text:
Soludo Recounts how he was Attacked
A former Central Bank of Nigeria (CBN) Governor Prof. Chukwuma Soludo has narrated how he was attacked by gunmen at a political gathering in Anambra state on Wednesday.
The Professor of econometrics, who is also an aspirant in the November 6 governorship election in Anambra, was meeting with youths at Isuofia community when the gunmen stormed the area and opened fire, The Nation learnt.
Three police officers attached to Soludo were killed during the attack
Speaking on Thursday during an interview on Channels TV Politics Today monitored by The Nation, Soludo said he would not like to conclude the attack was 'politically motivated'.
Recalling the incident, he said: "We were having an interactive town-hall meeting. We finished the diocesan crescent mass at about 2pm, and by 4:00pm, they scheduled an interactive town-hall meeting with the Isuofia youths
"And by the way Isofia happens to be my hometown… I'm a village boy; this is where I feel safest anywhere in the world. I walk the streets, I jog on the streets, I take long walks, I mean I'm everywhere as a village boy as it were.
"With this interactive session with over a thousand youths seated, and I was just making preliminary remarks to open up the interactive session with other dignitaries from the community, commissioners and members of the House of Assembly, former commissioners, former members of the house, and several other dignitaries were in attendance
"And just as I was about to wrap up, and for them now to open up comments, questions and answers, I heard gunshots. At first I thought maybe the policemen were shooting in the air and concurring with the kind of exuberance in the hall itself…
"And until I began to hear people screaming, and all of that, and everyone scampered for safety, but then when it was all over, it took several minutes about 10 to 15 minutes of continuous gun battle, going back and forth…stopping and starting again.
"By the time we all came back, we saw three dead bodies lying in the pool of their blood – they were the three policemen… and like I said earlier, may God grant their soul eternal rest… we'll continue to pray and support the families.
"And then, the commissioner was abducted. I understand that when they ran and jumped the fence, some people grabbed and took him away… our prayers are with him. We pray for his safe return."
He said: "The DSS, police, army and civil defense were all in my house… I was totally unhurt in anyway."
When asked if the attack was politically motivated to abduct or attack him, he said: "I think there are all kinds of theories as it were, I do not want to prejudge or preempt the investigation that is ongoing because I understand some persons have been apprehended and also providing useful information."
Soludo affirmed he is a friend to all the other contestants in the state.
Source: www.thenationonlineng.net
Tags #Soludo
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text:
The managerial and development issues of nurses and midwives in new roles
McKenna, Hugh, Richey, Roberta, Keeney, Sinead, Hasson, Felicity, Poulton, Brenda and Sinclair, Marlene (2008) The managerial and development issues of nurses and midwives in new roles. Scandinavian Journal of Caring Sciences, 22 (2). pp. 227-235. [Journal article]
URL: http://www3.interscience.wiley.com/journal/120084898/issue
DOI: 10.1111/j.1471-6712.2007.00519.x
Rationale: The number of innovative roles in nursing and midwifery has expanded considerably; however, the evidence base for the introduction of these roles is limited.Aim: This study aimed to identify the managerial and developmental issues affecting those in innovative roles.Methodological design: A self-completion postal questionnaire was distributed to all innovative role holders in Northern Ireland, 450 responses were analysed. This survey was the second phase of a larger investigation into innovative roles in nursing and midwifery.Instrument: The questionnaire that had been developed for the 'Exploring New Roles in Practice' project was adapted for use in this study.Ethical issues: The explanatory letter sent to potential participants clearly detailed that informed consent was assumed on receipt of a completed questionnaire and gave contact details for the research team. The confidentiality and anonymity of responses were assured.Results: Most participants (65%) stated that their innovative role had commenced since the year 2000. There was evidence of preparation for these roles including the development of job descriptions and protocols. Nevertheless, there were issues noted relating to resource allocation and replacement when the role holder was absent. The importance of role evaluation was highlighted, with some participants noting their role had not been assessed since its inception. Support provided by managers and colleagues was key to ensuring role effectiveness. Participants reported high levels of job satisfaction.Limitations: Innovative role holders were identified by managers in their employing organization: therefore the research team could not verify that all existing innovative roles were included in the study. Innovative roles in private and voluntary sectors were not included.Conclusions: The evolution of innovative nursing and midwifery roles has been considerable worldwide and they engender job satisfaction to those occupants them. Results support international literature showing that adequate support, especially from administrators, is significant to ensuring the success of new roles. Furthermore, the careful planning for innovative role development and evaluation are important in ensuring the role is effective.
Dr Felicity Hasson

redpajama_set_name: RedPajamaCommonCrawl


Programming Note: Episode 172 of the Brew / Drink / Run podcast is a "big show", where we drink two beers and also talk about running and homebrewing. The previous episode and the next episode are mini episodes, where we focus on a single beer.
Our beers this week are Daily Miel from Blackberry Farm Brewery and Fullsteam and Red 5 Standing By from Revelry.
Lee, Keith and Raymond are back for another week of beer chat. Get caught up on the progress of Raymond's home brewing studio, Lee's quest to get back into a running rhythm, and the latest from Keith's gig at Service Brewing Co.
Raymond's Book Club does not want you to read 13 Reasons Why, but Lee is happy to recommend Last Train to Zona Verde.


text:
The unexplained has, for centuries, held fascination for us. A decade into the new millenium, that hasn't changed. The paranormal researcher is still at large, hopping from location to location in his Maserati in hopes of capturing evidence of some kind for the existence of what might be called spiritual reality. But modern parapsychologists have at their disposal an array of equipment that their predecessors lacked. Paranormal research has become more sophisticated, utilizing the latest technology to record, measure, and ultimately prove (or disprove) the theories that have made their discipline so controversial. Parapsychologists can no longer be accused of being glorified shamans. With their arsenal of video cameras, audio recorders, electromagnetic field (EMF) detectors, and the like, they have made the leap into serious science.
Modern ghost hunters seek concrete, tangible evidence, evidence that will lead to a reasonable conclusion about the phenomena they investigate. The biggest hurtle they face today is the same one they have always faced: acceptance. Many within the scientific community still regard this branch of psychology as a pseudo-science. The nearly total rejection of all things spiritual among scientists has left many who conduct paranormal research struggling to gain credibility for their work.
UK based? Check out short term car insurance or car insurance that you can pay for monthly. Need car insurance for 1 day? Try carinsurancefor1day.co.uk
But that is beginning to change, due perhaps in large part to the increased exposure and popularity that paranormal research is enjoying in popular culture. Television shows and mass market books on the subject have seen a boom in recent years, and, while some of these have sensationalized (and even, unfortunately, theatricized) the paranormal, the end result has been an increased interest in the subject. And, whether scientists are willing to admit it or not, the fact is that such an increase in interest invariably creates a ripple effect within their own specialized community.
Science is, after all, as much at the mercy of the laws of supply and demand as any other discipline. The increased interest among the laity inevitably creates a demand for answers from those they deem to be the experts. It should be no surprise, then, to see many "serious" scientists slowly starting to reverse their views on parapsychology, particularly with paranormal researchers now offering more naturalistic theories to explain the unexplained—such as the influence of the electromagnetic field on personal perception or the effect of color on human emotion.
With this barrier starting to fall, the possibilities of paranormal research seem limitless. And there is no better evidence of this than the varying methods of investigation in use today in this field. An internet search of various paranormal socieites reveals several diverse, often contradictory, approaches being employed. One group's aim may be to prove the existence of ghosts, while another may seek to disprove them, to find a definite physiological or natural cause to such incidents. Conflict and controversy abound in this ever-growing field, but, like other areas of scientific study, the sometimes volatile climate of debate will ultimately keep paranormal research alive. Its future looks promising.
Copyright © 2000-2011 Paraweb Inc


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

Complex networks (or graphs) are a fundamental object of study
in modern science, across domains as diverse as the social sciences,
biology, physics, computer science, and engineering~\cite{WaFa94,Ne-survey,EaKl-book}.
Designing good models for these networks is a crucial area of research,
and also affects society at large, given the role of online social networks
in modern human interaction~\cite{BaAl99,WaSt98,ChFa06}. Complex networks are massive, high-dimensional,
discrete objects, and are challenging to work with in a modeling
context. A common method of dealing with this challenge
is to construct a low-dimensional Euclidean embedding that tries
to capture the structure of the network (see~\cite{HaYiLe17} for a recent survey). Formally,
we think of the $n$ vertices as vectors $\vec{v}_1, \vec{v}_2, \ldots, \vec{v}_n \in \mathbb{R}^d$,
where $d$ is typically constant (or very slowly growing in $n$).
The likelihood of an edge $(i,j)$ is proportional to (usually a non-negative
monotone function in) $\vec{v}_i\cdot \vec{v}_j$~\cite{AhSh+13,CaLuXu16}. This gives a graph
distribution that the observed network is assumed to be generated from.

The most important method to get such embeddings is the Singular Value Decomposition (SVD)
or other matrix factorizations of the adjacency
matrix~\cite{AhSh+13}. Recently, there has also been an explosion of
interest in using methods from deep neural networks to learn such graph embeddings~\cite{PeAlSk14,TaQu+15,CaLuXu16,grover2016node2vec}
(refer to~\cite{HaYiLe17} for more references). Regardless of the
specific method, a key goal in building an embedding is to
keep the dimension $d$ small --- while trying to preserve the network
structure --- as the embeddings are used in a variety of downstream
modeling tasks such as graph
clustering, nearest neighbor search, and link
prediction~\cite{Twitter-embeddings}.  Yet a fundamental question remains unanswered:
to what extent do such low dimensional embeddings actually capture
the structure of a complex network?

These models are often justified by treating the (few) dimensions as ``interests" of individuals,
and using similarity of interests (dot product) to form edges.
Contrary to the dominant view, we argue that low-dimensional embeddings are
\emph{not} good representations of complex networks. We demonstrate
mathematically and empirically that they lose local structure,
one of the hallmarks of complex networks. This runs counter
to the ubiquitous use of SVD in data analysis. The weaknesses
of SVD have been empirically observed in recommendation tasks~\cite{BaChGo10,GuGo+13,KlUgKl17},
and our result provides a mathematical validation of these findings.

