text:
\section{Introduction}

 Random fixed points (FPs) are generalization of classical deterministic  FPs,  and arise  when one considers systems with uncertainty.
 Broadly one can consider two types of such fixed points. There is considerable literature that  considers  stochastic FP equations on the space of probability distributions
 (e.g., \cite{Urns,WeightedBranch}).   These equations typically arise as a limit of some iterative schemes, or as asymptotic (stationary) distribution of stochastic systems.  Alternatively, one might be interested in sample path wise FPs (e.g., \cite{Measure_FP,Measure_Approx}).  For each realization of the random quantities describing the system, we have one deterministic FP equation. These kind of equations can arise  when the performance/status of an  agent  depends upon that of a number of other agents. For example, a financial network with any given liability graph   is affected by individual/common random economic shocks received by the agents.
 The amount cleared (full/fraction of liability) by an agent
 depends upon: a) the   shocks  it receives; and  b) the liabilities cleared by the other agents.
 Our primary focus in this paper is on the second type of equations.  Existing literature primarily considers the existence  of measurable FP, given the existence of realization-wise   FPs (e.g., \cite{Measure_FP,Measure_Approx}). In \cite{Measure_Approx} (and reference therein) authors consider the idea of random proximity points.

 \vspace{-2mm}
 To the best of our knowledge, there are no (common) techniques that provide `good' solutions to (some special types of) these equations.
 We consider a special type of FP equations,  which are quite common, and provide a procedure to compute the approximate solutions.
 We have FP equations
 in which the performance/status of an agent is influenced only by the aggregate  performance/status of its neighbours.
 A random graph describes the neighbours, while a set of FP equations (one per realization of the random quantities) describe the  performance vectors.
 %
The key  idea is to study these FPs, asymptotically as the number of agents increase.    Towards this, we first study the aggregate influence factors,
with an aim to reduce the  dimensionality of the problem. But due to random connections,
the aggregate influence  factors can also depend upon the nodes.
  However   the  aggregates might converge towards the same limit (e.g., as in  law of large numbers).
 {We  precisely consider such scenarios and show that the random FPs}   converge to  that of  a limit system, under certain conditions.
\ForTR{ The performance of  the agents   in the limit system, depends upon finitely many  'aggregate'  limits. }{}We could also obtain   closed form expressions for   approximate  solutions of some   examples.

 \vspace{-2mm}
  \ForTR{The mean field theory (MFT) is close to this approach:
MFT approximates many body problem with a one body problem and our result is also similar in nature.  However there are significant differences.}{}
The mean field theory    also deals with a system of large number of agents, wherein the state/behaviour of an individual agent  is influenced by its own (previous) state and the mean (aggregate)  field  seen by it (e.g., \cite{Mean_WLAN} and reference therein).\ForTR{ The mean field is largely described in terms of  occupation (empirical) measures representing the fraction of agents in different states.}{}
 The theory shows the convergence of the state trajectories as well as the stationary (time limit) distributions of the  original system towards that of a limit deterministic system.

  The stationary distribution   can be described by  FP equations in the space of distributions (e.g., \cite{Mean_WLAN}).  While
we directly have a set of FP equations, which are defined realization-wise and depend upon the realization-wise \ `mean' performance.
  Further, as already mentioned the mean influence factor is not common to all the agents.  In \cite{Mean_rand} and references therein, authors consider the mean field analysis  with `random' aggregate influence factors like in our case.
  They consider the first-order   approximation, wherein the   joint expected values are approximated
by the product of the  marginal expected values etc.  \ForTR{Thus the moments of the joint distributions representing the FP solutions are asymptotically proven to be product of marginal moments.     Some authors also consider second-order approximations or moment closure techniques, where
the joint states of triplets are assumed to have a specific distribution (see \cite{Mean_rand} for relevant discussions).}
Our FP solutions are also proved to be asymptotically independent, however the asymptotic solutions are
  independent (infinite dimensional) random vectors.

 \vspace{-3mm}
We consider FP equations with possibly multiple solutions. We show that any sequence of the chosen FPs, converges   to the unique FP of the limit system almost surely\ForTR{ under sufficiently general conditions.}{.}

\vspace{-3mm}
 We   apply our results to study the systemic risk  in a  large financial network with many financial institutions.
The institutions borrow/lend money from/to other institutions, and will have to clear their obligations at a later time point.  These systems are subjected to economic shocks,
because of which some entities default (do not clear their obligations).  Because of interdependencies, this can lead to further defaults and the cascade of these reactions can lead to
(partial/full)  {\it collapse} of the system.
 After the financial crisis of $2007$-$2008$, there is a surge of activity towards studying systemic risk (e.g.,  \cite{allen2000financial},\cite{acemoglu2015systemic},\cite{eisenberg2001systemic}).  The focus in these papers has been on several aspects including, measures to capture systemic risk,   influence of network structure on systemic risk, phase transitions etc.  These papers primarily discuss homogeneous systems, although  heterogeneity
 is a crucial feature of real world networks.    As already mentioned
 the clearing vectors are represented by FP equations and may have multiple FP solutions.
Thus our asymptotic solution can be  useful in this context.    We consider one  stylized example of  \underline{heterogeneous financial } network, that of one big bank and numerous small banks.
Our key contribution is that we develop a methodology to arrive at simplified asymptotic representation to large bank networks.
This allows easy resolution of many practical what-if  scenarios. For instance, in a simple framework we observe that
having a big bank in an economy well connected to the small banks can stabilize the small banks even when the big bank itself faces shocks. However,
the reverse may not be true.
The proposed methodology can be similarly used to provide insights into many other practical scenarios. We  analyze these in future. To summarise,  our analysis helps identify important patterns in a complex structure, since the structure simplifies when large number of constituents are involved.

 \vspace{-3mm}
\section{System Model  }

Consider a  random graph with $n+1$ vertices $\{1, 2, \cdots, n, b\}$  whose directed   edges,   given  by random weights    $\{W_{i,j}\}$, represent the influence factors.
The node $b$ is a `big node',   and is highly influential.
There is an edge between any two of the `small' nodes (nodes in $\{1, 2, \cdots, n\}$) with probability $p_{ss}$ independently of the others
and let
 $\{ I_{i,j} \}_{i \le n, j\le n}$ be the corresponding indicators.   Then the weights from a small node $j$ are the fractions\footnote{Note that
 $\sum_{i} W_{j,i} + W_{j, b} = 1$ for all $j$.} defined as below:
 \vspace{-2.5mm}
\begin{eqnarray}
 \label{Eqn_Weights} \hspace{20mm}
W_{j, b} =  \eta_j^{sb} \mbox{ and }  W_{ j, i}  =  \frac{I_{j,i} (1-\eta_j^{sb}) } { \sum_{i' \le n} I_{j, i'}  } ,
 %
\end{eqnarray}where $\{\eta_j^{sb} \}_j$ are IID (independent, identically distributed) random variables with values between $0,1$. These fractions, for example,  can represent random fractions of some resources shared between various nodes. From small node $j$, there is a dedicated fraction $\eta^{sb}_j$ towards the b-node while the remaining  ($1-\eta^{sb}_j$ ) fraction is equally shared by the other connected small nodes.    The weights from $b$-node are the fractions, \vspace{-7mm}
$$
\hspace{19mm}
W_{b, j}  = \frac{\eta_j^{bs} }{ \sum_i \eta_i^{bs}},
$$ where $\{\eta_j^{bs} \}_j$ are IID random variables again.  We are interested in   some  performance of the nodes, which depends upon the weighted average of the performance  of   other nodes with weights as given by $\{W_{i, j}\}.$
We consider the following fixed point (FP) equation  (in $R^{n+1}$)  constructed using  functions $(f^s, f^b)$, which in turn depend
upon weighted averages $\{{\bar X}^s_i \}_i$ and ${\bar X}^b$,  and
whose FP  ($i$-th component)
 represents   important performance measure of the nodes (node-$i$) as below:

\vspace{-9mm}
{\small \begin{eqnarray}
\label{Eqn_FixedeqGen}
X_i^{s}  &=& f^s(G_i,   {\bar X}^s_i , \eta_i^{bs} X^b) \mbox{ {\normalsize  for each }} i \le n,  \\
\label{Eqn_Fixedeq2Gen}X^b  &=&    f^b (   {\bar X}^b ) \ \ \ \mbox{ {\normalsize with aggregates}}    \\
{\bar X}^s_i  & := &  \sum_{j \le n} X_j^s W_{j, i}  \  \mbox{ {\normalsize and} } \  \nonumber
{\bar X}^b :=  \frac{1}{n}  \sum_{j \le n} {X}^s_j W_{j, b} .
\end{eqnarray}}%

\vspace{-5mm}
In the above, $\{G_i\}$ is an IID sequence and the performance of the big node $X^b$ is defined  per small node (performance divided by $n$).
For any $n$ define mapping ${\bf  f} := (f^b, f^s,   \cdots f^s)$, with ${\bf x}:= {\bf x}^n := (x_1^n, x_2^n, \cdots, x_n^n)$,    component wise:

\vspace{-8mm}
{\small \begin{eqnarray*}
f_{1}( {\bf x}, x_b ) \hspace{-1mm}&\hspace{-1mm} :=\hspace{-1mm}& \hspace{-1mm} f^b ( {\bar x}_b),  \   {\bar x}_b  :=  \frac{1}{n}\sum_{j \le n} x_j     W_{j,b} \mbox{ and} \\
f_i ( {\bf x}, x_b ) \hspace{-1mm}&\hspace{-1mm} :=\hspace{-1mm}& \hspace{-1mm}   f^s(G_i,  {\bar x}_i, \eta_i^{bs}x_b ) , \   {\bar x}_i  :=  \sum_{j \le n} x_j     W_{j,i}
\ \forall  i > 1,
\vspace{-1mm}
\end{eqnarray*}}which represents the FP of the random  operator (\ref{Eqn_FixedeqGen})-(\ref{Eqn_Fixedeq2Gen}).
We assume the following:\\
{\bf A.1}
  The functions $f^s, f^b$ are non-negative, continuous and are bounded by an $y < \infty$,

  \vspace{-8mm}
{\small $$
0 \le f^s(g,    x, x_b), \ f^b (x_b) \le y \mbox{ for all }  g, x, x_b.
$$}
 Under the above assumption, by well known Brouwers fixed point theorem, FP solution exists for     almost all realizations of $\{G_i\}$,  $\{W_{j,i}\}$ and for any $x_b$.  Thus we have a random  (measurable) FP $( {\bf X}^{*}, X_b^*)$ for each $n$ for the random  operator (\ref{Eqn_FixedeqGen})-(\ref{Eqn_Fixedeq2Gen}) (see \cite{Measure_FP}). \ATR{To be precise we have:}{The precise details are as Lemma 1 of \cite{TR}.}
\ATR {
\begin{lemma}
\label{CLemma_exist}
For any $n$ define mapping ${\bf  f} := (f^b, f^s,   \cdots f^s)$, with ${\bf x}:= {\bf x}^n := (x_1^n, x_2^n, \cdots, x_n^n)$,    component wise:

\vspace{-9mm}
{\small \begin{eqnarray*}
f_{1}( {\bf x}, x_b ) \hspace{-1mm}&\hspace{-1mm} :=\hspace{-1mm}& \hspace{-1mm} f^b ( {\bar x}_b),  \   {\bar x}_b  :=  \frac{1}{n}\sum_{j \le n} x_j     W_{j,b} \mbox{ and} \\
f_i ( {\bf x}, x_b ) \hspace{-1mm}&\hspace{-1mm} :=\hspace{-1mm}& \hspace{-1mm}   f^s(G_i,  {\bar x}_i, \eta_i^{bs}x_b ) , \   {\bar x}_i  :=  \sum_{j \le n} x_j     W_{j,i}
\ \forall  i > 1.
\vspace{-1mm}
\end{eqnarray*}}
Each component
 is a mapping from $[0, y ]^{n+1}  \to [0, y ]$
  for almost all $\{G_i\}$,  $\{W_{j,i}\}$ and for any $x_b$.
 Further by continuity of ${\bf  f}$,  we have a deterministic fixed point   for almost all $\{G_i\}$   and $\{I_{j,i}\}$ under {\bf A}.1.  Then we have (almost sure) random fixed point $( {\bf X}^{*}, X_b^*)$ for each $n$ (see \cite{Measure_FP}).  {\hfill $\blacksquare$}
 \end{lemma}
}{}

 \vspace{-4mm}
{\bf Assumptions on the graph structure:}  We require that the number of  nodes  influencing any given  node,  grows asymptotically linearly  for almost all   sample paths:

 \vspace{-2mm}
\noindent {\bf A.2}   Consider $p_{ss} > 0$, and  only graphs for which,
 %

 {\small
\begin{eqnarray*}
  \lim_{n \to \infty }  \sum_{j \le n}    \left|
    \frac{1 }{   \sum_{i} I_{j, i}}  - \frac{1}{ n p_{ss}}    \right |  = 0   \mbox{ almost surely (a.s.),       for any } i.
\end{eqnarray*}}

\vspace{-5mm}
\subsection{ Aggregate fixed points}
One can rewrite the fixed point equations for the weighted averages  $\{ {\bar X}_i^s \}_i$, ${\bar X}^b$ and we begin with their analysis.
 Define the following random variables, that depend upon real constants $(x, x_b)$:\vspace{-5mm}
\begin{eqnarray}\hspace{15mm} \label{Eqn_xi}
\xi_i (x, x_b) :=   f^s (G_i,   x , \eta_{i}^{bs} x_b) ,

\end{eqnarray} and assume: \\
 {\bf A.3}   {\small $|\xi_i (x, x_b)  - \xi_i (u, u_b) | \le \sigma ( |x- u| + |x_b- u_b|)$} with  {\small $\sigma \le 1$}.

