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Nathan Borggren
2023-09-01 21:36:51 -04:00
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7
docs/snippets/RingTDA.tex
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@@ -18,6 +18,13 @@ In practice we will simply call \textit{Giotto's} Vietoris-Rips functionality an
For a given RingCT we'd like to be able to evaluate the likelihood subrings came from a decoy selection algorithm, find similar and comparable rings.
We can also develop summary statistics about the nature of these rings and representations appropriate for Machine Learning.
In \ref{fig:alpha} we show the persistence diagram of a single ring. A log scale is shown to separate the points on the graph. A persistence diagram is a concise representation of all the information shown in Tables \ref{tab:pers2}, \ref{tab:pers}, \ref{UnionFind}.
Persistence diagrams are great at capturing structure at large scales. In \ref{tab:pers2} we see large scale structures; this guides the search to just bands of interest we can ignore or at least postpone queries for intersections in a large number of blocks when these diagrams are compared. We see as we zoom in at \ref{tab:pers}, the structure reappear as the filtration parameter is reduced.
Usual histograms have washed away a lot of this information, and require choices of bin widths that this process can circumvent (or guide).
\begin{figure}[h]
\includegraphics[scale=0.5]{block_height}
\caption{Persistence diagram showing the birth-death pairs of a single ring. A log scale is shown to separate the points on the graph. A persistence diagram is a concise representation of all the information shown in Tables \ref{tab:pers2}, \ref{tab:pers}, \ref{UnionFind}. These diagrams can be analyzed in bulk to find means, anomalies, a basis for Machine Learning and More!}

53
docs/snippets/probTDA.tex
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@@ -1,12 +1,33 @@
\section{Persistent Homology by probability}
(preliminary)
While persistence by height allows us to do some basic accounting and comparisons, it is not capturing the graph connectivity questions we are after.
Nor does it allow us to explore the taint tree probabilistically.
All transactions at a given time occupy the same set, they are not distinguishable one from the other.
We introduce another construction that lets us try to connect with graph approaches to the analysis.
We will need a notion for distance, and we refer to the cdfs and fits computed in the fit section to do so.
In the Ring object instantiation we require a set of decoys \textit{and} a reference tx, we label txo for origin transaction.
This allows us to do a few things.
The ring needn't be required to actually exist somewhere on the blockchain, we can instantiate it with a different txo and place the ring in the context of a different txo.
It is the case that rings have been re-used for different transactions \footnote{Isthmus, Rucknium private communications}, but they will still differ by different txo (must also differ in the real input too), and different hash.
These different txos change the offset, how long one must integrate to get the proper cdf, and thus the probabilities will be shifted monotonically as well.
Furthermore, we can take as input a height persistogram, along with some parameter, to find a different tx that could be `confused' with our tx (as in occupy the same simplex, and thus point to the same representative).
This parameter when set to zero will force the sampled tx to have come from the same block as the target tx, and the probabilities will be identical.
The evaluations of the cdf in particular we are interested in are the integral of the pdf from time zero (the height of the txo) to the time of the height of the ring constituent.
These give us the probabilities of the constituents being the real transaction.
We can also consider the relative probabilities by normalizing; dividing by the sum of the evaluated cdfs.
This has the more intuitive intrpretation of a weighted (currently) 16 sided dice.
We pivot to a distance notion by taking $1-q$ rather than q, so more likely things are the ones closer together, and certaicornties resolve on top of each other.
The registry objects, basically just a dictionary with keys the tx hash and values the tx object, can be used to construct the distance matrices we need to compute the homologies, or other graph metrics.
We can recover spectra and other metrics for the corresponding graphs (1-skeletons) by setting the distance to one for each ring constituent.
A first attempt at constructing these matrices is included in the Taint-Explorer notebook.
\subsection{Taint Trees}
@@ -16,8 +37,24 @@ For example a path two-hops deep with .9 connecting the first and .9 connecting
\subsection{Sampling Paths}
To sample paths each ring has a pymc categorical distribution over the RingCT that we can draw from. This distribution is also called in calls to the value of a ring or tx.
We have considered all paths with equal opportunity at this stage.
Fig. \ref{fig:ptoc} shows a histogram of 3300 paths to coinbase from a transaction.
We haven't parameterized this histogram at this stage, but we expect it to be exponential with mean related to the probability of drawing a coinbase transaction out of the ring, which terminates the sampling path.
We can construct persistence diagrams for any of these paths, height paths are used to show the four diagrams in Table \ref{tab:pdsSP}.
For a given decoy selection algorithm, (or series, since this changes with block height), we can evaluate the likelihood of a given path to occur.
Dynamic partition functions, that are weighted path integrals like this here, are called Maximum Caliber and have utility in statistical mechanics when the observables observed are not the energy paramater, but a categorical state (folded/unfolded, orbiting stationary points A,B,C etc. ).
These diagrams are used to estimate the value of a given tx, and to probabilistically sample the taint tree.
We can also look at a distribution of the values of the coinbases at \ref{fig:valsc}.
We expect taint trees of different txes with common true source to have comparable statistics.
We need to check if transactions which could have been used interchangably as a decoy, also generate similar statistics.
As mentioned in the Value section, these distrubitons of coinbases can be used to generate a probabilistic notion of value of an unknown tx.
\includegraphics[scale=0.5]{pathstocoinbase}
\begin{figure}[h]
\includegraphics[scale=0.5]{pathstocoinbase}
\caption{A histogram of the length it takes to get to a coinbase, drawn from 3300 samples of a single transaction. These values can be used in the value expectation}
\label{fig:ptoc}
\end{figure}
\begin{center}
\begin{table}
@@ -27,8 +64,14 @@ To sample paths each ring has a pymc categorical distribution over the RingCT th
\includegraphics[scale=0.3]{pds_0} & \includegraphics[scale=0.3]{pds_1} \\ \hline
\includegraphics[scale=0.3]{pds_2} & \includegraphics[scale=0.3]{pds_3} \\ \hline
\end{tabular}
\caption{As the filtration progresses, holes are filled, j}
\caption{Persistence diagrams of four sampled paths to coinbase. Diagrams with a few points have short trips to coinbase, diagrams with a lot of points have a lot of transactions prior to making it to coinbase. The spacings within the diagram specifies how large of block jumps were required to make it there.}
\label{tab:pdsSP}
\end{table}
\end{center}
\includegraphics[scale=0.5]{valscoinbase}
\begin{figure}[h]
\includegraphics[scale=0.5]{valscoinbase}
\caption{A distribution of the values at coinbase of the separate paths.
An estimate of the value of an unknown tx is the mean of this distribution times the number of inputs divided by the number of outputs.}
\label{fig:valsc}
\end{figure}

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overseer.txt Normal file
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@@ -0,0 +1,17 @@
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sandbox/Taint_Explorer.ipynb Normal file
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3
src/MoneroAna.py
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@@ -7,6 +7,7 @@ import gtda
from gtda import homology
import pymc as pm
import random
import pandas as pd
VR = homology.VietorisRipsPersistence(homology_dimensions=[0])
@@ -106,11 +107,13 @@ class ring:
#self.txo.sinks.append(j)
if j in registry:
registry[j].sources.append(self.txo.tx_hash)
registry[self.txo.tx_hash].sinks.append(j)
mixins.append(registry[j])
else:
registry[j] = tx(j, registry=registry)
registry[j].sources.append(self.txo.tx_hash)
registry[self.txo.tx_hash].sinks.append(j)
mixins.append(registry[j])
return mixins