Let us define the setting formally. Consider a set of vectors $\vec{v}_1, \vec{v}_2, \ldots, \vec{v}_n \in \mathbb{R}^d$
(denoted by the $d \times n$ matrix $V$) used to represent the $n$ vertices in a network.
Let $\mathcal{G}_V$ denote the following distribution of graphs over the vertex set $[n]$.
For each index pair $i,j$, independently insert (undirected) edge $(i,j)$ with probability
$\max(0,\min(\vec{v}_i \cdot \vec{v}_j, 1))$. (If $\vec{v}_i \cdot \vec{v}_j$ is negative,
$(i,j)$ is never inserted. If $\vec{v}_i \cdot \vec{v}_j \geq 1$, $(i,j)$ is always inserted.)
We will refer to this model as the ``embedding" of a graph $G$, and focus
on this formulation in our theoretical results.
{This is a standard model in the literature,
and subsumes the classic Stochastic Block Model~\cite{HoLa83}
and Random Dot Product Model~\cite{YoSc07,AtFi+18}. There are
alternate models that use different functions of the dot product
for the edge probability, which are discussed in Section~\ref{sec:variants}.}
Matrix factorization is a popular method to obtain such a vector representation: the original adjacency matrix $A$
is ``factorized" as $V^TV$, where the columns of $V$ are $\vec{v}_1, \vec{v}_2, \ldots, \vec{v}_n$.

\begin{figure}
        \includegraphics[scale=0.5]{ca-HepPh_tri_distro.pdf}
				\captionof{figure}{\small Plots of degree $c$ vs $\Delta$: For a High Energy Physics
        coauthorship network,
        we plot $c$ versus the total number of triangles only involving vertices of
        degree at most $c$. We divide the latter by the total number of vertices $n$,
        so it corresponds to $\Delta$, as in Def.~\ref{def:foundation}. We plot these
        both for the original graph (in thick blue), and for a variety of embeddings (explained in Section~\ref{sec:variants}).
        For each embedding, we plot the maximum $\Delta$ in
        a set of 100 samples from a 100-dimensional embedding. The embedding analyzed by
        our main theorem (TDP) is given in thick red. Observe how the embeddings generate
        graphs with very few triangles among low degree vertices. The gap in $\Delta$ for low degree is
        2-3 orders of magnitude. The other lines correspond to alternate embeddings,
        using the {\sc node2vec} vectors and/or different functions of the dot product.} \label{fig:intro-triangle-distro}
\end{figure}

Two hallmarks of real-world graphs are: (i) Sparsity: The average degree is typically constant with respect
to $n$, and (ii) Triangle density: there are many triangles incident to low degree vertices~\cite{WaSt98,SaCaWiZa10,SeKoPi11,DPKS12}.
The large number of triangles is considered a local manifestation of community structure. Triangle counts have a rich history
in the analysis and algorithmics of complex networks. Concretely, we measure these properties
simultaneously as follows.

\begin{definition} \label{def:foundation} For parameters $c > 1$ and $\Delta > 0$, a graph $G$ with $n$ vertices
has a \emph{$(c,\Delta)$-triangle foundation} if there are at least $\Delta n$ triangles contained among vertices of degree at most $c$.
Formally, let $S_c$ be the set of vertices of degree at most $c$. Then, the number of triangles
in the graph induced by $S_c$ is at least $\Delta n$.
\end{definition}

Typically, we think of both $c$ and $\Delta$ as constants.
{We emphasize that $n$ is the total number of vertices in $G$, not the number
of vertices in $S$ (as defined above).}
Refer to real-world graphs
in Table~\ref{tab:datasets}. In Figure~\ref{fig:intro-triangle-distro}, we plot the value
of $c$ vs $\Delta$. (Specifically, the $y$ axis is the number of triangles divided by $n$.)
This is obtained by simply counting the number of triangles contained in the set of vertices of degree at most $c$.
Observe that for all graphs,
for $c \in [10,50]$, we get a value of $\Delta > 1$ (in many cases $\Delta > 10$).

Our main result is that \emph{any} embedding of graphs that generates graphs with $(c,\Delta)$-triangle foundations,
with constant $c,\Delta$, must have near linear rank. This
contradicts the belief that low-dimensional embeddings capture the structure of real-world complex networks.

\begin{theorem} \label{thm:main} Fix $c > 4, \Delta > 0$. Suppose the expected number of triangles
in $G \sim \mathcal{G}_V$ that only involve vertices of expected degree $c$ is at least $\Delta n$.
Then, the rank of $V$ is at least $\min(1,\mathrm{poly}(\Delta/c)) n/\lg^2n$.
\end{theorem}

Equivalently, graphs generated from low-dimensional embeddings cannot contain many
triangles only on low-degree vertices. {We point out an important
implication of this theorem for Stochastic Block Models. In this model, each vertex is modeled
as a vector in $[0,1]^d$, where the $i$th entry indicates the likelihood
of being in the $i$th community. The probability of an edge is exactly
the dot product. In community detection applications, $d$ is thought of as a constant,
or at least much smaller than $n$. On the contrary, \Thm{main} implies
that $d$ must be $\Omega(n/\lg^2n)$ to accurately model the low-degree triangle behavior.}

\subsection{Empirical validation} \label{sec:emp}

We empirically validate the theory on a collection of complex
networks detailed in Table~\ref{tab:datasets}. For each real-world graph, we compute a 100-dimensional embedding through SVD
(basically, the top 100 singular vectors of the adjacency matrix).
We generate $100$ samples of graphs from these embeddings, and compute
their $c$ vs $\Delta$ plot. This is plotted with the true $c$ vs $\Delta$ plot.
(To account for statistical variation, we plot
the \emph{maximum} value of $\Delta$ observed in the samples, over all graphs. The variation
observed was negligible.)
Figure~\ref{fig:intro-triangle-distro} shows such a plot for a physics coauthorship network.
More results are given in Section~\ref{sec:emp-details}.

Note that this plot
is significantly off the mark at low degrees for the embedding. Around the lowest degree,
the value of $\Delta$ (for the graphs generated by the embedding)
is 2-3 order of magnitude smaller than the original value. This demonstrates
that the local triangle structure is destroyed around low degree vertices.
Interestingly, the total number of triangles is preserved well, as shown
towards the right side of each plot. Thus, a nuanced view of the triangle
distribution, as given in \Def{foundation}, is required to see the shortcomings
of low dimensional embeddings.

\subsection{Alternate models} \label{sec:variants}
{We note that several other functions of dot product have been
  proposed in the literature, such as the softmax
  function~\cite{PeAlSk14,grover2016node2vec} and linear
models of the dot product~\cite{HaYiLe17}. Theorem~\ref{thm:main} does not have
direct implications for such models, but our empirical validation holds for
them as well. The embedding in Theorem~\ref{thm:main} uses the \emph{truncated
dot product} (TDP) function $max(0,\min(\vec{v}_i \cdot \vec{v}_j, 1))$ to model
edge probabilities. We construct other embeddings that compute edge probabilities
using machine learning models with the dot product and Hadamard product
as features. This subsumes linear models as given in~\cite{HaYiLe17}. Indeed, the TDP
can be smoothly approximated as a logistic function.
We also consider (scaled) softmax functions, as in~\cite{PeAlSk14}, and standard
machine learning models (LRDP, LRHP).
(Details about these models are given in Section~\ref{sec:emp-details}\ref{sec:alt}.)

For each of these models (softmax, LRDP, LRHP), we perform the same experiment described above.
Figure~\ref{fig:intro-triangle-distro} also shows the plots for these other models.
Observe that \emph{none} of them capture the low-degree triangle structure,
and their $\Delta$ values are all 2-3 orders of magnitude lower than the original.

In addition (to the extent possible), we compute vector embeddings
from a recent deep learning based method (node2vec~\cite{grover2016node2vec}).
We again use all the edge probability models discussed above, and perform
an identical experiment (in Figure~\ref{fig:intro-triangle-distro}, these
are denoted by ``n2v"). Again, we observe that the low-degree triangle behavior
is not captured by these deep learned embeddings.
}

\subsection{Broader context} \label{sec:context}
{The use of geometric embeddings for graph analysis has a rich history, arguably
going back to spectral clustering~\cite{Fi73}. In recent years, the Stochastic Block Model
has become quite popular in the statistics and algorithms community~\cite{HoLa83}.
and the Random Dot Product Graph model is a generalization of this notion
(refer to recent surveys~\cite{Ab18,AtFi+18}). As mentioned earlier, Theorem~\ref{thm:main}
brings into question the standard uses of these methods to model social networks.
The use of vectors to represent vertices
is sometimes referred to as \emph{latent space models}, where geometric proximity models
the likelihood of an edge. Although dot products are widely used, we
note that some classic latent space approaches use
Euclidean distance (as opposed to dot product) to model edge probabilities~\cite{HoRa+02},
and this may avoid the lower bound of Theorem~\ref{thm:main}.
Beyond graph analysis, the method of Latent Semantic Indexing (LSI) also falls
in the setting of Theorem~\ref{thm:main}, wherein we have a low-dimensional embedding
of ``objects" (like documents) and similarity is measured by dot product~\cite{lsi-wiki}.
}

\section{High-level description of the proof} \label{sec:highlevel}

In this section, we sketch the proof of Theorem~\ref{thm:main}.
{The sketch provides
sufficient detail for a reader who wants to understand the reasoning
behind our result, but is not concerned with technical details.
We will make the simplifying assumption that all $v_i$s have the same
length $L$.  We note that this setting is interesting in
its own right, since it is often the case in practice that all vectors
are non-negative and normalized. In this case, we get a stronger rank
lower bound that is linear in $n$. Section~\ref{sec:vary}
provides intuition on how we can remove this assumption.
The full details of the proof are given in Section~\ref{sec:full-proof}.
}

First, we lower bound $L$. By Cauchy-Schwartz, $\vec{v}_i \cdot \vec{v}_j \leq L^2$. Let $X_{i,j}$
be the indicator random variable for the edge $(i,j)$ being present. Observe that all $X_{i,j}$s
are independent and $\hbox{\bf E}[X_{i,j}] = \min(\vec{v}_i \cdot \vec{v}_j, 1) \leq L^2$.

The expected number of triangles in $G \sim \mathcal{G}_V$ is:
\begin{eqnarray}
& & \hbox{\bf E}[\sum_{i\neq j \neq k} X_{i,j} X_{j,k} X_{i,k}] \\
& \leq & \sum_i \sum_{j,k} \hbox{\bf E}[X_{j,k}] \hbox{\bf E}[X_{i,j}] \hbox{\bf E}[X_{i,k}] \\
& \leq & L^2 \sum_i \sum_{j,k} \hbox{\bf E}[X_{i,j}] \hbox{\bf E}[X_{i,k}] = L^2 \sum_i (\sum_j \hbox{\bf E}[X_{i,j}])^2
\label{eq:tri1}
\end{eqnarray}
Note that $\sum_j \hbox{\bf E}[X_{i,j}] = \hbox{\bf E}[\sum_j X_{i,j}]$ is at most the degree of $i$,
{which is at most $c$. (Technically, the $X_{i,i}$ term creates a self-loop, so the correct
upper bound is $c+1$. For the sake of cleaner expressions, we omit the additive $+1$
in this sketch.)}

 The expected number of triangles is at least $\Delta n$. Plugging these bounds
in:
\begin{equation}
\Delta n \leq L^2 c^2 n \Longrightarrow L \geq \sqrt{\Delta}/c \label{eq:tri2}
\end{equation}
Thus, the vectors have length at least $\sqrt{\Delta}/c$. Now, we lower bound the rank of $V$.
It will be convenient to deal with the Gram matrix $M = V^TV$, which has the same rank as $V$.
Observe that $M_{i,j} = \vec{v}_i \cdot \vec{v}_j \leq L^2$.
We will use the following lemma stated first by Swanapoel, but has appeared
in numerous forms previously~\cite{Sw14}.

\begin{lemma} [Rank lemma] Consider any square matrix $M \in \mathbb{R}^{n \times n}$. Then
$$ \mathrm{rank}(M) \ge \frac{| \sum_i M_{i,i}|^2}{\left( \sum_i \sum_j |M_{i,j}|^2 \right)} $$
\end{lemma}

Note that $M_{i,i} = \vec{v}_i \cdot \vec{v}_i = L^2$, so the numerator
$|\sum_i M_{i,i}|^2 = n^2 L^4$. The denominator requires more work. We split
it into two terms.
\begin{equation} \label{eq:small-denom}
\sum_{\substack{i,j\\\vec{v}_i \cdot \vec{v}_j \leq 1}} (\vec{v}_i \cdot \vec{v}_j)^2
\leq \sum_{\substack{i,j\\\vec{v}_i \cdot \vec{v}_j \leq 1}} \vec{v}_i \cdot \vec{v}_j \leq cn
\end{equation}
If for $i\neq j$, $\vec{v}_i \cdot \vec{v}_j > 1$, then $(i,j)$ is an edge with probability $1$.
Thus, there can be at most $(c-1)n$ such pairs. Overall, there are at most $cn$ pairs
such that $\vec{v}_i \cdot \vec{v}_j > 1$. So, $\sum_{\substack{i,j\\ \vec{v}_i \cdot \vec{v}_j > 1}} (\vec{v}_i \cdot \vec{v}_j)
\leq cnL^4$. Overall, we lower bound the denominator in the rank lemma by $cn(L^4+1)$.