Consider  the  following  operators   on infinite  sequence space\footnote{
\label{footnote_sinf}
 Here $s^\infty$ is the space (subset) of  bounded sequences equipped with $l^\infty$ norm  $ |{\bar {\bf x}} |_\infty :=  \sup_i |x_i|$,
$$
s^\infty := \{  {\bar {\bf x}} =  (x_1, x_2, \cdots ) : x_i \in [0, y ]  \mbox{ for all }  i    \}.
$$} $s^\infty$,   one for each $ n$:

\vspace{-8mm}
{\small \begin{eqnarray}
\label{Eqn_bar_fixedpoint_randGen}
{\bar {\bf f}}^n ({{\bar {\bf x}}}, {\bar x}_b)  &=&  ({\bar f}^n_b, {\bar f}^n_1, {\bar f}^n_2 \cdots) \mbox{ where} \\ \nonumber
{\bar f}^n_i ( {\bar {\bf x}}, {\bar x}_b ) &=&
\left \{  \begin{array}{lll}
 \sum_{j \le n}   \xi_{j} ({\bar x}_j, x_b)  W_{j, i}  &\mbox{ if }  i \le n  \\
 0 &\mbox{ else,}
\end{array} \right .  \\
{\bar f}^n_b ({{\bar {\bf x}}}, {\bar x}_b)& := &
 %
 \frac{1}{n}
 \sum_{j \le n}   \xi_{j} ({\bar x}_j, x_b) W_{j, b}  \ \mbox{ with } \  x_b     :=  f^b (  {\bar x}_b).
 %
\nonumber
\end{eqnarray}} It is clear that the  fixed points   of the above operators  equal  the   aggregate vectors, $(\{{\bar X}_i^s \}_{i \le n} , {\bar X}_b )$.
Define the  'limit' operator $\bar {\bf f}^\infty ({\bar {\bf x}}, {\bar x}_b)  =  ({\bar f}^\infty_b, {\bar f}^\infty_1, {\bar f}^\infty_2 \cdots )$:
\begin{eqnarray}
\label{Eqn_bar_fixedpoint_limtGen}
 {\bar f}^\infty_i ( {\bar {\bf x}}, {\bar x}_b ) :=
\lim \sup_n {\bar f}^n_i ({\bar {\bf x}}, {\bar x}_b) \mbox{ for all } i \in \{b, 1, 2, \cdots \}.
\end{eqnarray}
The idea is to show that the fixed point of this operator equals that of   a  `limit' system and that the fixed points of the original system converge towards  these fixed points.
Recall that the weights sum up to one, i.e., $\sum_{i} W_{j, i} = 1$ for any $i$.
Thus we require the fixed point of the  operator:
$$
{\bar {\bf f}}^n   \mbox{ where }  {\bf f}^n  :  {[0, y]}   \times  {s}^\infty \to   {[0, y]}   \times  {s}^\infty,
$$
where  the $s^\infty$ is defined in footnote \ref{footnote_sinf}.
Idea is to derive a kind of mean field analysis where the aggregates will be approximated by their expected values.

 \vspace{-2mm}
When we consider constant sequence, i.e., if ${\bar {\bf x}} = ({\bar x},  {\bar x},\cdots)$ the limit superiors in the definition of the  limit system ${\bar {\bf f}}^\infty$ are actually limits
by {\bf A}.2 and Law of large numbers (LLN) and equal (with $x_b$ as in (\ref{Eqn_bar_fixedpoint_randGen}))

\vspace{-8mm}
{\small \begin{eqnarray}
{\bar f}_i^\infty  ({\bar {\bf x}},  {\bar x}_b)  \hspace{-1.5mm}&  \hspace{-1.5mm}=  \hspace{-1.5mm}& \hspace{-1.5mm}     E_{G_i, \eta_i^{bs}} \left [ \xi_i ({\bar x}, x_b)   \right ]  (1-E[\eta_1^{sb}]) \mbox{ and }  \nonumber \\
{\bar f}_b^\infty  ({\bar {\bf x}}, {\bar x}_b)  \hspace{-1.5mm}&  \hspace{-1.5mm}=  \hspace{-1.5mm}& \hspace{-1.5mm}     E_{G_i, \eta_i^{bs}}  [\xi_i ({\bar x}, x_b)  ] E[\eta_1^{sb}]  .  \label{Eqn_barf_limit}
\end{eqnarray}}
In the above, $E_{X,Y}$ represents the  expectation with respect to  $X,Y$.
The random variables are IID and hence the first equation is the same function for all $i$.
By Theorem \ref{Thm_MainGen},  given below,
one such constant sequence would be the almost sure limit of the solutions of the aggregate fixed point equations (\ref{Eqn_bar_fixedpoint_randGen}).   Thus one will have to solve a two-dimensional fixed point equation corresponding to the above function (\ref{Eqn_barf_limit}). And then random fixed points  (\ref{Eqn_FixedeqGen})-(\ref{Eqn_Fixedeq2Gen}) are asymptotically independent depending upon the other nodes only via  the aggregate fixed point, as given by the theorem below\ATR{.}{ (proof is in \cite{TR}).}
 %
\vspace{-4mm}
\begin{theorem}
\label{Thm_MainGen}
Assume either  $0 < E[\eta_1^{sb}]   < 1$    or $\sigma < 1$ in {\bf A.3}.
The aggregates  of the random system,  which are   FPs of    (\ref{Eqn_xi})-(\ref{Eqn_bar_fixedpoint_randGen}), denoted by $({{\bar {\bf X}}}^{*}, {\bar X}_b^*) (n) := ( \{{\bar X}_i^s\}_i, {\bar X}_b^* ) (n) $ converge   as $n \to \infty$:
$$
 {\bar X}_i^s  \to {\bar x}^{\infty*} \mbox{ for all } i \mbox{ and }  {\bar X}_b^*  \to {\bar x}_b^{\infty*} \mbox{ almost surely (a.s.),}
 $$
 where $({\bar x}_b^{\infty*}, {\bar {\bf x}}^{\infty*})$ with ${\bar {\bf x}}^{\infty*} := ({\bar x}^{\infty*}, {\bar x}^{\infty *} , \cdots)$ is the FP of
  the limit system given by  (\ref{Eqn_barf_limit}).
Further    (any sequence of) FPs of the  original system (\ref{Eqn_FixedeqGen})- (\ref{Eqn_Fixedeq2Gen}) converge almost surely:
\begin{eqnarray}
X^b (n)  &\to&  X_b^{\infty* } :=  f^b (  {\bar x}_b^{\infty*}) \mbox{ as $n \to \infty$    and } \hspace{3mm}\label{Eqn_Act_Fixed} \\
X^s_i (n)  &\to& f^s (G_i,  {\bar x}^{\infty*}, \eta_i^{bs} X_b^{\infty*})
.  \hspace{20mm}\mbox{ {\hfill $\blacksquare$}  } \nonumber
\end{eqnarray}
\end{theorem}
\ATR{
  {\bf Proof:}  The proof  is provided in section \ref{sec_proof}. {\hfill $\blacksquare$} }{}\ATR{}{\vspace{-5mm}}
  Thus  the fixed points of the finite $n$ system converge to that of the limit system.  The fixed points are asymptotically independent  and depend upon the other nodes only via an almost sure constant ${\bar x}^{\infty*}_i$ which is common for all $i$. Another important point to observe here is that,  \underline{the aggregate fixed}  \underline{points need not be unique}, however any sequence of fixed points (one for each $n$) converges towards that of the limit system (when it has   unique fixed point).

 \vspace{-4mm}
 {\bf Remarks:}  Another interesting observation is that the result does not depend upon the precise probability $p_{ss}$ of the connection between small nodes.  It only depends upon the fact that every node can potentially influence every other node directly or indirectly (i.e., $p_{ss} > 0$).
  It is straight forward to generalize  to the case   where\ATR{ $\{I_{i,j}\}$  are any IID random variables and  the weights  are formed in a similar way. Further}{} one may have finite number of groups of small nodes, nodes within a group are identical stochastically (identical $\{G_i\}$, $\{\eta_i^{sb}\}$ and $\{\eta_i^{bs}\}$), and any typical small node can be of group $i$ with probability $q_i$ independent of others.  With this one can study a wider variety of heterogeneous situations. For example one can consider a financial network with many big and small banks.  One can also  generalize to the case when we have more than one (but finite) distinct limits for  the aggregate fixed points.  The results are true even  when $\{X_i^s\}$ are  finite dimensional,   with dimension greater than one. One can then consider banks with different levels of connectivity.

\section{Financial Network}

  Consider a huge financial  network with $n$  small banks and one big bank. The assets (shares, bonds etc.) of the big bank are   large compared to any small bank and the small banks
  are similar in nature.
   At time  $T =0$    the banks invest in a project  by taking loans from one another or from outside the financial network. At time   $T=1$, the banks
anticipate some returns from their investments, which is used
   to  clear their obligations (e.g.,   \cite{eisenberg2001systemic,acemoglu2015systemic}). But the investments are risky,  there are chances of economic shocks,  the returns might be lower than anticipated, because of which   some banks may not be able to (fully/partially) pay the liability.  We say these banks \underline{defaulted.} The defaulted banks increase the shocks to other connected banks, because of which we may have more defaults. And this can continue and the system can `collapse'.  Systemic risk precisely studies these aspects.
\ForTR{The defaulted banks  break the bonds, those that they invested  at time $T=0$, and try to clear their obligations using the partial returns obtained after breaking.
At the end of second time period $T=2$ the survived banks obtain a return $A^s$  (respectively $nA^b$ for the big bank) while the defaulted ones obtain
 obtain $\rho^s A^s / \rho^b n A^b$ (at time period $T=1$) where $\rho^s/\rho^b  < 1$ (usually much less than 1).}{}Our main aim is to study the influence of economic shocks on the stability of the network, wherein stability is understood in terms of the fraction of defaults, for a given  realization of the  shocks\ForTR{ or in terms of the expected surplus after the shocks etc.}{.}

    \vspace{-2mm}
\underline {Liabilities:} We model the financial network using a directed weighted graph, with the banks as the nodes and the weighted edges   represent the liability fractions and directions.
The weight $W_{i,j} := l_{i,j} /  Y_i$  represents the fraction of the liability, where  $l_{i,j}$ is the amount the bank $i$ is liable to bank $j$ and   $Y_i := \sum_j l_{i,j} + l_{i,b}$ is the total liability of the small bank $i.$
The small banks are liable to  big bank with  proportionality factors $\{\eta^{sb}_i\}_i $ and,  a small bank is liable to another  small bank  with probability $p_{ss} >0$. Let $I_{i,j} $ be 1 if small bank $i$ is liable to   small bank $j$ and then the fractions of liability would be:

\vspace{-11mm}
{\small $$
\hspace{5mm}
W_{i, j} := \frac{I_{i,j} (1-\eta^{sb}_i) }{ \sum_{j'} I_{i,j'} }   \mbox{ and }     W_{i,b} = \eta^{sb}_i. \vspace{-0.5mm}
$$}
The fraction of liability of  big   bank towards  small  bank $i$   equals $W_{b,i}=\eta^{bs}_i / {\bar \eta}$, \ForTR{and $W_{b,o} =\eta^o/ {\bar \eta}$ is the fraction that it is liable to sources outside the network,  with
 ${\bar \eta} := \sum_{j \le n } \eta^{bs}_j +  n \eta^o  $.}{ with ${\bar \eta} := \sum_{j \le n } \eta^{bs}_j .$}
 Let $nY^b $ represent the total liability of big bank.

 \vspace{-3.5mm}
\underline{Shocks:}  Let  $K^s_{i}$ be the amount of money small bank $i$ is expecting as return (plus its liquid  cash) at time $T=1$, let
 $Z^s_i$ represent the individual/independent  shock experienced by small bank $i$, and
let $Z_c$ represent the shock that is commonly received by all the small banks.  The big bank receives a  shock of magnitude $n \delta Z_c$ along with its independent shock
$n Z_b$.  After the shocks the small bank $i$ receives
    $(K^s_{i}- Z_c- Z^s_{i})^+$ at $T=1$ while the big bank receives $n(K^b - \delta Z_c- Z^b )^+$, where   $nK^b$ is the shock free return anticipated at time $T=1$.

\begin{figure*}
 \vspace{-22mm}

       \centering
       \hspace{-7mm}
              \begin{subfigure}
               \centering
               \includegraphics[width=5.cm,height=6cm]{SB_nofail_BB_failExp}
            %

       \end{subfigure}
\hspace{.1mm}
       \begin{subfigure}
               \centering
               \includegraphics[width=4.9cm,height=6cm]{SB_fail_BB_nofailExp}

       \end{subfigure}   \hspace{.1mm}
              \begin{subfigure}
               \centering
               \includegraphics[width=5.cm,height=5.7cm]{SB_fail_BB_failExp}

       \end{subfigure}
       \vspace{-19mm}
       \caption{ First sub-figure: only BB defaults;  Second sub-figure: only SBs default;  Third sub-figure: both default.
        \label{Fig_All_retuns} }

         \vspace{-4mm}
\end{figure*}

\ForTR{
 \newpage

\begin{figure*}
 \vspace{-22mm}

       \centering
       \hspace{-9mm}
       \begin{subfigure}
               \centering
               \includegraphics[width=5.2cm,height=8cm]{GoodBBLossySB_ES2Exp}

       \end{subfigure} \hspace{-1mm}
       \begin{subfigure}
               \centering
               \includegraphics[width=5.2cm,height=8cm]{GoodBBLossySB_PDsExp}

       \end{subfigure}
       \vspace{-17mm}
       \caption{ Small bank-major shocks ($Z_i \sim Bin (0.4, 8)$), big bank-minor shocks ($Z_c = 2, Z_b= 2$)  connection with big bank (best at 0.9778) improves the surplus.
        \label{Fig_SmallSB_retuns} }

