's already being done in real life, they sure don't know what they're doing at all.
https://oaklandpostonline.com/33098/opinion/madden-21-still-flawed/#comment-924


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text:
#ifndef DATACHUNK_H
#define DATACHUNK_H

#include "../qrencode.h"

typedef struct _DataChunk {
	QRencodeMode mode;
	int size;
	int bits;
	unsigned char *data;
	struct _DataChunk *next;
} DataChunk;

DataChunk *DataChunk_new(QRencodeMode mode);
void DataChunk_free(DataChunk *chunk);
void DataChunk_freeList(DataChunk *list);
void DataChunk_dumpChunkList(DataChunk *list);
int DataChunk_totalSize(DataChunk *list);
unsigned char *DataChunk_concatChunkList(DataChunk *list, int *retsize);

#endif /* DATACHUNK_H */

redpajama_set_name: RedPajamaGithub


The reason why our car services are some of the most sought after in the area, is down to our professional attitude, affordable prices and premium vehicles. Whether you are simply in need of a leisure taxi or an airport transfer from Newport Longville, Admiral express are always available.
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Let Admiral provide you with the most exquisite cars and wedding chauffeurs in Newton Longville for your special day.
No matter what the occasion is, our leisure taxis in Newton Longville can transport groups of every shape and size in comfort and on time.
As experts in leisure and corporate chauffeuring, Admiral Express have honed each service to perfection, in order to provide every customer with a top class experience. Whatever the event or occasion, from weddings to long distance trips, we will have a car and driver to fit your requirements.
From the booking process right up to arriving at your final destination, our aim is to take you on a journey of impeccable standards from start to finish. Our friendly team are waiting to take your call, and talk you through every available option to suit your needs.
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We needed a taxi to the airport from Newton Longville in the middle of the night. With a large family, it was tricky finding a car big enough that didn't cost the earth. Fortunately, Admiral offered us a really nice car that fit everyone in comfortably, along with our luggage. The process was so easy from start to finish. Would definitely recommend them.


Unlike me, the husband gets pimples more often. Normally, they start to show up as a result of stress...mostly at work. So if he's stressed out about something and not saying anything about it, I could still tell by just looking at his face and/or forehead. Luckily, his pimples are usually the small ones and not nodular acne which can really be painful. Aside from being painful, nodular acne can get worse if they're popped intentionally and sometimes leaving some unsightly scar on the affected area. Hubby loves popping those little buggers because he couldn't help himself. He says it feels good to do that that's why he does it. Anyway, it seems that over the counter pimple cream helps get rid of his pimples so there's not much to worry about them. I worry not because of the pimples but the cause of the pimples. I guess that's inevitable considering how tough the economy is now.


Groove Distribution :: Stock :: Singles :: House :: Carlos Rodriquez/SUMMER IN BRAZIL 12"
Home : Stock : Singles : House : Carlos Rodriquez/SUMMER IN BRAZIL 12"
Carlos Rodriquez/SUMMER IN BRAZIL 12"
Summery Latin house jam with samples from GEORGE DUKE's "BRAZILIAN LOVE AFFAIR". Vocals that sound like EWF or KOOL AND THE GANG ontop of some deep but pumpin' house beats (like the beats used in many of those STEVIE WONDER bootlegs).
Psychemagik/VALLEY OF PARADISE PIC 12" Original Sin/MAD WORLD EP D12" Pascal Rioux/DON'T OUTSTAY OUT RMX 2 12"
Better Lost Than Stupid/ALTO 12" Ripple & Skyy/THE BEAT GOES ON..12" Toasters, The/SKA JERK EP (ORANGE) 7"


Thermoseal Ra toothpaste is a good paste as a desensitizer. in fact it is available over the counter also in grocery shops. initially when the sensitivity is more u could use THERMOSEAL RA i.e. rapid acting, later switch on to the ordinary one. but do not substitute for your children or others who are not indicated this unnecessarily and make it a family paste.
Page created in 1.602 seconds with 45 queries.