We plug these bounds into the rank lemma. We use the fact that $f(x) = x/(1+x)$ is decreasing for positive $x$,
and that $L \geq \sqrt{\Delta}/c$.
\begin{eqnarray*}
\mathrm{rank}(M) \geq \frac{n^2 L^4}{cn(L^4 + 1)} \geq \frac{n}{c} \cdot \frac{\Delta^2/c^4}{\Delta^2/c^4 + 1}
= \frac{\Delta^2}{c(\Delta^2 + c^4)} \cdot n
\end{eqnarray*}

\subsection{Dealing with varying lengths} \label{sec:vary}

{The math behind \Eqn{tri2} still holds with
the right approximations. Intuitively, the existence of at least $\Delta n$ triangles
implies that a sufficiently large number of vectors have length at least $\sqrt{\Delta}/c$.
On the other hand, these long vectors need to be ``sufficiently far
away" to ensure that the vertex degrees remain low. There
are many such long vectors, and they can can only be far away
when their dimension/rank is sufficiently high.

The rank lemma is the main technical tool that formalizes this
intuition. When vectors are of varying length,
the primary obstacle is the presence of extremely long vectors that create triangles.
The numerator in the rank lemma sums $M_{i,i}$, which is the length of the vectors.
A small set of extremely long vectors could dominate the sum, increasing the numerator.
In that case, we do not get a meaningful rank bound.

But, because the vectors inhabit low-dimensional space, the long vectors from \emph{different}
clusters interact with each other. We prove a ``packing" lemma (\Lem{dot}) showing that
there must be many large positive dot products among a set of extremely long vectors.
Thus, many of the corresponding vertices have large degree, and triangles incident
to these vertices do not contribute to low degree triangles.
Operationally, the main proof uses the packing lemma to show that there are few long vectors.
These can be removed without affecting the low degree structure. One can then perform
a binning (or ``rounding") of the lengths of the remaining vectors, to implement
the proof described in the above section.
}

\section{Proof of Theorem~\ref{thm:main}} \label{sec:full-proof}

For convenience, we restate the setting.
Consider a set of vectors $\vec{v}_1, \vec{v}_2, \ldots, \vec{v}_n$ $\in \mathbb{R}^d$, that represent
the vertices of a social network.
We will also use the matrix $V \in \mathbb{R}^{d \times n}$ for these vectors,
where each column is one of the $\vec{v}_i$s.
Abusing notation, we will use $V$ to represent
both the set of vectors as well as the matrix. We will refer to the vertices by the index in $[n]$.

Let $\mathcal{G}_V$ denote the following distribution of graphs over the vertex set $[n]$.
For each index pair $i,j$, independently insert (undirected) edge $(i,j)$ with probability
$\max(0,\min(\vec{v}_i \cdot \vec{v}_j, 1))$.

\subsection{The basic tools} \label{sec:tools}

We now state some results that will be used in the final proof.

\begin{lemma} \label{lem:rank} [Rank lemma~\cite{Sw14}] Consider any square matrix $A \in \mathbb{R}^{n \times n}$. Then
$$ | \sum_i A_{i,i}|^2 \leq \mathrm{rank}(A) \left( \sum_i \sum_j |A_{i,j}|^2 \right) $$
\end{lemma}

\begin{lemma} \label{lem:neg} Consider a set of $s$ vectors $\vec{w}_1, \vec{w}_2, \ldots, \vec{w}_s$ in $\mathbb{R}^d$.
$$ \sum_{\substack{(i,j) \in [s] \times [s] \\ \vec{w}_i \cdot \vec{w}_j < 0}} |\vec{w}_i \cdot \vec{w}_j|
\leq \sum_{\substack{(i,j) \in [s] \times [s] \\ \vec{w}_i \cdot \vec{w}_j > 0}} |\vec{w}_i \cdot \vec{w}_j| $$
\end{lemma}

\begin{proof}  Note that $(\sum_{i \leq s} \vec{w}_i)\cdot (\sum_{i \leq s} \vec{w}_i) \geq 0$.
Expand and rearrange to complete the proof.
\end{proof}

Recall that an independent set is a collection of vertices that induce no edge.

\begin{lemma} \label{lem:ind} Any graph with $h$ vertices and maximum degree $b$
has an independent set of at least $h/(b+1)$.
\end{lemma}

\begin{proof}  Intuitively, one can incrementally build an independent set,
by adding one vertex to the set, and removing at most $b+1$ vertices from the graph.
This process can be done at least $h/(b+1)$ times.

Formally, we prove by induction on $h$. First we show
the base case. If $h \leq b+1$, then the statement is trivially true.
(There is always an independent set of size $1$.) For the induction step, let us construct
an independent set of the desired size. Pick an arbitrary vertex $x$ and add it to the independent set.
Remove $x$ and all of its neighbors. By the induction hypothesis, the remaining graph
has an independent set of size at least $(h - b - 1)/(b+1) = h/(b+1) - 1$.
\end{proof}

\begin{claim} \label{clm:var} Consider the distribution $\mathcal{G}_{V}$.
Let $D_i$ denote the degree of vertex $i \in [n]$.
$\hbox{\bf E}[D^2_i] \leq \hbox{\bf E}[D_i] + \hbox{\bf E}[D_i]^2$.
\end{claim}

\begin{proof} (of Claim~\ref{clm:var}) Fix any vertex $i \in [n]$. Observe that $D_i = \sum_{j \neq i} X_j$, where $X_j$ is the indicator random variable
for edge $(i,j)$ being present. Furthermore, all the $X_j$s are independent.
\begin{eqnarray*}
    \hbox{\bf E}[D^2_i] & = & \hbox{\bf E}[(\sum_{j \neq i} X_j)^2] = \hbox{\bf E}[\sum_{j \neq i} X^2_j + 2\sum_{j\neq j'} X_j X_{j'}] \\
    & = & \hbox{\bf E}[\sum_{j \neq i} X_j] + 2\sum_{j \neq j'} \hbox{\bf E}[X_j] \hbox{\bf E}[X_{j'}] \\
    & \leq & \hbox{\bf E}[D_i] + (\sum_{j \neq i} \hbox{\bf E}[X_j])^2 = \hbox{\bf E}[D_i] + \hbox{\bf E}[D_i]^2
\end{eqnarray*}
\vspace{-5pt}
\end{proof}

A key component of dealing with arbitrary length vectors is the following dot product lemma.
This is inspired by results of Alon~\cite{Al03} and Tao~\cite{Ta-post},
who get a stronger lower bound of $1/\sqrt{d}$ for \emph{absolute values} of the dot products.

\begin{lemma}  \label{lem:dot} Consider any set of $4d$ unit vectors $\vec{u}_1, \vec{u}_2, \ldots, \vec{u}_{4d}$
in $\mathbb{R}^d$. There exists some $i \neq j$ such that $\vec{u}_i \cdot \vec{u}_j \geq 1/4d$.
\end{lemma}

\begin{proof} (of \Lem{dot})  We prove by contradiction, so assume $\forall i \neq j, \vec{u}_i \cdot \vec{u}_j < 1/4d$.
We partition the set $[4d] \times [4d]$ into
${\cal N} = \{(i,j) | \vec{u}_i \cdot \vec{u}_j < 0\}$ and $\mathcal{P} = \{(i,j) | \vec{u}_i \cdot \vec{u}_j \geq 0\}$.
The proof goes by providing (inconsistent) upper and lower bounds for $\sum_{(i,j) \in {\cal N}} |\vec{u}_i \cdot \vec{u}_j|^2$.
First, we upper bound $\sum_{(i,j) \in {\cal N}} |\vec{u}_i \cdot \vec{u}_j|^2$ by:
\begin{eqnarray}
    & \leq &  \sum_{(i,j) \in {\cal N}} |\vec{u}_i \cdot \vec{u}_j| \ \ \ \ \ \ \ \textrm{($\vec{u}_i$s are unit vectors)} \nonumber \\
    & \leq & \sum_{i \leq 4d} \|\vec{u}_i\|^2_2 + \sum_{\substack{1 \leq i \neq j \leq 4d\\ (i,j) \in \mathcal{P}}} |\vec{u}_i \cdot \vec{u}_j| \ \ \ \ \ \ \ \textrm{(\Lem{neg})} \nonumber \\
    & < & 4d + 16d^2/4d = 8d \ \ \ \ \textrm{(since $\vec{u}_i \cdot \vec{u}_j < 1/4d$)} \label{eq:dot-upper}
\end{eqnarray}
For the lower bound, we invoke the rank bound of \Lem{rank} on the $4d \times 4d$ Gram matrix $M$ of $\vec{u}_1, \ldots, \vec{u}_{4d}$.
Note that $\mathrm{rank}(M) \leq d$, $M_{i,i} = 1$, and $M_{i,j} = \vec{u}_i \cdot \vec{u}_j$.
By \Lem{rank}, $\sum_{(i,j) \in [4d] \times [4d]} |\vec{u}_i \cdot \vec{u}_j|^2 \geq (4d)^2/d = 16 d$.
We bound
\begin{eqnarray}
\sum_{(i,j) \in \mathcal{P}} |\vec{u}_i \cdot \vec{u}_j|^2 & = & \sum_{i \leq 4d} \|\vec{u}_i\|^2_2 + \sum_{(i,j) \in \mathcal{P}, i \neq j} |\vec{u}_i \cdot \vec{u}_j|^2 \nonumber \\
& \leq & 4d + (4d)^2/(4d)^2 \leq 5d
\end{eqnarray}
Thus, $\sum_{(i,j) \in {\cal N}} |\vec{u}_i \cdot \vec{u}_j|^2 \geq 16d - 5d = 11d$. This contradicts the bound of \Eqn{dot-upper}.