         \vspace{-5mm}
\end{figure*}
 }{}

  \vspace{-3mm}
\underline{Clearing Vectors:}
 Let $X_i^s$ represent the maximum possible part of the total liability, eventually cleared by small bank $i$, and let $X_b$ (per small bank) represent the same for big bank.
 The vector $(X^b,  \{X_i^s \}_s)$ is referred to  as clearing vector (e.g., \cite{eisenberg2001systemic,acemoglu2015systemic}) and we make the following commonly made assumptions (as in \cite{eisenberg2001systemic,acemoglu2015systemic} etc.) for computing the same.
  %

  \vspace{-3mm}
  When a bank (say bank $i$)  defaults, it may not be able to clear its liabilities completely. However  it repays   the maximum  possible, and the amount cleared to another  bank  (say bank $j$) is proportional to the fraction $W_{i,j}$.
  Thus small bank $i$ receives ${\bar X}_i^s := \sum_{j} W_{j,i} X_j^s$  at time period $T=1$ when the other small banks try to clear their liabilities. In a similar way, it receives
  $X^b W_{b, i}$ from big bank.
  It also   receives  $(K^s_{i}- Z_c- Z^s_{i})^+$ at time period $T=1$ (after shock) from outside investments.   The liabilities are paid, only after paying the operational costs/taxes $v^s$ ($nv^b$ for big bank).  Thus the bank $i$  at maximum can clear $( (K^s_{i}- Z_c- Z^s_{i})^+ + {\bar X}_i^s - v^s)^+$\ForTR{ and if this amount is less than $Y_i$ it breaks its bonds which are supposed to mature at time period $T=2$.}{.}
The amount cleared by big bank  is also computed in a similar manner.
  Thus, in all, the total amount cleared by small bank $i$ and big bank  respectively equals,

 \vspace{-6mm}
{\small
\begin{eqnarray*}
X_i^s  \hspace{-1.5mm}&  \hspace{-1.5mm}=  \hspace{-1.5mm}& \hspace{-1.5mm} \min  \Big \{ \Phi_i^s ({\bar X}^s_i, X^b) ,
    Y_i \Big \}   ,  \mbox{ and }\\
 %
  X^b \hspace{-1.5mm}&  \hspace{-1.5mm}=  \hspace{-1.5mm}& \hspace{-1.5mm} \frac{1}{{\bar \eta}} \min \left \{    n \Phi^b ({\bar X}^b)  ,  \    nY^b \right \}   \mbox { with  aggregates }
    \\
{\bar X}^b  \hspace{-1.5mm}&  \hspace{-1.5mm}=  \hspace{-1.5mm}& \hspace{-1.5mm}  \frac{  \sum_{j \le n} X_j^s W_{j,b} }{n}  , \ {\bar X}_i^s := \sum_{j} W_{j,i} X_j^s ,  \mbox{ where} \\
  \Phi_i^s ({\bar X}^s_i, X^b) \hspace{-1.5mm}&  \hspace{-1.5mm}:=  \hspace{-1.5mm}& \hspace{-1.5mm} \left (   (K^s_{i}-Z_c - Z^s_{i})^++ {\bar X}_i^s   +\ForTR{ \rho^s A^s+ }{} \eta_{i}^{bs} X^b - v^s\right )^+,\\
   %
   \Phi^b ( {\bar  X}_b) \hspace{-1.5mm}&  \hspace{-1.5mm}:=  \hspace{-1.5mm}& \hspace{-1.5mm}
  \left (   (K^b-Z_c\delta- Z^b)^+ \ForTR{+  \rho^b A^b}{} - v^b    +  {\bar X}^b \right )^+ .
  \end{eqnarray*}}
By Law of large numbers (LLN),
${\bar \eta} / n \to E[\eta^{bs}] \ForTR{+ \eta^o}{}$   a.s.
The rest of the system is exactly like the general system   (\ref{Eqn_FixedeqGen})-(\ref{Eqn_Fixedeq2Gen}),
for any given realization of $(Z_c, Z_b, K^b, Y^b).$  Assumptions {\bf A.1} and {\bf A.3}  are clearly satisfied and Theorem \ref{Thm_MainGen} is applicable if we assume {\bf A.2}.
By  Theorem \ref{Thm_MainGen} and equation (\ref{Eqn_barf_limit}), for any given realization $(Z_c, Z_b, K^b, Y^b) = (z_c, z_b, k^b, y^b)$,
the aggregate vectors are approximately (accurate for large $n$) the solutions of
  the following   fixed point equations:

  \vspace{-7mm}
  {\small
\begin{eqnarray}
 {\bar x}^{\infty*} \hspace{-1.5mm}&  \hspace{-1.5mm}=  \hspace{-1.5mm}& \hspace{-1.5mm}
  E_{Z_i, Y_i, K_i, \eta_i^{sb}} \hspace{-.5mm} \Big [ \hspace{-1.mm}  \min \Big \{  \Phi^{s}_i ({\bar x}^{\infty*} \hspace{-.5mm}, { x}^{\infty *}_b),  Y_i   \Big  \} \Big ]  \hspace{-.5mm}(1 - E[\eta_1^{sb}]) , \nonumber \\
  %
  %
   {\bar x}^{\infty *}_b  \hspace{-1.5mm}&  \hspace{-1.5mm}=  \hspace{-1.5mm}& \hspace{-1.5mm}
      {\bar x}^{\infty*}  \frac{E[\eta_1^{sb}]}{ 1 - E[\eta_1^{sb}]}, \ \
    %
    { x}^{\infty *}_b  =
       \frac{ \min \left \{   \Phi^b ({\bar x}^{\infty *}_b),

    \ y^b \right \}  }{E[\eta_1^{bs}] \ForTR{+ \eta^o}{}} . \label{Eqn_Finanace_FP}
 \end{eqnarray}}
 Once these fixed point equations are solved, the clearing vectors are approximately given by (\ref{Eqn_Act_Fixed}) of Theorem \ref{Thm_MainGen}.  These are \underline{asymptotically independent}, as now the aggregates ${\bar X}^s_i$ are almost sure constants and are common for all $i$.  We now compute relevant asymptotic performance measures.
\vspace{-4mm}

\ForTR{

\subsection{Performance measure to study Systemic risk}
Once these fixed point are available for each pair of shock, initial value, total liability realizations $(k^b, y^b, z_c, z_b)$,  one can obtain various performance measures as below:

1) \underline{Expected surplus till time $T=1$ per small bank:} This is the total income of the network (big bank as well as small banks) after paying away the liabilities and taxes $(v)$ divided by number of small banks.  We are currently using the following expression which has to be proved as in \cite{eisenberg2001systemic,acemoglu2015systemic}
\begin{eqnarray}
\label{Eqn_ES1}
E [S(1)]  \hspace{-1.5mm}&  \hspace{-1.5mm} :=  \hspace{-1.5mm}& \hspace{-1.5mm}   E \left [  \left ( \Psi^s  \right )^+  +  \left (\Psi^s +   \rho^s A^s  \right )^+ ;  \Psi^s < 0  \right ]  \nonumber \\
   &&  \hspace{1mm} +  (\Psi^b )^+ +     \left (\Psi^b + \rho^b A^b  \right )^+  1_{ \{  \Psi^b < 0   \} },
  \mbox{ with} \nonumber \\
    \Psi_i^s   \hspace{-1.5mm}&  \hspace{-1.5mm} :=  \hspace{-1.5mm}& \hspace{-1.5mm}
  (K_i^s -Z_c- Z^s_{i})^++    {\bar x}^{\infty*}  +     { x}^{\infty*}_b    \eta^{bs}_i -v^s  -  Y_i^s \nonumber \\
    \Psi^b  \hspace{-1.5mm}&  \hspace{-1.5mm} :=  \hspace{-1.5mm}& \hspace{-1.5mm}   (K^b - Z_c  \delta - Z_b)^+ +   {\bar x}^{\infty*}   \frac{E[\eta_1^{sb}]}{ 1 - E[\eta_1^{sb}]}  \nonumber \\ && \hspace{10mm}  -v^b   -  Y^b .
 \end{eqnarray}
 The expectations are with respect to $(Z_i, \eta_{i}^{bs}, Y_i, K_i^s)$ and are   conditioned over $(K^b, Z_c, Z_b, Y^b)$.

\ignore{
\begin{figure*}
\textcolor{blue}{
When big bank does not default and when $\rho = 0$, then
\begin{eqnarray}
\Psi^b &=& (K^b - Z_c  \delta - Z_b)^+ +   E[ \min \left  \{ \Psi^s +y, \ y\right \} ]   E[\eta_1^{sb}]    -v^b   -  y   E[\eta_1^{sb}]  \\
&=&
(K^b - Z_c  \delta - Z_b)^+ +    E \left [   (\Psi^s )^- \right ]   E[\eta_1^{sb}]   - v^b
\end{eqnarray}
And hence when $P_D^s  = (1- p_{bs}) = 1 - E[\eta]$ we have,
$$
{\bar x}^{\infty *}  =  \frac{  (  E[(k^s - Z_c - Z_i)^+ ] - v^s )  (1- E[\eta])^2 + y E[\eta]  (1- E[\eta]) }{ 1 -  (1- E[\eta])^2  }
$$
and then
\begin{eqnarray}
E[S(1)] &=&  E \left [   (\Psi^s )^+ \right ]   +    E \left [   (\Psi^s )^- \right ]   E[\eta_1^{sb}]    + (K^b - Z_c  \delta - Z_b)^+ - v^b \\
&=&   E[(k^s - Z_c - Z_i)^+ ] - v^s + (K^b - Z_c  \delta - Z_b)^+ - v^b \mbox{,   on simplification we get this. }
\end{eqnarray}
When $  P_D^s  =1$, then $\Psi^+ = 0$ and $\Psi^- = \Psi$ almost surely and then
\begin{eqnarray*}
E \left [   (\Psi^s )^+ \right ]   +    E \left [   (\Psi^s )^- \right ]   E[\eta_1^{sb}]  &=&  E [ \Psi^s ] E[\eta_1^{sb}] \\
&& \hspace{-30mm} = \left ( E[(k^s - Z_c - Z_i)^+ ] - v^s  + y  E[\eta]  - y  + \Big (  E[(k^s - Z_c - Z_i)^+ ] - v^s  + y E[\eta]  \Big  ) \frac{ 1 - E[\eta]  }{E[\eta]}    \right ) E[\eta] \\
\mbox{ as   with $P_D^s  =1$ } {\bar x}^{\infty *} \mbox{ satisfies}\hspace{-40mm} \nonumber \\
{\bar x}^{\infty *} &= & \Big (  E[(k^s - Z_c - Z_i)^+ ] - v^s  + y E[\eta]    \Big ) (1 - E[\eta] )  +   {\bar x}^{\infty *}  (1 - E[\eta] )
\end{eqnarray*}
Thus  with $P_D^s  =1$ also
\begin{eqnarray*}
E \left [   (\Psi^s )^+ \right ]   +    E \left [   (\Psi^s )^- \right ]   E[\eta_1^{sb}]  &=&  E[(k^s - Z_c - Z_i)^+ ] - v^s   .
\end{eqnarray*}
On the other hand when $P_D^s = 0$ we have then $\Psi^- = 0$ and $\Psi^+ = \Psi$ almost surely, $ {\bar x}^{\infty *}  = y (1 - E[\eta] )$  and then
\begin{eqnarray*}
E \left [   (\Psi^s )^+ \right ]   +    E \left [   (\Psi^s )^- \right ]   E[\eta_1^{sb}]  &=&  E [ \Psi^s ] =
  E[(k^s - Z_c - Z_i)^+ ] - v^s  + y  E[\eta]  - y  +   y (1 - E[\eta] ) \\
   &=&  E[(k^s - Z_c - Z_i)^+ ] - v^s   .
\end{eqnarray*} }
\end{figure*}
\begin{figure*}
\textcolor{blue}{
Below b1 no defaults, between b1 and b2 failure only with maximum shock and no connection to bb.
\begin{eqnarray*}
({\underline k}^s-v^s +x) w(1-pbs)   + y  (1 -w (1-pbs) )& =&\frac{ x} { 1-pbs} \mbox{ which implies }  \end{eqnarray*}
\begin{eqnarray*}
({\underline k}^s-v^s +x) P_D^s  + y  (1 -P_D^s )& =&\frac{ x} { 1-pbs} \mbox{ which implies }  \end{eqnarray*}
Hence
\begin{eqnarray}
\label{Eqn_xpbs}
x (pbs) =  \frac{y  (1 -P_D^s )  (1-pbs) + c(P_D^s)  } { 1 - (1-pbs) P_D^s}  =   y - \frac{  ypbs  - c(P_D^s) } { 1 - (1-pbs) P_D^s}
\end{eqnarray}
$$\mbox{ where }  c(P_D^s) =  ({\underline k}^s-v^s) P_D^s \mbox{ when } P_D^s =  w(1-pbs).
$$
The bound b2 can be  computed when  $pbs$ satisfies the following equation
$$
{\overline k}^s - v^s +  x(pbs)     = y
$$
Thus  $$ b2 =  c(P_D^s)   + ({\overline k}^s - v^s) (1 - (1-pbs) P_D^s) .$$
Between b2 and b3 the equation to be satisfied is
\begin{eqnarray*}
   (E[Z]+ x -v^s) P_D^s   + y  (1 -P_D^s) & =&\frac{ x} { 1-pbs} \mbox{ which implies }  \end{eqnarray*}
Thus $x(pbs)$ is again given by (\ref{Eqn_xpbs}) with $$
c(P_D^s) = (E[Z_i] - v^s) P_D^s  \mbox{ when }  P_D^s = (1-pbs).
$$
The bound b3 is obtained by
$$
{\underline k}^s - v^s + y +  x(pbs)     = y
$$
Thus  $$ b3 =  c(P_D^s)   + ({\underline k}^s - v^s + y) (1 - (1-pbs) P_D^s) .$$
Between b2 and b3 the equation to be satisfied is  (now with $P_D^s = 1 - pbs (1-w) $, $c(P_D^s) := {\underline k}^s  w +{\overline k}^s (1-pbs)(1-w)$)
\begin{eqnarray*}
   c(P_D^s)  + y  (1 -P_D^s) & =&\frac{ x} { 1-pbs} \mbox{ which implies }  \end{eqnarray*}
   Thus $x(pbs)$ is again given by (\ref{Eqn_xpbs}) with $$
c(P_D^s) = {\underline k}^s  w +{\overline k}^s (1-pbs)(1-w)   \mbox{ when }  P_D^s = 1 - pbs (1-w)).
$$
The bound b4 is obtained by
$$
{\overline k}^s - v^s + y +  x(pbs)     = y
$$
Thus  $$ b4 =  c(P_D^s)   + ({\overline k}^s - v^s + y) (1 - (1-pbs) P_D^s) .$$
\newpage
---------------------------
\\
\begin{eqnarray*}
  x &=&  \frac{  y (1 - w (1-pbs) )  (1-pbs) } {1 - w  (1-pbs)^2  }  = y  +  y \left ( \frac{(1 - w (1-pbs) )  (1-pbs) } {1 - w  (1-pbs)^2  }  - 1 \right ) = y  -  y  \frac{ pbs} {1 - w  (1-pbs)^2  }  \\
 & =&    \frac{  y (1-pbs)  - y w (1-pbs)^2  } {   1 - w  (1-pbs)^2  } \\
&=&  y (1-pbs) +   y(1-pbs) \left ( \frac{ 1}{ 1 - w  (1-pbs)^2 }  - 1 \right )
  - \frac{ y w (1-pbs)^2   }{  1 - w  (1-pbs)^2 } \\
&=&
 y (1-pbs) + \frac { y (1-pbs) w (1-pbs)^2 - y w  (1-pbs)^2  } { 1-w(1-pbs)^2 } \\
&=&
= y (1-pbs) - \frac {  y   w (1-pbs)^2   pbs }{ 1-w(1-pbs)^2 }
\end{eqnarray*}
The bound b2 can be  computed when  $pbs$ satisfies the following equation
$$
{\overline k}^s - v^s + y  -  y  \frac{ pbs} {1 - w  (1-pbs)^2  }  = y
$$
That is when
$$
ypbs  \le b2 \mbox{ with } b2 =   ({\overline k}^s - v^s )(1 - w  (1-pbs)^2 )
$$
For bound b3,
Next is when
\begin{eqnarray*}
x (1-pbs)   + y   pbs & =&\frac{ x} { 1-pbs} \mbox{ which implies }  \end{eqnarray*}
which implies
\begin{eqnarray*}
  x &=&   \frac{y pbs (1-pbs)} {1 - (1-pbs)^2}  = y  + y  \left (\frac{(1 - (1- pbs) ) (1-pbs)} {1 - (1-pbs)^2}   - 1 \right )  \\
  &=& y  -  y  \frac{pbs} {1 - (1-pbs)^2}
  \end{eqnarray*}
  }
\end{figure*}
\begin{figure*}
\textcolor{blue}{
  Bound b3 can be computed when
  $$
  {\underline k}^s - v^s  + y  + y  -  y  \frac{pbs} {1 - (1-pbs)^2}  = y
  $$
  Which implies the region is specified
  $$
  ypbs \le  b3 \mbox{ with } b3 = \left ( {\underline k}^s - v^s  + y \right  ) (1 - (1-pbs)^2 )
  $$
  %
  %
  %
  For bound b4,
Next is when
\begin{eqnarray*}
x (1-pbs(1-w))   + y   pbs(1-w) & =&\frac{ x} { 1-pbs} \mbox{ which implies }  \end{eqnarray*}
which implies
\begin{eqnarray*}
  x &=&   \frac{y pbs(1-w) (1-pbs)} {1 - (1-pbs) (1-pbs(1-w))}   = y  + y  \left (\frac{ pbs(1-w) (1-pbs) } {1 - (1-pbs) (1-pbs(1-w))}    - 1 \right )  \\
  &=& y  -  y  \frac{pbs} {1 - (1-pbs) (1-pbs(1-w))}
  \end{eqnarray*}
   Bound b4 can be computed when
  $$
  {\overline k}^s - v^s  + y  + y  -  y  \frac{pbs}   {1 - (1-pbs) (1-pbs(1-w))}  = y
  $$
  Which implies the region is specified
  $$
  ypbs \le  b4 \mbox{ with } b4 = \left ( {\overline k}^s - v^s  + y \right  ) (1 - (1-pbs) (1-pbs(1-w)))$$
}
\end{figure*}
}

