text:
\section{Introduction}

Networks that describe real-world relationships often have different types of links that connect their nodes. For example, links in a social network may represent friendships, marriages, or collaborations, while links in a transportation network may represent planes, trains, or automobiles. Such networks are called multitype or multilayer networks since each type of link can be separated into its own layer, thereby connecting the same set of nodes in multiple ways. However, understanding the large-scale structure of multilayer networks is made difficult by the fact that the patterns of one type of link may be similar to, uncorrelated with, or different from the patterns of another type of link. These differences from layer to layer may exist at the level of individual links, connectivity patterns among groups of nodes, or even the hidden groups themselves to which each node belongs. Therefore, finding community structure in multilayer networks requires simultaneously considering three related problems: (i) the \emph{multilayer community detection problem}, in which we seek a description of the network that divides the nodes according to groups hidden in the link patterns of multiple layers; (ii) the \emph{layer interdependence problem}, in which we seek a description of the relationships between the layers containing different types of links; and (iii) the \emph{link prediction problem}, in which we seek to accurately predict missing link data by making use of all relevant layers of the network.

These three problems are fundamentally intertwined. Multilayer community detection requires knowing which layers have related structure and which layers are unrelated, since redundant information across layers may provide stronger evidence for clear communities than each layer would on its own. However, measuring layer interdependence requires a working definition of interdependence and a method to measure it. The performance of a model or algorithm on the link prediction task provides exactly such a measure: specifically, whether it is possible to integrate information across layers, using links of one type to help predict those of another type. Thus, in settings where data are well-represented as a multilayer network, as is relevant in inferring genetic and protein-protein interactions in cells~\cite{costanzo2010}, characterizing interdependent infrastructures~\cite{rinaldi2001} and understanding the impact of different social ties in human relationships~\cite{verbrugge1979,wasserman1994}, we desire a model capable of recognizing interdependencies between layers, integrating and merging information between them, and using this information both to classify nodes and to predict links.

In this paper we propose an approach that performs all three of these tasks. We define a generative model for multilayer community detection that is applicable to both directed and undirected networks, as well as networks with integer-weighted links. We provide a highly-scalable expectation-maximization algorithm that fits this model to data, taking as its input a multilayer network and yielding a ``mixed-membership'' partition: nodes are divided into communities or groups, but each node may belong to some extent to multiple groups. While this partition is shared by all the layers, the model allows for different connectivity patterns in each layer, including arbitrary mixtures of assortative, disassortative, and directed structure.

In addition to classifying nodes, our model also makes link prediction possible: given an incomplete dataset where not all links are known, it assigns probabilities to each pair of nodes that they have an unobserved link of each type. Finally, by sequentially fitting the model to single layers and multiple layers, we show how to determine whether the information provided by additional layers improves link prediction performance, thereby quantifying layer interdependence. We show how to use this method to identify which layers of the network are redundant, and which provide independent information. For instance, we can identify small sets of layers which together capture most of the information about the network. This may be useful in contexts where gathering layers requires an investment of limited resources in the laboratory or the field---for instance, if a social scientist can ask their subjects a limited number of questions, identifying a limited number of types of social relationships.

Community detection is a fundamental element of network science, yet most community detection algorithms have been developed for single-layer networks. We can use these methods to analyze multilayer networks, either by aggregating all layers to create a single-layer network~\cite{taylor2016,taylor2016enhanced} or by analyzing each layer independently. Multilayer-specific methods that maximize community quality functions such as modularity~\cite{mucha2010} provide non-overlapping partitions, but inherit the issues of their single-layer counterparts, namely a dependence on proper choice of a null model~\cite{bazzi2016,paul2016}. Moreover, these methods do not typically provide a framework to perform link prediction, since without additional assumptions they do not assign probabilities to the presence or absence of a link.

More recent methods for multilayer community detection are based on fitting various generative models via Bayesian inference or maximum likelihood estimation~\cite{paul2015,peixoto2015,schein2014,schein2015,schein2016bayesian,valles2016multilayer}. Our algorithm falls into this category, but differs from most of these models by not assuming \textit{a priori} any specific network structure. Many network models assume \emph{assortative} or \emph{homophilic} community structure, meaning that nodes are more likely to be connected to others in the same community.
This assumption is often incorrect in food webs, technological networks, and social networks where links consist of nominations, for instance, of a trustworthy or powerful person that they might ask for advice, help or work. Our model avoids this assumption: while it looks for an (overlapping) community structure that is consistent with every layer, it deals happily with networks where some layers are assortative, others are disassortative, yet others have core-periphery or directed structure, and so on. In the case of directed links, it also recognizes that nodes might play different roles, and thus effectively belong to different groups, when forming incoming or outgoing links.

By measuring the extent to which one layer helps us predict links in another layer, our method also gives a mathematically principled way to measure the relationships between the layers of a multilayer network, including identifying layers which are redundant with others or highly independent from them. This problem arises even in naive algorithms that aggregate layers into a single-layer network. For instance, the multilayer versions of eigenvector centrality or modularity~\cite{mucha2010,sola2013} use weighted averages over layers, requiring them to infer or choose a weight for each layer. A number of recent works have taken a compression approach, aggregating layers with similar structure~\cite{de2015,stanley2015}; in particular, Ref.~\cite{stanley2015} uses a generative model that jointly assigns community memberships to nodes and groups of layers, which they call \emph{strata}. This is similar in spirit to our approach, but our model handles overlapping communities, directed networks, and weighted networks in a unified way.

In Section~\ref{sec:model} we describe our model and in Section~\ref{sec:em} give an efficient algorithm that fits its parameters to network data. In Section \ref{sec:synthetic}, we provide performance results on synthetic benchmarks and compare with other algorithms, and in Section \ref{sec:linkpr} we discuss how to perform link prediction and measure layer interdependencies. We then apply these concepts to two real world networks, drawn from anthropology and biology, and discuss the results in Section \ref{sec:results} before concluding.

\section{The multilayer mixed-membership stochastic block model}
\label{sec:model}

In this section we describe our model and fix our notation. The network consists of $N$ nodes and has $L$ layers. Each layer has an adjacency matrix $A^{(\alpha)}$, where $A_{ij}^{(\alpha)}$ is the number of edges from $i$ to $j$ of type $\alpha$; alternatively, we can think of $A$ as an $N \times N \times L$ tensor. Our model generates these networks probabilistically, assuming an underlying structure consisting of $K$ overlapping groups.

Each node belongs to each group to an extent described by a $K$-dimensional vector. Since we are interested in directed networks, we give each node $i$ two membership vectors, $\vec{u}_i$ and $\vec{v}_i$, which determine how $i$ forms outgoing and incoming links respectively. (When modeling undirected networks, we set $\vec{u}=\vec{v}$.) Each layer $\alpha$ has a $K \times K$ affinity matrix $w^{(\alpha)}$ describing the density of edges between each pair of groups. The expected number of edges in layer $\alpha$ from $i$ to $j$ is then given by a bilinear form,
\be
	M_{ij}^{(\alpha)} = \sum_{k,\ell=1}^K u_{ik} \,v_{j\ell} \,w_{k\ell}^{(\alpha)} \, .
\ee
Finally, for each $i, j$ and $\alpha$, we choose $A_{ij}^{(\alpha)}$ independently from the Poisson distribution with mean $M_{ij}^{(\alpha)}$.

Note that while we assume that the membership vectors have nonnegative entries, we do not normalize them. This allows us to account easily for heterogeneous degree distributions, since multiplying $\vec{u}_i$ or $\vec{v}_i$ by a constant increases the expected out- or in-degree without changing the distribution of neighbors to which a given one of $i$'s edges connects.

Note also that while this model supposes that nodes have the same group membership in all layers, it allows the structure of each layer to vary arbitrarily with respect to these groups. For instance, some layers could be assortative and others disassortative, with affinity matrices $w^{(\alpha)}$ which are large on or off the diagonal; other layers could have strongly directed structure, with asymmetric $w^{(\alpha)}$, or core-periphery structure, where $w^{(\alpha)}$ has one large entry on the diagonal.

For a single layer, our model is similar to existing mixed-membership block models~\cite{newman2007,airoldi2008,ramasco2008,ball2011,gopalan2013,yang2014,zhou2015}. Some of these use mixtures of Bernoulli random variables; we follow~\cite{ball2011} in using the Poisson distribution since it leads to a tractable and efficient expectation-maximization algorithm. The Poisson distribution also allows us to model multigraphs or integer-weighted networks. However, in our applications here we will focus on the sparse case where $M_{ij}^{(\alpha)}$ is small, and assume for simplicity that $A_{ij}^{(\alpha)}$ is $0$ or $1$.

Our model also bears a close mathematical relationship to topic models~\cite{PLSA,LDA}, which generate bipartite weighted graphs of documents and words based on their relevance to mixtures of topics. More generally, it can be viewed as a variant of non-negative tensor factorization (see e.g.~\cite{Kolda2009} for a review) and in particular of Poisson tensor factorization~\cite{lee1999,canny2004,dunson2005,cemgil2009,gopalan-rec,schein2014,schein2015,chi2012tensors}. However, the affinity matrices $w^{(\alpha)}$, which allow different layers to be assortative or disassortative, correspond to a kind of Tucker decomposition~\cite{TUCKER2} of tensor rank $K^2$. This is more general than the PARAFAC/CANDECOMP decomposition~\cite{PARAFAC,CANDECOMP}, which corresponds to the special case of our model where $w^{(\alpha)}$ is diagonal for each $\alpha$. In the undirected case where $\vec{u}=\vec{v}$, PARAFAC thus assumes a purely assortative structure, where a link between two nodes can only exist if their membership vectors overlap.

Various kinds of Poisson Tucker decomposition for dynamic and multilayer networks have also been proposed very recently in the machine learning community, particularly in~\cite{schein2014,schein2015,schein2016bayesian}. Indeed, our model is very nearly a special case of that of~\cite{schein2016bayesian}, which has additional parameters intended to model datasets with both multiple types of links and multiple time steps. The main difference between these works and our approach is that they impose priors on the ``core tensor'' parameters [analogous to $w^{(\alpha)}_{k\ell}$] and they use Monte Carlo sampling for Bayesian inference. In contrast, we find point estimates of these parameters using an expectation-maximization algorithm, detailed in the next section. Below we compare our results to the algorithm of~\cite{schein2015}, which is designed for the same kinds of datasets as ours.

