# Non-random network connectivity comes in pairs

Felix Z. Hoffmann  Frankfurt Institute for Advanced Studies (FIAS), Johann
Wolfgang Goethe University, Frankfurt am Main, Germany  International Max
Planck Research School for Neural Circuits, Max Planck Institute for Brain
Research, Frankfurt am Main, Germany  Email<EMAIL_ADDRESS>Jochen Triesch  Frankfurt Institute for Advanced Studies (FIAS), Johann
Wolfgang Goethe University, Frankfurt am Main, Germany

Overrepresentation of bidirectional connections in local cortical networks has
been repeatedly reported and is in the focus of the ongoing discussion of non-
random connectivity. Here we show in a brief mathematical analysis that in a
network in which connection probabilities are symmetric in pairs,
$P_{ij}=P_{ji}$, the occurrence of bidirectional connections and non-random
structures are inherently linked; an overabundance of reciprocally connected
pairs emerges necessarily when the network structure deviates from a random
network in any form.

## Introduction

Increasing evidence shows that cortical microcircuitry is highly structured
[1, 2]. Not every connection is equally likely to be established, rather some
pairs of neurons are more likely connected than others. In this context, the
relative occurrence of bidirectionally connected pairs has been of particular
interest. Using data obtained from paired whole-cell recordings in cortical
slices, the amount of bidirectionally connected pairs was compared to the
number of reciprocal pairs as one would expect in a random network with the
same overall connection probability. Connectivity of layer 5 pyramidal neurons
in the rat visual cortex [1] and somatosensory cortex [3, 2] was shown to have
a significantly stronger reciprocity than expected.

The prevalence of bidirectional connectivity has since been established as an
important indicator for the non-randomness of a network [4, 5]. However, the
exact relationship between non-randomness and relative reciprocity has not
been explained. Here, we model cortical circuitry as random networks in which
each possible connection has a separate probability to exist. Using this model
we’re able to show that any non-random connectivity, expressed as higher
connection probabilities in some edges and lower probabilities in others,
necessarily induces a relative overrepresentation of bidirectional connections
as long as connection probabilities remain symmetric within pairs.
Quantitatively, we analyze reciprocity in networks with a discrete and a
continuous distribution in connection probabilities to demonstrate that a
relative occurrence of bidirectional connections as reported from experimental
studies can be easily obtained in these models.

## Results

The emergence of non-random connectivity patterns can be modeled by assigning
each possible connection in a random graph a separate probability to exist. In
such a model some connections are more likely to be realized than others,
allowing for the encoding of patterns within the specific probabilities of
each connection. In the limiting case each connection either exists or is
absent with certainty, representing a blueprint for the network architecture.

To analyze the effect of non-random structures within a network, specifically
on the statistics of bidirectionally connected pairs found in the network, we
consider a random graph model of $N$ neurons in which the probability of node
$i$ to connect to node $j$ is modeled by a random variable $P_{ij}$. For this
we assume $P_{ij}$ for $i,j=1,\dots,N$ with $i\neq j$ to be identically
distributed random variables in $[0,1]$, yielding a probability of connection
for each ordered pair of nodes in the graph. Outside of pairs the random
variables $P_{ij}$ are assumed to be independent, that is non-equal $P_{ij}$
and $P_{kl}$ are independent as long as $i\neq l$ or $j\neq k$. Finally, we
explicitly exclude self-connections in this model and assume at all times that
$i\neq j$.

Given the distributions of connection probabilities, what is then the
probability in this model for a randomly selected node to have a projection to
another randomly selected node? As the random variables $P_{ij}$ are
identically distributed, we compute this overall connection probability $\mu$
easily as the expected value of $P_{ij}$,

$\displaystyle\mu=\operatorname{\mathbf{E}}(P_{ij}).$ (1)

For example, if the $P_{ij}$ have a probability density function $f$ with
essential support in $[0,1]$, we can compute the connection fraction as

$\displaystyle\mu=\int_{0}^{1}xf(x)\,dx.$ (2)

In this work we are interested in the probability $P_{\mathbf{bidir}}$ of a
bidirectional connection to exist in a random pair of neurons. We determine
$P_{\mathbf{bidir}}$ as the expected value of the product of $P_{ij}$ and
$P_{ji}$,

$\displaystyle P_{\mathbf{bidir}}=\operatorname{\mathbf{E}}(P_{ij}P_{ji}).$
(3)

The relative occurrence $\varrho$ of such reciprocally connected pairs
compares $P_{\mathbf{bidir}}$ with the occurrence of bidirectional pairs in an
Erdős-Rényi graph, in which each unidirectional connection is equally likely
to occur with probability $\mu$ [6, 7]. The probability of a particular
bidirectional connection to exist in such a random graph is simply $\mu^{2}$
and we obtain the relative occurrence as the quotient

$\displaystyle\varrho=\frac{P_{\mathbf{bidir}}}{\mu^{2}}=\frac{\operatorname{\mathbf{E}}(P_{ij}P_{ji})}{{\operatorname{\mathbf{E}}\left(P_{ij}\right)}^{2}}.$
(4)

Experimental studies in local cortical circuits of rodents have repeatedly
reported a relative occurrence of bidirectional connections $\varrho>1$ [3, 1,
2]. To understand in which cases such an overrepresentation occurs, we
consider two cases. In the first case, assume that connection probabilities
are independently determined in pairs as well, meaning that the random
variables $P_{ij}$ and $P_{ji}$ are independent. Then, as $P_{ij}$ and
$P_{ji}$ are identically distributed,

$\displaystyle\operatorname{\mathbf{E}}(P_{ij}P_{ji})=\operatorname{\mathbf{E}}(P_{ij})\,\operatorname{\mathbf{E}}(P_{ji})=\operatorname{\mathbf{E}}(P_{ij})^{2},$
(5)

and we expect to observe no overrepresentation of reciprocal connections,
$\varrho=1$. In the second case, assume that connection probabilities are
symmetric in pairs, $P_{ij}=P_{ji}$. In this case,

$\displaystyle P_{\mathbf{bidir}}=\operatorname{\mathbf{E}}(P_{ij}^{2}),$ (6)

and the expected relative occurrence of reciprocal connections becomes

$\displaystyle\varrho=\frac{\operatorname{\mathbf{E}}(P_{ij}^{2})}{{\operatorname{\mathbf{E}}\left(P_{ij}\right)}^{2}}.$
(7)

We note that now any distribution of $P_{ij}$ with a nonvanishing variance
will lead to a relative occurrence that deviates from the Erdős-Rényi graph,
as

$\displaystyle\operatorname{\mathbf{Var}}(P_{ij})=\operatorname{\mathbf{E}}(P_{ij}^{2})-\operatorname{\mathbf{E}}\left(P_{ij}\right)^{2}.$
(8)

Moreover, since $x\mapsto x^{2}$ is a strictly convex function, Jensen’s
inequality [8, 9] yields

$\displaystyle\operatorname{\mathbf{E}}(P_{ij}^{2})\geq\operatorname{\mathbf{E}}(P_{ij})^{2},$
(9)

and we find that $\varrho\geq 1$ in networks with symmetric connection
probabilities. Jensen’s inequality further states that equality in (9), and
thus $\varrho=1$, holds if and only if $P_{ij}$ follows a degenerate
distribution, that is if all $P_{ij}$ take the identical value $\mu$. In the
other case, where the $P_{ij}$ take on more than one value with non-zero
probability, we speak of a non-degenerate distribution.

