\newtheorem{defi}{Definition}[section]
\label{sec:comput_c}
\section{Computation of the concentration c}

\begin{table}[H]
        \caption{Notation for Riemannian Geometry tools}

    \centering
    \label{tab:riemannian_symbols}
    \begin{tabular}{|c|l|}
        \hline
        \textbf{Symbol}        & \textbf{Description} \\ \hline
        $\boldsymbol{\varphi}$              & Unit vector on the hypersphere representing embedding direction \\ \hline
        $\Phi$                 & Opening angle of the spherical cap \\ \hline
        $S^{D-1}$              & Unit hypersphere in $D$ dimensions \\ \hline
        $C_\Phi(\boldsymbol{\varphi})$      & Spherical cap of radius $\Phi$ centered around $\boldsymbol{\varphi}$ \\ \hline
        $\boldsymbol{\alpha}_i$             & Random normalized embedding direction on $S^{D-1}$ \\ \hline
        $g$     & Riemannian metric induced by the Euclidean space \\ \hline
        $d\sigma(\boldsymbol{x})$           & Riemannian volume element on $S^{D-1}$ \\ \hline

        $t = \cos \psi$        & Projection of the embedding onto the primary direction $\boldsymbol{\varphi}$ \\ \hline
        $p(t)$                 & Probability density of the projection $t$ on the spherical cap \\ \hline
        $c$                    & Concentration parameter $c = E[\alpha_i^\top \boldsymbol{\varphi}]$, quantifying dispersion \\ \hline
    \end{tabular}

\end{table}



For the reader's convenience, we redefine the necessary tools in Table. \ref{tab:riemannian_symbols} to make the subsequent proofs transparent and self-contained.
\subsection{Preliminaries}

\begin{definition}[Manifold]
An \textbf{$n$-dimensional manifold} $M$ is a topological space such that every point $p \in M$ has an open neighborhood $U$ that is homeomorphic to an open subset of $\mathbb{R}^n$. That is, there exists a collection of charts $\{(U, \phi)\}$ where $\phi : U \mapsto \phi(U) \subset \mathbb{R}^n$ is a homeomorphism.
\end{definition}

\begin{definition}[Riemannian Manifold]
A \textbf{Riemannian manifold} $(M, g)$ is a smooth manifold $M$ equipped with a Riemannian metric $g$, which is a smoothly varying inner product on the tangent space $T_pM$ at every point $p \in M$. In local coordinates $(x^1,\dots,x^n)$, the metric is represented by a positive-definite symmetric matrix $(g_{ij}(x))$, so that
\begin{equation}
g = g_{ij}(x) \, dx^i \otimes dx^j.
\end{equation}
\end{definition}

\begin{definition}[Integration on a Riemannian Manifold]
Let $(M, g)$ be an $n$-dimensional Riemannian manifold and let $\Omega$ be a differential $n$-form on $M$. In local coordinates, if
\begin{equation}
\Omega = f(x)\, dx^1 \wedge dx^2 \wedge \cdots \wedge dx^n,
\end{equation}
the \textbf{Riemannian volume form} is defined as
\begin{equation}
d\sigma(x) = \sqrt{\det(g(x))}\, dx^1 \wedge dx^2 \wedge \cdots \wedge dx^n.
\end{equation}
Here, the symbol $\wedge$ denotes the \textbf{exterior product}.  The integral of $\Omega$ over $M$ is defined by
\begin{equation}
\int_M \Omega = \int_{\phi}(U) f(x) \sqrt{\det(g(x))}\, dx^1 \cdots dx^n,
\end{equation}
where a partition of unity is used to cover $M$ by coordinate charts.
\end{definition}

\begin{definition}[Probability Density]
Let $p: M \to [0, +\infty[$ be a measurable function such that
\begin{equation}
\int_M p(x) \, d\mu(x) = 1.
\end{equation}
The function $p(x)$ is called the \textbf{probability density function} (pdf) of a random variable $X$ taking values in $M$ with respect to the measure $\mu$.  
The \textbf{differential probability} is then given by
\begin{equation}
dP(x) = p(x) \, d\mu(x).
\end{equation}
\end{definition}

\begin{definition}[Unit Sphere]
The \textbf{unit sphere} in $\mathbb{R}^D$ is defined by
\begin{equation}
S^{D-1} = \{ \textbf{x} \in \mathbb{R}^D \mid \|\textbf{x}\| = 1 \}.
\end{equation}
It can be shown that $S^{D-1}$ is a smooth manifold of dimension $D-1$. Moreover, $S^{D-1}$ inherits a Riemannian metric from the Euclidean metric on $\mathbb{R}^D$; thus, $S^{D-1}$ is a Riemannian manifold.
\end{definition}