\section{The Expectation-Maximization algorithm}
\label{sec:em}

Given an observed multilayer network with adjacency tensor $A$, our goal is to simultaneously infer the nodes' membership vectors and the affinity matrices for each layer. In this section, we describe an efficient algorithm which does this by the method of maximum likelihood.

Let $\Theta$ be shorthand for all $2NK+K^2 L$ model parameters, i.e., the $u_{ik}$, $v_{i\ell}$, and $w_{k\ell}^{(\alpha)}$. Assuming that all $\Theta$ are equally likely \textit{a priori}, the probability of $\Theta$ given $A$ is proportional to the probability of $A$ given $\Theta$. Using the Poisson distribution function gives
\be
	P(\Theta \mid A) \propto P(A \mid \Theta)
	= \prod_{i,j=1}^{N} \prod_{\alpha=1}^L \frac{\mathrm{e}^{-M_{ij}^{(\alpha)}} (M_{ij}^{(\alpha)})^{A_{ij}^{(\alpha)}}}{A_{ij}^{\alpha}!}
	\label{eq:likelihood}
\ee
(One could also impose a prior $P(\Theta)$ and perform maximum \textit{a posteriori} inference~\cite{dunson2005,cemgil2009,schein2015}, but we have not done this here.) The log-likelihood is then
\be
	\L(\Theta)
	 = \sum_{i,j,\alpha}
	\left[ A_{ij}^{(\alpha)} \log \sum_{k\ell} u_{ik} v_{j\ell} w_{k\ell}^{(\alpha)}
	 - \sum_{k\ell} u_{ik} v_{j\ell} w_{k\ell}^{(\alpha)} \right]
	 \label{logL}
\ee
where we omit the terms $\log A_{ij}^{(\alpha)}!$ since they depend only on the data.

We wish to find the $\Theta$ that maximizes Eq.~\eqref{logL}. This is computationally difficult, but we can make it more tractable with a classic variational approach. For each $i,j,\alpha$ with $A_{ij}^{(\alpha)} = 1$, consider a probability distribution $\rho_{ijk\ell}^{(\alpha)}$ over pairs of groups $k,\ell$: this is our estimate of the probability that that edge exists due to $i$ and $j$ belonging to groups $k$ and $\ell$ respectively. (If the network is a multigraph and $i,j$ have multiple links of the same type, we give each one its own distribution $\rho$; below we assume for simplicity that this does not occur.) Jensen's inequality $\log \overline{x} \ge \overline{\log x}$ then gives
\begin{align}
\log \sum_{k\ell} &u_{ik} v_{j\ell} w_{k\ell}^{(\alpha)}
= \log \sum_{k\ell} \rho_{ijk\ell}^{(\alpha)} \frac{u_{ik} v_{j\ell} w_{k\ell}^{(\alpha)}}{\rho_{ijk\ell}^{(\alpha)}} \nonumber \\
&\ge \sum_{k\ell} \rho_{ijk\ell}^{(\alpha)} \log \frac{u_{ik} v_{j\ell} w_{k\ell}^{(\alpha)}}{\rho_{ijk\ell}^{(\alpha)}} \nonumber \\
&= \sum_{k\ell} \rho_{ijk\ell}^{(\alpha)} \log u_{ik} v_{j\ell} w_{k\ell}^{(\alpha)}
- \sum_{k\ell} \rho_{ijk\ell}^{(\alpha)} \log \rho_{ijk\ell}^{(\alpha)} \, .
\label{eq:variational}
\end{align}

Moreover, this holds with equality when
\begin{equation}
\label{eq:boltzmann}
	\rho_{ijk\ell}^{(\alpha)} = \frac{u_{ik} v_{j\ell} w_{k\ell}^{(\alpha)} }{\sum_{k'\ell'} u_{ik'} v_{j\ell'} w_{k'\ell'}^{(\alpha)} } \, .
\end{equation}
Thus maximizing $\L(\Theta)$ is equivalent to maximizing
\begin{align}
\L(\Theta,\rho)
&= \sum_{i,j,\alpha,k,\ell}
	\Big[ A_{ij}^{(\alpha)}
	\left( \rho_{ijk\ell}^{(\alpha)} \log u_{ik} v_{j\ell} w_{k\ell}^{(\alpha)}
- \rho_{ijk\ell}^{(\alpha)} \log \rho_{ijk\ell}^{(\alpha)} \right) \nonumber \\
	& \quad - u_{ik} v_{j\ell} w_{k\ell}^{(\alpha)} \Big]
\end{align}
with respect to both $\Theta$ and $\rho$.

The expert reader will recognize that this variational argument is simply classical thermodynamics in disguise. Fix the parameters $\Theta$ and consider a spin system where each edge, i.e., each triple $(i,j,\alpha)$ with $A_{ij}^{(\alpha)} = 1$, has a state consisting of a pair of groups $(k,\ell)$. Define the Hamiltonian as
\begin{align*}
H &= - \sum_{(i,j,\alpha): A_{ij}^{(\alpha)}=1} \log u_{ik(i,j,\alpha)} v_{j\ell(i,j,\alpha)} w_{k(i,j,\alpha),\ell(i,j,\alpha)}^{(\alpha)} \\
& \quad - \sum_{i,j,\alpha,k',\ell'} u_{ik'} v_{j\ell'} w_{k'\ell'}^{(\alpha)}
\end{align*}
(note that the second term is constant). Then the Boltzmann distribution is a product distribution of the distributions $\rho_{ij}^{(\alpha)}$ given by Eq.~\eqref{eq:boltzmann} on each edge. Moreover, $-\L(\Theta,\rho)$ is the free energy $E-TS$ where $T=1$, and we recover the familiar fact that this is minimized by the Boltzmann distribution. In this context, finding the maximum-likelihood estimate of the parameters $\Theta$ corresponds to minimizing the free energy of this spin system.

We can maximize $\L(\Theta,\rho)$ by alternatively updating $\rho$ and $\Theta$. This general approach is called an expectation-maximization (EM) algorithm: the expectation step computes the marginals of the Boltzmann distribution for the current estimate of the parameters, and the maximization step finds the most-likely value of the parameters given those marginals. The fact that the Boltzmann distribution takes a simple product form makes the expectation step especially simple, making the algorithm highly efficient.

The update equations for $\Theta$ in the maximization step can be can be derived by computing the partial derivative of $\L(\Theta,\rho)$ with respect to the various parameters. For instance,
\be
\frac{\partial \L(A,\rho)}{\partial u_{ik}}
= \sum_{j,\ell,\alpha} \left[ \frac{A_{ij}^{(\alpha)}\, \rho_{ijk\ell}^{(\alpha)}}{u_{ik}} - v_{j\ell} w_{k\ell}^{(\alpha)} \right] \, .
\ee
Setting this to zero, and doing the same for the partial derivatives with respect to $v_{j\ell}$ and $w_{k\ell}^{(\alpha)}$, gives
\begin{gather}
{u}_{ik}
= \frac{\sum_{j,\alpha} A_{ij}^{(\alpha)} \sum_\ell \rho_{ijk\ell}^{(\alpha)}}
{\sum_\ell \left( \sum_{j} v_{j\ell} \right) \left( \sum_{\alpha} w_{k\ell}^{(\alpha)} \right)}
\label{uik_dis} \\
{v}_{j\ell}
= \frac{\sum_{i,\alpha} A_{ij}^{(\alpha)} \sum_k \rho_{ijk\ell}^{(\alpha)}}
{\sum_k \left( \sum_i u_{ik} \right) \left( \sum_{\alpha} w_{k\ell}^{(\alpha)} \right)}
\label{vjk_dis} \\
{w}_{k\ell}^{(\alpha)}= \f{\sum_{ij}A_{ij}^{(\alpha)} \rho_{ijk\ell}^{(\alpha)}}{\left(\sum_i u_{ik} \right) \left( \sum_j v_{j\ell} \right)}
\label{omegaka_dis} \, .
\end{gather}

The EM algorithm thus consists of randomly initializing the parameters $\Theta$, and then repeatedly alternating between updating $\rho$ using Eq.~\eqref{eq:boltzmann} and updating $\Theta$ using Eqs.~\eqref{uik_dis}-\eqref{omegaka_dis} until it reaches a fixed point. This fixed point is a local maximum of $\L(\Theta,\rho)$, but it is not guaranteed to be the global maximum. Therefore, we perform multiple runs of the algorithm with different random initializations for $\Theta$, taking the fixed point with the largest value of $\L(\Theta,\rho)$. The computational complexity per iteration scales as $O(MK^2)$ where $M$ is the total number of edges summed over all layers and $K$ is the number of groups. In practice we find that our algorithm converges within a fairly small number of iterations. Thus it is highly scalable, with a total running time roughly linear in the size of the dataset.

Once we converge to a fixed point, we can assign nodes to communities by normalizing the membership vectors to $\b{u}_i = u_i / \sum_k u_{ik}$, so that for each $i$ we have $\sum_k \b{u}_{ik} = 1$. This approach was used in~\cite{ball2011} as an method for classifying nodes in overlapping communities; however, since we allow $\vec{u}_i$ and $\vec{v}_i$ to be distinct, the ``outgoing'' and ``incoming'' assignments of a node might differ. These are \emph{soft} assignments, meaning that nodes can belong to more than one community. If one wishes to obtain a \emph{hard} assignment, one can assign each node to the single community corresponding to the maximum entry of $\vec{u}$ or $\vec{v}$, but the overlapping character of the community structure is then lost.

We call our model and its associated algorithm
 \mbox{{\small MULTITENSOR}}. A numerical implementation is available for use under an open source license \footnote{Code available at \href{https://github.com/cdebacco/{\small MULTITENSOR}}{github.com/cdebacco/{\small MULTITENSOR}}}.

\section{Results on synthetic networks}
\label{sec:synthetic}

We tested {\small {\small MULTITENSOR}}'s ability to detect community structure synthetic networks using the multilayer benchmark proposed in Ref.~\cite{bazzi2016}. This model is somewhat different from ours: rather than having mixed-membership vectors that are the same in every layer, they use hard partitions where each node belongs to a single group, but they allow these partitions to vary from layer to layer. The partitions in different layers are correlated by a so-called layer interdependence tensor. For simplicity, we use a one-parameter version of this tensor with dependency $p \in [0,1]$: when $p=0$ the partitions between layers are independent, and when $p=1$ they are identical. Once these partitions are chosen, they generate edges in each layer according to a degree-corrected block model~\cite{karrer2011}, with a user-specified degree distribution and affinity matrix.

