Volumetric Manifolds

A constructed mechanism for facilitation of efficient and effective contract

${}^{[}1]$ trading

$$G:=(V,E,w)$$

weighted by the function

$w:E\to {\mathbb{R}}^{+}$ corresponding to the price of some trade

$e\in E$

$$V,|V|\le |\mathbb{N}|$$

Consider an abstract liquid trading (ALT) system as a weighted directed graph

$G:=(V,E,w),$ where set of vertices

$V,|V|\le |\mathbb{N}|$ contains digital representation of all tradeable assets in

$G,$ set of edges

$E=\{e\in V\times V:$

$w(e)>0\}$ represents all possible atomic

${}^{79}$ asymmetric

${}^{80}$ trades, which are weighted by the function

$w:E\to {\mathbb{R}}^{+}$ corresponding to the price of some trade

$e\in E$

Liquidity Point/Vertex

$$\text{ Any liquid vertex }v\in V\text{ has both }de{g}^{-}(v)\ge 1\text{ and }{\mathrm{deg}}^{+}(v)\ge 1$$

Liquidity Indifference Preference Paths

${}^2$

$$C_B(v):=\sum _{s\ne t\ne v\in V}\frac{{\sigma }_{st}(v)}{{\sigma }_{st}}:\mathrm{\forall }(s,t)\in S,\text{ where }{\sigma }_{st}:=\sum _{(s,t)\in S}\sum _{e\in (s,t)}w(e)$$

Vertex

$v\in V$ represents half-liquid asset when either

${\mathrm{deg}}^{-}(v)=0$ (source)

$\mathrm{or}{\mathrm{deg}}^{+}(v)=0(\mathrm{sink}),$ where

${\mathrm{deg}}^{(-1+)}:V\to \mathbb{N}$ is respectively a number
of tail ends (indegree) and a number of head ends (outdegree) from vertices adjacent to

$v$.

Liquidity Preference Path

Let

$S\subset V\times V$ contain all shortest paths from vertex

$s$ to vertex

$t:\mathrm{\forall }s,t\in V$ Also let vertex

$v\in V$ have the maximal

${}^{82}$ betweenness centrality measure

$C_B(v):=\sum _{s\ne t\ne v\in V}\frac{{\sigma }_{st}(v)}{{\sigma }_{st}}:\mathrm{\forall }(s,t)\in S,$ where

${\sigma }_{st}:=\sum _{(s,t)\in S}\sum _{e\in (s,t)}w(e)$
and

${\sigma }_{st}(v)$
is a sum of only those shortest paths in

$S$ which contain

$v.$ We say that

$(s,t)\in S$ is a path with preferable liquidity if it ends with

$v$ in essence,

$t=v$

So that we may compute the approximate list of preferable assets, we highlight two distinct properties that are both necessary and sufficient.

In order to capture a desired hyper-manifold liquidity property of an always preferable asset,

$G,$ , we identify such asset not only as a preferable "exit" (that is to say, the position can be unwound within a resonable amount of blocks without materially affecting price of the asset)

$vertex$, but also as the one that can be consequently traded for any other liquid asset in

$G$ at the most best settlement price.

List

$(\mathrm{manifold})$

$(\mathrm{hyper-manifold})$

$(\mathrm{supra-manifold})$

$(\mathrm{inter-manifold})$

References

cont…

Let

$S\subset V\times V$ contain all shortest paths from vertex

$s$ to vertex

$t:\mathrm{\forall }s,t\in V$ Also let vertex

$v\in V$ have the maximal

${}^{82}$ betweenness centrality measure

$C_B(v):=\sum _{s\ne t\ne v\in V}\frac{{\sigma }_{st}(v)}{{\sigma }_{st}}:\mathrm{\forall }(s,t)\in S,$ where

${\sigma }_{st}:=\sum _{(s,t)\in S}\sum _{e\in (s,t)}w(e)$

and

${\sigma }_{st}(v)$ is a sum of only those shortest paths in

$S$ which contain

$v.$

We say that

$(s,t)\in S$ is a path with preferable liquidity if it ends with

$v,$ i.e.

$t=v$

In order to capture a desired liquidity perference curve ("hyper manifold") property of an always preferable asset in an
"ALT"-system

$G,$ we must first identify such asset. Such an asset must exhibit the properties of not only having genuine liquidity (that is to say, "a preferable "exit"

$(\mathrm{sink})$ vertex"), but also as the one that can be consequently traded for any other liquid asset in

$G$ at the most attractive price.

Flash Loan led path-finding

A "liquid"

${}^{[}3]$ vertex

$v\in V\left(G^{\prime }\right)$ of a complete liquid subgraph

$G^{\prime }\subseteq G$ is called a hyper-liquid vertex when any preferable liquidity path

$p=(s,v)$

can be almost surely continued with an efficient trade for any other liquid

$u\in V\left(G^{\prime }\right),u\ne v$ in such a way that

$\sum _{e\in (s,u)}w(e)\le \sum _{e\in (s,v)}w(e)+\sum _{e\in (v,u)}w(e)$ and

$(s,u)$ is
a shortest path.

Footnotes

${}^{[}1]$ Contracts meaning Smart Contracts

${}^{[}3]$ Pockets of Liquidity is a well known microstructure in markets. Further research in the microstructrual differences between AMM's and traditional settlement procedures (e.g. LMM, FIFO, etc) must be pursued.