\begin{definition}[Density on a Manifold]
Let $(M,g)$ be an $n$-dimensional Riemannian manifold with Riemannian volume form $d\sigma(x)$. A function $p: M \to [0,\infty)$ is called a \textbf{density} w.r.t to $d\sigma$ if, for any measurable subset $A \subset M$, the \emph{probability} of $A$ is defined as
\begin{equation}
\mathbb{P}(A) = \frac{\displaystyle \int_A p(x)\, d\sigma(x)}{\displaystyle \int_M p(x)\, d\sigma(x)}.
\end{equation}
In particular, if $p(x)$ is constant, the distribution is said to be \textbf{uniform} on $M$.
\end{definition}

\subsection{The Concentration Parameter C for a Spherical Cap}

\begin{theorem}[Expression for the Concentration Parameter] \label{thm:d1}
Let $\boldsymbol{\varphi} \in S^{D-1}$ be a fixed unit vector and let
\begin{equation}
C_\Phi(\boldsymbol{\varphi}) = \{ \boldsymbol{\alpha} \in S^{D-1} \mid \boldsymbol{\alpha}^\top \boldsymbol{\varphi} \ge \cos\Phi \},
\end{equation}
be the spherical cap of opening angle $\Phi \in (0,\pi]$. Assume that the random variable $\alpha$ is uniformly distributed on $C_\Phi(\boldsymbol{\varphi})$ and define the concentration parameter by
\begin{equation}
c = \mathbb{E}[\boldsymbol{\alpha}^\top\boldsymbol{\varphi}].
\end{equation}
Then, 
\begin{equation}
c = \frac{\displaystyle \int_{\cos\Phi}^{1} t\,(1-t^2)^{\frac{D-3}{2}}\, dt}{\displaystyle \int_{\cos\Phi}^{1} (1-t^2)^{\frac{D-3}{2}}\, dt}.
\end{equation}
\end{theorem}
This result shows that integrating over the hypersphere involves a term that depends on the dimensionality $D$ and the angular separation of the embeddings. This is crucial because it explains how variance in the embedding space is linked to geometric dispersion.

We start by a lemma : 
\subsubsection{A first lemma to compute the riemannian metrics on the unit sphere}
\begin{lemma} \label{lem:d2}
Let $F: [0,\pi]\times S^{D-2}\to S^{D-1}$ be given by  
\begin{equation}
F(\psi,\boldsymbol{\omega}) = (\cos\psi,\;\sin\psi\,\boldsymbol{\omega}), \quad \psi\in[0,\pi],\ \boldsymbol{\omega}\in S^{D-2}.
\end{equation}
Then the Riemannian volume element on $S^{D-1}$ induced by the Euclidean metric on $\mathbb{R}^D$ is  
\begin{equation}
\boxed{d\sigma(\psi,\boldsymbol{\omega})= \sin^{D-2}\!\psi\, d\psi\, d\sigma_{S^{D-2}}(\boldsymbol{\omega}),}
\end{equation}
where $d\sigma_{S^{D-2}}(\boldsymbol{\omega})$ is the area form on $S^{D-2}$.
\end{lemma}

\begin{proof}
Our goal is to compute the expression for the Riemannian volume form $d\sigma$ on $S^{D-1}$ by computing the induced metric via the parameterization
\begin{equation}
F(\psi,\boldsymbol{\omega}) = (\cos\psi,\;\sin\psi\,\boldsymbol{\omega}),
\end{equation}
where $\psi\in[0,\pi]$ and $\boldsymbol{\omega}\in S^{D-2}$. This mapping $F$ serves as a coordinate chart (in the sense of the definition of a manifold) for $S^{D-1}$. We introduce local coordinates $\psi$ and $\{y^1,\dots,y^{D-2}\}$ on $S^{D-2}$ (so that $\boldsymbol{\omega}=\boldsymbol{\omega}(y^1,\dots,y^{D-2})$).