We used this benchmark to generate synthetic networks with $N=300$ nodes, $L=4$ layers, and $K=5$ communities. For each layer we used a truncated power-law degree distribution with exponent $\gamma =-3$, minimum degree $k_{min}=3$ and maximum degree $k_{max}=30$. We varied the affinity matrix of the block model according to a mixing parameter $\mu$: if $\mu=0$ all edges lie within communities, and if $\mu=1$ edges are assigned regardless of the community structure.

We define our algorithm's accuracy on these benchmarks in terms of how well the inferred membership vectors match the ground-truth distribution of memberships across layers. That is, we hope that, after normalizing so that $\sum_k \b{u}_{ik}=1$, each $u_{ik}$ is close to the fraction of layers in which node $i$ belongs to group $k$, which we denote $\b{u}^0_{ik}$. We quantify the similarity between these two distributions using two measures. The first is their inner product, often called the \emph{cosine similarity}  (CS), averaged over all the nodes:
\be\label{CS}
	\mathrm{\mathrm{CS}}
	= \frac{1}{N} \sum_{i=1}^{N} \frac{\b{u}^0_i \cdot \b{u}_i}{| \b{u}^0_i | | \b{u}_i |} \, ,
\ee
where in the denominator $| \b{u} |$ denotes the Euclidean norm. Here $\mathrm{CS}=1$ corresponds to perfect accuracy. The second is the $L_1$ error between the two distributions, also known as their statistical distance or total variation distance, averaged over all the nodes:
\be\label{L1}
	L_1 = \frac{1}{2N} \sum_{i=1}^{N} \left\| \b{u}^0_i - \b{u}_i \right\|_1
	= \frac{1}{2N} \sum_{i=1}^{N} \sum_{k=1}^{K} \left| \b{u}^0_{ik} - \b{u}_{ik} \right| \, .
\ee
The factor of $1/2$ is used so that this distance ranges from $0$ for identical distributions to $1$ for distributions with disjoint support. In both measures, we give ourselves the freedom to permute the groups, so that the inferred groups of our model correspond to the groups of the benchmark. Thus we maximize the cosine similarity $\mathrm{CS}$, and minimize the $L_1$ error, over all $K!$ permutations of the $K$ groups.

For comparison, we use two other algorithms that infer overlapping multilayer partitions.
The first is the restricted diagonal version of our model, which only allows diagonal affinity matrices $w^{(\alpha)}$; as discussed above, this is equivalent to the Poisson version of PARAFAC tensor factorization~\cite{chi2012tensors} and in the undirected case $\vec{u}=\vec{v}$ this corresponds to assume an assortative network structure. The second algorithm is a fully Bayesian Poisson tensor factorization (BPTF)~\cite{schein2015}. The main differences between these two models are the prior information and the optimization approach. The former considers a uniform prior and calculates point estimates of the parameters using an iterative algorithm similar to ours. The BPTF algorithm instead assumes Gamma-distributed parameters and updates the parameters of these distributions instead of the point estimates; in the end it uses the geometric mean of these distributions as its estimate of the parameters.

\begin{table*}[ht]

 \begin{tabular}{lccccccccccccccccccccccc}
	\toprule
\multirow{2}{4em}{$\mu=0.0$ } & \multicolumn{4}{c}{$p=0.5$} && \multicolumn{4}{c}{$p=0.8$} && \multicolumn{4}{c}{$p=0.9$} \\
 \cmidrule(l){2-5} \cmidrule(l){7-10} \cmidrule(l){12-15}
& CS & $\sigma$ &$L_{1}$ &$\sigma$ && CS & $\sigma$ &$L_{1}$ &$\sigma$ && CS & $\sigma$ &$L_{1}$ &$\sigma$ \\ \hline
{\footnotesize MULTITENSOR} & \bf{ 0.66} & 0.06 & 0.58 & 0.07 && \bf{ 0.93} & 0.07 & \bf{ 0.19} & 0.09 && \bf{ 0.99} & 0.01 & \bf{ 0.07} & 0.01 \\
Diagonal & 0.65 & 0.05 & 0.60 & 0.06 && 0.84 & 0.06 & 0.31 & 0.08 && 0.97 & 0.04 & 0.10 & 0.05\\
BPTF & \bf{ 0.66} & 0.06 & \bf{ 0.57} & 0.06 && 0.89 & 0.07 & 0.24 & 0.08 && 0.96 & 0.03 & 0.10 & 0.04 \\
 	\end{tabular}

 \vspace{0.01in}

 \begin{tabular}{lccccccccccccccccccccccc}
	\toprule
\multirow{2}{4em}{$\mu=0.1$ } & \multicolumn{4}{c}{$p=0.5$} && \multicolumn{4}{c}{$p=0.8$} && \multicolumn{4}{c}{$p=0.9$} \\
 \cmidrule(l){2-5} \cmidrule(l){7-10} \cmidrule(l){12-15}
& CS & $\sigma$ &$L_{1}$ &$\sigma$ && CS & $\sigma$ &$L_{1}$ &$\sigma$ && CS & $\sigma$ &$L_{1}$ &$\sigma$ \\ \hline
{\footnotesize MULTITENSOR} &\bf{ 0.63} & 0.05 & 0.62 & 0.05 && \bf{ 0.92} & 0.06 & \bf{ 0.22} & 0.08 && \bf{ 0.98} & 0.01 & \bf{ 0.11} & 0.01 \\
Diagonal & \bf{ 0.63} & 0.04 & 0.62 & 0.04 && 0.87 & 0.06 & 0.29 & 0.07 && 0.94 & 0.07 & 0.17 & 0.08\\
BPTF & 0.62 & 0.06 &\bf{ 0.61} & 0.05 && 0.84 & 0.07 & 0.32 & 0.08 && 0.93 & 0.06 & 0.18 & 0.07 \\
 	\end{tabular}

 \vspace{0.01in}

 \begin{tabular}{lccccccccccccccccccccccc}
	\toprule
\multirow{2}{4em}{$\mu=0.5$ } & \multicolumn{4}{c}{$p=0.5$} && \multicolumn{4}{c}{$p=0.8$} && \multicolumn{4}{c}{$p=0.9$} \\
 \cmidrule(l){2-5} \cmidrule(l){7-10} \cmidrule(l){12-15}
& CS & $\sigma$ &$L_{1}$ &$\sigma$ && CS & $\sigma$ &$L_{1}$ &$\sigma$ && CS & $\sigma$ &$L_{1}$ &$\sigma$ \\ \hline
{\footnotesize MULTITENSOR} &\bf{ 0.55} & 0.03 & 0.70 & 0.02 && \bf{ 0.55} & 0.05 & \bf{ 0.67} & 0.04 && \bf{ 0.59} & 0.07 & 0.59 & 0.07 \\
Diagonal & \bf{ 0.55} & 0.03 & 0.71 & 0.02 && 0.53 & 0.03 & 0.69 & 0.03 && 0.58 & 0.07 & 0.63 & 0.06 \\
BPTF & 0.52 & 0.03 & \bf{ 0.69 }& 0.03 && 0.49 & 0.05 & 0.68 & 0.04 && \bf{ 0.59} & 0.07 & \bf{ 0.58} & 0.06 \\ \bottomrule
 	\end{tabular}
	\caption{Performance in detecting overlapping partitions on synthetic networks, using our {\small MULTITENSOR} algorithm, the diagonal special case which corresponds to the Poisson version of PARAFAC tensor decomposition~\cite{chi2012tensors}, and Bayesian Poisson Tensor Factorization~\cite{schein2015}. Benchmark networks were generated with the model of~\cite{bazzi2016} with interdependence $p$ and mixing parameter $\mu$: the community structure is stronger when $p$ is large and $\mu$ is small. For each $(\mu,p)$ pair we measure the cosine similarity (CS) and the average $L_1$ error between the planted and inferred structure averaged over $50$ independently generated benchmark networks. For each network, we run each network $50$ times with independently random initializations, and use the parameters given by the highest-likelihood fixed point. Good performance corresponds to high cosine similarly and low $L_1$ error, and the best performance for each pair of parameter values is indicated by boldface. The errors $\sigma$ are the standard deviation over the 50 benchmarks.}
\label{table:sim}
\end{table*}

In Table~\ref{table:sim} we report the best results in terms of cosine similarity and $L_1$ error obtained by three algorithms: {\small MULTITENSOR}, its diagonal special case (or Poisson PARAFAC), and BPTF. By varying the layer interdependence $p$ and the mixing parameter $\mu$, we range over cases where the community structure is relatively easy to infer to those where it is much harder. Specifically, inference is easier when $p$ is large, so that the layers are strongly correlated, and $\mu$ is small, so that most links are within communities. For each pair $(\mu,p)$ we generated $50$ independent benchmark networks, and for each network and each algorithm we performed 50 independent runs with independently random initial conditions, taking the fixed point with highest likelihood. We defined convergence numerically by testing whether $\L(\Theta,\rho)$ has not improved by more than $0.1$ for $10$ iterations.

In every case our algorithm achieves the highest cosine similarity, and in the majority of the cases the smallest $L_1$ error, indicated in boldface. All three algorithms perform poorly in the hard regime where one of the two parameters introduces a high level of stochasticity in the network, i.e. when either $\mu=0.5$ or $p=0.5$. In the other cases our algorithm is significantly better according to both measures.

The benchmarks of~\cite{bazzi2016} are assortative in every layer, and the diagonal and BPTF algorithms work fairly well. To illustrate the greater flexibility of our algorithm, we also generated synthetic networks with different kinds of structure in different layers.. Specifically, we generated layers whose structures are
assortative ($w_{11} = w_{22} > w_{12} = w_{21}$),
disassortative ($w_{11} = w_{22} < w_{12} = w_{21}$),
core-periphery ($w_{11} > w_{12} = w_{21} > w_{22}$),
and directed with a bias from the first group to the second one ($w_{12} > w_{11} = w_{22} > w_{21}$).

We considered three types of networks, all having $K=2$ groups and $N=300$ nodes, but with different numbers and kinds of layers (see Table \ref{table:AucSynthetic}). In each one the groups are of equal size, with un-mixed group memberships (i.e., $u_i = v_i = (0,1)$ or $(1,0)$). The first type of network has $L=2$ layers, one assortative and one disassortative; the second network has $L=4$, with two assortative and two disassortative layers; the last one has $L=4$, with one layer each with assortative, disassortative, core-periphery, and biased directed structure.