Want to explore a new show? Explore highlight clips from this popular drama and discover your next favorite show! Or if you've already watched or are watching this show, you can relive all of your favorite moments with these unforgettable..
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text:
Q: Magento2: How to get simple products Assosiate to configurable products I want to get simple products Associate to configurable products in custom module using factory method.
(not using object manager)

A: You can use this below code using factory method :
protected $productFactory;

public function __construct(
    ....
    \Magento\Catalog\Model\ProductFactory $productFactory,
    ....
)
{
   ....
   $this->productFactory = $productFactory;
   ....
}

public function yourFunction()
{
   $configProduct = $this->productFactory->create()->load($product_id);

   $_children = $configProduct->getTypeInstance()->getUsedProducts($configProduct);
   foreach ($_children as $child){
    echo "Simple Product Ids ".$child->getID()."<br/>";
   }
}

redpajama_set_name: RedPajamaStackExchange


Have you found that when you spend time in nature you feel refreshed and a renewed sense of well-being? I can definitely feel the health benefits of nature.
I always feel better after relaxing or exercising in nature.
My personal experience is evidence in my own life of these research findings.
The study on the health benefits of nature also found that feeling a part of nature has been shown to significantly correlate with life satisfaction, meaningfulness, happiness, and lower cognitive anxiety. I hope you feel the researched benefits of getting outside for some fresh air.
I have noticed that breathing in fresh air for a few minutes in the morning dramatically enhances and stabilizes my mood for the day.
If I'm having a "good day" with minimal vertigo symptoms, I like to exercise outside but I often bring my cane or a walking stick for balance on uneven surfaces and hiking trails.
If I'm having a "bad day," just sitting quietly in the freshness of the outdoors soothes me and relaxes me.
I am always amazed when I see the plants in my yard with a new flower bloom or my avocado tree growing a new branch. Observing nature reminds me that things are constantly changing, so it helps me not to get mentally stuck. This just comes to show me all the health benefits of nature.
Fall is the perfect time to get out so let's take advantage of the health benefits of nature and experience all of the beauty that nature has to offer. Sometimes I take a rest on a solid boulder or a fallen log, noticing the drops of dew on nearby plants and stones. The birds chirping and the squirrels gathering nuts for the winter help me expand my view to witness the bigger picture and the cyclical nature of life.
From hiking in nature to sitting on the front porch, nature has a lot to offer us to support our well-being, regardless of our level of mobility.
This blog post is provided for informational purposes only. The content is not intended to be a substitute for professional medical advice, diagnosis, or treatment. Always seek the advice of your physician or other qualified health provider with any questions you may have regarding a medical condition. The details of any case mentioned in this post represent a typical patient that I might see and do not describe the circumstances of a specific individual.


Kina Repp is excited to announce a clip of her upcoming video is available! This video shows a little about her story and samples of her speaking. A full-length video will soon be for sale. If you are interested in pre-orders, contact us!


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Needless to say, little kids need to be kept busy when traveling with older members of the family.
Fun and engaging, these are the toys need to keep babies and toddlers entertained on vacation.
1. Tablet screens. Widely available and offers a variety of game options, tablets like iPad almost guarantee kids will never get bored. Game apps, songs and videos should occupy most of their time to feel bored and start bothering other people around them. There are portable gaming consoles that feature educational games and other entertaining activities. The iPod Touch or iPad is also a great choice for babies. There are several apps complete with full-color moving images and sounds that will educate and entertain infants and toddlers throughout the whole trip. Toddlers can wear earphones so they can listen to music or play games quietly.
2. Musical toys. These are generally priced anywhere between $20 and $40. There are simple ones like rattles and xylophones and other electronic toys that play music and lullabies. Babies can press buttons, spin balls or twirl handles to initiate the sounds. There are toys that are ideal for particular age ranges like 0 to12 months or 12 to 24 months.
3. Sticker and coloring books. These also fall well within the category of travel toys. Children can color several pages or match stickers to spaces. These will hone their mental abilities as well as keep them occupied for 1 to 2 hours. These are very ideal when traveling in a train or airplane. The toddler can stay seated and behave for the duration of the trip.
4. Pillow pets and stuff toys. These toys are very basic and will keep children feeling safe and comfortable for long rides. There are several stuff animals available like pandas, hippos, monkeys and teddy bears. There are also pillow toys that feature cartoon characters. Babies like to play with these anywhere.
5. Slinky. The classic favorite will entertain babies and young children well. They can play it with their siblings or on a portable desk.
6. Board games. There are board games intended for children 2 to 4 years old. These are great for the whole family and also boost mental and computing abilities. Some of the titles are classics although there are also new ones complete with electronic devices. Some versions of popular favorites are available for young players. Here are suggestions on best travel board games you can choose from.
7. Alphabet blocks. Building blocks are proven to enhance mental prowess among young children. Even babies can play with plastic models and create towers. Toddlers can practice the alphabet.
All these games will be helpful in keeping children happy and comfortable when traveling while making adults focus on making the trip enjoyable for the whole family.