As a central result of this study we thus find that any non-degenerate
distribution of symmetric connection probabilities ($P_{ij}=P_{ji}$)
necessarily induces an overrepresentation of bidirectional connections in the
network, $\varrho>1$. In other words, in a network in which both directions of
connection are equally likely within any given pair, but where some pairs are
more likely to be connected than others, the count of expected reciprocally
connected pairs is strictly underestimated by the statistics of an Erdős-Rényi
graph with same the overall connection probability
$\operatorname{\mathbf{E}}(P_{ij})=\mu$.

### Upper bound for $\varrho$

The overrepresentation of bidirectional connections $\varrho$ in a network is
maximal when every connected pair is already a reciprocally connected pair. In
terms of the model defined above, this is the case when

$\displaystyle\operatorname{\mathbf{E}}(P_{ij}P_{ji})=\operatorname{\mathbf{E}}(P_{ij}).$
(10)

The relative occurrence of reciprocal connections from (4) then becomes

$\displaystyle\varrho=\frac{1}{\operatorname{\mathbf{E}}(P_{ij})}=\frac{1}{\mu}$
(11)

Thus, for local cortical circuits of L5 pyramidal neurons with a typical
connection probability of $\mu=0.1$ [10, 1], the network model yields a
maximal overrepresentation of $\varrho=10$. While this theoretical maximum is
unlikely to exist in actual cortical networks, the precise degree of
overrepresentation will depend on the specific distribution of connection
probabilities in the network. In the following, we study two generic examples.

### Two-point distribution

The simplest non-degenerate distribution of connection probabilities is a
distribution that takes two values $x$, $y$ with probability $p$ and $1-p$,
respectively, as illustrated in Figure 1A. This distribution may be seen as a
crude approximation to the connection probabilities recently observed in
visual cortex as a function of the neurons’ absolute difference in orientation
preference, where a “high” connection probability was reported for a
difference between 0°–45° and a “low” probability was seen for cells with a
difference of 45°–90° in orientation tuning [11]. Formally, let $x,y\in[0,1]$
with $x>y$ and $0<p<1$. A random variable $X$ follows the two-point
distribution $\mathcal{T}(p,x,y)$ if $P(X=x)=p$ and $P(X=y)=1-p$.

In our network model let then the $P_{ij}$ be $\mathcal{T}(p,x,y)$
distributed. The overall connection probability $\mu$ is

$\displaystyle\mu=\operatorname{\mathbf{E}}(P_{ij})=px+(1-p)y.$ (12)

Assume again that $P_{ij}=P_{ji}$. The relative occurrence of bidirectional
connections is given by

$\displaystyle\varrho=\frac{\operatorname{\mathbf{E}}(P_{ij}^{2})}{\mu^{2}}=\frac{px^{2}+(1-p)y^{2}}{\mu^{2}}.$
(13)

Solving (12) for $p$ as

$\displaystyle p=\frac{\mu-y}{x-y}$ (14)

and inserting into equation (13) yields an expression for the relative
overrepresentation depending on $x$, $y$ and $\mu$ (see Supplementary
Information SI1),

$\displaystyle\varrho=\frac{x+y}{\mu}-\frac{xy}{\mu^{2}}.$ (15)

Figure 1: Relative overrepresentation $\varrho$ of bidirectional connections
in networks with a fraction of pairs connected with a high probability $x$ and
the rest of the pairs connected with a low probability $y$. A Diagram
illustrating the targets connecting with a high chance $x$ (thick arrows) and
targets connecting with a low probability $y$ (thin arrows) for a single
source node (hatched). B Different pairings of $x$ and $y$ can induce a high
relative overrepresentation $\varrho$ in a network with two-point distributed
connection probabilities, $P_{ij}\sim\mathcal{T}(\frac{\mu-y}{x-y},x,y)$, and
a fixed overall connection probability $\mu=0.1$. The dashed line marks an
overrepresentation of bidirectional connections of $\varrho=4$ as observed for
layer 5 pyramidal neurons in the rat visual cortex [1].

Here we fix $\mu=0.1$ in accordance with the overall connection probability
found in local circuits of pyramidal cells in the rat visual cortex [1] and
obtain the relative occurrence dependent on the two connection probability
values $x$ and $y$. Given $x\geq\mu$ it follows that $y\leq\mu$ (see
Supplementary Information SI2) and the possible values for $x$ and $y$ are
$0.1\leq x\leq 1$ and $0\leq y\leq 0.1$. Figure 1B shows contours of $\varrho$
for the $(x,y)$ pairings illustrating how different values for the relative
overrepresentation of reciprocal connections can be induced by two-point
distributed connection probabilities. We find that in such networks higher
values of $\varrho$ are easily obtained with reasonable network
configurations. For example, a relative overrepresentation of $\varrho=4$
could be achieved by a two-point distribution of connection probabilities
where one group of neuron pairs is highly connected with probability $x=0.7$,
while the other group of neuron pairs is sparsely connected with probability
$y=0.05$. Collectively, the highly connected pairs then make up less than
$8\%$ of all neuron pairs, showing that it is sufficient to have a small
subgroup of highly connected neuron pairs to induce a high overrepresentation
of bidirectionally connected pairs in the network. For more densely connected
networks, $\mu>0.1$, the effect that two distinct connection probabilities
have on the overrepresentation of reciprocal connections is reduced (c.f.
Figure S1), as one would intuitively expect from the dependence of the maximal
overrepresentation on $\mu$ in (11).

### Gamma distribution

Next, we analyze the relative overrepresentation of bidirectional connections
in a network with continuously distributed connection probabilities. The gamma
distribution $\Gamma(\alpha,\beta)$ with probability density function

$\displaystyle
f_{\alpha,\beta}(x)=\begin{cases}\frac{1}{\beta^{\alpha}\Gamma(\alpha)}\,x^{\alpha-1}\,e^{-x/\beta}&x\geq
0\\\ 0&\text{otherwise},\end{cases}$ (16)

allows the variation of the variance
$\operatorname{\mathbf{Var}}(X)=\alpha\beta^{2}$ of a gamma distributed random
variable $X\sim\Gamma(\alpha,\beta)$, while keeping its mean
$\operatorname{\mathbf{E}}(X)=\alpha\beta$ constant [12]. The exponential
distribution emerges as a special case of the gamma distribution ($\alpha=1$).