We generated $10$ independent samples of each of these types of network and calculated the CS and the $L_{1}$ norm between the inferred membership and the ground truth using the maximum-likelihood fixed point over $10$ runs of each algorithm with different random initial conditions. As shown in Table \ref{table:synCSL1}, {\small MULTITENSOR} achieves significantly greater performance than the diagonal or BPTF algorithms in all three cases, due to its flexibility in modeling mixtures of these different types of structure.

\begin{table}[htbp]
\resizebox{1.0\linewidth}{!}{
\begin{tabular}{lccccc} \toprule
 G & K & L & $\langle E \rangle$ & Structure \\ \hline
1 & 2 & 2 & 1980 & \{assort, disassort\} \\
2 & 2 & 4 & 3960,2250 & \{assort, disassort, core-per, dir. disassort\} \\
3 & 2 & 4 & 1980 & \{2 assort, 2 disassort\} \\ \bottomrule
\end{tabular}
}
\caption{ Description of synthetic network structures. $\langle E \rangle$ is the average number of edges per layer, all networks have $N=300$ nodes. For networks $G=1,3$ we used affinity matrices $W^{a}$ and $W^{d}$ for the assortative and disassortative layers respectively, with entries $w_{11}^{a}=w_{22}^{a}=w_{12}^{d}=w_{21}^{d}= 0.04$, $w_{12}^{a}=w_{21}^{a}=w_{11}^{d}=w_{22}^{d}=0.004$ so that $\langle E \rangle=1980$; for $G=2$ the first two layers (1 assortative and 1 disassortative) have affinity matrices $W^{a}$ and $W^{b}$ with entries $w_{11}^{a}=w_{22}^{a}=w_{12}^{b}=w_{21}^{b}= 0.08$, $w_{12}^{a}=w_{21}^{a}=w_{11}^{b}=w_{22}^{b}=0.008$ so that $\langle E \rangle=3960$; the third and fourth (1 core-periphery and 1 directed disassortative) have affinity matrices $W^{c}$ and $W^{d}$ with entries $w_{11}^{c}=w_{12}^{d}=0.08$, $w_{12}^{c}=w_{21}^{c}=w_{11}^{d}=w_{22}^{d}=0.008$ and $w_{22}^{c}=w_{21}^{d}=0.004$ so that $\langle E \rangle=2250$.}
\label{table:AucSynthetic}
\end{table}

\begin{table*}[ht]
 \begin{tabular}{lcccccccccccccccccccccccc}
	\toprule
 \multirow{2}{4em}{ } & \multicolumn{4}{c}{$G=1$} && \multicolumn{4}{c}{$G=2$} && \multicolumn{4}{c}{$G=3$} \\
 \cmidrule(l){2-5} \cmidrule(l){7-10} \cmidrule(l){12-15}
& CS & $\sigma$ &$L_{1}$ &$\sigma$ && CS & $\sigma$ &$L_{1}$ &$\sigma$ &&CS & $\sigma$ &$L_{1}$ &$\sigma$ \\ \hline
{\footnotesize MULTITENSOR} & \textbf{0.984} & 0.008 & \textbf{0.06} & 0.01 && \textbf{0.990} & 0.001 & \textbf{0.058} & 0.003 && \textbf{0.989} & 0.001 & \textbf{0.056} & 0.002 \\
Diagonal & 0.61 & 0.01 & 0.49 & 0.01 && 0.91 & 0.01 & 0.22 & 0.01 && 0.63 & 0.02 & 0.48 & 0.01 \\
BPTF & 0.60 & 0.02 & 0.48 & 0.01&& 0.90 & 0.03 & 0.21 & 0.05 && 0.63 & 0.03 & 0.48 & 0.01 \\ \bottomrule
 	\end{tabular}
	\caption{Cosine similarity and $L_{1}$ norm for mixed structure synthetic networks. CS as in Eq. (\ref{CS}) and $L_{1}$ norm as in Eq. (\ref{L1}) are calculated between membership vectors $\b{u}$, $\b{v}$ of the inferred partitions and the ground truth [$\b{u}_{0}=\b{v}_{0}=(1,0),\,(0,1)$ for nodes in group 1 and 2, respectively]. Results are averages and standard deviations of the results on the two memberships over 10 networks sampled from each network type $G$, as described in Table \ref{table:AucSynthetic}. The best performance is indicated in boldface. }
\label{table:synCSL1}
\end{table*}

\section{Learning layer interdependence via link prediction}
\label{sec:linkpr}

Despite the fact that a network may have multiple layers, there is no guarantee that the structure of one layer is related to the structure of any other. In fact, depending on the context, it may even be desirable that two layers are entirely uncorrelated, since then they reveal different kinds of information about the underlying structure. The \emph{layer interdependence problem} consists of identifying which sets of layers are structurally related, and quantifying the strengths of those interrelationships.

We are not the first to provide a solution to the layer interdependence problem. One intuitive approach is to independently infer the community structure of each layer and then simply compute pairwise correlations between the community partitions of each layer~\cite{larremore2013}. While this approach is straightforward, it is unable to use the information in some layers to assist with inference of structure in other layers, since every layer is treated independently. Another approach has been to cluster the inferred affinity matrices, analogous to our $w^{(\alpha)}$, during the parameter inference and estimation procedures, thus gathering layers into ``strata''~\cite{stanley2015}.

However, while we also explore clustering the $w^{(\alpha)}$ below, neither of these methods captures the general kind of independence we are interested in. For instance, if two layers use the same group labels for the vertices, but one layer is assortative while the other is disassortative, then knowing one of them is very helpful in predicting the other. In this sense they are closely related, even though they are statistically very different, and indeed anticorrelated with each other.

Our proposal to capture this more general kind of interdependence is based on the idea that two layers are interdependent if and only if the structure of one layer provides meaningful knowledge about the structure of the other. Specifically, by performing link prediction in one layer with or without information about another layer, we quantify the extent to which these two layers are related.

Since our model is generative, it naturally includes a framework to predict links given partially-observed data: simply use the known data to estimate the parameters, and then use these estimated parameters to compute $M_{ij}^{(\alpha)}$, i.e., the expected number of links of type $\alpha$ between each pair of nodes $i,j$. We then rank each missing entry in the adjacency tensor according to $M_{ij}^{(\alpha)}$. The method succeeds to the extent that true missing links are given a higher estimate of $M_{ij}^{(\alpha)}$ than false ones. We follow~\cite{Clauset2008} in defining the accuracy as the area under the receiver-operator curve or AUC~\cite{hanley1982}. This is the probability that a random true positive is ranked above a random true negative; thus the AUC is $1$ for perfect prediction, and $1/2$ for chance.

To test our ability to predict layer $\alpha$, we perform experiments with $5$-fold cross validation. That is, we hold out $20\%$ of its adjacency matrix $A^{(\alpha)}$, hiding those entries from the algorithm. We infer the model parameters using the remaining $80\%$ as a training dataset, with or without knowledge of other layers of the network, and measure the inferred model's accuracy on the held-out entries in $A^{(\alpha)}$.
The independence of the training and test datasets makes cross-validation a robust method against overfitting.
Note that holding out $20\%$ of a layer does not mean removing 20\% of the nodes or 20\% of the links, but rather hiding 20\% of the entries of its adjacency matrix, including both zeros and ones. This means that we just the accuracy of our link prediction algorithm on both links and non-links.

The final AUC is the average obtained over the $5$ folds, each of which holds out a different subset of $20\%$. Clearly this AUC depends both on the layer $\alpha$ we are trying to predict, and on what set of other layers we give the algorithm access to.

Given this framework, we can define the pairwise interdependence between two layers $\alpha, \beta$ as follows. We perform link prediction on layer $\alpha$, with $20\%$ of the entries of $A^{(\alpha)}$ hidden; but we do this first by giving the algorithm access only to the rest of $A^{(\alpha)}$, and then by giving it access to all of $A^{(\beta)}$ as well. We then measure the difference in the AUC between these two experiments, determining how much knowledge about layer $\beta$ helps us predict layer $\alpha$. We call this the \emph{two-layer AUC}. Similarly, the \textit{three-layer AUC} tells us how much knowledge about two layers $\beta$ and $\gamma$ help us predict $\alpha$, and so on. Notice that if different layers have independent structure, without common underlying communities, then including one in the training set much actually decreases our ability to predict the other, causing the AUC to go down.

Computing all \textit{$\ell$-layer AUCs} would require us to try ${L \choose \ell}$ subsets of the layers, which becomes computationally expensive as $\ell$ increases. To avoid this computational bottleneck, we use a greedy bottom-up procedure in which we add one layer at a time to the training dataset, whichever one most increases the AUC, until as many layers as desired have been added.
This allows us to find a small set of layers which together make it possible to predict links accurately. While this greedy procedure is not guaranteed to find the best possible subset of a given size, it is computationally efficient.

Alternatively, if the goal is to decide which layers are less informative in predicting the others, in order for instance to compress information by discarding less informative layers, we can use a top-down procedure which starts with all $L$ layers, then iteratively removes whichever one decreases the AUC the least, until a small informative subset of layers remains. We did not pursue this here.