\end{proof}

\subsection{The main argument} \label{sec:proof}

We prove by contradiction. We assume that the expected number
of triangles contained in the set of vertices of expected degree at most $c$,
is at least $\Delta n$. {We remind the reader that $n$
is the total number of vertices.} For convenience, we simply remove the vectors
corresponding to vertices with expected degree at least $c$.
Let $\hat{V}$ be the matrix of the remaining vectors, and we focus
on $\mathcal{G}_{\hat{V}}$. The expected number of triangles in $G \sim \mathcal{G}_{\hat{V}}$
is at least $\Delta n$.

The overall proof can be thought of in three parts.

{\em Part 1, remove extremely long vectors:} Our final aim is to use the rank lemma (Lemma~\ref{lem:rank})
to lower bound the rank of $V$. The first problem we encounter is that
extremely long vectors can dominate the expressions in the rank lemma,
and we do not get useful bounds. We show that the number of such long vectors
is extremely small, and they can removed without affecting too many triangles.
In addition, we can also remove extremely small vectors, since they cannot participate
in many triangles.

{\em Part 2, find a ``core" of sufficiently long vectors that contains enough triangles:} The previous step
gets a ``cleaned" set of vectors. Now, we bucket these vectors by length. We show
that there is a large bucket, with vectors that are sufficiently long, such that
there are enough triangle contained in this bucket.

{\em Part 3, apply the rank lemma to the ``core":} We now focus on this core of vectors,
    where the rank lemma can be applied.
    \main{At this stage, the mathematics shown
    in Section~\ref{sec:highlevel} can be carried out almost directly.}

Now for the formal proof.
For the sake of contradiction, we assume that $d = \mathrm{rank}(\hat{V}) < \alpha (\Delta^4/c^9) \cdot n/\lg^2n$
(for some sufficiently small constant $\alpha > 0$).

{\bf Part 1: Removing extremely long (and extremely short) vectors}

We begin by showing that there cannot be many long vectors in $\hat{V}$.

\begin{lemma} \label{lem:long} There are at most $5cd$ vectors of length at least $2\sqrt{n}$.
\end{lemma}

\begin{proof} Let ${\cal L}$ be the set of ``long" vectors, those with length at least $2\sqrt{n}$.
Let us prove by contradiction, so assume there are more than $5cd$ long vectors.
Consider a graph $H = ({\cal L},E)$, where vectors $\vec{v_i},\vec{v_j} \in {\cal L}$ ($i \neq j$) are connected
by an edge if $\frac{\vec{v_i}}{\|\vec{v}_i\|_2} \cdot \frac{\vec{v_j}}{\|\vec{v}_j\|_2} \geq 1/4n$.
We choose the $1/4n$ bound to ensure that all edges in $H$ are edges in $G$.

Formally, for any edge $(i,j)$ in $H$, $\vec{v_i}\cdot\vec{v_j} \geq \|\vec{v}_i\|_2 \|\vec{v}_j\|_2/4n
\geq (2\sqrt{n})^2/4n = 1$. So $(i,j)$ is an edge with probability $1$
in $G \sim \mathcal{G}_V$. The degree of any vertex
in $H$ is at most $c$. By \Lem{ind}, $H$ contains an independent set $I$ of size at least $5cd/(c+1) \geq 4d$.
Consider an arbitrary sequence of $4d$ (normalized) vectors in $I$  $\vec{u}_1, \ldots, \vec{u}_{4d}$.
Applying \Lem{dot} to this sequence, we deduce the existence of $(i,j)$ in $I$ ($i \neq j$) such that
$\frac{\vec{v_i}}{\|\vec{v}_i\|_2} \cdot \frac{\vec{v_j}}{\|\vec{v}_j\|_2} \geq 1/4d \geq 1/4n$.
Then, the edge $(i,j)$ should be present in $H$, contradicting the fact that $I$ is an independent set.
\end{proof}

Denote by $V'$ the set of all vectors in $\hat{V}$ with length in the range $[n^{-2}, 2\sqrt{n}]$.

\begin{claim} \label{clm:prune} The expected degree of every vertex in $G \sim \mathcal{G}_{V'}$ is
at most $c$, and the expected number of triangles in $G$ is at least $\Delta n/2$.
\end{claim}

\begin{proof} Since removal of vectors can only decrease the
degree, the expected degree of every vertex in $\mathcal{G}_{V'}$ is naturally at most $c$.
It remains to bound the expected number of triangles in $G \sim \mathcal{G}_{V'}$.
By removing vectors in $V\setminus V'$, we potentially lose some triangles. Let us
categorize them into those that involve at least one ``long" vector (length $\geq 2\sqrt{n}$)
and those that involve at least one ``short" vector (length $\leq n^{-2}$) but no long vector.

We start with the first type.
By \Lem{long}, there are at most $5cd$ long vectors. For any vertex, the expected number
of triangles incident to that vertex is at most the expected square of the degree.
By \Clm{var}, the expected degree squares is at most $c + c^2 \leq 2c^2$.  Thus,
the expected total number of triangles of the first type is at most $5cd\times 2c^2 \leq \Delta n/\lg^2n$.

Consider any triple of vectors $(\vec{u},\vec{v},\vec{w})$ where $\vec{u}$ is short
and neither of the others are long. The probability that this triple forms
a triangle is at most
\begin{eqnarray*}
& & \min(\vec{u}\cdot\vec{v},1) \cdot \min(\vec{u}\cdot\vec{w},1) \\
& \leq & \min(\|\vec{u}\|_2\|\vec{v}\|_2,1) \cdot \min(\|\vec{u}\|_2 \|\vec{w}\|_2,1) \\
& \leq & (n^{-2} \cdot 2\sqrt{n})^2 \leq 4n^{-3}
\end{eqnarray*}
Summing over all such triples, the expected number of such triangles is at most $4$.

Thus, the expected number of triangles in $G \sim \mathcal{G}_{V'}$ is at least $\Delta n - \Delta n/\lg^2n - 4 \geq \Delta n/2$.
\end{proof}

{\bf Part 2: Finding core of sufficiently long vectors with enough triangles}

For any integer $r$, let $V_r$ be the set of vectors $\{\vec{v} \in V' | \|\vec{v}\|_2 \in [2^r, 2^{r+1})\}$.
Observe that the $V_r$s form a partition of $V'$. Since all lengths in $V'$
are in the range $[n^{-2},2\sqrt{n}]$, there are at most $3\lg n$ non-empty $V_r$s.
Let $R$ be the set of indices $r$ such that $|V_r| \geq (\Delta/60c^2)(n/\lg n)$.
Furthermore, let $V''$ be $\bigcup_{r\in R} V_r$.

\begin{claim} \label{clm:prune2} The expected number of triangles in $G \sim \mathcal{G}_{V''}$ is at least $\Delta n/8$.
\end{claim}

\begin{proof} The total number of vectors in $\bigcup_{r \notin R} V_r$ is at most
$3\lg n \times (\Delta/60c^2) (n/\lg n)$  $\leq (\Delta/20c^2) n$. By \Clm{var} and linearity of expectation,
the expected sum of squares of degrees of all vectors in $\bigcup_{r \notin R} V_r$
is at most $(d+c^2) \times (\Delta/20c^2) n$ $\leq \Delta n/10$. Since the expected
number of triangles in $G \sim \mathcal{G}_{V'}$ is at least $\Delta n/2$
(\Clm{prune}) and the expected number of triangles
incident to vectors in $V' \setminus V''$ is at most $\Delta n/10$,
the expected number of triangles in $G \sim \mathcal{G}_{V''}$ is at least $\Delta n/2 - \Delta n/10 \geq \Delta n/8$.
\end{proof}

We now come to an important claim. Because the expected number of triangles in $G \sim \mathcal{G}_{V''}$ is large,
we can prove that $V''$ must contain vectors of at least constant length.

\begin{claim} \label{clm:len} $\max_{r \in R} 2^r \geq \sqrt{\Delta}/4c$.
\end{claim}

\begin{proof} Suppose not. Then every vector in $V''$ has length at most $\sqrt{\Delta}/4c$.
By Cauchy-Schwartz, for every pair $\vec{u}, \vec{v} \in V''$, $\vec{u}\cdot\vec{v} \leq \Delta/16c^2$.
Let $I$ denote the set of vector indices in $V''$ (this corresponds to the vertices in $G \sim \mathcal{G}_{V''}$).
For any two vertices $i \neq j \in I$, let $X_{i,j}$ be the indicator random variable for
edge $(i,j)$ being present.
The expected number of triangles incident to vertex $i$ in $G \sim \mathcal{G}_{V''}$
is
$$ \hbox{\bf E}[\sum_{j\neq k \in I} X_{i,j}X_{i,k}X_{j,k}]  =  \sum_{j\neq k \in I} \hbox{\bf E}[X_{i,j} X_{i,k}] \hbox{\bf E}[X_{j,k}] $$
Observe that $\hbox{\bf E}[X_{j,k}]$ is at most $\vec{v_j} \cdot \vec{v_k} \leq \Delta/16c^2$.
Furthermore, $\sum_{j \neq k \in I} \hbox{\bf E}[X_{i,j} X_{i,k}] = \hbox{\bf E}[D^2_i]$
{(recall that $D_i$ is the degree of vertex $i$.)}
By \Clm{var}, this is at most $c+c^2 \leq 2c^2$. The expected number of triangles in $G \sim \mathcal{G}_{V''}$
is at most $n \times 2c^2 \times \Delta/16c^2 = \Delta n/8$. This contradicts \Clm{prune2}.
\end{proof}

{\bf Part 3: Applying the rank lemma to the core}

We are ready to apply the rank bound of \Lem{rank} to prove the final result. The following
lemma contradicts our initial bound on the rank $d$, completing the proof. We will omit
some details in the following proof, and provide
a full proof in the SI.

\begin{lemma} \label{lem:main} $\mathrm{rank}(V'') \geq (\alpha \Delta^4/c^9) n/\lg^2 n$.
\end{lemma}

\begin{proof} It is convenient to denote the index set of $V''$ be $I$.
Let $M$ be the Gram Matrix $(V'')^T(V'')$, so for $i,j \in I$,
$M_{i,j} = \vec{v}_i \cdot \vec{v}_j$
By \Lem{rank}, $\mathrm{rank}(V'') = \mathrm{rank}(M) \geq (\sum_{i \in I} M_{i,i})^2/\sum_{i,j \in I} |M_{i,j}|^2$.
Note that $M_{i,i}$ is $\|\vec{v_i}\|^2_2$, which is at least $2^{2r}$
for $\vec{v_i} \in V_r$. Let us denote $\max_{r \in R} 2^r$ by $L$,
so all vectors in $V''$ have length at most $2L$.
By Cauchy-Schwartz, all entries in $M$ are at most $4L^2$.