To ensure that the randomly drawn connection probabilities lie within the
interval $[0,1]$, we here consider a modification to the traditional gamma
distribution in the form of a truncated version. Let $\alpha,\beta>0$. A
random variable $X$ follows the truncated gamma distribution
$\Gamma^{T}(\alpha,\beta)$ if it has the probability density function

$\displaystyle
f_{\alpha,\beta}^{T}(x)=\begin{cases}K_{\alpha,\beta}\,\frac{1}{\beta^{\alpha}\Gamma(\alpha)}\,x^{\alpha-1}\,e^{-x/\beta}&0\leq
x\leq 1\\\ 0&\text{otherwise}.\end{cases}$ (17)

The factor $K_{\alpha,\beta}$ is the inverse of the cumulative probability
that $x\leq 1$ of the untruncated gamma distribution,

$\displaystyle
K_{\alpha,\beta}=\left(\int_{0}^{1}f_{\alpha,\beta}(x)\,dx\right)^{-1},$ (18)

and is needed to ensure that

$\displaystyle\int f_{\alpha,\beta}^{T}(x)\,dx=1.$ (19)

Consider then the above network model in which the connection probabilities
$P_{ij}^{T}$ are $\Gamma^{T}(\alpha,\beta)$ distributed and
$P_{ij}^{T}=P_{ji}^{T}$. We compute the relative overrepresentation $\varrho$
numerically from

$\displaystyle\mu=\operatorname{\mathbf{E}}\left(P_{ij}^{T}\right)$
$\displaystyle=\int_{0}^{1}xf_{\alpha,\beta}^{T}(x)\,dx,$ (20)
$\displaystyle\operatorname{\mathbf{E}}\left({P_{ij}^{T}}^{2}\right)$
$\displaystyle=\int_{0}^{1}x^{2}f_{\alpha,\beta}^{T}(x)\,dx.$ (21)

Pairings of the shape parameter $\alpha$ and the scale parameter $\beta$ were
chosen such that the overall connection probability reflects connectivity
statistics in local cortical networks, $\mu=0.1$ [1, 10]. Probability density
functions and resulting relative overrepresentation of reciprocal connections
$\varrho$ for four representative $\alpha,\beta$ pairs are shown in Figure 2A.
Here, $\beta$ was determined to yield $\mu=0.1$ for the given $\alpha$,
following the relationship shown in Figure 2B (solid curve).

Figure 2: Relative occurrence of bidirectional connections $\varrho$ in
networks with gamma distributed connection probabilities. A Probability
density functions of the truncated gamma distribution
$\Gamma^{T}(\alpha,\beta)$ for different shape parameters $\alpha$ and the
induced relative overrepresentation $\varrho$ in a network with such
distributed connection probabilities $P_{ij}$. For a given $\alpha$, the scale
parameter $\beta$ was chosen such that $\mu=0.1$. Plot to the right continues
the density functions on a different scale. B Contour of $\alpha$, $\beta$
pairings that yield an overall connection probability of $\mu=0.1$. The dashed
line shows the approximation $\beta=\frac{\mu}{\alpha}$, where $\mu=0.1$. C
Relative occurrence $\varrho$ as a function of $\alpha$ for fixed $\mu=0.1$.
For $\alpha\geq 1$ this relationship is well approximated by $\varrho\approx
1+\frac{1}{\alpha}$.

In the sparse networks we modeled, the tail of the gamma distribution is near
zero at $P_{ij}^{T}=1$ (see Figure 2A). Thus $K_{\alpha,\beta}\approx 1$ and
the truncated gamma distribution can be well approximated by the untruncated
version. Assuming connection probabilities to be standard gamma distributed,
$P_{ij}\sim\Gamma(\alpha,\beta)$, we have

$\displaystyle\operatorname{\mathbf{E}}(P_{ij}^{2})=\operatorname{\mathbf{Var}}(P_{ij})+\operatorname{\mathbf{E}}(P_{ij})^{2}=\alpha\beta^{2}+\alpha^{2}\beta^{2},$
(22)

and thus

$\displaystyle\varrho=\frac{\operatorname{\mathbf{E}}\left({P_{ij}^{T}}^{2}\right)}{\operatorname{\mathbf{E}}\left(P_{ij}^{T}\right)^{2}}\approx\frac{\operatorname{\mathbf{E}}\left({P_{ij}}^{2}\right)}{\operatorname{\mathbf{E}}\left(P_{ij}\right)^{2}}=\frac{\alpha^{2}\beta^{2}}{\alpha^{2}\beta^{2}}+\frac{\alpha\beta^{2}}{\alpha^{2}\beta^{2}}=1+\frac{1}{\alpha}=:\tilde{\varrho}.$
(23)

The approximation $\varrho\approx\tilde{\varrho}=1+\frac{1}{\alpha}$ works
well for $\alpha\geq 1$ as shown in Figure 2C.

To induce a high overrepresentation of reciprocal pairs in the network, the
gamma distribution of connection probabilities takes a highly skewed shape. In
order to obtain $\varrho=4$, only $57\%$ of pairs are expected to have a
higher connection probability than $0.01$ ($\alpha=0.248$, $\beta=0.487$).
Such a situation in which a large part of all neuron pairs have a small
connection probability while some few pairs have a high chance to be connected
is likely if, e.g., the connection probability strongly depends on the spatial
separation of the neuron, as it was found in layer 5 excitatory circuits of
the rat somatosensory cortex [2]. Then only nearby neurons are likely to be
connected, while the larger part of more distant neurons has a low probability
of connection.

### Symmetry of connection probabilities in neural circuits

In neural circuits, connection probabilities that are equal within pairs but
differ across the network are plausible from both an anatomical and a
functional perspective. From the anatomical point of view, the distance
dependency of connection probabilities mentioned above is a characteristic of
cortical circuits that necessarily leads to symmetric probabilities: the
distance from the first neuron’s soma to the second neuron’s soma is the same
as the distance from the second to the first, resulting in equal probabilities
within a pair of neurons when inter-neuron distance determines connection
probabilities. Regarding the functional perspective, connection probabilities
may also depend on functional properties of the cells in the network. For
example, the probability of connection of orientation tuned cells in the mouse
primary visual cortex depends on their absolute difference in orientation
tuning [11, 13]. Since the absolute difference in orientation tuning will be
the same in both directions, connection probabilities can be expected to be
equal within a pair of orientation tuned cells.