In addition to this link prediction approach, we also cluster the affinity matrices $w^{(\alpha)}$ in a way similar to~\cite{stanley2015}. That is, we treat the inferred $w^{(\alpha)}$ as $K^2$-dimensional vectors, and cluster them in $K^2$-dimensional space using the $k$-means algorithm.
Results for both notions of layer interdependence are shown in the following section.

\begin{table}[htbp]
\begin{tabular}{lcccccccccc} \toprule
 Network & K & {\small MULTITENSOR} & Diagonal & BPTF \\ \hline
Te\underbar npa\d t\d ti Village & 4 & \textbf{0.89}& 0.88 & 0.87\\
A\underbar lak\= apuram Vill. & 6 & \textbf{0.93} &	0.92 & 0.91 \\
Malaria & 3 & \textbf{0.83} &	0.82 & 0.82 \\
Malaria & 5 & \textbf{0.86}&	0.85 & 0.85 \\
Malaria & 8 & 	\textbf{0.88}& \textbf{0.88} & \textbf{0.88} \\ \bottomrule
\end{tabular}
\caption{AUC for link prediction on real networks using our {\small MULTITENSOR} model, the diagonal/PARAFAC algorithm and Bayesian Poisson Tensor Factorization (BPTF)~\cite{schein2015}. Here we look at the entire dataset at once, and define the AUC as the probability a random link is ranked above a random non-link. Results correspond to the maximum likelihood fixed point over 100 runs of each algorithm with random initial conditions. The best performance for each network is indicated by boldface. All three algorithms perform quite well when shown all layers at once, although {\small MULTITENSOR}'s performance is the best by a small margin.}
\label{table:AllAuc}
\end{table}

\begin{figure*}[ht]%
 \centering
 \subfloat{{\includegraphics[width=4cm]{caste_full_1} }}%
 \subfloat{{\includegraphics[width=0.19\linewidth]{K4_1_c0_dis} }}%
 \subfloat{{\includegraphics[width=0.19\linewidth]{K4_1_c1_dis} }}%
 \subfloat{{\includegraphics[width=0.19\linewidth]{K4_1_c2_dis} }}%
 \subfloat{{\includegraphics[width=0.19\linewidth]{K4_1_c3_dis} }}%
 \caption{Te\underbar npa\d t\d ti Village community partition. On the left we show the division by caste membership. To the right we show the membership in each of the 4 communities for each node (each figure represents one community), with color ranging from white if the normalized out-going membership $u_{ik}=0$ to black if $u_{ik}=1$. Values in between denote overlapping membership (grey). The fact that caste membership partially overlaps with the communities identified by our algorithm suggests a relationship between topological structure and caste, a topic that will be investigated in a future paper.} %
 \label{fig:partition1}%
\end{figure*}

\begin{figure}[htbp]
	\centering
	\includegraphics[width=1.0\linewidth]{AUC_DL.pdf}
	\caption{Layer interdependence in the Indian social support networks. On the $x$-axis are the layers' labels used in the test dataset, and the $y$-axis shows the AUC obtained through the cross-validation schemes for measuring layer interdependence. Bold lines are for Te\underbar npa\d t\d ti Village, dashed for A\underbar lak\= apuram Village. $L=1$ refers to single-layer AUC, where the algorithm is only given access to that layer. $L=2, 3, 12$ show the increase in the AUC for that layer when the algorithm is given access to $L$ layers; for $L=2$ and $L=3$ we choose the best set of $L-1$ additional layers using the greedy procedure described in the previous section.}
	\label{fig:ellylayers}
\end{figure}

\begin{figure}[htbp]
	\centering
	\includegraphics[width=\linewidth]{AUC_malaria_DL.pdf}
	\caption{Layer interdependence in the malaria network. Each of the 9 layers corresponds to a so-called ``highly variable region'' (HVR) of the malaria parasite genes, indicated on the $x$-axis, and the $y$-axis shows the AUC obtained through the cross-validation schemes for measuring layer interdependence. $L=1$ refers to single-layer AUC, where the algorithm is only given access to that layer. $L=2, 3, 9$ show the increase in the AUC for that layer when the algorithm is given access to $L$ layers; for $L=2$ and $L=3$ we choose the best set of $L-1$ additional layers using the greedy procedure described in the previous section. Points and error bars are the average and standard deviation over the $5$ folds of cross-validation. Unlike the social support networks, we see that the accuracy of predicting one layer actually decreases when we include others in the training set, indicating that the different layers have independent structure.}
	\label{fig:malarialayers}
\end{figure}

\begin{figure}[htbp]%
	\centering
	\includegraphics[width=0.48\linewidth]{pca1a}
	\includegraphics[width=0.48\linewidth]{pca2a}
	\caption{Clusters of the affinity matrices in the layers of the Indian village networks, for Te\underbar npa\d t\d ti on the left and A\underbar lak\= apuram on the right. Cluster labels were obtained using the $k$-means algorithm, treating each $w^{(\alpha)}$ as a $K^2$-dimensional vector, and we use PCA to visualize them in two dimensions.}
	\label{fig:ellypca}
\end{figure}

\section{Link prediction and layer interdependence in real networks}
\label{sec:results}

To demonstrate our {\small MULTITENSOR} model and algorithm beyond synthetic data, we apply it to two real-world multilayer networks.
In one network, the {\small MULTITENSOR} model finds that many layers are interdependent, revealing a shared community structure among them. However, in the other network, the models finds the layers to be independent, concluding that there exists no shared structure among them. Together, these two different scenarios illustrate the concrete use of our method in both positive result and negative result scenarios, both of which are likely to arise when analyzing real-world data.

First, we analyze social support networks from two villages in the Indian state of Tamil Nadu, which we call by the pseudonyms~\footnote{Pseudonyms for villages are used for privacy reasons.} ``Te\underbar npa\d t\d ti'' and ``A\underbar lak\= apuram''~\cite{power2015, power2017}. As part of a survey questionnaire, village residents were asked to name those individuals who provided them with 12 different types of support, ranging from lending them household items to helping them navigate government bureaucracy. The resulting directed networks have $N=362$ and $N=420$ nodes, respectively. Each type of support corresponds to a layer in these networks, giving each of them $L=12$ layers, with average degrees ranging from $2.0$ to $4.4$.

Second, we analyze the patterns of shared genetic substrings among a set of malaria parasite virulence genes~\cite{larremore2013}. Each of the $N=307$ nodes represents a single gene, and an edge connects two genes if they share a substring of significant length. Due to the fact that the same set of genes was analyzed at nine different genetic loci (i.e., locations on the genes themselves) which are called ``highly variable regions'' (HVRs), this undirected network has $L=9$ layers, with average degrees ranging from $5.1$ to $76.4$.

The scientifically interesting questions for both networks revolve around the mechanisms driving edge formation. Hypothesized factors include kinship and caste in the Indian social support networks, and upstream promotor sequence or parasite origin in the malaria genetic networks. However, addressing these questions is beyond the scope of this paper, where we instead wish to evaluate the effectiveness of our algorithm.

One option would be to use our algorithm to cluster the nodes, and compare the resulting group assignments with metadata such as gender, caste, or geographical location. Indeed, in Figure~\ref{fig:partition1} we show the community assignment for Te\underbar npa\d t\d ti predicted by our model, and compare it with the division of individuals into castes. Although the figure suggests that the partition might be correlated with caste membership, we do not expect this to be the only type of metadata correlated with the community structure, and we do not consider this correlation to be a good measure of accuracy.
Here we focus instead on link prediction, and in particular on the extent to which knowledge of some layers helps us predict links in others, as described in the previous section.

As for the synthetic networks, our {\small MULTITENSOR} algorithm, the Diagonal/PARAFAC algorithm, and the BPTF algorithm each provide a framework for link prediction. Table~\ref{table:AllAuc} reports the AUC over each entire network and for each algorithm. The algorithms' performance are roughly similar, although our algorithm has slightly higher performance. This suggests that these networks are primarily assortative; this is certainly true of the malaria network, since it is defined in terms of similarity.

To measure layer interdependence we implemented the method described in the previous section, where we attempt to predict the adjacency matrix of a given layer with 20\% of its entries held out, and give the algorithm access to a subset of other layers as part of its training dataset. Interestingly, we obtain opposite results in these two cases.

For the social networks, we find that increasing the number of layers in the training dataset does indeed improve link prediction, with a performance that increases monotonically with the number of additional layers. In Figure~\ref{fig:ellylayers} we show that the AUC for each layer as a function of the number of layers the algorithm is given access to. We found that the best number of groups for link prediction was $K=4$ for the first village and $K=6$ for the second one.

Many layers viewed on their own ($L=1$) are difficult to predict, with AUCs just above $0.5$, i.e., only slightly better than chance. By giving the algorithm access to one more layer ($L=2$) the AUC typically improves by only about $0.05$. However, if we give it access to two additional layers ($L=3$) the AUC improves significantly for almost all of the layers, and this is even more true when we give it access to the entire dataset. (For $L=2, 3$ we use the greedy procedure to choose which $L-1$ layers to add to the training dataset.)

Thus in these social networks, the {\small MULTITENSOR} algorithm is able to usefully apply knowledge from some layers to others. Interestingly, we also see consistency between the two villages with regard to which layers are the hardest to predict, and which layers are the most helpful to include in the training dataset. In particular, the ImpIss layer (``Who do you discuss important matters with?'') is helpful in predicting many layers, while Position, Work, Loan, and Babysit are much less so, and in some cases even decrease the AUC.

We can compare this with the clustering of the $L$ affinity matrices we obtained using standard clustering algorithms, in a spirit similar to~\cite{stanley2015}. In Figure~\ref{fig:ellypca} we use Principal Component Analysis~\cite{jolliffe2002} to visualize the $L$ matrices $w^{(\alpha)}$, projecting them along two principal directions in $K^2$-dimensional space, and we give them cluster labels using the $k$-means algorithm~\cite{macqueen1967}. Indeed we see that Position, Work, Loan, and Babysit are farther from the others, suggesting that these layers are structurally quite different from the others; note also in Figure~\ref{fig:ellylayers} that these layers are among the hardest to predict. In contrast, ImpIss is closer to the other layers, at least for the second village, consistent with the fact that it often helps predict other layers. We also find for $L=2$, that the Borrow layer is the most helpful when predicting the Talk layer in both villages, which is consistent with the fact that these two layers are clustered close together.

In contrast, for the malaria network we find that the best performance is obtained when no other layer is added to the dataset, meaning that prediction actually worsens monotonically as we increase the number of added layers, as shown in Figure~\ref{fig:malarialayers}. This seems to corroborate past findings~\cite{larremore2013} in an important way. Specifically, the standing hypothesis about these genes is that they are maximally diverse in order to most effectively evade the immune system. If there were correlations between loci, which we would see here as the ability of one layer to help in the link prediction of another layer, then this would diminish these genes' overall diversity. This would diminish the amount of ``immune evasion space'' that is spanned by the parasites, and would therefore result in an overall fitness decrease for the parasites.

\section{Conclusions}
\label{sec:conclusion}

We have proposed a generative model for multilayer networks that extends and generalizes the mixed-membership stochastic block model. It assumes that the layers share a common community structure, but allows links in different layers to be correlated with the community memberships in different ways, such as assortative, disassortative, core-periphery, or hierarchical structure, or arbitrary mixtures thereof.  It explicitly allows the communities to overlap, and can be applied to networks with directed, undirected, or integer-weighted links.  We showed that it can be fit to large datasets using a scalable expectation-maximization algorithm, whose running time per iteration is linear in the total size of the dataset and which converges quickly in practice. Due to its ability to describe a wide variety of graph structures, it performs well on synthetic and real data, in terms of both community detection and link prediction.