 2) \underline{Expected number of defaults:}}
 {\underline{Fraction of defaults:}}  The  fraction of small banks that defaulted  (by bounded convergence theorem) and the indicator that the big bank defaults
asymptotically equal:

 \vspace{-8mm}
 {\small
\begin{eqnarray}
\label{Eqn_PDs}
\hspace{-2mm}
P_D^s \hspace{-1.mm} := \hspace{-1.mm} P  \left ( \Phi}_i^s ({\bar x}^{\infty*}  , { x}^{\infty *}_b)  <  Y_i   \right ),      \
P_D^b  \hspace{-1.mm} := \hspace{-1.mm}   1_{ \{ \Phi}^b (  {\bar x}^{\infty *}_b)  < y^b   \} } \mbox{a.s.}   \hspace{-1mm}
 \end{eqnarray}}
 \vspace{-5mm}

\ForTR{
 3) \underline{Expected surplus till time $T=2$ per small bank:}  The banks invest and the liability structure is defined at time period $T=0$. The first installment is returned  to the banks at $T=1$ (as in \cite{eisenberg2001systemic,acemoglu2015systemic}) and we are studying the situation when there are shocks to these returns.  The surplus per small bank till time period $T=1$ is given by $E[S(1)]$.
 The banks that defaulted at time $T=1$, break their bonds/investments. If this did not happen a small bank receives amount $A^s$ on maturity at $T=2$, while the big bank   receives $nA^b$ amount (if it did not default). Thus the surplus at $T=2$ equals:
 \begin{eqnarray}
 \label{Eqn_ES2}
E[S(2)] =   E[S(1)]  + (1-P_D^s ) A^s + (1-P_D^b) A^b . \hspace{2mm}
 \end{eqnarray}
}{}

\ignore{
\subsubsection*{Non zero returns on breaking bonds}
If a bank is not able to clear its obligations, for example if,
$
\Psi^s  <  0,
$ then it breaks the bonds that were supposed to mature at time period $T=2$  and then it receives (only) $\rho A$ (where $\rho \ll 1$). The bank tries to clear its obligations using the money so returned. Hence the fixed point equations would now be:
\begin{eqnarray*}
\nonumber
X_i^s& = &
\min \left \{
  (K^s_{i}-Z_c - Z^s_{i})^++ \rho^s A^s + \sum_{j \le n} X_j^s \frac{I_{ji} (1-\eta^{sb}_j) }{ |A_j |}    +  \eta^{bs}_i  X^b - v^s  ,  \ Y_i \right \}  , \\
  X^b  &=&   \min \left \{   \frac{n}{ {\bar \eta}} (K^b-Z_c\delta- Z^b)^+ + \rho^b A^b + \frac{1}{ {\bar \eta}}  \sum_{j \le n} X_j^s \eta^{sb}_j  - \frac{n}{ {\bar \eta}}  v^b ,  \  \ \frac{n}{ {\bar \eta}}  Y^b \right \}  .

  \end{eqnarray*}
  Thus the limit fixed point equations are altered to the following:
  \begin{eqnarray*}
 {\bar x}^{\infty*} &=&  \boxed{ E \left [  \min \left \{
  (K^s_{i}-z_c- Z^s_{i})^+ + \rho^s A^s  +    {\bar x}^{\infty*} l  +     { x}^{\infty*}_b  \eta^{bs}_i - v^s,  \ Y_i \right \} \right ]} \\
   {\bar x}^{\infty *}_b &=&  \frac{1}{E[\eta^{bs}] + \eta^o}   {\bar x}^{\infty*}(1- l) \\
    { x}^{\infty *}_b &=&  \boxed{   \frac{1}{E[\eta^{bs}] + \eta^o} \min \left \{
  (k^b - z_c \delta - z_b)^+ + \rho^b A^b +    {\bar x}^{\infty*}(1- l)  - v^b   ,  \ y^b \right \} },
 \end{eqnarray*}when $K^b \equiv k^b.$
  However the probability of default remains (\ref{Eqn_PDs}).   But the expected surplus at time period $T=2$ changes to the following

  \vspace{-4mm}
  {\small
   \begin{eqnarray}
   \label{Eqn_ES1_rho}
E^\rho[S(1)] &:=&  E_{Z_c, Z_b}\left [  E_{Z_i^s, \eta^{bs}_i} \left [  \left ( \Psi^s  \right )^+  +  \left (\Psi^s +
   \rho^s A^s  \right )^+ ;  \Psi^s < 0
  \right ]   +  (\Psi^b )^+ +     \left (\Psi^b +
   \rho^b A^b  \right )^+ ;  \Psi^b < 0 \right ]  \hspace{4mm}\\
 \label{Eqn_ES2_rho}
E^\rho[S(2)] &=&   E^\rho[S(1)]  + (1-P_D^s ) A^s + (1-P_D^b) A^b .
 \end{eqnarray}}
This again needs }

 \ForTR{
\begin{figure*}
 \vspace{-22mm}
       \centering
       \hspace{-9mm}
       \begin{subfigure}
               \centering
               \includegraphics[width=5.9cm,height=8cm]{BigBankBigSmallBankSmallES2Exp}

       \end{subfigure} \hspace{3mm}
       \begin{subfigure}
               \centering
               \includegraphics[width=5.9cm,height=8cm]{BigBankBigSmallBankSmallPDsExp}

       \end{subfigure}
       \vspace{-20mm}
       \caption{ When  big bank has major shocks ($Z_c = 2, Z_b= 8$),  risk of small bank is minor  ($Z_i \sim Bin (0.2, 2)$). Connection with big bank does not help,
       surplus at $p_{bs} = 0$ equals  that at $p_{bs}=1$.
        \label{Fig_Big_BB} }

        %
        %

        %
        %
       \centering
       \vspace{-9mm}
       \begin{subfigure}
               \centering
               \includegraphics[width=5.2cm,height=8cm]{BigZbBigZcES2Exp}

       \end{subfigure} \hspace{-1mm}
       \begin{subfigure}
               \centering
               \includegraphics[width=5.2cm,height=8cm]{BigZbBigZcPDsExp}

       \end{subfigure}
       \vspace{-17mm}
       \caption{ With big common shock, setting as in Figure \ref{Fig_Big_BB} but for $z_c = 8$.  Now the connection with big bank  improves the performance as in Figure \ref{Fig_SmallSB_retuns}. The improvement is more pronounced, but big-bank performance remains the same.
        \label{Fig_ZcZbBig} }

         \vspace{-5mm}
\end{figure*}}{}

\vspace{-4mm}
\underline{Regular networks:}  In these   networks,  the total claim of any bank equals  its total liability, i.e.,  $ \sum_{j} l_{ji}+l_{bi} = \sum_{j} l_{ij} +l_{ib}  $ for all $i$.
We consider these networks for further study, as they ensure the initial  wealth\footnote{This is proportional to the amount anticipated without shocks at time $T=1$ plus the amount anticipated by the returns from other banks minus the amount it has to pay to other banks, all at time $T=1$.
For small banks and big bank (per small bank) it is proportional respectively to:

\vspace{-6mm}
{\small $$
K^s_i + \sum_{j} l_{j, i} + l_{b,i}  - \sum_{j} l_{i,j} - l_{i, b}  \mbox{ and }  K^b +  \sum_{j} l_{j, b}    - \sum_{j} l_{b,j}.
$$}} of the network remains the same
once the characteristics of $K^b, K^s, Z_i, Z_c, Z_b$ remain  the same (stochastically).
 This allows fair comparison of  stability of the  network for different values of the parameters\ForTR{, importantly the big bank-small bank connection parameter $E[\eta^{sb}]$. }{.}
 In our network  the liabilities are  random and equal:

\vspace{-8mm}
{\small \begin{eqnarray*}
l_{ji}  = Y_j \frac{I_{ji} (1-\eta^{sb}_j) }{\sum_{i'} I_{j,i'}} \mbox{,  } \  l_{jb}  = \eta^{sb}_j Y_j   \mbox{ and }
l_{bj}=  \frac{1}{{\bar \eta}}  \eta^{bs}_j nY^b
\end{eqnarray*}}
Thus clearly  $\sum_{i} l_{ji} +  l_{jb} = Y_j$, and  as $n \to \infty$ by LLN and {\bf A.2} (\cite[Section 5]{TR})

\vspace{-8mm}
{\small\begin{eqnarray*}
\sum_{i} l_{ij}   \hspace{-1.mm}&  \hspace{-1.mm}=  \hspace{-1.mm}& \hspace{-1.mm} \sum_i   Y_i \frac{I_{i,j} (1-\eta^{sb}_i) }{ \sum_{j}I_{i,j} } \to E[Y_i]  E[ 1-\eta^{sb}_i ] \mbox{,  thus} \\
\sum_{i} l_{ij} + l_{bj}  \hspace{-1.mm}&  \hspace{-1.mm}  \to  \hspace{-1.mm}& \hspace{-1.mm}   E[Y_i] (1 - E[\eta^{sb}_i ] )+  \frac{1}{E[\eta^{bs}] \ForTR{+ \eta^o}{}}  \eta^{bs}_j Y^b  . \vspace{-2mm}
\end{eqnarray*} }
Since the liabilities are random, we require equality in  stochastic sense or atleast in expected sense ($\stackrel{m}{=}$), i.e., we need:
{\small $$
 E[Y_i] (1 - E[\eta^{sb}_i ] )+  \frac{1}{E[\eta^{bs}]  \ForTR{+ \eta^o}{}}  \eta^{bs}_i Y^b  \stackrel{m}{=}  Y_i
$$}
We also need that  the liabilities and claims  of the big bank match, i.e.,
$
\sum_j l_{bj}  = \sum_j l_{jb} .
$ That is we need (at limit),
{\small $$
\lim_{n \to \infty} \frac{\sum_j l_{bj}}{n}  = \lim_{n \to \infty}  \frac{ \sum_j \eta^{bs}_j  }{nE[\eta^{bs}] \ForTR{+\eta^o}{}  } Y^b
\  \stackrel {m}{=} \
\lim_{n \to \infty} \frac{ \sum_j  \eta^{sb}_j Y_j }{n} .
$$}

\vspace{-7mm}
\subsection{Example Case studies}

\ForTR{
We consider two scenarios of regular networks and compute the performance measures (\ref{Eqn_ES1})-(\ref{Eqn_ES2}).