We lower bound the numerator.
\begin{eqnarray*}
    & & \big(\sum_{i \in I} \|\vec{v_i}\|^2_2\big)^2 \geq \big(\sum_{r \in R} 2^{2r} |V_r|\big)^2 \\
    & \geq & \big(\max_{r \in R} 2^{2r} (\Delta/60 c^2)(n/\lg n)\big)^2 \\
    & = & L^4(\Delta^2/3600 c^4)(n^2/\lg^2 n)
\end{eqnarray*}

Now for the denominator. We split the sum into four parts and bound each separately.
\begin{eqnarray}
    \sum_{i,j \in I} |M_{i,j}|^2 & = & \sum_{i \in I} |M_{i,i}|^2 + \sum_{\substack{i,j \in I \\ i \neq j, M_{i,j} \in [0,1]}} |M_{i,j}|^2 \nonumber \\
    & + & \sum_{\substack{i,j \in I \\ i \neq j, M_{i,j} > 1}} |M_{i,j}|^2 + \sum_{\substack{i,j \in I \\ M_{i,j} < 0}} |M_{i,j}|^2 \nonumber :!
\end{eqnarray}
Since $|M_{i,i}| \leq L^2$, the first term is at most $4nL^4$.
For $i \neq j$ and $M_{i,j} \in [0,1]$, the probability that edge $(i,j)$ is present is precisely $M_{i,j}$.
Thus, for the second term,
\begin{equation}
\sum_{\substack{i,j \in I \\ i \neq j, M_{i,j} \in [0,1]}} |M_{i,j}|^2
\leq \sum_{\substack{i,j \in I \\ i \neq j, M_{i,j} \in [0,1]}} M_{i,j}
\leq 2cn
\end{equation}
For the third term, we observe that when $M_{i,j} > 1$ (for $i \neq j$),
then $(i,j)$ is an edge with probability $1$. There can be at most $2cn$ pairs $(i,j)$, $i \neq j$,
such that $M_{i,j} > 1$. Thus, the third term is at most $2cn \cdot (4L^2)^2 = 32cnL^4$.

Now for the fourth term. Note that $M$ is a Gram matrix, so we can invoke \Lem{neg} on its entries.
\begin{eqnarray}
    \sum_{\substack{i,j \in I \\ M_{i,j} < 0}} |M_{i,j}|^2
    & \leq & L^2\sum_{\substack{i,j \in I \\ M_{i,j} < 0}} |M_{i,j}| \nonumber \\
    & \leq & L^2(\sum_{i \in I} |M_{i,i}| + \sum_{\substack{i,j \in I \\ M_{i,j} > 0}} |M_{i,j}|) \nonumber \\
    & \leq & 4nL^4 + L^2 \sum_{\substack{i,j \in I \\ M_{i,j} \in [0,1]}} |M_{i,j}|  +
4L^4 \sum_{\substack{i,j \in I \\ M_{i,j} > 1}} 1 \nonumber \\
    & \leq & 4nL^4 + 2cn L^2 + 8cnL^4
\end{eqnarray}
Putting all the bounds together,
we get that $\sum_{i,j \in I} |M_{i,j}|^2 \leq n(4L^4 + 2c + 32cL^4 + 4L^4 + 2cL^2 + 8cL^4)
\leq 32n(L^4 + c(1 + L^2 + L^4))$. If $L \leq 1$, we can upper bound
by $128cn$. If $L \geq 1$, we can upper bound by $128cnL^4$. In either case,
$128cn(1+L^4)$ is a valid upper bound.

Crucially, by \Clm{len}, $L \geq \sqrt{\Delta}/4c$. Thus, $4^4 c^4 L^4/\Delta^2 \geq 1$.
Combining all the bounds (and setting $\alpha < 1/(128\cdot 3600 \cdot 4^4)$),
\begin{eqnarray*}
    \mathrm{rank}(V'') & \geq & \frac{L^4(\Delta^2/3600 c^4)(n^2/\lg^2 n)}{128cn(1+16L^4)} \\
    & \geq &
\frac{L^4(\Delta^2/3600 c^4)(n/\lg^2 n)}{128cn(4^4 c^4 L^4/\Delta^2+16L^4)} \\
& \geq & (\alpha \Delta^4/c^9)(n/\lg^2 n) \nonumber
\end{eqnarray*}
\end{proof}

\section{Details of empirical results} \label{sec:emp-details}

 {\bf Data Availability:}   The datasets used are summarized in \Tab{datasets}.
   We present here four publicly available datasets from different domains. The \texttt{ca-HepPh} is a co-authorship network
   \texttt{Facebook} is a social network, \texttt{cit-HepPh} is a citation network, all obtained from the SNAP graph database~\cite{Snap}.
   The \texttt{String\_hs} dataset is a protein-protein-interaction network obtained from~\cite{string}. (The citations
   provide the link to obtain the corresponding datasets.)

   {We first describe the primary experiment, used to validate Theorem~\ref{thm:main}
   on the SVD embedding.}
	We generated a $d$-dimensional embedding for various values of $d$
    using the SVD.
    Let $G$ be a graph with the $n \times n$ (symmetric) adjacency matrix $A$,
    with eigendecomposition $\Psi \Lambda \Psi^T$.  Let $\Lambda_d$ be the matrix with the $d \times d$ diagonal matrix
    with the $d$ largest magnitude eigenvalues of $A$ along the diagonal. Let $\Psi_d$ be the $n \times d$ matrix
    with the corresponding eigenvectors as columns. We compute the matrix $A_d = \Psi_d \Lambda_d \Psi_d^T$ and
    refer to this as the $d$ spectral embedding of $G$. This is the standard PCA approach.

	From the spectral embeddings, we generate a graph from $A_d$ by considering every pair of vertices $(i, j)$ and generate a random value in $[0, 1]$. If the $(i, j)^{\textrm{th}}$ entry of $A_d$ is greater than the random value generated, the edge is added to the graph. Otherwise the edge is not present. This is the same as taking $A_d$ and setting all negative values to 0, and all values greater than 1 to 1 and performing Bernoulli trials for each edge with the resulting probabilities.
    In all the figures, this is referred to as the ``SVD TDP" (truncated dot product) embedding.

\begin{figure*}
\begin{center}
\begin{tabular}{|c |c |c |c|}
	\hline
		\textbf{Dataset name} & \textbf{Network type} & \textbf{Number of nodes} & \textbf{Number of edges} \\
		\hline
		Facebook \cite{Snap} & Social network & 4K & 88K \\
		\hline
		cit-HePh \cite{Arnet-data}& Citation & 34K & 421K \\
        \hline
		String\_hs \cite{string} & PPI & 19K & 5.6M \\
		\hline
		ca-HepPh \cite{Snap}& Co-authorship & 12K & 118M \\
		\hline
\end{tabular}
\captionof{table}{\small Table of datasets used}
\label{tab:datasets}
\end{center}
\end{figure*}

\begin{figure*}[h!]
        \includegraphics[scale=.37]{facebook_tri_distro.pdf}
        \includegraphics[scale=.37]{String_hs_tri_distro.pdf}
        \includegraphics[scale=.37]{cit-HepPh_tri_distro.pdf}
				\captionof{figure}{\small Plots of degree $c$ vs $\Delta$: For each network,
        we plot $c$ versus the total number of triangles only involving vertices of
        degree at most $c$. We divide the latter by the number of vertices,
        so it corresponds to $\Delta$, as in the main definition.
        In each subfigure, we plot these
        both for the original graph, and the maximum $\Delta$ in
        a set of 100 samples from a 100-dimensional embedding. Observe how the embeddings generate
        graphs with very few triangles among low degree vertices. The gap in $\Delta$ for low degree is
        2-3 orders of magnitude in all instances.} \label{fig:triangle-distros}
\end{figure*}

\begin{figure*}[h!]
        \includegraphics[scale=.3]{ca-HepPh_ccdh.pdf}
        \includegraphics[scale=.3]{facebook_ccdh.pdf}
        \includegraphics[scale=.3]{String_hs_ccdh.pdf}
        \includegraphics[scale=.3]{cit-HepPh_ccdh.pdf}
				\captionof{figure}{\small Plots of degree distributions: For each network,
                we plot the true degree distribution vs the expected degree distribution of a 100-dimensional
                embedding. Observe how the embedding does capture the degree distribution quite accurately
                at all scales.} \label{fig:degdist}
\end{figure*}

\subsection{Triangle distributions}
  To generate Figure~\ref{fig:intro-triangle-distro} and Figure~\ref{fig:triangle-distros}, we calculated the number of triangles incident to vertices of different degrees in both the original graphs and the graphs generated from the embeddings. Each of the plots shows the number of triangles in the graph on the vertical axis and the degrees of vertices on the horizontal axis. Each curve corresponds to some graph, and each point $(x, y)$ in a given curve shows that the graph contains $y$ triangles if we remove all vertices with degree at least $x$. We then generate
  100 random samples from the 100-dimensional embedding, as given by SVD (described above). For each value of $c$,
  we plot the maximum value of $\Delta$ over all the samples. This is to ensure that our results are not affected
  by statistical variation (which was quite minimal).

\subsection{Alternate graph models} \label{sec:alt}

{ We consider three other functions of the dot product, to construct graph distributions
from the vector embeddings. \new{Details on
parameter settings and the procedure used for the optimization are given in the SI.}

{\em Logistic Regression on the Dot Product (LRDP):} We consider the probability
    of an edge $(i,j)$ to be the logistic function $L(1+\exp(-k(\vec{v}_i \cdot \vec{v}_j - x_0)))^{-1}$,
    where $L, k, x_0$ are parameters. Observe that the range of this function is $[0,1]$,
    and hence can be interpreted as a probability. We tune these parameters to fit the expected
    number of edges, to the true number of edges. Then, we proceed as in the TDP experiment.
    We note that the TDP can be approximated by a logistic function, and thus the LRDP
    embedding is a ``closer fit" to the graph than the TDP embedding.

{\em Logistic Regression on the Hadamard Product (LRHP):} This is inspired
    by linear models used on low-dimensional embeddings~\cite{HaYiLe17}. Define the Hadamard
    product $\vec{v}_i \odot \vec{v}_j$ to be the $d$-dimensional vector where the $r$th
    coordinate is the product of $r$th coordinates. We now fit a logistic function over
    linear functions of (the coordinates of) $\vec{v}_i \odot \vec{v}_j$. This is a significantly
    richer model than the previous model, which uses a fixed linear function (sum). Again,
    we tune parameters to match the number of edges.
    The fitting of LRDP and LRHP was done
    using the Matlab function {\tt glmfit} (Generalized Linear Model Regression Fit)
    \cite{matlab}. The distribution parameter was set to ``binomial", since
    the total number of edges is distributed as a weighted binomial.

{\em Softmax:} This is inspired by low-dimensional embeddings for random walk
    matrices~\cite{PeAlSk14,grover2016node2vec}. The idea is to make the probability
    of edge $(i,j)$ to be proportional to softmax,
    $\exp(\vec{v}_i \cdot \vec{v}_j)/\sum_{k \in [n]} \vec{v}_i \cdot \vec{v}_k$.
    This tends to push edge formation even for slightly higher dot products, and one might
    imagine this helps triangle formation. We set the proportionality constant separately for
    each vertex to ensure that the expected degree is the true degree. The probability matrix
    is technically undirected, but we symmetrize the matrix.