In addition to performing community detection, the methods in this paper naturally incorporate a framework for link prediction, which we use as a quantitative definition of interdependency between the network's layers.  Namely, we measure how much knowledge of one layer, or a set of layers, improves the accuracy of link prediction in another layer.  This measure is quite general, and goes beyond approaches that cluster layers into strata with similar parameters (e.g.~\cite{stanley2015}); for instance, if two layers both depend strongly on the underlying communities, they will be interdependent in this sense even if one is assortative and the other is disassortative, making their affinity matrices very different.  In addition to providing hints about causal or structural relationships between the layers, this notion of interdependence may be useful to choosing weights for multilayer versions of common network measures, such as eigenvector centrality~\cite{sola2013} and modularity~\cite{mucha2010}. The same link prediction and cross-validation framework used to quantify layer interdependence can also be used to identify and avoid overfitting.

Beyond establishing high performance on synthetic datasets, we also applied our methods to two real-world datasets. We found patterns of interdependence between layers of social networks from two Indian villages, indicating correlations between different kinds of social ties, and confirmed that these patterns are largely consistent between the two villages. In contrast, when we applied our methods to a multilayer network of sequence sharing among malaria's virulence genes, we found that the layers were essentially unrelated. This suggests that similarities at different loci of the amino acid sequences are evolving under uncorrelated constraints, rigorously confirming a result based on independent analyses of each layer~\cite{larremore2013}. In both cases, our {\small MULTITENSOR} approach provided information not revealed in previous studies of these datasets, and proved to be useful in identifying not only the presence of meaningful structure, but its absence as well.

The solution we provide for the layer interdependence problem may find application beyond the analysis of extant datasets. Because our method can be used to aggregate layers into clusters, or to compress a dataset by identifying especially relevant or redundant layers, it can direct experimentalists or field researchers in learning which data to collect or prioritize. For example, if two layers of a social network are found by our methods to be redundant during a pilot study, the redundant layer need not be collected at scale. Particularly in cases where data collection is labor intensive, expensive, or generally difficult, robust solutions to the layer interdependence problem can help maximize the impact of studies constrained by limited resources in the laboratory or the field. On the other hand, when layers are found to be independent of each other, our methods provide justification for comprehensive data collection of the relevant layers.

\section*{Acknowledgements}
This work was supported by the John Templeton Foundation (CDB, CM), the Army Research Office under grant W911NF-12-R-0012 (CM), a National Science Foundation Doctoral Dissertation Improvement grant (BCS-1121326), a Fulbright-Nehru Student Researcher Award, the Stanford Center for South Asia, and Stanford University (EAP), and the Santa Fe Institute Omidyar Fellowship (DBL, EAP). We are grateful to Hanna Wallach and Aaron Schein for helpful discussions.

\bibliographystyle{apsrev4-1}

redpajama_set_name: RedPajamaArXiv


Charlene joined the Marine Corps in 1963. Assigned as a PersO MOS, this 21-year career Marine saw assignments as Company XO & CO (women's unit), Adjutant, XO/PersO MSG Company (Manila, Philippines), StaffSec/S-3 Woman Officers School, G-1 and Human Resources Officer. Last duty was at Camp Pendleton establishing the first Substance Abuse Recovery Program, receiving a Master's Degree in Counseling Psychology. Retiring from the Corps in 1984, Charlene established a counseling private practice in Carlsbad, CA. Eventually Charlene was ordained (recovery ministry) and on the church staff as Director of Celebrate Recovery (part-time volunteer), eventually passing leadership of the program on.
The purpose of the Scholarship Committee is to solicit applications and make recommendations to the Executive Board for awarding the various WMA Chapter CA-7 scholarships for attendance at accredited four-year colleges, universities, or community colleges. To qualify for the award, the student must have demonstrated dependability, leadership, and service in the home, school, and community. Academic requirements (minimum combined SAT or ACT scores and High School GPA) are published annually. He or she must be a graduate of a North San Diego County high school and must be a legal resident of the United States. Military affiliation is not required.
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Each February the chapter celebrates the anniversary of the United States Marine Corps Women's Reserve. The Women's Reserve was established February 13, 1943. The chapter celebrates this event the second Saturday of February with a special luncheon. We honor the courage and esprit de corps of each woman who volunteered to serve her country during a time of great need and became the pioneer who paved the way for all women who have served and are serving today. See our USMCWR Anniversary Event Webpage.
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Welcome to a Golden Globes edition of The Grind, where we've got a hot script idea that the Hollywood Foreign Press is going to love. A professional golfer toils in obscurity for more than a decade before magically morphing into his sport's best player. Not juicy enough? How about we give him a really cool name, like, oh, Jimmy Walker? Yeah, that sounds good. Leo? Scorsese? Any takers?
Jimmy Walker: What a run this guy is on! Every time I see/hear his name, I think of the Ben Folds Five song, "Eddie Walker," and I change it to Jimmy. "Ohhhh, Jimmy Walker, this is your life! You never had a coat or a tie! You never had a reason to cry!" I'm not sure all of that's true, but that will be the title song for the aforementioned upcoming biopic in which audiences will fall in love with a man who is an astrophotographer (true story) and whose wife, Erin, is a nationally-ranked horse rider/show jumper (again, true story). They'll also enjoy the scene from the 2014 Sony Open in which Walker shoots a final-round 63 to win for the second time in six starts.
Don't look now, but Walker's, um, meteoric rise has him first on both the FedEx Cup and Ryder Cup points lists. "Ohhhh, Jimmy Walker, ahhh, ahhh, ahhh, ahhh, this is your life!" We'll have to work on the rest of the plot/song lyrics, but we've got a good start.
Albatrosses: Or double eagles if you want to rile up certain people. Regardless of what you call them, being three under on a single hole is awesome and very rare. Well, on any day other than last Friday. For what is believed to be the first time ever, the feat was recorded on both the PGA Tour (James Hahn) and the European Tour (Joost Luiten) on the same day. These guys are too good.
Harris English: One of the newest members of the Callaway equipment family finished solo fourth at Waialae CC in just his second start with his new sticks. In other words, maybe we shouldn't go crazy the next time a high-profile player changes gear. . .
Louis Oosthuizen: The South African successfully defended his title at the Volvo Champions. And to top off another trophy and a big check, Oosthuizen got to celebrate by riding a tractor! We're not kidding. The man loves tractors.
Tim Clark: On a serious note, we hope his latest elbow injury isn't serious. That being said, when we picked him as our "one-and-done" player of the week, we didn't think he'd take it so literally and withdraw after the first round. At least we also tabbed Jimmy Walker ("Ohhhh Jimmy Walker, this is your life!" Sorry, couldn't help it.) as a top sleeper before the start of the season. How's that working out?
James Hahn: How do we sell a guy who made an albatross? When that guy only manages to finish T-46. Joost Luiten didn't win in South Africa, but at least he parlayed his double eagle into a T-3. Oh, and Hahn, who is best known for doing the Psy "Gagnam Style" dance at last year's Phoenix Open, didn't even whip that out after holing the unlikely shot. A real opportunity missed there, James.
Nominating yourself for SportsCenter: On Monday night, Bubba Watson posted this video of him holing a putt on a private jet and doing the ESPN SportsCenter "dadada, dadada" theme to get that show's attention. We're not sure if it worked yet, but we prefer non-manufactured sports moments getting air time. You know, like a certain lefty golfer hooking a wedge out of the woods and onto the green to win the Masters. . .
Tan lines: Specifically, really, really bad tan lines. On your head. A grainy (and probably somewhat distorted) picture of Stewart Cink's distinct two-toned dome taken off of Thursday's TV coverage of the Sony Open wound up getting more attention than the tournament itself. To Cink's credit, he handled the situation incredibly well on Twitter: "Apparently my glowing head has received some attention today. It's nice to be (in) the spotlight." Well played, Stewart. Now go buy some spray tan.
After two weeks in Hawaii, the PGA Tour heads to the mainland for the Humana Challenge, aka the old Bob Hope. Unfortunately, the field in Palm Springs isn't particularly strong -- not even for its once famous celebrity pro-am. Oh, Holly Sonders is playing? Never mind. That's pretty strong.
Random tournament fact: This event's three courses were ranked the three easiest on the PGA Tour last year. In other words, the word "Challenge" should be taken lightly.
DJ is teaching Paulina how to play golf! And it actually looks like it's going well. No, seriously, Paulina Gretzky has a good golf swing. Check it out.
What? We can double dip with Paulina since this wasn't technically a public display of affection between our two favorite lovebirds! This photo does include DJ, but also The Great One, Wayne Gretzky, and Mrs. Great One, Janet Jones. The four posed while celebrating Jones' birthday. Wait, that's what Paulina wears to celebrate her mom's birthday? Poor Wayne. We guess it shouldn't be too surprising, though, considering she wore this last year to her little brother's birthday bash.
See Bubba, that's a SportsCenter-worthy shot.
Nobody seems to know all the details, but a man fell through the ceiling of a pro shop -- and nobody there made a big deal about it.
Peter Kostis announced he's healthy again and that he plans to return to work roaming the fairways for CBS soon. Great news. Now get that Konica Minolta BizHub SwingVision machine cranking! . . . The daughter of former college basketball star Teddy Dupay has her own golf business. She's 11. We're impressed. . . . And speaking of the Golden Globes, my new go-to dessert (above) was inspired by recently binge-watching "The Sopranos" for the first time. I call it the "Tony Soprano Sundae," because it's what James Gandolfini (rest in peace) would often make in the show after a tough day of money laundering, gambling, murdering, etc. Ice cream (preferably chocolate and mint chocolate chip), whipped cream and M&Ms. MMM, those are good.
Why hasn't Will Ferrell ever won a Golden Globe?
Will a golf movie ever win a Golden Globe?
Why have I gained a few pounds lately? Oh. Right.


text:
Alfred Briggs Irion (February 18, 1833 – May 21, 1903) was a U. S. Representative for Louisiana's 6th congressional district.