However, even when connection probabilities within pairs do not match exactly,
an overrepresentation of reciprocal connections is still likely to be observed
when connection probabilities follow a non-degenerate distribution. To see
this, consider that connection probabilities $P_{ij}$ are distributed
according to some probability density function $f_{P_{ij}}(x)$. As before we
assume that the $P_{ij}$ are independent outside of pairs. In the following we
also assume that $i>j$ without loss of generality. The expected probability of
a reciprocal connection within a pair can then be expressed as

$\displaystyle\operatorname{\mathbf{E}}(P_{ij}P_{ji})=\int_{0}^{1}\int_{0}^{1}xy\,f_{P_{ij},P_{ji}}(x,y)\,dx\,dy,$
(24)

where $f_{P_{ij},P_{ji}}(x,y)$ is the joint probability density function of
$P_{ij}$ and $P_{ji}$,

$\displaystyle f_{P_{ij},P_{ji}}(x,y)=f_{P_{ji}|P_{ij}}(y\mid
x)f_{P_{ij}}(x).$ (25)

In the case that $P_{ij}$ and $P_{ji}$ are independent we have
$f_{P_{ji}|P_{ij}}(y\mid x)=f_{P_{ji}}(y)$ and in the case of $P_{ij}=P_{ji}$
it is $f_{P_{ji}|P_{ij}}(y\mid x)=\delta(y-x)$. Here we propose a model for
the conditional density function that transitions between the two extreme
cases by multiplying $f_{P_{ji}}(y)$ with the density function of a normal
distribution centered around $x$,

$\displaystyle f_{P_{ji}|P_{ij}}(y\mid
x)=\frac{1}{N_{\sigma}(x)}f_{P_{ji}}(y)\,\frac{1}{\sigma\sqrt{2\pi}}\,e^{\frac{(y-x)^{2}}{2\sigma^{2}}},$
(26)

where the additional factor $N_{\sigma}(x)^{-1}$ makes sure that
$f_{P_{ji}|P_{ij}}(y\mid x)$ integrates to one,

$\displaystyle
N_{\sigma}(x)=\int_{0}^{1}f_{P_{ji}}(z)\,\frac{1}{\sigma\sqrt{2\pi}}\,e^{\frac{(z-x)^{2}}{2\sigma^{2}}}\,dz.$
(27)

Indeed, as the standard deviation $\sigma$ of the modulating normal
distribution increases $f_{P_{ji}|P_{ij}}(y\mid x)$ approaches $f_{P_{ji}}(y)$
and in the limit $\sigma\to 0$ we have

$\displaystyle\lim_{\sigma\to 0}f_{P_{ji}|P_{ij}}(y\mid x)=\delta(y-x).$ (28)

In Figure 3A, conditional density functions for various $\sigma$ are shown for
the truncated gamma distribution. For low values of $\sigma$ the conditional
density function resembles a narrow Gaussian around $x$, reflecting
approximately symmetric connection probabilities. For $\sigma>1$ on the other
hand $f_{P_{ji}|P_{ij}}(y\mid x)$ becomes virtually indistinguishable from
$f^{T}_{\alpha,\beta}(y)$, reflecting independence of $P_{ji}$ from $P_{ij}$.

Figure 3: Relative overrepresentation of bidirectional connections $\varrho$
is sustained when connection probabilities are only approximately symmetric in
pairs. A Illustration how the conditional density function
$f_{P_{ji}|P_{ij}}(y\,|\,x)$ of (26) transitions from equality of the random
variables $P_{ij}$ and $P_{ji}$ to independence with increasing $\sigma$. We
use $f_{P_{ij}}(y)=f^{T}_{\alpha,\beta}(y)$, with $\alpha=0.248$ and $\beta$
such that $\operatorname{E}(P_{ij})=0.1$. For the illustration $P_{ij}$ was
fixed as $x=0.15$. Already for $\sigma=1$ the conditional density function
becomes visually indistinguishable from $f^{T}_{\alpha,\beta}(y)$. B Relative
occurrence of reciprocally connected pairs $\varrho$ as a function of
$\sigma$. The curves for $\alpha=1$ and $\alpha=2$ show numerical solutions of
(29) with $f_{P_{ij}}(y)=f_{\alpha,\beta}^{T}(y)$, where $\beta$ was chosen
such that $\operatorname{E}(P_{ij})=0.1$. Relative reciprocal pair counts from
generated networks following the model matched these theoretical curves (data
not shown). For $\alpha=0.248$ random variables with the respective
probability density functions were sampled and the average $\varrho$ was
computed via (29) using the sample means. Error bars show SEM, the curve for
$\alpha=0.248$ (solid line) was fitted to the data points and is purely for
illustrative purposes.

Finally we employ the model to examine how the relative overrepresentation of
bidirectional connections $\varrho$ changes with the degree of symmetry in the
connection probabilities within a pair of neurons. For this $\varrho$ is
computed as a function of $\sigma$ as for a given distribution of $P_{ij}$ as

$\displaystyle\varrho=\frac{\operatorname{\mathbf{E}}(P_{ij}P_{ji})}{\mu^{2}},$
(29)

where the numerator is given by (24) with (26) and the overall connection
probability $\mu$ is calculated as

$\displaystyle\mu=\frac{1}{2}\int_{0}^{1}xf_{P_{ij}}(x)\,dx+\frac{1}{2}\int_{0}^{1}f_{P_{ij}}(x)\int_{0}^{1}y\,f_{P_{ji}|P_{ij}}(y\mid
x)\,dy\,dx.$ (30)

Figure 3B shows the change of $\varrho$ with $\sigma$ for connection
probabilities $P_{ij}$ following a truncated gamma distribution
$\Gamma^{T}(\alpha,\beta)$. For the three parameter sets chosen we see that a
strong overrepresentation of bidirectional connections is sustained when
connection probabilities are only approximately symmetric in pairs.
Furthermore, as long as $P_{ji}$ is at least somewhat biased to take similar
values to $P_{ij}$ an overrepresentation of $\varrho>1$ can be observed,
implying that effects such as distance-dependency or the dependence on the
absolute difference in orientation tuning of connection probabilities will
tend to increase the relative occurrence of bidirectional connections, even
when other effects are also influencing the neurons’ connection probabilities.

## Discussion

Experimental evidence suggests that any pair of excitatory cells within a
cortical column has contact points between axon and dendrite close enough to
support a synaptic connection between the cells [14, 15]. Despite this
potential “all-to-all” connectivity, only a small fraction of the contacts are
realized as functional synapses. Uncovering the underlying principles of which
contact points get utilized for synaptic transmission is crucial for our
understanding of the structure and function of the local cortical circuits in
the mammalian brain.