{\bf {\sc node2vec} experiments}:
\new{We also applied {\sc node2vec},
a recent deep learning based graph embedding method~\cite{grover2016node2vec}, to generate vector representations
of the vertices.
We used the optimized C++
implementation~\cite{n2vcpp} for {\sc node2vec}, which is equivalent
to the original implementation provided by the authors~\cite{n2v}. For all our experiments, we use the default
settings of walk length of 80, 10 walks per node, p=1 and q=1.
The {\sc node2vec} algorithm tries to model the random walk
matrix associated with a graph, not the raw adjacency matrix. The dot products
between the output vectors $\vec{v}_i \cdot \vec{v}_j$ are used to model
the random walk probability of going from $i$ to $j$, rather than the presence of an edge. It does not make
sense to apply the TDP function on these dot products, since this will
generate (in expectation) only $n$ edges (one for each vertex). We apply
the LRDP or LRHP functions, which use the {\sc node2vec} vectors as inputs to a machine learning
model that predicts edges.
}

In Figures~\ref{fig:intro-triangle-distro} and~\ref{fig:triangle-distros}, we show results for all the datasets.
We note that for all datasets and all embeddings, the models fail to capture the low-degree triangle behavior.

\subsection{Degree distributions}

{We observe that the low-dimensional embeddings obtained from SVD and the truncated dot product
can capture the degree distribution accurately. In Figure~\ref{fig:degdist}, we plot the degree
distribution (in loglog scale) of the original graph with the expected degree distribution
of the embedding. For each vertex $i$, we can compute its expected degree by the sum $\sum_i p_{ij}$,
where $p_{ij}$ is the probability of the edge $(i,j)$. In all cases, the expected degree distributions
is close to the true degree distributions, even for lower degree vertices.
The embedding successfully captures the ``first order"
connections (degrees), but not the higher order connections (triangles). We believe that this
reinforces the need to look at the triangle structure to discover the weaknesses of low-dimensional
embeddings.}

\subsection{Detailed relationship between rank and triangle structure} \label{sec:relation}

For the smallest {\tt Facebook} graph, we were able to compute the entire set of eigenvalues.
This allows us to determine how large a rank is required to recreate the low-degree triangle structure.
In Figure~\ref{fig:full-spectrum}, for varying rank of the embedding,
we plot the corresponding triangle distribution.
In this plot, we choose the embedding given by the eigendecomposition
(rather than SVD), since it is guaranteed to converge to the correct triangle distribution
for an $n$-dimensional embedding ($n$ is the number of vertices). The SVD and eigendecomposition
are  mostly identical for large singular/eigenvalues, but tend to be different (up to a sign)
for negative eigenvalues.

We observe that even a 1000 dimensional embedding does not capture the $c$ vs $\Delta$ plots
for low degree. Even the rank 2000 embedding
is off the true values, though it is correct to within an order of magnitude.
This is strong corroboration of our main theorem, which says that near linear
rank is needed to match the low-degree triangle structure.

\begin{figure}
        \includegraphics[scale=0.5]{facebook_spectrum.pdf}
				\captionof{figure}{\small Plots of degree $c$ vs $\Delta$, for varying rank:
                For the {\tt Facebook} social network, for varying rank of embedding, we plot $c$ versus the total number of triangles only involving vertices of
        degree at most $c$. The embedding is generated by taking the top eigenvectors. Observe how
        even a rank of 2000 does not suffice to match the true triangle values for low degree.} \label{fig:full-spectrum}
\end{figure}

\bibliographystyle{alpha}

redpajama_set_name: RedPajamaArXiv


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helps foreign nationals to live and work lawfully in the United States. He is the Managing Attorney of the JQK Law Firm, a boutique style immigration law firm headquartered in Los Angeles, CA, with satellite offices throughout the west coast of the United States.
He is also Adjunct Professor of Immigration Law at Loyola Law School, in Los Angeles. As a first-generation American of immigrant parents, John Khosravi is known as a defender of Immigrant Rights and believes all people should have the same opportunities in the United States. John Q. Khosravi's law practice focuses on Family, Business (Investment & Employment) and Talent (Artists, Athletes, Scientists/Professors/Researchers & Individuals with Extraordinary Abilities) based immigration to the United States. Visa Petitions include EB-1, EB-2, EB-5, E-2, L-1, H1-B and TN NAFTA Visas.
He also helps individuals Naturalization, obtaining travel documents (White Passport) and other immigration-related needs of his clients.
John works with individuals and clients of various sizes, from start-ups and small businesses to multinational corporations, in a broad range of industries, including academic, advertising, medical and research, new media, biotechnology, aerospace, fashion and the arts and entertainment fields. Mr Khosravi is a member of the American Immigration Lawyers Association (AILA) and currently serves as the Southern California Chapter's Law School Liaison and on the AILA National Law Student Committee.
He is also an Adjunct Professor of Immigration Law at Loyola Law School, Los Angeles. John earned his Juris Doctor degree from Loyola Law School, after receiving his B.A. with Honors from UCLA.
JQK Immigration Law Firm: Mr Khosravi has extended under Judge Debre K. Weintraub and Marvin Lager (Appellate Division of the California Superior Court), Alexander Williams (Mediation Division) and the Federal Bankruptcy Court of the Honorable Judge Sheri Bluebond. Mr Khosravi also extern for the Business and Legal Affairs Department of the Home Entertainment Division of NBC Universal.
He is licensed by the California State Bar. John is also an avid guitar player, songwriter, singer, runner, and bicyclist.
Active Member of AILA's Southern California Chapter Law School Liaison of AILA's Southern California Chapter (2014-2018) Founder of the Immigration Lawyers Listserv (www.
Co-Founder of the Los Angeles Immigration Clinic Staff Member of the Loyola Law School's International and Comparative Law Review (2008-2009) Selected as one of the Top 40 Under 40 USA Immigration Lawyers in the State of California by the American Society of Legal Advocates for 2016, 2017 and 2018 (top 1.5%) Chosen Top 40 Under 40 In California by the National Advocate for 2017 & 2018 Published by the Immigration Daily, the Leading Iranian immigration Lawyer Publisher and the American Bar Association (ABA) Published in the American Immigration Lawyers Association (AILA) Journals: "[Immigration Law] Is This What You Want To Do?


Born the six of seven kids to parents of modest means, Conn's CEO Norm Miller knows firsthand the challenges facing working families. That's why he's proud of Conn's approach to serving customers and believes the retailer's unique business model offers staggering growth potential.
Conn's recently opened its 100th store in Las Vegas, but CEO Norm Miller is convinced the retailer could eventually operate as many as 400 to 500 stores nationwide by providing quality, value priced merchandise and convenient financing options to the millions of Americans with less than perfect credit. Miller joined the company in September 2015 and to capitalize on the growth potential he's intent on establishing what he calls the right foundation for growth. That means fine-tuning the product assortment of furniture and mattresses, home appliances, consumer electronics and home office products while making sure the company continues to drive sales by extending credit to underserved customers who are most likely to make installment payments in a timely manner.
Conn's has been fine-tuning its lending procedures and refining its product assortment to become more of a furniture retailer while exiting some volatile consumer electronics categories like video games, digital cameras and tablet. Over time, Miller sees the rapidly growing furniture and mattress category, which produced same store sales of 18% in the third quarter, accounting for between 42% and 45% of the company's product sales, up from about 33% currently. The appliance category is the other major contributor to sales, accounting for about 27% of total sales with the de-emphasized consumer electronic categories accounting for another 22%.
Because transaction sizes can be large, Conn's offers in-house financing which is quite popular with customers but exposes the retailer to losses. It's also a risky proposition to investors which is why Conn's provides frequent and detailed disclosures into a credit portfolio where 10.2% of balances were 60 days past due during the company's most recent quarter. This despite the fact the company is selective about those to whom it extends credit. For example, of the more than 300,000 credit applications processed during the quarter, only 42% were approved and the average household income of the applicants was about $41,000. The company has nearly 752,000 active accounts with an average credit score of 594 and average balance of $2,370 giving it a total receivables portfolio of $1.5 billion.
Miller, whose background appears tailor-made to lead Conn's, joined the company after the struggling retailer concluded a strategic business review and determined its best path forward was to remain an independent, publicly held company. Miller had previously served as president of Sear Automotive and president and COO of DFC Global Corp., a leading financial service company focused on serving unbanked and underbanked consumers. In addition, he held management roles at Aramark, Nestle, Kraft and Pepsi. And he graduated from West Point and served as an officer in the U.S. Army.
Conn's should be well served by an executive with such a diverse background considering the company's unique operating model and reliance on extending credit to subprime customers to drive sales.
The value proposition the company provides is yielding respectable improvement, but not yet remarkable results. In the third quarter, total revenues increased 6.8% to $395.2 million due to retail revenues that increased 5.9% to $323.1 million and credit revenues that increased 11.2% to $72.2 million. Despite top line growth, Conn's posted a net loss of $2.4 million, or seven cents a share, which was better than a prior year loss of $3.1 million, or eight cents a share. The company's performance would have been stronger but it incurred charges equal to six cents a share related to the review of strategic alternatives which preceded Miller's arrival and it also made a prudent decision to increase reserves for bad debt expense.


text:
Technical and Commercial Management
Due Digilence and transaction support
Tailor-made specialized services
Four areas of expertise to offer a multidisciplinary service aimed at supporting the client in all management, contractual and administrative issues.
Our Asset Management services support the owner of the photovoltaic or wind plant not only for what concerns the supervision and monitoring of performances but also for every technical, administrative, contractual and financial issue as well as for compliance to current regulation. An experienced "control room" designed to manage the relations with every player involved in the energy market, such as maintenance technicians, insurance companies, surveillance companies, financial institutions, local authorities and real estate owners.
Overview and monitoring of plant perfomance;
Management of relations with counterparts in the contractual sector, with local authorities and real estate owners;
Administrative, financial, fiscal and accounting consultancy;
ICT Management.
We provide services for the measurement and sale of the energy produced by the plants: from the selection of the best Power Purchase Agreement to the handling of the relations with GSE and compliance checks with the producers, including the management of the relations with customs, electric energy, gas and hydric authority.
GSE relations managament;
Compliance check;
Energy sale via the best Power Purchase Agreement;
Optimized handling of energy purchases.
We evaluate the adequacy and proper mode of operation of renewable energy plants, with a particular focus on the analysis of technical, contractual, commercial and management issues, but also extending to assistance in transactions as well as selection of plants of potential interest for the investor. A service designed to monitor and manage balance sheets and corporate structures.
Evaluation of the adequacy and proper functioning of plants;
Analysis of technical, contractual, commercial and management issues;
Transaction support and purchase of renewable energy plants;
Consultancy for the identification of plants of potential interest for the investor.
We provide services for the preparation to plant evaluation visits or plant purchase: we put in effect a precautionary evaluation of the inclusiveness of technical, administrative and legal documentation, we offer support for the preparation or management of GSE inspections; we also organize extraordinary maintenance and repowering; furthermore we provide operative backing to financial institutions in cases of distressed assets. Proxima helps accountants, lawyers, curators and consultants engaged in the repossesment of non-performing plants or in case of bankruptcy of the owner.
Technical services for compliance to sector regulations;
Evaluation of inclusiveness of technical, administrative and legal documentation;
Preparation of documentation for GSE inspections;
Operative management support to inverstors in case of distressed assets.
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These are very interesting recipe. Pimm's Cupcakes combine many different flavors, and some of them are so unusual for the cake, like cucumber. Cucumber frosting sounds very intriguing . Actually, these delicious cupcakes are frosted with four different frosting : orange, strawberry, mint and cucumber frosting.
Four colours of frosting swirled together on cupcakes and garnished with fresh slices of orange, cucumber, mint or strawberry and some sprinkles make these cupcakes look like cocktail in cupcake form.