\subsection{No external links, $\eta^o = 0$}}{
We consider an example scenario of a regular network  and compute the performance measures.
}To keep things simple  yet sufficiently interesting, we consider a deterministic $  Y_i^s \equiv y, Y^b \equiv y^b $ and $K_i^s \equiv k^s$.  We then consider a given scenario
$(z_c, z_b, k^b)$ as discussed before. This is the case with identical small banks in terms of initial wealth, investments at time $T=0$ and when they receive common shock $z_c$
as well individual independent shocks $\{Z_i^s\}$. The total shock of the big bank equals $\delta z_c +z_b$. To have regular networks  we set: \vspace{-5mm}
$$
\hspace{22mm}\eta^{bs}_i  \stackrel{d}{=}  \eta^{sb}_i  \mbox{ and }  y^b   =   y  E[\eta^{bs}]  \mbox{ for all } i,
$$and so $E[\eta_{i}^{sb}] =E[\eta_{i}^{bs}]= p_{bs}$.
\ignore{We also need that  the liabilities and claims  of the big bank match, i.e.,
$$
\sum_j l_{bj}  = \sum_j l_{jb}
$$But this is readily satisfied since,
$$
\sum_j l_{bj}  =  \frac{1}{E[\eta^{bs}]  }  \sum_j \eta^{bs}_j Y^b    \mbox{ and } \sum_j l_{jb}  = \sum_j  \eta^{sb}_j Y_j .
$$

The fixed point equations for this special case are the following. For any given realization of $(z_c, z_b)$ the common shock and big bank shock, the liability vectors in the limit are given by solving:
\begin{eqnarray*}
 {\bar x}^{\infty*} (z_c, z_b) &=&  \boxed{ E_{Z_i^s, \eta^{bs}_i} \left [  \min \left \{
  (k^s -z_c- Z^s_{i})^++    {\bar x}^{\infty*} l  +     {\bar x}^{\infty*}_b  \eta^{bs}_i -v,  \ y \right \} \right ]} \\
    {\bar x}^{\infty *}_b (z_c, z_b) &=& \frac{1}{E[\eta^{bs}_1]} \boxed{   \min \left \{
  (k^b - z_c  \delta - z_b)^+ +   {\bar x}^{\infty*}(1- l) -v    ,  \ y^b \right \} }
 \end{eqnarray*} with $y^b = y(1-l)$ and $E[\eta^{bs}_1] = (1-l)$ for regular networks.
}We immediately have the following   for the limit system  when $\eta_{i}^{bs}$ are indicators.
Let ${\underline k}^s$, ${\overline k}^s$ respectively represent the worst and best returns ($(k^s_i-z_c-Z_i^s)^+$) of a small bank, given $z_c$. Then
\ATR{}{\vspace{-5mm}}
\begin{lemma}
\label{Lemma_First}
(i) If $y p_{bs} \le  \big ({\underline k}^s    - v^s   \big ) $  then none of the small banks default, i.e, {\small $P_D^s = 0$}.  \\
(ii) If  {\small $yp_{bs} > \big  ( {\overline k}^s +    x^{\infty *}_b - v^s \big )  $}, then  all   small banks default, i.e., {\small $P_D^s = 1$}. Thus if
{\small $y p_{bs} > \big  ( {\overline k}^s  +  y - v^s \big )  $}, then {\small $P_D^s = 1$}.
\ForTR{\\ (iii) If  {\small $0 < yp_{bs} < \big  ( {\overline k}^s +    x^{\infty *}_b - v^s \big ) $}, then atleast some small banks do not default,  as {\small $P_D^s \le 1- (1-w)pbs < 1$}.}{}
\end{lemma}\vspace{-6mm}
{\bf Proof:}  The  small banks never default if for all  scenarios (realizations of $Z_i, \eta_i^{bs}$):

\vspace{-6mm}
  {\small $$ (k^s - z_c - Z_i)^+ + y (1-p_{bs}) - v^s + \eta_i^{bs} x_b \ge y. $$}
  The worst scenario is with  worst shock ${\underline k}^s$ and with $\eta_i^{bs}=0$ and hence $P_D^s = 0$ when  $y p_{bs} \le {\underline k}^s - v^s $  proving (i). In a similar way consider the best scenario   to obtain part (ii).  {\hfill $\blacksquare$}

Thus we identified the conditions for zero and all defaults.
As long as  $p_{bs} < ({\underline k}^s-v^s)/y$, none of the small banks default.  But (for example) when $p_{bs}$ increases beyond $({\underline k}^s-v^s)/y$,   there can be a   `phase transition'  in
the fraction of defaults,   $P_D^s$. At this point it probably would jump from $0$ to some non-zero value. One need more analysis to understand this possible `phase transition'.
We derive more such details for the special case with binary shocks.

Consider that  {\small $Z_i  \sim  Bin (w, \epsilon)$}, i.e.,   binary $(0, \epsilon)$ shocks with {\small $P(Z_i = \epsilon) = w$}.
    Then {\small ${\underline k}^s =  (k^s - z_c -\epsilon)^+$} and {\small ${\overline k}^s = (k^s -z_c)^+$}.
   \ForTR{
 \subsubsection*{With two time periods}
 We begin with analysis with two time period, $T = 0, 1$. Thus $\rho^s\ \rho^b$ and $A^s \ A^b$ are not applicable.  We compute only the expected fraction of defaults.
   }{
  %
   We have the closed form expressions for the clearing vectors  as well as the asymptotic fraction of defaults, for the sub-case when the big bank does not default. These expressions approximately equal the corresponding quantities  for system with large number of small banks.

   \ForTR{
     \subsection*{When the big bank does not default}}{}\vspace{-2mm}
\begin{lemma}
\label{Lemma_BinaryNoRho}
Let  $y > \epsilon.$   With binary shocks,  the a.s.  limit fraction of defaults equal (with {\small ${\bar K}^s_Z := E[(k^s-z_c-Z_i)^+]$}):

\vspace{-8mm}
{\small    \begin{eqnarray*}
\hspace{-3mm}
P_D^s (p_{bs})\hspace{-1.2mm} &\hspace{-1.2mm} =\hspace{-1.2mm} &\hspace{-1.2mm}  \left [ \hspace{-2mm} \begin{array}{lllll} P_{D1} \\
P_{D2} \\
P_{D3} \\
P_{D4} \\
P_{D5} \end{array} \hspace{-2mm}\right ]  \hspace{-1.2mm}  = \hspace{-1.2mm}  \left \{ \hspace{-1.2mm}
\begin{array}{lllll}
0           & \mbox{if } \  b_0< yp_{bs}  \le   b_1  \\
w (1-p_{bs}) \hspace{-1.2mm}  & \mbox{if } \  b_1 <yp_{bs} \le b_2\\
1-p_{bs}            & \mbox{if } \  b_2 < yp_{bs} \le   b_3   \\
1- p_{bs}(1-w)  \hspace{-1.2mm}      & \mbox{if } \  b_3  < yp_{bs} \le   b_4\\
1                          & \mbox{else}  ,
\end{array} \right . \mbox{{\normalsize with,} } \\
&& \hspace{-15mm}b_i\   := \ c_i  (1-p_{bs}) P_{Di} + d_i  (1 - (1-p_{bs}) P_{Di}),  i > 0, \ b_0 = 0,
\end{eqnarray*}}
   %
    \vspace{-7mm}
{\small    \begin{eqnarray*}
\left [ \hspace{-2mm} \begin{array}{lllll} c_1 \\
c_2 \\
c_3 \\
c_4 \end{array} \hspace{-2mm}\right ]   \hspace{-1mm} =\hspace{-1mm} \left [\hspace{-2mm}
\begin{array}{lllll}
0            \\
 {\underline k}^s-v^s    \\
 {\bar K}^s_Z  - v^s \\
  \frac{ {\bar K}^s_Z   (1-p_{bs}) + ({\underline k}^s +y) w p_{bs} }{1- p_{bs}(1-w) } - v^s  \hspace{-1mm}
  %
\end{array}  \hspace{-1mm} \right ],
\left [ \hspace{-1mm} \begin{array}{lllll} d_1 \\
d_2 \\
d_3 \\
d_4 \end{array}  \hspace{-1mm} \right ]   \hspace{-1mm}  = \hspace{-1mm} \left [\hspace{-2mm}
\begin{array}{lllll}
 {\underline k}^s  - v^s              \\
 {\overline k}^s - v^s     \\
 {\underline k}^s - v^s + y   \\
{\overline k}^s - v^s + y
  %
\end{array}  \hspace{-1mm} \right ]\hspace{-1mm} .
\end{eqnarray*}}
Let $b_5 :=y$, $c_5 := {\bar K}^s_Z  - v^s + y p_{bs}$ and $P_{D5} (p_{bs}) = 1$.
The common clearing  aggregate   (when {\small $b_{i-1} < yp_{bs} < b_{i} $}):
{\small $$
\bar{x_s}^\infty  =
   y (1-p_{bs}) -\frac{ \left ( yp_{bs} - c_i \right )  (1-  p_{bs})P_{Di}(p_{bs}) }{1 - (1-p_{bs})P_{Di}(p_{bs})} \ \forall i \le 5.
$$}
The above is true when the big bank does not default, i.e., if
{\small $$(k^b-\delta z_c - z_b)^+ - v^b + \bar{x_s}^\infty \frac{p_{bs}}{1-p_{bs}} > y p_{bs} \mbox{.  {\hfill $\blacksquare$} }$$}
\end{lemma}
{\bf Proof :} is available in Appendix, in pages 9-10. {\hfill $\blacksquare$}

  The above result indicates the `phase transitions' with respect to the connection parameter $ p_{bs}$. This lemma is true
for the sub-case when big bank does not default (one example scenario,  when $(k^b - \delta z_c - z_b)^+ - v^b > y$).  However some of the `phase transition' results mentioned below are also true for the other case, by Lemma \ref{Lemma_First}.

   As already discussed  when the connectivity parameter $p_{bs}$   is below $b_{1}/ y$, the network of all  small banks remains stable (the fraction of  defaults is zero).
  %
  But as soon as the connection parameter crosses  $b_{1}/ y = ({\underline k}^s - v^s)/y$  the  small banks start defaulting,
and we see a sharp jump of size   (see $P_{D1}$ and $P_{D2}$ in Lemma \ref{Lemma_BinaryNoRho}):
$$w(1-p_{bs}) =\frac{ w(y - {\underline k}^s  + v^s)}{y} \mbox{  \underline{at  exactly} } p_{bs } = \frac{{\underline k}^s - v^s}{ y}.$$
  These kind of phase-transitions can also be seen in Figures \ref{Fig_All_retuns}-\ref{Fig_ZbZc}.
  When  $p_{bs}$ is increased further, when it crosses the threshold $ b_{2}/y$,   $P_D^s$ has another jump/phase-transition.
  At   $p_{bs} = b_2 / y$, we notice a sharp jump of size  (see $P_{D2}$ and $P_{D3}$ in Lemma \ref{Lemma_BinaryNoRho}):
  $$\left (1 - \frac{b_2}{y} \right ) (1-w). $$
  %
 For this case, one needs to solve the equation  (Lemma \ref{Lemma_BinaryNoRho})
\begin{eqnarray*}
p_{bs} &=& \frac{b_2} { y} \\
& & \hspace{-10mm}=   \frac{({\underline k}^s-v^s) (1-p_{bs})^2w + ({\overline k}^s -v^s) ( 1 - (1-p_{bs})^2w) }{y}
\end{eqnarray*} to get the exact point of phase transition.
 In a similar way from  Lemma \ref{Lemma_BinaryNoRho}, we see four possible phase transitions with respect to parameter $p_{bs}$. Since all the co-efficients also depend upon the shock
 realizations ($z_c, z_b$) one can also obtain phase transitions with respect to shock sizes. Same is the case with other parameters.

 More general observation from   Lemma \ref{Lemma_BinaryNoRho} is that, we have a possibility of small fraction of defaults when $p_{bs}$ is near 0 ($P_D^s $ can be 0)
or when $p_{bs}$ is near 1 with $b_4 > 1$ ($P_D^s \propto (1-p_{bs})$ or $\propto (1-p_{bs}(1-w))$).  The coefficient $b_4 $ with  $p_{bs}$ close to one, approximately
equals $d_4 = {\overline k}^s - v^s + y$, is mostly bigger than one.    Thus either small connectivity or large connectivity is better, but   intermediate connectivity may  not be good.  This can also be observed in the figures.