Biography
Born near rural Evergreen in Avoyelles Parish, Irion attended the common schools, Franklin College in Opelousas, and graduated from the University of North Carolina at Chapel Hill in 1855. He studied law privately and was admitted to the bar in 1857. He launched his practice  in Marksville, the seat of government of Avoyelles Parish. He served as delegate to the Louisiana secession convention in 1860 and opposed the secession of the southern states.

During the Civil War, Irion served in the Confederate States Army. He was in the 28th Texas Cavalry Regiment under Colonel Horace Randal, part of General John George Walker's Greyhound Division.
He served as member of the Louisiana House of Representatives in 1864 and 1865. when he returned to his law practice. He was the editor of a local newspaper in Marksville from 1866 to 1874. He moved to his native Evergreen in 1870 and engaged there in planting. He continued the practice of law and also engaged in literary pursuits. He was a member of the Louisiana Constitutional Convention in 1879. From 1880 to 1884, he served as judge of the Louisiana Court of Appeal for the Third Circuit.

Irion was elected as a Democrat to the Forty-ninth Congress (March 4, 1885 – March 3, 1887) but was an unsuccessful candidate for renomination. He died in New Orleans and is interred in the Baptist Cemetery in Evergreen, Louisiana.

Notes

References

1833 births
1903 deaths
University of North Carolina at Chapel Hill alumni
Confederate States Army personnel
Democratic Party members of the United States House of Representatives from Louisiana
Louisiana state court judges
Circuit court judges in the United States
Democratic Party members of the Louisiana House of Representatives
Journalists from Louisiana
Burials in Louisiana
19th-century American politicians
19th-century American judges

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text:
HomeAbout Neil CahnServicesContact
Published By: Neil Cahn, PLLC
Divorce: New York
Home > Discovery > What Exactly is an "Order of Preclusion?"
What Exactly is an "Order of Preclusion?"
Posted in Discovery, Evidence
If a spouse wilfully fails to provide financial information during the discovery phase of a divorce action, one remedy may be an order of preclusion under C.P.L.R. §3126. Thus, an August, 2010 decision of the Appellate Division, Second Department, in Raville v. Elnomany, affirmed the preclusion of the husband "from offering financial evidence at the hearing on equitable distribution based on his willful failure to comply with discovery." 76 A.D.3d 520, 906 N.Y.S.2d 586.
But what, precisely, is an order of preclusion from offering evidence? Is it really what you want? Is it everything you need? A brief review of some decisions granting "orders of preclusion" may highlight the open issues even after preclusion is granted.
For example, is precluding the husband from offering evidence the same as precluding "testimony?" Casey v. Casey, 39 A.D.3d 579, 835 N.Y.S.2d 277 (2nd Dept. 2007).
Is preclusion the same as eliminating the right to cross-examine? Grande v. Grande, 129 A.D.2d 612, 514 N.Y.S.2d 250 (2nd Dept. 1987); cf., Settembrini v. Settembrini, 270 A.D.2d 408, 704 N.Y.S.2d 641 (2nd Dept. 2000).
If the husband is allowed to cross-examine, may cross-examination be curtailed if the subject matter involves the very information which the husband failed to produce? Cohen v. Cohen, 228 A.D.2d 961, 644 N.Y.S.2d 831 (3rd Dept. 1996).
Does it require that there be an "inference" in favor of the non-offending party? Dolny v. Dolny, 32 A.D.3d 818, 820 N.Y.S.2d 520 (2nd Dept. 2006).
Does preclusion from offering evidence necessarily require that financial issues of fact be deemed resolved in favor of the non-offending party? Pearl v. Pearl, 266 A.D.2d 366, 698 N.Y.S.2d 160 (2nd Dept. 1999).
Is it the same as permitting the wife "to proceed at trial solely upon her proof of the financial matters." Is the wife entitled to "a default equitable distribution?" Reed v. Reed, 93 A.D.2d 105, 462 N.Y.S.2d 73 (3rd Dept. 1983).
"Preclusion" can be a very imprecise remedy. Counsel should be careful to request, and the Court should be careful to detail the metes and bounds of the remedy. However, for counsel, a detailed request may be unavailable until there is a clear understanding of the fact issues to be determined and the available proof. This may require delaying the motion; or seeking leave to obtain additional or different relief once more is known.
Tags: Default, Inferences, Preclude, Preclusion, Raville, Remedies
Pro Se Father Denied Copy of Forensic Report Although Received in Evidence
Being Bound by Statements in Tax Returns and Court Papers
Filing Tax Returns as "Single" May Not Estop Claim to Be Decedent's Widow
About Neil Cahn
A graduate of Yale College and a Law Review graduate of the Hofstra University School of Law, Neil Cahn has practiced law on Long Island…
A graduate of Yale College and a Law Review graduate of the Hofstra University School of Law, Neil Cahn has practiced law on Long Island for more than 40 years.
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THE WELL-ESTABLISHED French contralto Nathalie Stutzmann has had a second career as a conductor. Her work has been very impressive, as is this new CD, on which she both sings and leads the superb, stylistically flexible ensemble Orfeo 55. The trove of what we term "arie antiche"—Baroque vocal pieces including opera arias, songs and the rare prayer—was researched and adapted for piano by Alessandro Parisotti (1853–1913), a Vatican-trained composer and scholar who studied voice alongside the last castratos. He intended the three-volume collection—which included his own pastiche "Se tu m'ami," masquerading as a Pergolesi song—as teaching materials for singers in training. They are still familiar as such, though in public performance Parisotti's versions tend to be relegated to warm-ups and encores.


So interesting. Hawking is a perfect example of the specialist, corporate capitalism's perfect citizen. The specialist knows nothing outside his field of expertise. In Hawkings' case, this means he does not even know of himself as a being within the company of other beings. He has become the perfect cog in a seamlessly working machine, and as the machine has worked so well for him, he assumes it is a good machine. It is not for most.
Susan, I read the article by O'Brien a very long time ago and had totally forgotten about it. Thanks for the reminder and link. It reminded me of McBryde Johnson's essay in the NYT magazine section.
Julie, Hawking is about as specialized as a specialist can be. I have no idea what his ties are to the corporate world. I would like to think he is simply an ivory tower scholar with little ability to interact with us regular folk. What I find remarkable is the number of people who read his Brief History of Time. I could not get past page 25.
The Ivory Towers are all funded by corporations. Whether that is public knowledge is another story.
Julie, Yes the Ivory Tower is funded by corporations. But universities are supposedly non profit institutions. Flawed as they are, universities do support dissent.
i recommend reading the chapter entitled "The Illusion of Wisdom" in Chris Hedges' "Empire of Illusion."
Universities do allow dissent, up to a point. Identity politics dominate over a class based understanding of oppression, which contributes to the oppressed continuing to fight amongst themselves instead of those with real power. Those who offer seriously dissenting views that pose a threat to the assumption of capitalism's supremacy face challenges including passive censorship (not getting published, not being given tenure) unless they are as well known as Noam Chomsky.
Hawking actually provides a really interesting case study in the decoupling of disability from its usual social repercussions. It is a rare person who has lived the experience of severe disability, yet has *not* experienced having the value of his life questioned or outright dismissed by others, has not been treated as if his opinion carried less weight (quite the opposite, as we're seeing), and has not had to fight for the right to speak for himself in spite of communication barriers. The great respect that is afforded him on account of his intellect and his accomplishment is deserved; but it has insulated him from what most people with disabilities experience in society. It's no wonder that he cannot relate to the implicit threat that is so obvious to many whose lives are overtly devalued by those around them. A person who always been surrounded by respect and support for his own self-determination cannot fully appreciate where his own autonomy leaves off and the societal affirmation that so many others lack begins. He has little reason to feel vulnerable when "death with dignity" is discussed, and he cannot speak for those whose visceral experience of that vulnerability needs to be heard.
Lynn, As usual spot on comments. Hawking will indeed never have a person question his quality of life or status as a human being. This has insulated him from the gritty reality most people with a disability live with.
Middlechild, I was not surprised at all. Hawking's observations r.e. disability have never been insightful. I doubt he has given assisted suicide much thought. Steve Drake really took Hawking to task the other day. I thought he was too harsh. I was wrong.
Wow, the comments this blog gets are as good as the posts. Fascinating!
Claire, One of the reasons I so enjoy writing this blog is the outstanding comments that are routinely posted.
Since this, I have had people say to me and I feel it is pointed - as they know my thoughts - that if Hawking said its okay then he would know... just because you are intelligent in one aspect gives you no more intelligence in other areas than the rest of us.
There is something very ugly and cold about purposely setting up to kill someone and making it out to be mercy... mercy for extreme pain is adequate pain relief - not to be pressured to die.
I completely agree about him not understanding the position of privilege he speaks from, and I was struck by something similar in this piece http://abcnews.go.com/Health/stephen-hawking-flip-assisted-suicide-divides-die-movement/story?id=20291577 where a proponent of assisted suicide says "we couldn't imagine a person with more experience with medical care who is more vulnerable." The view is of disability, vulnerability and assisted suicide as all purely medical things. Stephen Hawking has access to all the care and equipment he needs and a recognition of the value of his existence from society at large: it's very frightening that people can be so blinkered as to see nothing of that and how different things are for other people.
But for me the most obvious problem with what he said is the simple logical flaw. How can you possibly think "We don't let animals suffer, so why humans?" is a sound argument while also saying it must be the person's choice? It's precisely the argument many people would've used for turning off his ventilator. His view is being given special weight because of his position as an extraordinary thinker and yet his reasoning on this is utterly dire.
Hawking seems to be imagining a fictional world in which the safeguards he lays out as the condition for supporting assisted suicide do indeed exist.
In his imaginary world, Hawking is able to justify support for a right to assisted suicide. In the real world, his conditions for such support cannot be met and his comments actually form an argument against assisted suicide.


text:
Diecezja Kribi – diecezja rzymskokatolicka w Kamerunie. Powstała w 2008 przez podział diecezji Ebolowa-Kribi.