The emerging local networks in the rat visual and somatosensory cortex have
been shown to feature non-random structure [1, 2] and much attention was given
to bidirectionally connected neuron pairs that are occurring more often than
expected from random connectivity [5, 16, 17]. In this study we have shown a
condition under which non-random network structure and the occurrence of
reciprocally connected pairs are inherently linked; a relative
overrepresentation of bidirectional connections arises necessarily in networks
with a non-degenerate distribution of symmetric connection probabilities.
Absence of an overabundance of reciprocal pairs on the other hand, as for
example found in the intra-layer connectivity of the mouse C2 barrel column
[4], points towards either a truly random network or an asymmetry in the
connection probabilities.

Quantitatively, a network in which connection probabilities take on one of two
values is easily able to account for even the highest values of
overrepresentation reported. A network with such a two-point distribution of
connection probabilities might occur naturally, where the probability of
connection depends on whether a given pair of neurons shares a certain
feature, for example has a similar orientation preference or not [11].

A continuous distribution in connection probabilities on the other hand might
occur when pair connectivity depends on a continuous parameter such as the
inter-neuron distance or the neurons’ age. We showed that networks in which
connection probabilities follow a gamma distribution can as well have a high
relative occurrence of reciprocally connected pairs, however in this case a
larger fraction of pairs remain unconnected with a very high probability.

It is likely that a combination of such effects determines the connection
probabilities in local cortical networks. Importantly, we showed that as long
as this probability is symmetric for pairs, any such effect that creates a
non-degenerate distribution of probabilities will cause an increase of the
reciprocity in the network.

Our results confirm the intuitive notion that reciprocity is favored in
symmetric networks, whereas asymmetric probabilities of connection inhibit the
occurrence of bidirectionally connected pairs. Network models with symmetric
connectivity such as Hopfield nets generally excel at memory storage and
retrieval through fixed point attractor dynamics [18], while asymmetric
network models such as synfire chains are suitable for reliable signal
transmission [19, 20]. This suggests the intriguing possibility that one may
be able to infer the nature of the computations in a neural circuit based on
certain statistics of its connectivity such as the abundance of
bidirectionally connected pairs.

In conclusion, the present study puts the overrepresentation of bidirectional
connections found in local cortical circuits in a new light. If connection
probabilities are symmetric in pairs, the overrepresentation emerges as a
symptom of any form of non-random connectivity. It is thus crucial for both
future experimental and modeling studies to develop a more refined view of
non-random network connectivity that goes beyond simple pair statistics.
Focusing on higher order connectivity patterns and taking into account the
actual synaptic efficacies seem promising avenues for future research into the
non-random wiring of brain circuits.

## Supplementary Material

The supplementary information document for references SI1 and SI2 and Figure
S1 is available online at doi: 10.6084/m9.figshare.3501860. Python code for
the numerical computations is available as a GitHub repository and was
archived including the generated data at doi:
https://doi.org/10.5281/zenodo.200368. A website documenting the code is found
at https://non-random-connectivity-comes-in-pairs.github.io/.

## Acknowledgements

The authors would like to thank the anonymous reviewers for their helpful and
constructive comments on earlier versions of this article. JT is supported by
the Quandt foundation.

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# Focus Point from Direct Gauge Mediation

Sibo Zheng  Department of Physics, Chongqing University, Chongqing 401331, P.
R. China

###### Abstract

This paper is devoted to reconcile the tension between theoretic expectation
from naturalness and the present LHC limits on superpartner mass bounds. We
argue that in SUSY models of direct gauge mediation the focusing phenomenon
appears, which dramatically reduces the fine tuning associated to 126 GeV
Higgs boson. This type of model is highly predictive in mass spectrum, with
multi-TeV third generation, $A_{t}$ term of order 1 TeV, gluino mass above LHC
mass bound, and light neutralinos and charginos beneath 1 TeV.

##  I I. Introduction

As the LHC keeps running, the searches of supersymmetry (SUSY) signals such as
stop/gluino, sbottom and Higgs mass discovered at 126 GeV 125  continue to
push their mass bounds towards to multi-TeV range ATLAS2013 ; CMS2013 . On the
other hand, the argument of naturalness requires the masses of third
generation scalars, the Higgsinos and gluinos should be $\sim$ 1 TeV. This is
the present status of SUSY.

To reconcile the experimental limits and expectation of naturalness, either of
them needs subtle reconsiderations. In this paper, we consider relaxing the
upper bounds from argument of naturalness. The upper bounds on above soft
breaking parameters arise from the significant contribution to renormalization
group (RG) running for up-type Higgs mass squared $m^{2}_{H_{\mu}}$, which
connects to the electroweak (EW) scale through electroweak symmetry breaking
(EWSB) condition (for $\tan\beta>10$ in the context of the minimal
supersymmetric model (MSSM)),

$\displaystyle{}m^{2}_{Z}\simeq-2\mu^{2}-2m^{2}_{H_{\mu}},$ (1)

Naively, low fine tuning implies the value of $\mu$ and $\mid m_{H_{\mu}}\mid$
at EW scale should be both near EW scale. But there exists an exception. In
some cases, there is significantly cancellation among the RGE corrections
arising from soft breaking parameters to $m^{2}_{H_{\mu}}$, although their
input values are far beyond 1 TeV. This is known as focusing phenomenon
FPSUSY1 ; FPSUSY2 .

The early attempts in FPSUSY1 ; FPSUSY2 ; 1201.4338 ; 1303.1622  were mainly
restricted to SUSY models near grand unification scale (GUT). One recent work
related to focus point SUSY deals with gaugino mediation gaugino . In this
text, we consider gauge mediated (GM) SUSY models with intermediate or low
messenger scale $M$ (for a review see, e.g., Giudice ). Since the focusing
phenomenon can be analytically estimated only if the gaugino masses dominate
over all other soft breaking masses, or they are small in compared with the
third-generation scalar masses (with FPSUSY3  or without FPSUSY1 ; FPSUSY2
$A$ terms ), following this observation, in this paper we study direct GM
model, in which the gaugino masses are naturally small due to the fact that
gaugino masses of order $\mathcal{O}(F)$ vanishes Yanagida .

Another rational for employing direct GM models is that focusing phenomenon
can be understood as a result of hidden symmetry. Because without directly
gauging global symmetries of the model, there would be larger symmetries
maintained in the hidden theory. Otherwise, without the protection of symmetry
tiny deviation for model parameters from their focus point values leads to
significant fine tuning again, and the model is actually unnatural.

As we will see, there are three free input parameters in our model. Two of
them are fixed so as to induce focusing phenomenon, leaving an overall mass
parameter $m_{0}$. The fit to 126 GeV Higgs boson discovered at the LHC then
determines the magnitude of this parameter, with $m_{0}\sim 4-7$ TeV. Thus,
our model is highly predictive in mass spectrum.