text:
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Overnight Merchandise Associate in Atlanta, Georgia.
Floor & Decor is like no other flooring company. Founded in 2000 in Atlanta, we have quickly become one of the country's leading flooring retailers, with dozens of stores nationwide – and more to come. A strong entrepreneurial spirit flows through our young company.
Time, Talent & Teamwork.
The remarkable success we have enjoyed could not have happened without the time, talent, and teamwork of our extremely dedicated associates. Working with equal amounts of passion and grit, we've become the "category killer of flooring" by revolutionizing the way people buy flooring. Our customers enjoy the largest in-stock selection of tile, wood, stone, and flooring accessories at prices no one can beat.
Opportunities to advance.
To continue our amazing growth, Floor & Decor is actively seeking gifted people with a solid work ethic and a real passion for helping others. Are you ready to make a difference at a truly groundbreaking company?
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Overnight Merchandise Associate
Atlanta | Georgia | United States | 30317
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It's great being part of a culture where entrepreneurship and team spirit are not just buzzwords. If you love working with a great group of people and desire the opportunity to grow, this is the place for you.
This position will maintain the store merchandising standards and store pricing integrity. This role is responsible for managing price changes, execution of product SKU labeling and processing returned merchandise.
Minimum Eligibility Requirements
(1) year of retail sales experience required or an equivalent combination of education and experience sufficient to perform the essential functions of the job, as determined by the company.
Excellent communication skills (verbal and written).
Excellent interpersonal skills with the ability to work with all levels of store management and store associates.
Strong computer skills and internet project coordination experience.
Must be a self-starter and work well in a fast-paced environment.
Sensitivity to confidential information.
Utilize and maintain store pricing using Tags-On-Demand (TOD) mobile printing station.
Identify and label merchandise that does not have vendor provided SKUs using TOD.
Support merchandise returns by returning saleable product to sales floor.
Demonstrate a thorough understanding and comply with the company's safe lifting practices standard operating procedures (SOP).
As a product knowledge expert, assist customers with product selection and answer product related questions.
Create price tags and signage for all products.
Replace outdated signage based upon updated price changes.
Return products to the correct home locations after customer returns.
Clean and stock products according to Floor & Decor's standards.
Communicate in-store moves and non-scanning SKU products via email.
Partner with Department Managers to determine which products need SKUs, price changes and need to be returned.
Working Conditions (travel, hours, environment)
While performing the duties of this job, the employee is frequently exposed to a warehouse environment and moving vehicles. The noise level in the work environment is typically moderate to noisy.
Physical/Sensory Requirements
Physical Work – Ability exert 30 – 60 pounds of force occasionally, and or 20-30 pounds of force frequently, and/or greater than negligible up to 20 pounds of force constantly to lift, carry push, pull, or otherwise move objects. The employee is often required to use their hands and fingers, to handle or feel. The employee is frequently required to stand, walk, reach with arms and hands, climb or balance, and to stoop, kneel, crouch or crawl.
Floor & Decor is an equal opportunity employer and is committed to equal opportunity for all associates and applicants. F&D recruits, hires, trains, promotes, compensates and administers all personnel actions without regard to race, color, religion, sex, sex stereotyping, pregnancy (which includes pregnancy, childbirth and medical conditions related to pregnancy, childbirth or breastfeeding), gender, gender identity, gender expression, national origin, age, mental or physical disability, ancestry, medical condition, marital status, military or veteran status, citizenship status, sexual orientation, genetic information or any other status protected by applicable law.
This policy applies to all areas of employment, including recruitment, testing, screening, hiring, selection for training, upgrading, transfer, demotion, layoff, discipline, termination, compensation, benefits and all other privileges, terms and conditions of employment. This policy and the law prohibit employment discrimination against any associate or applicant on the basis of any legally protected status outlined above.
Overnight Warehouse Associate
Atlanta Georgia United States Atlanta, Georgia, United States, 30329 Store Operations Store Operations
It's great being part of a culture where entrepreneurship and team spirit are not just buzzwords. If you love working with a great group of people and desire the opportunity to grow, this is the place for you. What You'll Do As a Receiving Associ...
Morrow Georgia United States Morrow, Georgia, United States Store Operations Store Operations
Atlanta Georgia United States Atlanta, Georgia, United States, 30317 Warehouse Warehouse
It's great being part of a culture where entrepreneurship and team spirit are not just buzzwords. If you love working with a great group of people and desire the opportunity to grow, this is the place for you. What You'll Do As a Warehouse Associ...
Atlanta Georgia United States Atlanta, Georgia, United States, 30317 Customer Service Customer Service
It's great being part of a culture where entrepreneurship and team spirit are not just buzzwords. If you love working with a great group of people and desire the opportunity to grow, this is the place for you. PurposeThis position is responsible ...
It's great being part of a culture where entrepreneurship and team spirit are not just buzzwords. If you love working with a great group of people and desire the opportunity to grow, this is the place for you. What You'll Do We work to ensure tha...
Receiving Associate
First Name f1b80bfe
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Email e125607f
Department and Location c4a48257
Departments efb216fb Departments 659f12b1 Accounts Payable Admin Customer Care Operations Customer Care Services Customer Service Design Services Distribution Operations Distribution Operations Leadership Distribution Warehouse Ecommerce Inventory Merchandising Leadership Merchandising Operations Operations Pro Services Product Review Store Operations Store Operations Leadership Store Operations Specialist Store Sales Store Sales Specialist Supply Chain Systems Support & Enhancements Technology Warehouse
Locations ccae7a09 Locations f773e5d0 Albuquerque, New Mexico, United States Alexandria, Virginia, United States Arlington, Texas, United States Arlington Heights, Illinois, United States Arvada, Colorado, United States Atlanta, Georgia, United States Aurora, Illinois, United States Austin, Texas, United States Avon, Massachusetts, United States Birmingham, Alabama, United States Bloomingdale, Georgia, United States Boynton Beach, Florida, United States Bridgeton, Missouri, United States Brookfield, Wisconsin, United States Buford, Georgia, United States Burlingame, California, United States Carson, California, United States Charlotte, North Carolina, United States Chicago, Illinois, United States Cincinnati, Ohio, United States Clearwater, Florida, United States Concord, North Carolina, United States Countryside, Illinois, United States Dallas, Texas, United States Denver, Colorado, United States Devon, Pennsylvania, United States Downey, California, United States Draper, Utah, United States Edgemere, Maryland, United States El Paso, Texas, United States Everett, Washington, United States Farmingdale, New York, United States Fort Lauderdale, Florida, United States Fort Myers, Florida, United States Fort Worth, Texas, United States Fountain Valley, California, United States Fullerton, California, United States Gaithersburg, Maryland, United States Glendale, Arizona, United States Greensboro, North Carolina, United States Gretna, Louisiana, United States Gurnee, Illinois, United States Hampton, Virginia, United States Henderson, Nevada, United States Henrico, Virginia, United States Hialeah, Florida, United States Hilliard, Ohio, United States Hollywood, Florida, United States Houston, Texas, United States Indianapolis, Indiana, United States Jacksonville, Florida, United States Katy, Texas, United States Kennesaw, Georgia, United States Knoxville, Tennessee, United States Lakeland, Florida, United States Las Vegas, Nevada, United States Levittown, Pennsylvania, United States Littleton, Colorado, United States Lombard, Illinois, United States Louisville, Kentucky, United States Marietta, Georgia, United States McDonough, Georgia, United States Memphis, Tennessee, United States Mesa, Arizona, United States Mesquite, Texas, United States Miami, Florida, United States Milpitas, California, United States Mission Viejo, California, United States Moorestown, New Jersey, United States Moreno Valley, California, United States Morrow, Georgia, United States Nashville, Tennessee, United States New Orleans, Louisiana, United States Norco, California, United States North Charleston, South Carolina, United States North Richland Hills, Texas, United States Oklahoma City, Oklahoma, United States Orlando, Florida, United States Overland Park, Kansas, United States Paramus, New Jersey, United States Pasadena, Texas, United States Phoenix, Arizona, United States Plano, Texas, United States Pompano Beach, Florida, United States Port St. Lucie, Florida, United States Reno, Nevada, United States Reynoldsburg, Ohio, United States Riverdale, Utah, United States Rocklin, California, United States Roswell, Georgia, United States Saint Petersburg, Florida, United States San Antonio, Texas, United States San Diego, California, United States Sanford, Florida, United States Santa Ana, California, United States Sarasota, Florida, United States Saugus, Massachusetts, United States Savannah, Georgia, United States Seattle, Washington, United States Skokie, Illinois, United States Smyrna, Georgia, United States St. Louis, Missouri, United States Sugar Land, Texas, United States Tampa, Florida, United States Tempe, Arizona, United States The Colony, Texas, United States Thornton, Colorado, United States Tucson, Arizona, United States Utica, Michigan, United States Virginia Beach, Virginia, United States Wayne, New Jersey, United States West Palm Beach, Florida, United States Wichita, Kansas, United States Woodbridge, Virginia, United States Woodland Hills, California, United States
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Floor & Decor is like no other flooring company. Founded in 2000 in Atlanta, we have quickly become one of the country's leading flooring retailers, with dozens of stores nationwide – and more to come. A strong entrepreneurial spirit flows through our young company. The remarkable success we have enjoyed could not have happened without the time, talent, and teamwork of our extremely dedicated associates. Working with equal amounts of passion and grit, we've become the "category killer of flooring" by revolutionizing the way people buy flooring. Our customers enjoy the largest in-stock selection of tile, wood, stone, and flooring accessories at prices no one can beat.
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text: Firehouse.com: http://www.firehouse.com/terrorist/1022_Pbutt.html Fireman Nails Osama "Butt" Good LARRY CELONA NY Post Online October 22, 2001 -- A New York City firefighter's fightin' Irish words for Osama bin Laden were the surprise hit of the benefit Concert for New York. Wild applause shook Madison Square Garden when Michael Moran, 38, delivered his in-your-face patriotic message to the terror chief: "Kiss my royal Irish ass!" But the rave reviews didn't stop there. Moran's fiancée, Donna, was moved to tears. And when he got home, the Ladder Co. 3 veteran found a congratulatory note on his door. His answering machine was full - with messages from people who watched the live event on television. "Mike, you did my Irish heart proud," said a doctor from Maryland who has never met Moran but was touched by his remarks. Moran's brother, FDNY Battalion Chief John Moran, perished in the Sept. 11 terrorist strike. While Moran was watching the concert, he realized he wanted to get up on stage and say something. Actor Michael J. Fox, who was standing nearby, told him he could. Then fireman-turned-actor Steve Buscemi - who attended probationary firefighters school with John Moran - and comic star Adam Sandler gave him a thumbs-up on his hastily composed speech. As he was going up to the stage, someone handed him a card with a picture of FDNY Capt. Louis Modaferri, and Moran mentioned his name along with his brother's - and some choice words for Osama. Last night, Moran said his invitation to Osama wasn't just talk. "I meant every word of it," he said. And what if he found himself face to face in an alley with bin Laden? "I'd put my hands around his throat and bite off his nose - for starters," he said.