    \ForTR{
    \subsubsection*{With three time periods}
From  equation (\ref{Eqn_Finanace_FP}),  If ${\underline k}^s +\rho^sA^s > v^s + y p_{bs}$  then ${\bar x}^{\infty*} = y (1- p_{bs})$   and
 further $k^b > \delta z_c + z_b + v^b$ implies  $x_b^{\infty*} = y$.
Thus the banks (small) may default, but they are able to clear their obligations completely by breaking the bonds.  One can easily verify the following in this case.
\begin{lemma}
\label{Lemma_Binary2}
   Assume $k^b > \delta z_c + z_b + v^b$, $\epsilon < y$ and ${\underline k}^s +\rho^sA^s > v^s + y p_{bs}$. Then
\begin{eqnarray*}
P_D^s = \left \{
\begin{array}{lllll}
w (1-p_{bs})   & \mbox{if} &  \frac{ {\underline k}^s  - v^s    }{y} < p_{bs} \le \frac{ {\overline k}^s -   v^s   }{y}  \\
1-p_{bs}            & \mbox{if} &  \frac{ {\overline k}^s -   v^s   }{y} < p_{bs} \le   \frac{ {\underline k}^s + y - v^s    }{y}   \\
1- p_{bs}+w p_{bs}        & \mbox{if} &  \frac{ {\underline k}^s + y - v^s    }{y} < p_{bs} \le   \frac{ {\overline k}^s +y -   v^s   }{y}  \\
1                          & \mbox{if} &  \frac{ {\overline k}^s +y -   v^s   }{y}  < p_{bs}  .
\end{array} \right .
\end{eqnarray*}
\end{lemma}
  {\bf Proof:}
  When $y > \epsilon$ (i.e., $x_b > \epsilon$),  $ {\overline k}^s < {\underline k}^s + x_b $
   and
  one can derive the result considering various scenarios as in the previous lemma.
  {\hfill $\blacksquare$} \\
  %
We continue with  sub-case considered of Lemma \ref{Lemma_Binary2} for which $P_D^b = 0.$
Define,
\vspace{-4mm}
$$
P^{*s}_D  :=  \inf_{p_{bs} }  P^s_D,
$$
the  minimum `expected defaults'  possible.  It is clear that for the sub-case considered in Lemma \ref{Lemma_Binary2}  it equals the following:
\begin{eqnarray*}
P^{*s}_D = \hspace{-8mm} &  \\  &  \left \{  \begin{array}{lllll}
\min \left \{  w  \left (1- \frac{  {\overline k}^s-v^s )} {y} \right),  \  \frac{ (v^s- {\underline k}^s )^+} {y} \right \}  &\mbox{if }
 v^s < {\overline k}^s, \\
\min \left \{  w + (1-w) \frac{  v^s -{\overline k}^s} {y},  \  \frac{ (v^s- {\underline k}^s )} {y}, \ 1 \right \}  \hspace{-2mm} &\mbox{if }    v^s > {\overline k}^s.
\end{array} \right .
\end{eqnarray*}
Further  the assumptions of Lemma \ref{Lemma_Binary2} for any $p_{bs} = E[\eta^{bs}]$, we have  ${\bar x}^{\infty*} = y (1- p_{bs})$, $x_b^{\infty*} = y$,  and
 hence
$$
  \Psi^s  =
  (k^s -z_c- Z^s_{i})^++       y (    \eta^{bs}_i - E[\eta^{bs}]  ) -v^s  $$  and $
    \Psi^b  =  \psi^b =  k^b - z_c  \delta - z_b      -v^b > 0.
$
Further $\Psi^s + \rho^sA^s \ge 0$ almost surely and
thus  from equations (\ref{Eqn_ES1})-(\ref{Eqn_ES2}) the expected surplus equals
\begin{eqnarray*}
E[S(2)] &=&  E [ \Psi^s + \rho^s A^s  ; \Psi^s  <  0 + A^s ;  \Psi^s  \ge  0 ]  + \psi^b \\
& & \hspace{-20mm} \ =  \  E [   (k^s -z_c- Z^s_{i})^+   ] - v^s +  A^s  - (1-\rho^s)   A^s  P_D^s   + \psi^b.
\end{eqnarray*}
  Therefore the expected surplus is maximized at the same $p_{bs}^*$ which minimizes $P_D^s$  and then the optimal surplus is obtained by substituting  $P_D^s*$ in equation (\ref{Eqn_ES2}).

  }{}

\ignore { Check again
  When  $P_D^s = 1$ then
  $$
  {\bar K}^s_Z + {\bar x} - v^s+  y p_{bs}   = {\bar x} / (1-p_{bs})
  $$
  Thus
    $$
 {\bar x} =  \frac{(y p_{bs} +  {\bar K}^s_Z  - v^s ) (1-p_{bs} ) } { 1 - (1-p_{bs}) P_D^s} $$

And hence
  $$
 {\bar x} = y (1-p_{bs})   + \frac{( {\bar K}^s_Z  - v^s ) (1-p_{bs} ) } { 1 - (1-p_{bs}) P_D^s} $$

 }

  \vspace{-1mm}
\underline{Influence of connectivity, shocks:}
   We study the influence of connectivity parameter $p_{bs} = E[\eta^{bs}] = E[\eta^{sb}]$ for various shock scenarios.
We begin with the case when $p_{bs} = 0^+$, i.e., as $p_{bs}$ approaches 0 (or is  $\approx 0$). There is negligible connection between the small banks and big bank, and    the small banks are primarily liable to other small banks.   The limit FP equations are: $ x_b = 0^+ $ and

\vspace{-9mm}
{\small  \begin{eqnarray*}
  {\bar x}^{\infty*}  \hspace{-1.5mm} =\hspace{-.5mm} E \left [\hspace{-.5mm} \min \{   (k^s -z_c- Z^s_{i})^++    {\bar x}^{\infty*}  \hspace{-1mm} \ForTR{ +  \rho^s A^s}{} + \eta_i^{bs} 0^+ - v^s,  y \}  \right ].
  \end{eqnarray*}}
  When ${\underline k}^s > v^s$,  it is clear that  $ {\bar x}^{\infty*} = y$, i.e.,  the small banks do not default at  $p_{bs} = 0^+$.  If  the big bank  encounters  big shock, i.e., if $( k^z - \delta z_c - z_b )^+ < v^b$, then
  it  defaults near $p_{bs} = 0^+.$  In fact from equation (\ref{Eqn_Finanace_FP}),   the big bank defaults   for any $p_{bs} > 0$  under these conditions\footnote{
  Clearly {\small $ {\bar x}^{\infty*} \le  y (1-p_{bs})$},  {\small $y^b =  y (1-p_{bs}$)} and  so
  {\small $x_b < y$}.}.
 Therefore we conclude,  {\it`when the big bank receives large shocks, it always defaults, the connection with the small banks does not help'}.
Further  some of the small banks can also default leading to an increased fraction of defaults as $p_{bs}$ increases.
  We considered one such example in the  first sub-figure of Figure \ref{Fig_All_retuns}, which reaffirms our observation: the fraction of defaults $P_D^s = 0$  for small $p_{bs}$ (till 0.1)
  and near  $p_{bs} = 1$.  The  defaults  are more  for intermediate  $p_{bs}$.

  \vspace{-3mm}

 We consider the reverse situation now. Say the big bank does not default at $p_{bs} = 0^+$.  This is because it received small shocks such that  $( k^z - \delta z_c - z_b )^+ > v^b$.
  Say  some small banks receive big shocks such that
    ${\underline k}^s < v^s$.  Then from  Lemma \ref{Lemma_BinaryNoRho}, $P_D^s \ge w$ at $p_{bs } = 0^+$.
 The connection with big bank can improve the fraction of defaults,
for example,  if we manage to chose a large enough $p_{bs}$ which  is between  $b_2/y$ and $b_3/y$ of  Lemma \ref{Lemma_BinaryNoRho} (and if further the big bank does not default).  In this case the asymptotic fraction of defaults could be smaller than $w$.
We consider one such example   in the second sub-figure of Figure \ref{Fig_All_retuns}. We notice  that
$P_D^s$ reduces as $p_{bs}$ increases beyond 0.8, in fact
 $P_D^s = 0$ (i.e, no small bank defaults) at  $p_{bs} = 1$.
  Thus we conclude {\it `the big bank (with small shocks) can help the small banks'.}

\begin{figure}
  \vspace{-17mm}
       \centering
       \hspace{-13mm}
       \begin{subfigure}
               \centering
               \includegraphics[width=4cm,height=6cm]{PDsVsZbZcNoRho_exp}

       \end{subfigure} \hspace{-1mm}
       \begin{subfigure}
               \centering
               \includegraphics[width=4cm,height=6cm]{PDbVsZbZcNoRho_exp}

       \end{subfigure} \hspace{-15mm}
       \vspace{-22mm}
       \caption{For different shock realizations: {\small $ v^s \hspace{-1mm}= \hspace{-1mm}v^b \hspace{-1mm}=  \hspace{-1mm}12$,}  {\small $Z_i \hspace{-1mm}\sim \hspace{-1mm}Bin (.4,  20)$,     $(y, k^b, k^s) = (80, 55, 25)$,   $ (\delta, p_{bs}) = (.4 , .9)$}
  %
   \label{Fig_ZbZc}}
  \vspace{-5mm}
\end{figure}

\vspace{-3mm}
We consider a third example in Figure \ref{Fig_All_retuns}, where big bank and some small banks receive big shocks to default at $p_{bs} = 0^+$.
The big bank continues to default at all $p_{bs}$, however  $P_D^s$ decrease  with increase in   $p_{bs}$.
   Thus  for the chosen example of  economy with one big bank and many small banks, {\it `small banks can be stabilized by connecting to big bank, even if the later defaults, however they can't help the big bank'.}

\ignore{
\subsubsection*{Influence of shocks}
We plot the system risk parameters as a function of the realizations of the two major shocks in Figure \ref{Fig_ZbZc}. \kav{To write more later}}

  \ignore{
  \subsection{When BB is liable to external elements with $\eta^o = 1 - E[\eta^{bs}]$}
When big bank is liable to external nodes, i.e., if $\eta^o > 0$ and in particular if $\eta^o = 1 - E[\eta^{bs}]$ then  we will require (for example) the following for regular networks.
$$
\eta^{bs}_i  \stackrel{d}{=}  \eta^{sb}_i  \mbox{ and }  Y^b   \stackrel{d}{=}   Y_i    \mbox{ for all } i,
$$

Previous curves..  (Figure \ref{Fig_ToExt})
\begin{figure}[h]
\vspace{-15mm}
\includegraphics[scale=0.3]{ObserveSurplus}
\vspace{-25mm}
\caption{When BB is liable to external elements a fraction $l = 1-E[\eta^{bs}]$ \label{Fig_ToExt}}
\end{figure}

  \newpage
  Proof outline :

1) Existence of fixed point, Bruewer fixed point theorem ..
Need not be unique.

2)  Limit system Definition and !ness of fixed point using $l^\infty$ norm and that the fixed point is a constant vector.

3) Maximum theorem  and convergence using weak topology..

Final conclusion:

1) It is possible that the finite bank system have multiple aggregate-fixed points/clearing vectors. However as n increase they converge to an unique vector, which has equal components.

Organization of paper:

Finance part --

One suggestion

One is discuss the problem with rho = 0 and give all the results..  (Lemmas, ES constant, PDS etc)

Then include rho > 0 part and give further results..

Second suggestion :

Direcly give results with general rho ..

Specialize to rho= 0 (ES is constant etc).  }

\vspace{-2mm}
\underline{Influence of shocks:} The banks can face   two types of shocks.   The idiosyncratic shocks ($\{Z_i^s\}, Z^b$)  are bank specific shocks,  while the common shock ($Z_c$)  affects all the  banks.  We aim to study  the role of magnitude of these shocks on the  cascading of defaults  in Figure \ref{Fig_ZbZc} for a fixed $p_{bs}=0.9$. We observe  two phase transitions in the fraction of defaults, $P_D^s$ remains at 0.1  ($1-p_{bs}$) for some region of $(Z_b, Z_c)$, jumps to
0.46 ($1-p_{bs} (1-w)$)
 and then to 1, and, one phase transition with respect to big bank.   This behaviour can also be explained using the barrier constants $\{b_i \}$ of Lemma \ref{Lemma_BinaryNoRho}, which depend   upon these shocks.  One can observe significant sharp jumps at the phase transition points,  and these points are very important for any financial network. These jumps and transitions points can be studied either using Lemma \ref{Lemma_BinaryNoRho} or by studying  the   simplified FP equations (\ref{Eqn_Finanace_FP}) numerically.

\vspace{-4mm}
\section*{Conclusions}

 We considered a  random graph, with edges representing the influence factors between a big  (highly influential)  node and numerous  small nodes.
 The performance/status of individual nodes is resultant of these  influences, which are represented by  fixed point (FP) equations. We   showed that the solution of the random FP equations  converge almost surely to that of a limit system and these solutions are asymptotically independent. One may have  multiple solutions
for finite graphs, however any sequence of them  converge to the unique FP of the limit system (if it has unique FP).   Thus we have a procedure to solve the large dimensional FP equations, using mean-field kind of techniques.  The proposed solution requires solving of  'aggregate' FPs in a much smaller dimensional space and is accurate asymptotically.

\vspace{-2mm}
The clearing vectors (the fraction of liabilities eventually cleared) in a financial network are generally represented by  random FP equations and we studied the same using our results.
We study  an example  heterogeneous  financial network with one big bank and many small banks.
We have reduced the overall economy problem in this set-up to a two node problem - one big bank and one aggregate small bank, thus facilitating big picture analysis.
We observe some interesting phase transitions, one can easily study the nature of these phase transitions using the approximate solutions of the involved FPs.
When small banks invest more in big banks, lesser fraction of them default and this is true even when
all of them face  large idiosyncratic shocks.
 These observations could be specific to the example considered by us,  however we now have a procedure to study complex networks and a more elaborate  study would help us derive more concrete observations.
 %
One can easily generalize the results by relaxing many of the assumptions, one can   apply this approach to   more applications and these two would be  the topics of future interest.