Biskupi diecezjalni
 Joseph Befe Ateba (2008 – 2014)
 Damase Zinga Atangana (od 2015)

Bibliografia
 Gcatholic

Kribi
Kribi


text:
AEW-NJPW Relationship Has Reportedly Changed Over The Past Few Months
Gaurav Yadav
Image via Twitter
All Elite Wrestling and New Japan Pro Wrestling's relationship has been under scrutiny for the last few days, ever since NJPW referenced Chris Jericho and his AEW World Title reign for the first time ever.
It was previously reported that AEW-NJPW's relationship hasn't seen any advancements, and this was merely a tactic to promote the match between Hiroshi Tanahashi and Chris Jericho. However, according to Dave Meltzer of Wrestling Observer Radio, there have been certainly some new developments in the partnership.
According to Meltzer, NJPW's relationship with Anthem has soured recently, which has led them to a partnership with AEW instead. He noted that a few months ago NJPW officials didn't even allow the mention of AEW in the company, but that has obviously changed with the stipulation around the Tanahashi-Jericho match.
"There is absolutely a change. Just the fact that New Japan acknowledged on their website that Chris Jericho is the AEW Champion and if Tanahashi wins he gets a championship match. As we all know when I was in Dallas [for NJPW] any mention of Kenny Omega and AEW were forbidden."
NJPW's planned expansion into the USA didn't go down as planned, and now with no proper TV deal in place, the company is struggling to break into the western market. A partnership with AEW could thus, make things smoother for them and help both the companies equally.
"As far as I know there's no long-term deal or anything like that, but the door is open. Of course, New Japan now wants to run the United States, but they lost their television in the United States."
Reiterating on the previous report, Meltzer noted that there is no concrete deal in place, but things are much different from what they were months ago. AEW-NJPW working together will be a dream, but it's still a pipedream for the professional wrestling community.
How AEW Will Air Matches From Chris Jericho's Cruise On Dynamite This Week
Spoiler: AEW Dynamite Results From Chris Jericho's Rock N' Wrestling Rager At Sea
AEW 'Hires' Marty Jannetty After Marty Scrull's Snub, Dynamite Set To Debut In New York
Rumor Roundup Jan 13: WWE Superstars Refusing Saudi Arabia Trip, RAW Fist Fight, AEW's "Big Plays" For WWE Talent
Former Impact World Champion Brian Cage Signs With AEW
Reason Why WWE Never Signed Tessa Blanchard


text:
Q: How do I solve a sum of products optimization problem using DQCP with cvxPy? In cvxPy's examples on DQCP explainer page the following example is shown to be a type of problem solvable with DQCP:
import cvxpy as cp

x = cp.Variable(pos=True)
y = cp.Variable(pos=True)

product = cp.multiply(x, y)

assert product.is_dqcp()

Can someone give ideas on how to solve this for a vector variable? My problem has constraints, but here's the sample code that shows where the primary issue is:
import cvxpy as cp

x = cp.Variable(2, pos=True)
y = cp.Variable(2, pos=True)

product = cp.sum(cp.multiply(x, y))

assert product.is_dqcp() #this fails since sum of products isn't DQCP

Tried DGP and DCCP as well, but can't find the right way to reformulate the problem.
Thank you!

A: The objective function you have defined is bilinear. The traditional way to solve this problem is linearize the objective functions via McCormick envelopes. I personally havent used this package, but it looks promising

redpajama_set_name: RedPajamaStackExchange


text:
Das [030] Magazin (ehemals: [030] Magazin Berlin), dessen Name sich auf die Berliner Telefonvorwahl bezieht, wurde 1994 gegründet. Es bietet aktuelle Infos, Interviews, Beiträge und redaktionelle Tipps aus dem Berliner Stadt- und Nachtleben.

Geschichte
Das [030] Magazin Berlin – kurz [030] – erschien von 1994 bis 2015 alle zwei Wochen als kostenloses bzw. werbefinanziertes Stadtmagazin. Von 2008 bis 2015 gehörte es zur Zitty Verlag GmbH.

Seit dem 1. Januar 2016 erscheint es im Verlag der Medienagentur [030] Media. Verlags- und Redaktionssitz sind in Berlin-Prenzlauer Berg. Inhaber und Herausgeber ist der bereits zuvor für [030] als freier Musik- und Programmredakteur tätige Tim Schäfer. Mit der Übernahme zum 1. Januar 2016 erhielt der Online-Auftritt einen kompletten Relaunch. Seit dem 1. Dezember 2016 erscheint die [030] als monatliches Print-Magazin. Es liegt weiterhin kostenlos an über 150 Ausgabestellen in Berlin aus.

Weblinks
 Website

Einzelnachweise

Onlinemagazin
Online-Journalismus
Zeitschrift (Berlin)
Berlin-Prenzlauer Berg
Ersterscheinung 2016
Stadtmagazin
Ersterscheinung 1994
Deutschsprachige Monatszeitschrift


Soanes Cycles is one of the oldest bicycle shops in Britain, the shop was built in 1854 by Mr R Abrahams who was a local draper and tailor. Mr Abrahams closed his shop in 1902 and sold the premises to Charles and Gerald Richards from Manor Farm, Branscombe. The brothers went on to open Richard Bro's Liberty Cycles and Sidney J Soanes came to work as a cycle mechanic in 1910 in response to an advert.
Cycles were produced at the Colyton based shop with bicycle frames brazed up from bought-in fittings, the frames were then taken to a shed lined with galvanised sheets and stove enamelled. Cycle manufacture in Colyton stopped in the early 1920's as a result of the competition from factory made cycles, during the 1930's mass production meant cheap cycles became widely available.
A new cycle company, Norman Cycles, started in Ashford, Kent producing mass cycles. As there were remaining stocks of the original Swan Cycles (aka Soanes Cycles) transfers available, they were sent to the Norman Cycles company who applied them to their cycles. The Swan Cycles bikes sold for £3-15 shillings for a standard Ladies or Gents cycle, plus £1 for Sturmey Archer 3-speed. Other makes sold at the time were Raleigh, BSA, Elswick and Sunbeam.
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Your heating and cooling and overall comfort in Hertford, NC is important to Comfort First.
See how we show it in every service call and vist in Hertford, NC by taking a look at our reviews!
They sent out a technician named Ryan, he was on time and got us up and running in no time.
Worked on 3 ton Split Rheem heat pump. Found bad capacitor. Replace with truck stock part. Test operation. System is cooling properly at this time.
Arrived to perform heating maintenance. Checked all operations of the system, checked amp draw on fan motor, heater kit and compressor. Checked all capacities temperatures and refrigerant pressures. System working properly at this time.


long overdue for a post on our progress at the cabin: this Fall we did a good amount of demolition…I always say demo is the easy part: anyone can tear down a wall! it's fun, a good way to get in some exercise and get out some stress. reminder: the main living spaces of the cabin have beautiful open wood ceilings. but as seen above, the two bedrooms had dropped ceilings, making the rooms feel cramped. I was convinced that there was the same wood above, in an enclosed attic….there was no access so we just had to bust through. and… what was up there? we found .. knotty pine tongue & groove, woo-hoo!! exactly what we hoped to find. we continued to pull down the tiles. in one corner, we found what looked like an old hornet nest. I knocked it over, and heard the sound of dripping onto the floor…what the hell liquid could be dripping? it turned out to be….ANTS by the hundreds: Scott ran out to buy Raid at the local market while i sprayed them with clorox or whatever cleaner we had. not fun. but we bagged them up and pushed on. this is our first peek up into the 'attic' over the bedrooms. wood ceiling untouched since 1963. the gable exterior wall had no insulation in it, not surprised to find.


Would you like to know where the most vacancies related to what you want are hidden? The place that leads the ranking is certainly Tsim Sha Tsui, with 40% of offers. You should not overlook Central, since 1 of the jobs lie in that location. Hong Kong is in third place, with 1 job ads that may match your search criteria.


my travels. It was also my first time visiting a zoo!
The creepy Tilapia who lost one of its eyes!
Back outdoors to find a Red River Hog? playing with some sort of drum?
they had avocados, pineapples, among other fruits, growing in the gardens!
positive aspects of conservation efforts, research, and education were redeemable.
Wow, that looks like a great zoo!!


text:
With some flair along the way, St. Andrew's lacrosse caps perfect MAC season
By David J. Kim
David J. Kim
High School Sports Reporter
Video of St. Andrew's senior Chase Noah deking a defender and scoring the first goal of the Mid-Atlantic Athletic Conference semifinals picked up steam quickly. Multiple lacrosse websites posted it online, and people flooded the attackman's inbox after the team's 14-4 victory over Sidwell Friends on Friday.
But Coach Dane Smith understood that the biggest game of the season had yet to be played.
2019 @UAlbanyMLax commit Chase Noah (@SAESlacrosse @NextLevelLax) might be a nice fit on attack next year w/ @Tehoka1??? #REPS pic.twitter.com/NBOoNAZenT
— COMMITTED COMBINE™ (@coachpughlax) May 11, 2019
"I didn't expect Chase to go viral overnight," Smith said. "I had to do a little media control there. I had to tell our guys no more commenting on Instagram. We got another game to play."
After St. Andrew's throttled Flint Hill, 16-1, in the final Monday in Potomac to clinch the school's first conference title since 2002, Smith doesn't mind the attention now.
The Lions (16-3) were hungry for revenge after getting upset by the Huskies in the championship game last year, despite going undefeated in conference in the regular season. St. Andrew's swept the conference slate again this year, ready to prove that last season's hiccup wouldn't be a recurring event.
Noah had four goals and five assists Monday. Sophomore Liam Savage also scored four goals, and senior Stephen Bickel added three.
"There's not a lot of tradition here, and we're trying to build that," said Smith, who just completed his fifth year as coach. "When you talk to the lacrosse people of the world and especially in this area, they're starting to take notice of our players."
On the top of that list is Noah. The Albany commit is a do-it-all athlete who plays more like a midfielder. He finds just as much thrill finding his teammates for open shots as he does dodging and scoring. Despite sitting out the second half of many games, he finished the season with 56 goals and 54 assists.
Anchored by senior Tony Diallo on defense, the Lions allowed just six goals per game; no conference opponent scored more than five goals. Against Flint Hill in the final, the Lions didn't allow a goal until the final two minutes.
"You could point to a lot of directions for the reasons for our success," Smith said. "One thing I will always talk about is the creation and the investment that we had in the past four to five years. The step-by-step investment into our culture is why we have been successful."
That investment is paying off in the form of back-to-back regular season titles and now a tournament crown.
"A championship win like this and a statement season like this show people in this area that we don't plan on going away," Smith said. "This is hopefully just the beginning."
David J. Kim David Kim is a high school sports reporter and news aide for The Washington Post. He graduated from Medill School of Journalism at Northwestern University. Follow