Volumetric Manifolds

# Volumetric Manifolds > A constructed mechanism for facilitation of efficient and effective contract$^[1]$ trading $$ G:=(V, E, w) $$ weighted by the function $w: E \rightarrow \mathbb{R}^{+}$ corresponding to the price of some trade $e \in E$ $$ V,|V| \leq|\mathbb{N}| $$ Consider an abstract liquid trading (ALT) system as a weighted directed graph $G:=(V, E, w),$ where set of vertices $V,|V| \leq|\mathbb{N}|$ 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 $w: E \rightarrow \mathbb{R}^{+}$ corresponding to the price of some trade $e \in E$ ### Liquidity Point/Vertex $$ \text { Any liquid vertex } v \in V \text { has both } d e g^{-}(v) \geq 1 \text { and } \operatorname{deg}^{+}(v) \geq 1 $$ ### Liquidity Indifference Preference Paths$^{2}$ $$C_{B}(v):=\sum_{s \neq t \neq v \in V} \frac{\sigma_{s t}(v)}{\sigma_{s t}}: \forall(s, t) \in S, \text { where } \sigma_{s t}:=\sum_{(s, t) \in S} \sum_{e \in(s, t)} w(e)$$ Vertex $v \in V$ represents half-liquid asset when either $\operatorname{deg}^{-}(v)=0$ (source) $\operatorname{or} \operatorname{deg}^{+}(v)=0(\operatorname{sink}),$ where $\operatorname{deg}^{(-1+)}: V \rightarrow \mathbb{N}$ 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 $t: \forall s, t \in V$ Also let vertex $v \in V$ have the maximal $^{82}$ betweenness centrality measure $C_{B}(v):=\sum_{s \neq t \neq v \in V} \frac{\sigma_{s t}(v)}{\sigma_{s t}}: \forall(s, t) \in S,$ where $\sigma_{s t}:=\sum_{(s, t) \in S} \sum_{e \in(s, t)} w(e)$ and $\sigma_{s t}(v)$ 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 $(\operatorname{manifold})$ $(\operatorname{hyper-manifold})$ $(\operatorname{supra-manifold})$ $(\operatorname{inter-manifold})$ ## References cont... ## Let $S \subset V \times V$ contain all shortest paths from vertex $s$ to vertex $t: \forall s, t \in V$ Also let vertex $v \in V$ have the maximal $^{82}$ betweenness centrality measure $C_{B}(v):=\sum_{s \neq t \neq v \in V} \frac{\sigma_{s t}(v)}{\sigma_{s t}}: \forall(s, t) \in S,$ where $\sigma_{s t}:=\sum_{(s, t) \in S} \sum_{e \in(s, t)} w(e)$ and $\sigma_{s t}(v)$ 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" $(\operatorname{sink})$ 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 $u \in V\left(G^{\prime}\right), u \neq v$ in such a way that $\sum_{e \in(s, u)} w(e) \leq \sum_{e \in(s, v)} w(e)+\sum_{e \in(v, u)} w(e)$ 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.