\vspace{-2mm}

redpajama_set_name: RedPajamaArXiv


LOCO20 can be used to change the name of an adult and LOCO22 can be used to change the name of a child.
Subscribers to this range are requested to use the latest versions of these two forms after the Government amended them last year.
LRPN1ID has received a minor amendment.


text:
Study in Australia and New Zealand
Ranking & Fees Comparison
Living Cost Calculator
Student Visa Checklist
Post Study Work Visa
JCU Singapore
Top Universities (Finance)
Student Visa Guide & Checklist
5 Types of Post Study Work Visa
Home » University in Singapore » Why Study in Singapore? » Best Private University in Singapore for MBA Programme
Best Private University in Singapore for MBA Programme
Singapore has been acknowledged as one of the world's best business hub, and it is the perfect study destination for international students pursuing their MBA programme.
Besides the six public universities in Singapore, there are numbers of world top universities and business schools are offering MBA programmes in Singapore.
INSEAD Business School
The European Institute of Business Administration (INSEAD) opened its Asia campus in Singapore in 1999. It offers a range of business programmes in master degree and PhD programmes, also offers a range of executive education for management and leadership programmes.
As one of the world's leading and largest graduate business schools, INSEAD is consistently ranked top among business schools in the world.
University Rankings for INSEAD Business School
QS Global MBA Rankings 2019 6th
ARWU Global Ranking of Academic Subjects 2019 - Business Administration 46th
Bloomberg Businessweek Best International MBA Programs 2017 1st
Financial Times MBA Programme 2016 & 2017 1st
Same with other INSEAD campuses, the Asia campus in Singapore offers three types of MBA programmes, full-time MBA programme, global executive MBA programme, and Tsinghua-INSEAD executive MBA programme. Below is the list of tuition fees for all INSEAD MBA programmes.
Name of Programme
Master in Business Administration $133,233
Global Executive MBA $191,000
Tsinghua-INSEAD Executive MBA $170,906
All fees listed are FULL programme tuition fee in Singapore Dollars inclusive of local tax (GST) and apply for the 2019/2020 academic year
In 2015, ESSEC Business School set up its first overseas college in Singapore. The ESSEC Business School Asia-Pacific campus is offering business courses in undergraduate, postgraduate and PhD levels, and also a range of executive programmes.
ESSEC is one of the 90 business schools in the world hold the "triple crown" accreditation (AACSB, AMBA, EQUIS). Their business programmes consistently recognised worldwide.
University Rankings for ESSEC Business School
QS Global MBA Rankings 2019 26th
International Experience, Financial Times Global MBA Ranking 2018 5th
Executive MBA, The Economics 2018 17th
QS Global Joint Executive MBA Ranking 2019 10th (ESSEC & Mannheim Executive MBA)
There are two types of MBA programmes offering in ESSEC Singapore. They are full-time Global MBA and ESSEC & Mannheim EMBA – Asia-Pacific.
The ESSEC one year full-time Global MBA programme aims to help graduates develop their creativity and problem-solving skills in a multicultural context business environment, and in several specific areas of expertise. The Global MBA programme has five majors, they are:
Strategies and Management
Digital Business & Innovation
Luxury Brand Management
The ESSEC & Mannheim EMBA – Asia-Pacific is designed to prepare senior leaders an international outlook with a pan-Asian core, with courses examining new business models in Asia. The course will be completed in 15 months.
Below are the tuition fees for ESSEC Business School MBA programmes:
Global MBA $73,700
ESSEC & Mannheim EMBA - Asia-Pacific $93,946
S P Jain School of Global Management
By the invitation of the Singapore government, S P Jain School of Global Management established their Singapore campus in 2006.
S P Jain School of Global Management Singapore campus offers business undergraduate and postgraduate programme. They also offer two types of MBA which are Global MBA programme and Executive MBA programme.
The 12 months Global MBA programme offered by S P Jain has been ranked 4th in the world by the Times Higher Education-Wall Street Journal Business School Rankings for one year MBA degrees.
Executive MBA (part-time) programme in S P Jain is designed for working professionals to develop strategic business and management perspective. This programme will take 18 months to complete.
Below are the MBA programme tuition fees for S P Jain School of Global Management.
Executive MBA $25,680
All fees listed are FULL programme tuition fee in Singapore Dollars inclusive of local tax (GST) and apply for the 2019 - 2020 academic year
James Cook University Australia established its Singapore campus in 2003 and offering a wild field of excellent degree programmes suits Singapore campus. Detailed information on JCU Singapore can be found here.
The Academic Ranking of World Universities has ranked James Cook University top 2% of universities worldwide for the past seven years (2013 – 2019). JCU Singapore is a member of The Association to Advance Collegiate Schools of Business (AACSB) and currently undergoing the process of accreditation.
In 2015, JCU Singapore is the 1st institution received the Singapore Education Trust Certification Star Rating (Edu Trust Star), and obtain it again in 2019. This is the highest level of award in the non-public university sector to the institutions who managing their institutions and providing an outstanding quality of education and welfare for their students.
JCU Singapore's Master of Business Administration (MBA) programme consists of 12 subjects chosen from a board spectrum covering all fields of contemporary management. Students can choose to complete it either full-time in 1year time or part-time in 16 months.
Besides, JCU Singapore is also offering MBA joint degree with Master of Information Technology, International Tourism and Hospitality Management and Master of Professional Accounting. Students can integrate their business theory while learning practical knowledge from another different discipline. These joint MBA programmes will enable students to gain two degrees when they graduate.
Here is the list of MBA programmes tuition fees for JCU Singapore.
Domestic Student Tuition Fees
international Student Tuition Fees
Master in Business Administration $37,236 $38,520
Master of Information Technology - Master of Business Administration $49,648 $51,360
Master of International Tourism and Hospitality Management - Master of Business Administration $49,648 $51,360
Curtin Singapore
Curtin University Singapore campus was officially open in 2008 and is managing by the Curtin Education Centre of the Navitas Education Group. It offers degrees in commerce, communication, management and health practices related degrees. Detailed information on Curtin Singapore can be found here.
The Academic Ranking of World Universities 2018 ranked Curtin University in the top 1% of university worldwide. Curtin University is accredited by The Association to Advance Collegiate Schools of Business (AACSB).
The MBA (Global) programme offered by Curtin Singapore is on full-time bases that need 12 to 16 months to complete. Throughout the course, students will have a deep understanding of business practices, at the same times integrate their leadership and management skills. Below is the MBA programme tuition fee.
Master in Business Administration (Global) $46,080
All fees listed are the FULL programme tuition fee in Singapore Dollars inclusive of local tax (GST) and apply for the 2019/2020 academic year
Copyright © 2023 GetUniOffer Pte. Ltd.
Cost of Living in Singapore for Students
A Diverse Student City
Work Opportunities for Students in Singapore
Singapore's Top-ranking Universities
Singapore's Amazing Choice of University Courses
Singapore: Study in a Global City
Top 9 Private Universities in Singapore
Singapore's Six Public Universities

redpajama_set_name: RedPajamaCommonCrawl


I had a very strange music request from a reader just recently who has asked to be kept anonymous but has passionately asked me to feature his favorite artist of all time. - Wolgang Amadeus Mozart.
Yup, I being a natural giver definitely said yes to the topic and am grateful that someone has yet again voiced out what they wanted to hear and see. Now I know a lot of you might think why Mozart? my answer is why not? :) Wolfgang Amadeus Mozart or Mozart as he more popularly known as is a genius and is definitely someone whose work and magic with music very few can rival even up to this day and age. Now I had to admit that as I did my research into his history and the kind of music he made I was amazingly left frozen in amazement after I started to hear for myself what he has done and right now am listening to everything about Mozart I can get. I literally had to stop what ever research I was conducting at that moment because I was so drawn into the mystery of the music - It was stimulating to the mind and the senses. If you never felt music before you are sure to feel it with his compositions.
Mozart always learned voraciously from others, and developed a brilliance and maturity of style that encompassed the light and graceful along with the dark and passionate—the whole informed by a vision of humanity "redeemed through art, forgiven, and reconciled with nature and the absolute". His influence on all subsequent Western art music is profound. Beethoven wrote his own early compositions in the shadow of Mozart, of whom Joseph Haydn wrote that "posterity will not see such a talent again in 100 years"
Wow, your pick caught me by surprise!
The man was simply a genious, it makes all sense to feature Mozart. His art is timeless and, if we think well, classic music is the root of many musics.
Although Bellomo Salomone has a great voice I prefer the instrumental pieces like in the second and third video (this concert is specially beautiful, like spring).
I really enjoyed the film of his life. Did you have the chance to see it?
Very good post Btw I am running a blog contest in My blog I would very glad if your Part of it.
It's so funny how my 4 year-old son was asking me why I'm listening to his song! I guess he watches "Little Einsteins" too much. They feature classical music ranging from Mozart to Edvard Grieg.
He is absolutely a genius!
After I watch the movie Amadeus, I feel so astonished by his fate!
Hi Metz! I grew up in the 80's and I'm so familiar with this song. Happy MM.


The use of social media marketing is getting very popular in today's business world because of the many benefits that businesses can gain from it. Social media is taking the marketing world by storm and now it surpasses outbound marketing as the best marketing strategy for businesses to use. Even B2B marketers are already using social media marketing of some form. Any kind of business can use social media marketing as a tool for growing their brand.
You get isolated from the business world and the consumer world if your business does not use social media marketing. You might be someone who has not yet learned how social media marketing can benefit your business immensely. There are many reasons why your business needs social media marketing.
Social media posts can drive great traffic to your site. Your social media posts ca make a huge difference on your new site pages and landing pages. You might now have a handful of daily visitors, but with social media posts, you can increase it to hundreds a day. It is important to take advantage of that.
Your site's SEO can be boosted by social media. Search engine crawlers know which pages are consistently earning traffic and which are just floating and ignored. If traffic is driven to your optimized pages with social media posts, then it can climb faster in search engine results pages.
Social media can help companies build real relationships with their customers. If you are on social media, then you have the privilege of interacting with your customers and knowing their comments and feedback on your products. You will have an idea of the products that they like and why they like it. With this you can find tune your strategy. You can use media to connect to influencers who cover your industry.
If your social media posts look like a social thing instead of a marketing scheme , the more people will most likely read your posts. Many will like to read you posts. This translates to web traffic when you link it to your site. When your posts are shared with the friends of your followers then your post is marketing itself.
Social media ads are highly customizable and this is why it is very important. These ads allow you to target users by things like location, education level, industry, and even purchase history and the pages that they like. Retargeting users that visit your site is possible and you can convert them into solid leads and sales.
There is immediate feedback in social media.


Miami is famous in the world for its vibrant beach culture, beautiful people, and general vibe of sunshine in the Sunshine State. Every year, thousands of tourists come seeking fun on its famous beaches and to enjoy the dazzling nightlife. If you are thinking about booking Miami vacation packages for your group or as the next corporate retreat, you will want to make informed decisions regarding who to get the job done for you. EKITravels is the #1 travel agency in Miami, and also hold offices all over the United States on both the East and West coasts and service clients from all around the world. With our combined experience of over forty years, we are the premier choice when it comes to negotiating the best vacation packages for your needs. Read on to learn more about the best Miami beaches you will want to visit. Contact EKITravels today for your travel needs!
One of the best part about Miami beaches is that they are all open to the public. Whether you're looking for a quiet spot to rest, or want to people-watch at a busy tourist beach, there are all different kinds of beaches to browse and our Miami vacation packages cater to all different tastes. Certain beaches in Miami are lined with palm trees to create a shady oasis, while other beaches are perfect for a golden summer tan. No matter what you're looking for, there is stunning natural beauty everywhere, as well as many designated public parks along the shore and various water activities, restaurants, bars, and other attractions. Lummus Park Beach is located on the Eastern side of Ocean Drive, and is a perfect destination for people-watching, volleyball, and sunbathing. Redesigned in the mid-1980's, this park beach features grassy areas with pull up bars for a work out and palm trees for shade seekers. This beautiful promenade is also a popular location for photo shoots and filming, and shows such as Miami Vice were shot from this location. Another great beach is the South Pointe Park Pier. The South Pointe Park Pier is part of a 17-acre park that hosts a playground and fishing pier, complete with washing stations and several picnic areas. If you want to get away from all the hustle and bustle of downtown Miami, this beach makes a lovely retreat.
For a small beach town vibe, Surfside is a small oceanfront community hosting about 6,000 residents. There are plenty of entertainment options for visitors, and attracts both locals and tourists, offering plenty of entertainment options for both. Haulover Beach is another beach to check out as one of the most scenic beaches that Miami offers. Featuring landscaped dunes, picnic areas, and dozens of lifeguard towers, this beach sits between Sunny Isles and Bal Harbour, and attracts travelers from all around the world with its stunning landscape.
We at EKITravels provide the best Miami vacation packages so that you can visit the best Miami beaches and have everything organized for your next outing. Contact EKITravels to put together your next corporate retreat or group vacation!


text:
Q: Pass values as query string in javascript In my application I want to pass values at the time of initialize JavaScript as follows. do not use any tag or other methods.
<script src="../js/carTravalling.js?uid=Bjkls&appid=1234"  type="text/javascript"></script>

How can I get uid and appid in  my JavaScript carTravalling.js  file. ?