In section IIA, we introduce the model in detail. In section IIB, we discuss
the focusing phenomenon, the boundary conditions for such structure and the
mass spectrum at EW scale. In section IIC, we discuss the possibility of
uplifting the gluino mass above LHC lower bound while keeping the focusing.
Finally we conclude in section III.

##  II II. The Model

###  II.1 A. Setup

In contrast to Agashe , in which non-minimal GM model was employed to discuss
focusing phenomenon, we study SUSY models that don’t spoil the grand
unification of SM gauge couplings and restrict to the context of direct GM.
The messenger fields include chiral quark superfields $q+q^{\prime}$ and their
bi-fundamental fields $\bar{q}+\bar{q^{\prime}}$, lepton superfields
$l+l^{\prime}$ and their bi-fundamental fields $\bar{l}+\bar{l^{\prime}}$, and
singlet $S$ and its bi-fundamental field $\bar{S}$. They transform under
$SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$ as, respectively,

$\displaystyle{}q,~{}q^{\prime}$ $\displaystyle\sim$
$\displaystyle\left(\mathbf{3},\mathbf{1},-\frac{1}{3}\right),$
$\displaystyle\bar{q},~{}\bar{q^{\prime}}$ $\displaystyle\sim$
$\displaystyle\left(\bar{\mathbf{3}},\mathbf{1},\frac{1}{3}\right),$
$\displaystyle l,l^{\prime}$ $\displaystyle\sim$
$\displaystyle\left(\mathbf{1},\mathbf{2},\frac{1}{2}\right),$ (2)
$\displaystyle\bar{l},~{}\bar{l^{\prime}}$ $\displaystyle\sim$
$\displaystyle\left(\mathbf{1},\bar{\mathbf{2}},-\frac{1}{2}\right)$
$\displaystyle S,~{}\bar{S}$ $\displaystyle\sim$
$\displaystyle\left(\mathbf{1},\mathbf{1},0\right)$

So, these messenger multiplets complete a $\mathbf{5}+\bar{\mathbf{5}}$
representation of SM gauge group. The renormalizable superpotential consistent
with SM gauge symmetry is given by 111It belongs to general Wess-Zumino model,
which can be completed as effective theory of strong dynamics at low energy
ISS . The direct gauge mediation arises after gauging the global symmetries in
the weak theory and identifying them as SM gauge groups.,

$\displaystyle{}W=fX+Xq\bar{q}+Xl\bar{l}+m(q^{\prime}\bar{q}+q\bar{q^{\prime}})+m(l^{\prime}\bar{l}+l\bar{l^{\prime}}).$

where $X=M+F\theta^{2}$, denotes the SUSY-breaking sector with nonzero $F$
term. In what follows, we will consider $N$ copies of such messengers
multiplets, with $N<6$ so as to maintain the grand unification of SM gauge
couplings.

For the purpose of focusing we add a deformation to superpotential Eq.(II.1),

$\displaystyle{}W=\lambda H_{u}S\bar{l}.$ (4)

This superpotential can be argued to be natural by either imposing a hidden
$U(1)_{X}$ symmetry soft2  or matter parity 1007.3323 . For example, we can
impose $U(1)_{X}$ charges of fields as,

$\displaystyle{}q_{X}(X,~{}\phi_{i},~{}\bar{\phi}_{i},~{}H_{u},H_{d})=(1,-1/2,-1/2,1,-1)$
(5)

where $\phi_{i}=\\{q,q^{\prime},l,l^{\prime},S\\}$. In addition, this hidden
symmetry forbids some operators such as $H_{d}Sl$.

In Eq.(II.1) we have assumed unified mass parameter $m$ and ignored the Yukawa
coefficients for simplicity. For $m<M$ which we adopt in this paper the soft
scalar mass spectrum induced by superpotential Eq.(II.1) is the same as that
of minimal GM at the leading order. Since the minimal GM can not induce
focusing phenomenon, the deformation to the scalar mass spectrum due to Eq.(4)
is thus crucial for our purpose. In particular, Eq.(4) gives rise to a
negative one-loop contribution to $m^{2}_{H_{u}}$ with suppression factor
$F/M^{2}$. Unless we take $\sqrt{F}<<M$, the sign of $m^{2}_{H_{u}}$ would be
negative, it will not lead to focusing (see explanation around Eq.(51)).
Therefore, we are restricted to choose

$\displaystyle{}m<M,~{}~{}\text{and}~{}~{}\sqrt{F}<<M.$ (6)

For detailed calculation of the deviation to scalar mass spectrum given by
Eq.(4), We refer the reader to soft1 ; soft2 . With small SUSY breaking given
by Eq.(6), $m^{2}_{H_{\mu}}$ will be uplifted as required for focusing.

One can verify that gaugino masses at one loop of order $\mathcal{O}(F)$
vanish due to the fact the mass matrix of messengers

$\displaystyle{}\mathcal{M}=\left(\begin{array}[]{ll}X&m\\\
m&0\end{array}\right)$ (9)

satisfies $\det\mathcal{M}=\text{const}$ as long as $m$ doesn’t vanish,
although $m$ is small in comparison with scale $M$. So we expect that the RGE
for $m^{2}_{H_{\mu}}$ is dominated by stop mass squared $m^{2}_{Q_{3}}$,
$m^{2}_{u_{3}}$, and Eq.(4) induced A term.

###  II.2 B. Focusing And Mass Spectrum

Following the observation FPSUSY1 ; FPSUSY2 ; FPSUSY3  that the REGs for
$A_{t}$ and scalar masses such as $m^{2}_{H_{\mu}}$ are affected by both
themselves and gluino masses, while the RGE for gluino mass is only affected
by itself, we can solve the RGEs for soft scalar masses,

$\displaystyle{}\left(\begin{array}[]{c}m^{2}_{H_{\mu}}(Q)\\\
m^{2}_{u_{3}}(Q)\\\ m^{2}_{Q_{3}}(Q)\\\ A^{2}_{t}(Q)\\\ \end{array}\right)$
$\displaystyle=$
$\displaystyle\kappa_{12}I^{2}(Q)\left(\begin{array}[]{ccccc}3\\\ 2\\\ 1\\\
6\\\ \end{array}\right)+\kappa_{6}I(Q)\left(\begin{array}[]{ccccc}3\\\ 2\\\
1\\\ 0\\\ \end{array}\right)$ (22) $\displaystyle+$
$\displaystyle\kappa_{0}\left(\begin{array}[]{ccccc}1\\\ 0\\\ -1\\\ 0\\\
\end{array}\right)+\kappa^{\prime}_{0}\left(\begin{array}[]{ccccc}0\\\ 1\\\
-1\\\ 0\\\ \end{array}\right).$ (31)

for small gluino masses (in compared with above scalar soft masses). Here,

$\displaystyle{}I(Q)=\exp\left(\int^{\ln Q}_{\ln
M}\frac{6y^{2}_{t}(Q^{\prime})}{8\pi^{2}}d\ln Q^{\prime}\right)$ (32)

which depends on $M$ and RGE for top Yukawa. In Fig. 1 we show the numerical
value of $I$ as function of $M$, with the context of MSSM below scale $M$.