Since $S^{D-1}$ is a submanifold of $\mathbb{R}^D$, it inherits a Riemannian metric from the Euclidean metric of $\mathbb{R}^D$ via the pull-back under $F$. In local coordinates $u^\alpha=(\psi, y^1, \dots, y^{D-2})$, the metric is defined by
\begin{equation}
g_{\alpha\beta} = \langle \frac{\partial F}{\partial u^\alpha}, \frac{\partial F}{\partial u^\beta} \rangle.
\end{equation}
Once the metric $g$ is computed, the Riemannian volume form on $S^{D-1}$ is given by
\begin{equation}
d\sigma(u) = \sqrt{\det(g(u))}\, d\psi\, dy^1 \cdots dy^{D-2},
\end{equation}
in accordance with the definition of integration on a Riemannian manifold. Let's compute $g$.  
First, compute the partial derivatives:
\begin{equation}
F_\psi = \frac{\partial F}{\partial \psi} = (-\sin\psi,\;\cos\psi\,\boldsymbol{\omega}),
\end{equation}
\begin{equation}
F_{y^i} = \frac{\partial F}{\partial y^i} = \Bigl(0,\;\sin\psi\,\frac{\partial \boldsymbol{\omega}}{\partial y^i}\Bigr).
\end{equation}
The induced metric $g$ is defined by
\begin{equation}
g_{\alpha\beta}=\langle F_{u^\alpha},F_{u^\beta}\rangle_{\mathbb{R}^D},\quad u^1=\psi,\; u^{i+1}=y^i.
\end{equation}

\textbf{Step 1:} For $\alpha=\beta=\psi$,
\begin{equation}
g_{\psi\psi} = \langle F_\psi,F_\psi\rangle = (-\sin\psi)^2 + \cos^2\psi\,\|\boldsymbol{\omega}\|^2 = \sin^2\psi+\cos^2\psi = 1.
\end{equation}

\textbf{Step 2:} For mixed indices $\alpha=\psi,\; \beta=y^i$,
\begin{equation}
g_{\psi,y^i} = \langle F_\psi,F_{y^i}\rangle = \langle (-\sin\psi,\;\cos\psi\,\boldsymbol{\omega}), (0,\;\sin\psi\,\boldsymbol{\omega}_{y^i})\rangle = \cos\psi\,\sin\psi\,\langle \boldsymbol{\omega},\boldsymbol{\omega}_{y^i}\rangle.
\end{equation}
Since $\boldsymbol{\omega}$ is a unit vector, it is normal to its tangent space, hence $\langle\boldsymbol{\omega},\boldsymbol{\omega}_{y^i}\rangle=0$ and $g_{\psi,y^i}=0$.

\textbf{Step 3:} For $\alpha=y^i,\; \beta=y^j$,
\begin{equation}
g_{y^i,y^j} = \langle F_{y^i},F_{y^j}\rangle = \sin^2\psi\,\langle \boldsymbol{\omega}_{y^i},\boldsymbol{\omega}_{y^j}\rangle.
\end{equation}
Define the metric on $S^{D-2}$ by
\begin{equation}
h_{ij}(\boldsymbol{\omega})=\langle \boldsymbol{\omega}_{y^i},\boldsymbol{\omega}_{y^j}\rangle,
\end{equation}
so that
\begin{equation}
g_{y^i,y^j} = \sin^2\psi\, h_{ij}(\boldsymbol{\omega}).
\end{equation}

Collecting these results, the metric in block matrix form is 
\begin{equation}
g = \begin{pmatrix} 1 & 0 \\ 0 & \sin^2\psi\,h_{ij}(\boldsymbol{\omega}) \end{pmatrix}.
\end{equation}
Its determinant is then
\begin{equation}
\det(g)=1\cdot(\sin^2\psi)^{D-2}\det(h_{ij}(\boldsymbol{\omega})).
\end{equation}
Taking the square root yields the volume element:
\begin{equation}
d\sigma = \sqrt{\det(g)}\,d\psi\,dy^1\cdots dy^{D-2} = \sin^{D-2}\!\psi\,\sqrt{\det(h_{ij}(\boldsymbol{\omega}))}\, d\psi\,dy^1\cdots dy^{D-2}.
\end{equation}
Since
\begin{equation}
\sqrt{\det(h_{ij}(\boldsymbol{\omega}))}\,dy^1\cdots dy^{D-2}=d\sigma_{S^{D-2}}(\boldsymbol{\omega}),
\end{equation}
we obtain the desired formula:
\begin{equation}
d\sigma(\psi,\boldsymbol{\omega})= \sin^{D-2}\!\psi\, d\psi\, d\sigma_{S^{D-2}}(\boldsymbol{\omega}).
\end{equation}
\end{proof}