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text:
from ..log_month import LogMonth
from ..log_year import LogYear
import unittest
from unittest.mock import patch, Mock

from kao_list import KaoList
from Test import BuildDate

class previous(unittest.TestCase):
    """ Test cases of previous """

    def test_previous(self):
        """ Test that when there is a previous year, it is returned """
        firstYear = BuildDate(year=2015)
        secondYear = BuildDate(year=2014)
        year = LogYear(firstYear)
        expected = LogYear(secondYear)
        items = [expected.date, year.date]

        itemsMock = Mock(return_value=items)

        with patch('devlog.dates.log_year.DateFiles') as DateFilesMock:
            DateFilesMock.Years = itemsMock
            actual = year.previous
            self.assertEqual(actual, expected)

    def test_noPrevious(self):
        """ Test that when there is no previous year, None is returned """
        yearDate = BuildDate(year=2015)
        year = LogYear(yearDate)
        expected = None
        items = [year.date]

        itemsMock = Mock(return_value=items)

        with patch('devlog.dates.log_year.DateFiles') as DateFilesMock:
            DateFilesMock.Years = itemsMock
            actual = year.previous
            self.assertEqual(actual, expected)

class next(unittest.TestCase):
    """ Test cases of next """

    def test_next(self):
        """ Test that when there is a next year, it is returned """
        firstYear = BuildDate(year=2015)
        secondYear = BuildDate(year=2014)
        year = LogYear(firstYear)
        expected = LogYear(secondYear)
        items = [year.date, expected.date]

        itemsMock = Mock(return_value=items)

        with patch('devlog.dates.log_year.DateFiles') as DateFilesMock:
            DateFilesMock.Years = itemsMock
            actual = year.next
            self.assertEqual(actual, expected)

    def test_noNext(self):
        """ Test that when there is no next year, None is returned """
        yearDate = BuildDate(year=2015)
        year = LogYear(yearDate)
        expected = None
        items = [year.date]

        itemsMock = Mock(return_value=items)

        with patch('devlog.dates.log_year.DateFiles') as DateFilesMock:
            DateFilesMock.Years = itemsMock
            actual = year.next
            self.assertEqual(actual, expected)

class all(unittest.TestCase):
    """ Test cases of all """

    def test_build(self):
        """ Test that the years are built properly and return """
        dates = [BuildDate(year=year) for year in [2012, 2013, 2014]]
        year = LogYear(dates[0])
        expected = KaoList([LogYear(date) for date in dates])

        itemsMock = Mock(return_value=dates)

        with patch('devlog.dates.log_year.DateFiles') as DateFilesMock:
            DateFilesMock.Years = itemsMock
            actual = year.all
            DateFilesMock.Years.assert_called_once_with()
            self.assertEqual(actual, expected)

class months(unittest.TestCase):
    """ Test cases of months """

    def test_build(self):
        """ Test that the months are built properly and return """
        dates = [BuildDate(month=month) for month in [8, 9, 10]]
        year = LogYear(dates[0])
        expected = KaoList([LogMonth(date, year) for date in dates])

        itemsMock = Mock(return_value=dates)

        with patch('devlog.dates.log_year.DateFiles') as DateFilesMock:
            DateFilesMock.Months = itemsMock
            actual = year.months
            DateFilesMock.Months.assert_called_once_with(year.date)
            self.assertEqual(actual, expected)


text:
Q: How to Invoke Admin Level Authorized Azure HTTP Trigger Function I've Created HttpTrigger Azure Java Functions with ADMIN LEVEL Authorization:

Reference Document Link: https://vincentlauzon.com/2017/12/04/azure-functions-http-authorization-levels/
For Invoking Admin Authorized Azure Function, we need Host Keys from Function App so, I have Three Host keys In Function App

Invoking Function URL in Postman with Function App Host key & Value in Header Section

*

*Invoking with _master key and Value

*Invoking with default key and Value

*Invoking with custom Host Key and Value

results: 404Not Found
please share solution or correct me if I am Wrong anywhere
Don't POST wrong Answers Please...

A: I have deployed an Admin Level Authorization Http Trigger to Azure Java Function app using VS Code:

As stated in the given vincentlauzon's article of Azure Function App keys:

It only works with the master (host) key to Admin authorized functions in the Azure Function app:

Yes, there is a fault only in Admin Authorized Level Java 8.0 Versioned Azure Function Http Trigger that I have tested along with the different Authorization Levels and Java Versions 11, 17.0 Azure Functions are also tested and working successfully.
Java 11.0 Version Azure Function Result:

Java 17.0 Version Azure Function Result:

I have raised the ticket (No. 9010) in GitHub Azure Functions Host repository with all the test results where the MSFT backend team will check and provide the fix to it.