Figure 1: $I$ as function of $M$ for the context of MSSM below scale $M$.

In particular, $I(1\text{TeV})\simeq 0.527$ for $M=10^{8}$ GeV.

The condition for focusing phenomenon can be derived from Eq.(22) by imposing
$m^{2}_{H_{\mu}}(1\text{ TeV})\simeq 0$. Define
$m^{2}_{H_{\mu}}(M)=+m^{2}_{0}$. Similar to FPSUSY3  we choose $x$ to
parameterize the splitting between $m^{2}_{Q_{3}}(M)$ and $m^{2}_{u_{3}}(M)$,
and $y$ to be directly related to $A_{t}(M)$. In the case of small SUSY
breaking the mass spectrum which induces focusing at scale $\mu=1\text{TeV}$
reads as,

$\displaystyle{}m^{2}_{0}\left(\begin{array}[]{c}1\\\ 1.41+x-1.58y\\\
1.82-x-3.16y\\\ 9y\\\ \end{array}\right)_{M}\rightarrow
m^{2}_{0}\left(\begin{array}[]{c}0\\\ 0.74+x-1.58y\\\ 1.48-x-3.16y\\\ 1.66y\\\
\end{array}\right)_{\mu}$ (41)

Alternatively we rescale parameter $x$ as in FPSUSY3  such that
$m^{2}_{Q_{3}}$ only depends on $x$. For $m^{2}_{H_{\mu}}(M)=-m^{2}_{0}$,
Eq.(41) is instead of,

$\displaystyle{}m^{2}_{0}\left(\begin{array}[]{c}-1\\\ -1.41+x-1.58y\\\
-1.82-x-3.16y\\\ 9y\\\ \end{array}\right)_{M}\rightarrow
m^{2}_{0}\left(\begin{array}[]{c}0\\\ -0.74+x-1.58y\\\ -1.48-x-3.16y\\\
1.66y\\\ \end{array}\right)$ (51)

This parameterization appears when $F/M^{2}\rightarrow 1$. In this limit,
$m^{2}_{H_{\mu}}$ is dominated by the one-loop negative contribution
proportional to Yukawa coupling $\lambda$. From Eq.(51), there is no
consistent solution to $x$ and $y$ in this case.

Soft masses in Eq.(22) at scale $\mu=1\text{TeV}$ are functions of Yukawa
coupling $\lambda$, number of messenger pairs $N$, ratio $F/M^{2}$ and SUSY-
breaking mediated scale $M$. From Eq. (41) one connects the variables $(x,y)$
and the model parameters $\lambda$ and $N$. For the three input parameters
$m_{0}$, $x$ and $y$ (with $M$ fixed) for focusing in the model, two of them
can be fixed by the choices of $\lambda$ and $N$. We choose $x$ and $y$ for
analysis. Fig.2 shows the plots of $x$ (dotted) and $y$ (solid) as function of
$\alpha_{\lambda}$ and $N$. For each $N$ the focus point values of $x$ and $y$
are read from the crossing points between vertical line and solid curve
(dotted curve ) for $y$ ($x$) . Therefore, there is only one free parameter
left in the model by imposing the focusing condition, which is very predictive
in the mass spectrum and signal analysis.

Since we perform our analysis in perturbative theory, in order to avoid Landau
pole up to GUT scale, the Yukawa coupling $\alpha_{\lambda}$ is upper bounded,
$\sim$0.1 for our choice of messenger scale. The dotted and solid horizontal
lines in fig. 2 refer to allowed ranges for $x$ and $y$, respectively. These
ranges are derived from the requirement that the stop soft masses aren’t
tachyon-like and the $A_{t}$ squared is positive. Following these we obtain,

$\displaystyle{}0<y<0.40,~{}-0.74<x<1.48,$ $\displaystyle
1.58y-0.74<x<1.48-3.16y,$ (52) $\displaystyle 1.58y-1.41<x<1.82-3.16y.$

It is easy to verify that for each $N$ the crossing points satisfy the
constraints above.

Figure 2: Plots of $x$ (dotted) and $y$ (solid) as function of
$\alpha_{\lambda}$ for $N=\\{1,2,3,4\\}$. The red, blue, purple and black
curves correspond to $N=1,2,3,4$, respectively. For each $N$ the focus point
value are read from the crossing points between vertical line and solid curve
for $y$ and dottoed curve for $x$, respectively. The dotted (solid) horizontal
lines refer to range allowed for $x$ ($y$).

With focusing phenomenon we have single free parameter, namely $m_{0}$ at
hand. It can be uniquely determined in terms of the mass of Higgs boson
observed at the LHC. Fig. 2 shows how $m_{h}$ changes as parameter $m_{0}$ for
different $N$s. The two-loop level Higgs boson mass in the MSSM is given by
Carena ,

$\displaystyle{}m_{h}^{2}$ $\displaystyle=$ $\displaystyle
m_{Z}^{2}\cos^{2}2\beta+\frac{3m^{4}_{t}}{4\pi^{2}\upsilon^{2}}\left\\{\log\left(\frac{M^{2}_{S}}{m^{2}_{t}}\right)+\frac{1}{2}\tilde{A}_{t}+\frac{1}{16\pi^{2}}\left(\frac{3}{2}\frac{m^{2}_{t}}{\upsilon^{2}}-32\pi\alpha_{3}\right)\left[\tilde{A}_{t}+\log\left(\frac{M^{2}_{S}}{m^{2}_{t}}\right)\right]\log\left(\frac{M^{2}_{S}}{m^{2}_{t}}\right)\right\\}$
(53)

Here $\upsilon=174$ GeV and
$\tilde{A}_{t}=\frac{2X^{2}_{t}}{M^{2}_{S}}\left(1-\frac{X^{2}_{t}}{12M^{2}_{S}}\right)$,
with $X_{t}=A_{t}-\mu\cot\beta$. We focus on large $\tan\beta$ region. For
$\tan\beta\geq 20$, the fit to Higgs boson mass doesn’t change much. From
fig.3 one observes that $m_{0}\sim 4.0-7.0$ due to the fit to 126 GeV Higgs
boson.

Figure 3: $m_{h}$ vs $m_{0}$ for different $N$s, with $N=1,2,3,4$ from bottom
to top, respectively. Multi-TeV $m_{0}$ is required by the 126 GeV Higgs
boson.