A: You could do something like this
create a file such as passArgLibrary.js

var passArgLibrary = passArgLibrary || (function(){
    var argumentObject = {};

    return {
        init : function(argumentValues) {
            argumentObject = argumentValues;
            // Add other init if needed
        },
        test : function() {
            alert('Test! -' + argumentObject[0]);
        }
    };
}());

<!-- Would usually use this first line to get the js file but i'm using dropbox for example
<script type="text/javascript" src="passArgLibrary.js"></script> -->
<script src="https://dl.dropboxusercontent.com/u/71030101/passArgLibrary.js"></script>
<script type="text/javascript">
   passArgLibrary.init(["testValue", 1, "testValueID"]);
   passArgLibrary.test();
</script>

A: I got the answer with help of http://feather.elektrum.org/book/src.htmlenter link description here,
function parseQuery ( query ) {
   var Params = new Object ();
   if ( ! query ) return Params; // return empty object
   var Pairs = query.split(/[;&]/);
   for ( var i = 0; i < Pairs.length; i++ ) {
      var KeyVal = Pairs[i].split('=');
      if ( ! KeyVal || KeyVal.length != 2 ) continue;
      var key = unescape( KeyVal[0] );
      var val = unescape( KeyVal[1] );
      val = val.replace(/\+/g, ' ');
      Params[key] = val;
   }
   return Params;
}

        var scripts = document.getElementsByTagName('script');
        var myScript = scripts[0];
        var queryString = myScript.src.replace(/^[^\?]+\??/,'');
        var params = parseQuery( queryString );

        console.log(params.uid+" "+params.appid);

redpajama_set_name: RedPajamaStackExchange


A Photoshopped Response Sticks Up For China's Plunging Necklines After a TV drama was yanked from Chinese airwaves so censors could cover up some low-cut outfits, snarky social media users have responded by Photoshopping coverings for cleavage everywhere.
After a TV drama was yanked from Chinese airwaves so censors could cover up some low-cut outfits, snarky social media users have responded by Photoshopping coverings for cleavage everywhere.
Good morning. I'm Renee Montagne. China's social media is obsessed with a new Photoshopping opportunity - covering cleavage from Venus de Milo to Scarlett Johansson. This after the country's popular drama "The Empress Of China" was yanked from the airwaves briefly so sensors could cover up the low-cut bodices of the characters. The deeply cut gowns were apparently authentic to fashions of the seventh century, leaving outraged viewers, though, in the 21st. It's MORNING EDITION.


text:
Suiting Up for Team Racist
Catholics Should Call It Out
By Mollie Wilson O'Reilly
The Trump family takes part in the "Make America Great Again! Welcome Celebration" in January 2017 (CNS photo/Jonathan Ernst, Reuters)
I thought I must be seeing things. That's how pessimistic I've become. When I read the opinion piece by Bishop John Stowe of Lexington, Kentucky, that ran in the Lexington Herald-Leader, I thought, "Can this be real? A Catholic bishop directly, with no irrelevant to-be-sure-ing, calling out the racist politics of President Donald Trump as an affront to the church's prolife witness?"
Stowe did exactly that. "It astonishes me," he wrote, "that any students participating in a prolife activity on behalf of their school and their Catholic faith could be wearing apparel sporting the slogans of a president who denigrates the lives of immigrants, refugees, and people from countries that he describes with indecent words and haphazardly endangers with life-threatening policies."
He's talking about so-called MAGA hats and gear—as in "Make America Great Again," Trump's inflammatory campaign slogan. What prompted Stowe to write was a highly publicized incident involving students from Covington Catholic, a high school in a neighboring diocese. Video footage circulated of a group of boys from that school, many clad in MAGA gear, facing off in (at best) a disrespectful manner with a Native American demonstrator, while in Washington, D.C., for the March for Life. The initial video clip prompted an outcry against the students, after which came more footage from that day, accompanied by an insistence in conservative circles that the boys had been unfairly tarred as racists and were the true victims.
Is Donald J. Trump personally prejudiced against nonwhites? It doesn't matter. What matters is that he traffics openly in racism as a means of riling up his supporters.
Stowe declines to get into that debate, and calls attention to the U.S. bishops' recent pastoral letter on combating racism, "Open Wide Our Hearts." The word "Trump" does not appear in that letter, but when the bishops list the factors that prompted them to address that subject, they write, "Extreme nationalist ideologies are feeding the American public discourse with xenophobic rhetoric that instigates fear against foreigners, immigrants, and refugees." It is startling to realize that this sentence is a fair and even restrained description of the current president of the United States. "Make America Great Again" is the rhetoric they're talking about.
Is Donald J. Trump personally prejudiced against nonwhites? It doesn't matter. What matters is that he traffics openly in racism as a means of riling up his supporters. Trump's presidential campaign was racist. His administration is racist. His ongoing campaign to win the next election is racist. Standing with Trump is racist. It is racist regardless of what motivates you to do it—your desire to see the Supreme Court filled with conservative judges; your enthusiasm for low taxes for the very wealthy; your earnest belief that Vladimir Putin is an ally worth courting.
Like many conservatives, National Review editor Rich Lowry holds this conclusion in contempt. After the Covington drama, he wrote a column in which he complained, "For much of progressive America, if you are wearing the hat, you are suiting up for Team Racist." Well, yes. Lowry fleshes out his column by quoting a number of people who had said as much on Twitter, including me: "As Commonweal magazine columnist Mollie O'Reilly wrote of the Covington controversy, 'You don't let your kid wear a MAGA hat and then act offended when they get taken for a racist.'"
I wrote that, and I stand by it, obviously. But he quoted only half of that tweet. Here's the first sentence: "Every person now claiming those MAGA teens have been unfairly judged as racists is trying, consciously or not, to place Trump gear and MAGA hats in a neutral category."
Which, of course, is exactly what Lowry was doing in that column: insisting that there's some wiggle room, some way to say "I'm Team Trump" but not "I'm Team Racist." There isn't. If you welcome some of what President Trump has done but abhor his racism, it's up to you to figure out how to express those convictions. Wearing MAGA gear says you're not bothered.
That fact may not be obvious to teenagers, especially to privileged white teenagers. Which is why it's important to remember that the question is not whether those boys are thoroughly racist individuals, but whether they are able to recognize racist policies and politics and their own responsibility to resist them. Can they see that the call to defend the vulnerable is not only about unborn children? The adults in their life owe it to them to clue them in, especially if those adults are supposed to be educating them and forming their characters according to the teachings of the church. Bishop Stowe spoke up, and spoke the simple truth. Let's hope it's the start of a trend.
Published in the February 22, 2019 issue:
Mollie Wilson O'Reilly is editor-at-large and columnist at Commonweal.
On sex abuse, asking the right questions
Yanqui, Stay Home
Speaking of New York
Donald TrumpBishopsRaceSocial Justice
Benjamin's Warning
By Alexander Stern
Know-Nothing Know-It-Alls
By Charles McNamara
The Downfall of Liz Cheney
By Griffin Oleynick


How long is going to take?
Every claim is unique and the duration of claim depends on a number of factors. We can however assure you that we will act fast to get your claim resolved as soon as we can.
Please click here for an idea of how long your claim might take.
Please click here for an idea of how much you will get.
We can take your claim on, on a no win no fee basis. Contact us to discuss the costs of making your claim.
What is a CFA (Conditional Fee Agreement) or 'No win no fee' agreement?
This is the agreement that you sign as our client and we sign as your solicitor. The agreement means that if you lose your case, we won't ask you to pay anything. If you do win, the other side will pay most of our costs. Call us to discuss the costs of making your claim if you win.
Apart from my injuries, what else am I entitled to claim for?
Click here to see what your claim is worth.
Is there a time limit for me to make a claim?
In almost all cases, for either an accident or diagnosis of a work-related illness you have three years to make a claim. If the accident happened when you were under the age of 18, you have three years from your 18th birthday to make your claim. On rare occasions you might still be able to make a claim outside of the three year time limit. We recommend that you make your claim as soon as possible in order to avoid any deadlines.


The abolition of the Swedish state alcohol retail monopoly, Systembolaget, would lead to an increase in binge drinking, alcohol-related deaths and sick leave, according to a new study from Stockholm University, reported SVT news programme Rapport.
According to the researchers the sale of alcohol products in grocery stores and other retail outlets would lead to 37 percent hike in alcohol sales, 2,000 more alcohol-related deaths, 20,000 more violent crimes, 6,600 more drink driving offences and 11 million more sick days per annum.
"Systembolaget's existence is questioned from time to time, so it is good for both politicians and the public to know what the consequences of such a change would be," said professor Thor Norström at the institute for social research to Rapport.
The study has also looked at an alternative scenario with Systembolaget's 400 stores replaced with 800 privately-owned, but controlled, stores licensed to sell alcohol and concluded that alcohol consumption would also increase, but only by 17 percent.
The study is, at least in part, based on experiences gained from when full-strength beer was sold in retail stores in the 1960s and SVT asked Norström if Swedes had not perhaps become more mature and informed about the adverse effects of alcohol since then.
"You never know. Drinking patterns and culture are sluggish processes," he said.


text:
Six hourly weather update for Tuesday 30th August, 2022 from Noon to 6pm
UNMA
Create Date 30 August 2022
Six hourly weather update for Tuesday 3 May, 2022 from Noon to 6pm. Six hourly weather update for Tuesday 3 May, 2022 from Noon to 6pm. Six hourly weather update for Tuesday 08 February, 2022 from 6am to Noon. Six hourly weather update for Tuesday 1st March, 2022 from Noon to 6pm. Six hourly weather update for Tuesday 24th May, 2022 from 6am to Noon Six hourly weather update for Tuesday 02nd July, 2022 from Noon to 6pm.


Debutants Jack Higgs and Ryan Ingrey sank 10-men Uxbridge in a thriller at the ProEdge Stadium on Non-League Day.
The visitors were looking particularly dangerous from set pieces and on 17 minutes a Wayne Carter corner was won in the air by Mark Dennison, but his goal-bound header was headed away by Lambert.
Another set piece saw the Reds take the lead on 21 minutes as a foul on Chris Moore by Lambert 25 yards from goal resulted in a well-placed free-kick by Carter which dipped over the wall and beat Lawrence on his near post.
The equaliser came on 33 minutes when Seymour-Shove was brought down by Dennison 25 yards out and Higgs opened his account by curling his free kick low around the wall to beat Pickford at his near post.
Only four minutes later and the Ives were in front as Seymour-Shove won the ball in midfield and fed in Stead. His powerful shot was blocked by Dennison but Higgs was following up and gleefully fired home the loose ball from just inside the box.
The game was tur ned on its head shortly after half-time. On 52 minutes Lewis Cole fed in the dangerous Woods and although his initial effort was well blocked by Lambert, the ball ran loose to Logon who made no mistake from 12 yards to bring the Reds level.
They followed this up by going in front only four minutes later as Moore played the ball wide to Logon, whose driven cross was turned in at close-range by the unmarked Cole.
The Ives were struggling to get back into the game, but manager Ricky Marheineke once again pulled off a master stroke, bringing on Jimmy Dean and Stuart on 67 minutes. Five minutes after coming on Eason was put through by Ingrey after good work by Seymour-Shove.
As he entered the box Eason was brought down by Otira, the referee pointed to the spot and produced a red card for Otira.
Higgs fired home the spot-kick for his hat-trick and level the game.
Two minutes later and the comeback was complete as a foul on Eason wide on the left led to a free-kick. Seymour-Shove swung the ball into the box, three Ives players went for it at the back post with Dean winning the ball and heading it back into the middle, where Ingrey firmly headed home from six yards.


text:
Q: WP_Query and WooCommerce orders with pending status issue I can't get order objects with status of wc-pending / Pending Payment. It simply returns ALL the order objects:
$my_course_query = new WP_Query( array(
    'post_type'     => 'shop_order',
    'post_status'   => 'wc-pending',
    'posts_per_page'      => -1
) );

A:
Your code Is just working perfectly as expected, in frontend, I have test it and it output only orders with **pending status. So I can't tell what is your issue as your question is not detailed.
I have found this note on WordPress WP_Query reference that could be useful:
  Note: Ticket #18408 For querying posts in the admin, consider using get_posts() as wp_reset_postdata() might not behave as expected.

In general, I don't use WP_Query() for customer orders but wc_get_orders() (or get_posts() too) this way:
$customer_orders = wc_get_orders( array(
    'limit'    => -1,
    'status'   => 'pending'
) );

// Iterating through each Order with pending status
foreach ( $customer_orders as $order ) {

    // Going through each current customer order items
    foreach($order->get_items() as $item_id => $item_values){
        $product_id = $item_values['product_id']; // product ID

        // Order Item meta data
        $item_meta_data = wc_get_order_item_meta( $item_id );

        // Some output
        echo '
Line total for '.wc_get_order_item_meta( $item_id, '_line_total', true ).'

';
    }
}

This works also just to get the orders objects.
Related documentation: wc_get_orders and WC_Order_Query

A:
I fixed this weird issue by simply using custom query.

Somehow adding 'post_status' => 'wc-pending' doesn't actually change the query, but if I use 'post_status' => 'pending', the query changes.
So what I did was using that custom query and modify pending to wc-pending.

A: I do have the same issue (returning ALL Orders) while debugging.
Wrapping the debug-code into an action helped outputting the expected data:
add_action( 'init', 'debug_init' );

function debug_init() {

    $custom_query_args = array(
        "fields" => "ids",
        "post_type" => "shop_order",
        "post_status" => array('wc-processing'),
        "posts_per_page" => "-1",
        "offset" => "0",
        "date_query" => [
                "before" => "2020-09-10 23:59",
                "after" => "1970-01-01 00:00",
                "inclusive" => "1"
        ],
        "order" => "DESC"
    );

    $debugQuery = new WP_Query( $custom_query_args );
    $order_ids   = $debugQuery->posts;

    print_r($order_ids);

    die();
}