text:
The Complete Nevadan
In-Depth Coverage of the Silver State
You are here: Home / Featured / News / Las Vegas Review-Journal: Lawsuit wants protections for 8 rivers, including Amargosa
Las Vegas Review-Journal: Lawsuit wants protections for 8 rivers, including Amargosa
April 9, 2018 By The Complete Nevadan Leave a Comment
by The Complete Nevadan
April 9, 2018 Filed under:
The Las Vegas Review-Journal reported on Sunday that environmentalists have filed suit against federal regulators over protections for eight rivers in California, including one that originates in Nevada.
Congress designated portions of the Amargosa and seven other rivers as wild and scenic in 2009, but the U.S. Forest Service and the Bureau of Land Management never completed comprehensive management plans for them as required by law, according to the Center for Biological Diversity.
The Tucson, Arizona-based group sued the two agencies in U.S. District Court in Los Angeles late last month, arguing that the Wild and Scenic Rivers Act required the development of management plans for the rivers within three years of their designation.
"These rivers are some of southern California's crown jewels and provide critical habitat for endangered birds and other imperiled species," said Ileene Anderson, a senior scientist for the environmental group, in a written statement. "They also provide respite and recreation for people, who rely on the government to protect these special places for their kids and grandkids. Without a management plan, a wild and scenic river designation is meaningless."
To read the full story, click here.
Amargosa River
Wild and Scenic Rivers Act
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Here is a screenshot of Kynaptic, a new experiment I'm working on. This is a KDE-based frontend for APT-RPM and APT. Kynaptic is based on Synaptic, and shares the same base code on top of the APT library.
On the right side you see a new toy: the interactive search dialog. This should definetly improve the user experience while looking for packages, and could eventually be ported to Synaptic itself.
Another interesting point is that during the development process of Kynaptic I'm also cleaning up the Synaptic code, something I had in my mind for some time already.
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Good. Now When Do We Get A Source Release And From Where?


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text:
Clinical 15 Social 13 Research 12 Education 10 Other 10 Behavioral 10
Cognitive 9 General 9 Health 8 Neuroscience 7 Postdoctoral 7 Experimental 6 Lifespan 5 Personality 5 Psychopathology 5 Quantitative 5 Biological 4 Cognitive Neuroscience 4
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Post-Doctoral Fellowship: Kent State University Student Life Study The Department of Psychological Sciences at Kent State University invites applications for a post-doctoral research position. The successful candidate will become part of a multi-disciplinary investigation to prospectively examine health and well-being in a new cohort of >10,000 young adults. Led by Drs. Karin Coifman and John Gunstad, the team of investigators will track mental, physical, academic, and social outcomes and deliver novel interventions using smart technology. The Post-Doctoral Fellow will help coordinate the KSU Student Life Study and have an opportunity to assist with other existing data collection efforts , related manuscript preparation, grant development, and develop new their own lines of inquiry based on current findings. They will work with integrated research program and enhance skills necessary for a successful research career. Opportunities exist for supervised clinical...
Kent State University Kent, OH, USA Postdoc
Postdoctoral Research Fellow in Cognitive Science
The Weinberg Institute seeks applicants willing to collaboratively and innovatively contribute to the developing Institute, the mission of which is to create a world class center for cognitive science research and teaching, at both the undergraduate and graduate levels. The Weinberg Institute for Cognitive Science at The University of Michigan invites applications from creative, highly qualified and motivated researchers for a two-year postdoctoral position. The position is primarily a research appointment, but also involves annually co-teaching a cognitive science graduate or undergraduate seminar with a UM faculty member, or the instruction of one Cognitive Science course. The desired start date of the appointment is September 2023, but this is negotiable. We welcome applications working in all areas of cognitive science spanning linguistics, philosophy, psychology, neuroscience, and related fields such as computer science / AI and psychiatry. Applications should...
The Weinberg Institute for Cognitive Science at The University of Michigan Ann Arbor, MI, USA Full time
Tenure-Track Assistant Professor of Human Factors Psychology
The Department of Human Factors and Behavioral Neurobiology at Embry-Riddle Aeronautical University, Daytona Beach, FL invites applications for a tenure-track Assistant Professorship for the academic year beginning in August 2023. The successful candidate will join the Human Factors program, which is fully accredited by the Human Factors and Ergonomics Society. Specialty area is open but candidates must have training in Human Factors and/or Psychology, and an applied research program. Preference will be given to candidates whose research and teaching interests include but is not limited to applied social and cognitive psychology, individual differences/personality, applied motivation, and factors influencing technology adoption. The ideal candidate must have a Ph.D in Human Factors or Applied Psychology or a related field by Fall 2023, a record of research productivity, an ability to attract external funding, a commitment to teaching both graduate students and...
Embry-Riddle Aeronautical University Daytona Beach, FL, USA Full time
Black Youth/Young Adult Mental Health, Clinical or Clinical Developmental Psychology
Black Youth/Young Adult Mental Health, Clinical or Clinical Developmental Psychology, Faculty of Health The Department of Psychology, Faculty of Health at York University invites highly qualified candidates to apply for a professorial stream tenure-track appointment in Black Youth/Young Adult Mental Health at the Assistant Professor level, to commence July 1, 2023. Salary will be commensurate with qualifications and experience. All York University positions are subject to budgetary approval. This position is open to qualified individuals who self-identify as Black peoples of African Descent (for example Africans and African heritage people from the Caribbean, Americas, Europe). Recognizing the underrepresentation of Black faculty, this opportunity supports the University's Affirmative Action program and has been developed based on the special program provisions of the Ontario Human Rights Code. To receive full consideration, candidates should complete and...
York University Toronto, ON, Canada Full time
Visiting Assistant Professor , Department of Psychological & Brain Sciences, Indiana University-Bloomington We seek to fill one position for a visiting assistant professor to teach undergraduate courses in the area of either Developmental Psychology or Industrial/Organizational Psychology. The candidate will also contribute to general introductory courses in our psychology or neuroscience majors. This will be a 1-year appointment, renewable for one additional year. The teaching load is five courses per year. Candidates with a doctorate in psychology or related field in hand at the time of appointment are preferred, but those who have completed all degree requirements except the dissertation defense (i.e., ABDs) will be considered. Applicants should have documented teaching experience and submit a letter of application that includes a statement of teaching philosophy and experience, evidence of teaching effectiveness, a diversity, equity, and inclusion statement (i.e., a...


Nothing can beat the beautiful display that nature puts on each evening. I love being down on the beach from golden hour until after sunset, enjoying the light show, whilst the waves crash on the shore. I love how different it looks each time, depending on the weather, the clouds, the tide.....and how it changes from minute to minute.
And it never ceases to amaze me how quickly the weather can change. All of these photos were taken on days that started out grey and rainy and full of thick cloud. For the last 3 years we have stayed in lovely apartments at the Waves complex (with Beach Retreats), only a couple of minutes away from the beach. I can't recommend it highly enough - it's so wonderful to look out the window and see the weather brighten up and then be able to leg it down to the beach with camera and tripod in hand. We had a very late dinner some evenings, but it was totally worth it!
There is no better way to get the city out of your hair, than a bracing cliff path walk. I love the epic views that make me feel so small and humble. There is so much space - perfect for clearing out your head and making room for fresh thinking. And the lungfuls of fresh air really help too.
Some firm favourites of ours are the walks from Watergate Bay to (a) the Bedruthan Steps, via Mawgan Porth and (b) going in the opposite direction to Newquay, via Porth. If you're feeling very energetic, the walk from Mawgan Porth to Harlyn is pretty spectacular - and there is nothing finer than dipping your walk weary feet in the sea at the end of that walk!!
It's often said there is something special about the light at St. Ives and I would absolutely agree - it seems to glow, even on grey days. I can see why it is such a draw for artists.
There are lots of wonderful places to eat in St. Ives. My absolute favourite is the Porthminster Beach Cafe - amazing food with amazing views. I love Porthminster Beach too - on a lovely day it can look tropical. Other favourite eateries are Blas burger works and the Seafood Cafe.
As well as exploring the many and beautiful beaches at St. Ives, I love pottering around the narrow cobbled streets and looking at all the pretty houses and doors, dreaming about living here... The Tate and Barbara Hepworth museums are also lovely places to visit - and you can enjoy a coffee in the lovely Porthmeor Cafe just outside the Tate on the beach - it has fabulous brightly coloured chairs and the best views!
Sandymouth beach is one of my all time favourite beaches: masses of lovely sand and space to reflect the sky (perfect for doggie walks), lots and lots of super smooth beautiful pebbles, and the most impressive cliffs and rocks - it's a real geological wonder.
And last, but not least, another favourite of mine is the beautiful beach at Chapel Porth (masses of sand when the tide is out and lots of caves) and the stunning cliffs above it, that are home to wheal coates, the old tin mines. It's a truly beautiful spot and a wonderful place for walking.
Apologies for such an epic post. Cornwall is just so photogenic, I can't help myself. If you haven't seen enough photos (!), feel free to have a browse through my Cornwall collection on Flickr.


Post 4 of your favorite games bellow.
I have like 50+ games on my pc but I'll try..
And tons of other junk games from Steam.
Gta 5 got released on pc?
...then time to time I install some good Classic game like X-Com, Doom, Duke Nukem 3D, Civilication 2' Settlers 2, Masters of Orion 2, Sierra adventures.... Stuuf like that.
I have 2 is that OK?
I probably play more games than that but those are the ones I can list off the top of my head.
Growtopia, spelunky, Terraria, and minecraft.
Random featured IOS App Store games.
Don't Play Other Games :3.. Please Recomend Me A Nice Game.. With Paid Or Free Games Anything Please..
Yea lemme just go boot up my windows 95 with my filled floppy disk to emulate dos and ill just update to directx 2.1 and there we go.
Hm.. I have quite a few, but okay.
I have loads more, but cba to list them.
They came from Xbox360, PC, xbox one, And my phone.
Spelunky is good, it's super hard but pretty fun, it will last a long time before you get bored of it.
I have so many more... but these are what I mostly play.


(HealthDayNews) -- Every year, thousands of children are scalded in bath water that's too hot, according to the Shriners Hospital for Children.
Run cold water into the tub first, then add hot water to reach a safe temperature. This will prevent a scald burn if your child should fall into the tub while it is being filled.
Before placing your child into the tub, check the water temperature by running your hand through the water for a few seconds. If it feels hot, it's too hot for your child.
Face your child away from the faucet, and closer to the other end of the tub.