Substituting the values of $m_{0}$ from fig.3 and $x$, $y$ from fig.2 into
Eq.(41) we find the mass spectrum, which is shown in table 1.

The choice on large $\tan\beta$ might be forbidden by possibly large B$\mu$
term induced by Eq.(4). As noted in 1007.3323 , B$\mu\sim\mu A_{t}$. In terms
of electroweak symmetry breaking condition, we have
$\sin(2\beta)\simeq\text{B}\mu/m^{2}_{0}\sim(A_{t}/m_{0})^{2}\cdot(\mu/A_{t})$.
With a small $\mu$ term of order $\sim 300-500$ GeV (as shown in table 1) at
messenger scale $M$, one does not have to worry about $\mu$ being made very
large by radiative correction involving heavy soft scalar masses (see e.g.,
soft1 ). So, one obtains $\sin(2\beta)$ of order $\sim(1/4)^{2}\cdot(1/4)$
from table 1, and the choice on large value of $\tan\beta$ is not violated by
B$\mu$ term.

###  II.3 C. Gaugino Mass

As mentioned above due to $\det\mathcal{M}=\text{const}$ gaugino masses vanish
at one-loop level of order $\mathcal{O}(F)$ and at the two-loop level of order
$\mathcal{O}(F)$. Their leading contributions appear at one-loop level of
order $\mathcal{O}(F^{3}/M^{5})$ Yanagida . Under small SUSY-breaking limit
the magnitude of gaugino mass relative to $m_{Q_{3}}$ at input scale is given
by 222We thank the referee for pointing out a critical error in estimation of
gaugino mass in the previous version of this manuscript.,

$\displaystyle{}\frac{m_{\tilde{g}_{i}}}{m_{Q_{3}}}\sim\left(\frac{F}{M^{2}}\right)^{2}\cdot\frac{\sqrt{N}\alpha_{i}}{\sqrt{2\times\left(\frac{4}{3}\alpha^{2}_{3}(M)+\frac{3}{4}\alpha^{2}_{2}(M)+\frac{1}{60}\alpha^{2}_{1}(M)\right)}}$

Using one-loop RGEs for gluino masses, we find their values at the
renormalization scale $\mu=1$ TeV. One observes from Eq.(II.3) that the gluino
mass is far below the 2013 LHC bound $\simeq 1.3$ TeV due to the suppression
by factor $F^{2}/M^{4}$.

Without extra significant modifications to the gaugino mass spectrum, LHC
bound would exclude this simple model, despite it provides a natural
explanation of Higgs boson mass and is consistent with present experimental
limits. Here, we propose a recipe 0612139  in terms of imposing small
modification to superpotential $\delta
W=m^{\prime}\left(\bar{l^{\prime}}l^{\prime}+\bar{q^{\prime}}q^{\prime}\right)$,
with small mass $m^{\prime}<m$. These mass terms are consistent with gauge
symmetries and matter parity of messenger sector.

If so, Eq.(9) will be instead of

$\displaystyle{}\mathcal{M}=\left(\begin{array}[]{ll}X&m\\\
m&m^{\prime}\end{array}\right)$ (57)

The correction to soft scalar mass spectrum is of order $\mathcal{O}(m^{\prime
4}/m^{4})$ and very weak. However, the correction to gaugino mass, which is of
order,

$\displaystyle{}m_{\tilde{g}_{i}}\simeq
N\cdot\frac{\alpha_{i}}{4\pi}\cdot\frac{F}{m}\cdot\frac{m^{\prime}}{m}$ (58)

can be large enough to reconcile with the LHC bound when $m^{\prime}/m$ is
larger than $F^{2}/M^{4}$. For example, we choose $N=1$, $M=10^{8}$ GeV and
$m=0.1M$. Then $m_{0}\sim 7$ TeV and $\sqrt{F}\sim 8.2\cdot 10^{6}$ GeV, and
further $m_{\tilde{g}_{3}}\sim 7\cdot 10^{-3}\cdot m^{\prime}$ from Eq.(58).
LHC gluino mass bound requires $m^{\prime}\geq 2\cdot 10^{5}$ GeV, which is
consistent with the constraint $m^{\prime}<m<M$. The bino and wino masses are
both near 1 TeV. So they are the main target of 14-TeV LHC.

| $N=1$ | $N=2$ | $N=3$ | $N=4$
---|---|---|---|---
$m_{0}$ | $7.0$ | $5.9$ | $4.0$ | $3.5$
$m_{\tilde{t}_{1}}$ | $3.12$ | $3.62$ | $4.54$ | 4.83
$m_{\tilde{t}_{2}}$ | $7.65$ | $4.98$ | $4.80$ | 6.0
$A_{t}$ | $1.64$ | $1.48$ | $1.50$ | 1.50
$\mu$ | $0.50$ | $0.42$ | $0.28$ | 0.24
Table 1: Given a focus point, input mass parameter $m_{0}$ (in unit of TeV)
required for $m_{h}=126$ GeV and corresponding soft mass spectrum (in unit of
TeV) at renormalization scale $\mu=1$ TeV in the context of MSSM, for
different values of messenger number $N$.

##  III III. Discussion

From mass spectrum of table 1, the main source for fine tuning arises from
$\mu$ term. The fine tuning parameter c, which is defined as
$c=\max\\{c_{i}\\}$, with

$\displaystyle c_{i}=\mid\partial\ln m^{2}_{Z}/\partial\ln a_{i}\mid$

where $a_{i}$ are the soft mass parameters involved, has been reduced from
$\sim 2000$ to $\sim 20$ due to the focusing phenomenon.

As for other indirect experimental limits such as flavor changing neutral
violation, the model feels comfortable. Because the masses of the three-
generation sleptons and first two-generation squarks are all of order $\sim$
multi-TeV, with highly degeneracy in each sector.

What about the sensitivity of our results to the messenger scale ? At first,
assuming that there exists a completion of strong dynamics at high energy
indicates that $M$ should be smaller than the GUT scale. Typically, we have
$M<10^{10}$ GeV in the context of direct gauge mediation. For the case of low-
scale mediation, i.e, $M<10^{8}$ GeV, the gluino mass is already close to the
2013 LHC mass bound. In other words, $M=10^{8}$ GeV as we studied in detail is
a reference value for intermediate scale SUSY model. The promising signals for
this simple and natural model include searching gluino, neutralinos and
charginos at the LHC.

Along this line it is of interest to extend the model-independent focusing
condition to the whole energy range below GUT scale Zheng , and construct
natural SUSY models in the context of either direct or non-direct GM.

$\mathbf{Acknowledgement}$ The work is supported in part by Natural Science
Foundation of China under grant No.11247031 and 11405015.

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