\subsubsection{Proof of theorem \ref{thm:d1} }
\begin{proof}[Proof of Theorem. \ref{thm:d1}]
Without loss of generality, we choose $\boldsymbol{\varphi}$ so that $\boldsymbol{\varphi} = (1,0,\dots,0)$. Then, any $\boldsymbol{x} \in S^{D-1}$ can be written as 
\begin{equation}
\boldsymbol{x} =(\cos\psi,\,\sin\psi\,\boldsymbol{\omega}), 
\quad \psi \in [0,\pi], 
\quad \boldsymbol{\omega}\in S^{D-2}.
\end{equation}
Then we have from Lemma. \ref{lem:d2}:
\begin{equation}
d\sigma(\boldsymbol{x}) \;=\;\sin^{D-2}\!\psi\,d\psi\,d\sigma_{S^{D-2}}(\boldsymbol{\omega}).
\end{equation}
Set $t=\cos\psi$. Then $dt = -\sin\psi\,d\psi$, so 
\begin{equation}
d\psi = \frac{dt}{\sqrt{1-t^2}}.
\end{equation}
Hence
\begin{equation}
d\sigma(\boldsymbol{x}) 
= \sin^{D-2}\!\psi\,\frac{dt}{\sqrt{1-t^2}}\,d\sigma_{S^{D-2}}(\boldsymbol{\omega})
= (1-t^2)^{\frac{D-3}{2}}\,dt\,d\sigma_{S^{D-2}}(\boldsymbol{\omega}).
\end{equation}
A uniform distribution on $S^{D-1}$ is constant with respect to this measure, so integrating out $\boldsymbol{\omega}$ implies the marginal density of $t$ is 
\begin{equation}
p_0(t) :=(1-t^2)^{\frac{D-3}{2}}A_{D-2}, 
\quad t\in[-1,1].
\end{equation}

\noindent
\textbf{Spherical cap.} Consider the cap 
\begin{equation}
C_\Phi(\boldsymbol{\varphi}) = \{\boldsymbol{\alpha}\in S^{D-1}\mid \boldsymbol{\alpha}^\top \boldsymbol{\varphi} \ge \cos\Phi\}.
\end{equation}
Since $\boldsymbol{\varphi}=(1,0,\dots,0)$, this condition is $t=\cos\psi\ge \cos\Phi$. The normalized marginal density on $[\cos\Phi,1]$ is 
\begin{equation}
p(t) = \frac{p_0(t)}{\int_{\cos{\Phi}}^1p_0(t)dt} =\frac{(1-t^2)^{\frac{D-3}{2}}}{\displaystyle \int_{\cos\Phi}^{1}(1-t^2)^{\frac{D-3}{2}}\,dt}.
\end{equation}

Then the concentration parameter is:
\begin{equation}
c 
\;=\;\int_{\cos\Phi}^1 t\,p(t)\,dt
\;=\;\frac{\displaystyle \int_{\cos\Phi}^1 t\,(1-t^2)^{\frac{D-3}{2}}\,dt}{\displaystyle \int_{\cos\Phi}^1 (1-t^2)^{\frac{D-3}{2}}\,dt}.
\end{equation}
\end{proof}

\begin{theorem}[Asymptotic Concentration]
Let $c$ be defined as above for the spherical cap $C_\Phi(\boldsymbol{\varphi})$ on $S^{D-1}$. We have:
\begin{equation}
\lim_{D\to\infty} c 
\;=\;\cos\Phi.
\end{equation}
\end{theorem}

\begin{proof}[Sketch]
Because $\frac{D-3}{2}$ becomes large, the function $(1-t^2)^{\frac{D-3}{2}}$ peaks near $t=\cos\Phi$. Thus both numerator and denominator in
\begin{equation}
c \;=\;\frac{\displaystyle \int_{\cos\Phi}^1 t\,(1-t^2)^{\frac{D-3}{2}}\,dt}{\displaystyle \int_{\cos\Phi}^1 (1-t^2)^{\frac{D-3}{2}}\,dt},
\end{equation}
are dominated by values $t\approx \cos\Phi$, so $c\approx \cos\Phi$.
\end{proof}

This results show us that for $D \to +\infty$, $c \to \cos(\Phi)$, where $\Phi$ is the opening of the spherical cap around $\boldsymbol{\varphi}$. This indicates that in high-dimensional spaces, the perturbed embedding directions are effectively determined by the cap’s opening angle, providing a way to tune DropConnect rate. 