compnet 0.0.2

A library for sampling correlated binary variates.

compnet — Compression for Market Network data

About

compnet is a package for market compression of network data.

It is based on xxx.

How to get started

Given a dataframe el containing a network's edge list, start by constructing the graph representation $G$ via the class compnet.Graph:

import pandas as pd
import compnet as cn

el = pd.DataFrame([['A','B', 10],
                   ['B','C', 15],
                   ['B','A', 5],
                   ],
                  columns=['SOURCE', 'TARGET' ,'AMOUNT'])
g = cn.Graph(el)

If the dataframe does not contain columns named 'SOURCE', 'TARGET', and 'AMOUNT', the corresponding column names should be passed as well to compnet.Graph via the parameters source, target, and amount.

For example:

el = pd.DataFrame([['A','B', 10],
                   ['B','C', 15],
                   ['B','A', 5],
                   ],
                  columns=['bank', 'counterpart' ,'notional'])
g = cn.Graph(el, source='bank', target='counterpart', amount='notional')

Once the graph object g is created, it is possible to quickly inspect its properties as

g.describe()

which returns the gross, compressed, and excess market sizes of the graph

┌─────────────────┬──────────┐
│                 │   AMOUNT │
├─────────────────┼──────────┤
│ Gross size      │       30 │
│ Compressed size │       15 │
│ Excess size     │       15 │
└─────────────────┴──────────┘

Denoting by $A$ the weighted adjacency matrix of the network with elements $A_{ij}$, the gross, compressed, and excess market sizes are respectively defined as

$$ GMS = \sum_{i}\sum_{j} A_{ij} $$

$$ CMS = \frac{1}{2}\sum_i\left|\sum_j \left(A_{ij} - A_{ji}\right) \right| $$

$$ EMS = GMS - CMS $$

Notice in particular that $\sum_j \left(A_{ij} - A_{ji}\right)$ represents the net position of node $i$.

---

At this point, it is possible to run a compression algorithm on g via the method Graph.compress. For any two graphs one can further compute the compression efficiency

$$CE = 1 - \frac{EMS_2}{EMS_1} $$

with $EMS_j$ the excess market size of graph $j$. Moreover, the compression ratio of order p for two adjacency matrices $A$ and $A^c$ is defined as

$$CR_p(A, A^c) = \frac{2}{N(N-1)}\frac{||L(A^c, N)||_p}{||L(A, N)||_p} $$

with $N$ the number of nodes and $||L(A, N)||_p$ the $p$-norm of xxx

$$||L(A, N)||_p = \left( \frac{2}{N(N-1)} \sum_i \sum_{j=i+1} |A_{ij}|^p \right)^{1/p}$$

and the compression factor of order p for two adjacency matrices $A$ and $A^c$ as

$$CF_p(A, A^c) = 1 - CR_p.$$

Four options are currently available: bilateral, c, nc-ed, nc-max.

Bilateral compression

Bilateral compression compresses only edges between pairs of nodes. In our example above there exists two edges (trades) in opposite directions between node A and node B, which can be bilaterally compressed.

Running

g_bc = g.compress(type='bilateral')
g_bc

returns the following bilaterally compressed graph object

compnet.Graph object:
┌──────────┬──────────┬──────────┐
│ SOURCE   │ TARGET   │   AMOUNT │
├──────────┼──────────┼──────────┤
│ A        │ B        │        5 │
│ B        │ C        │       15 │
└──────────┴──────────┴──────────┘

with compression efficiency and factor

Compression Efficiency CE = 0.667
Compression Factor CF(p=2) = 0.718

Conservative compression

Under conservative compression onlyexisting edges (trades) are reduced or removed. No new edge is added.

The resulting conservatively compressed graph is always a sub-graph of the original graph. Moreover, the resulting conservatively compressed graph is always a directed acyclic graph (DAG).

The conservatively compressed graph can be obtained as

g_cc = g.compress(type='c')
g_cc

which in our example above returns

compnet.Graph object:
┌──────────┬──────────┬──────────┐
│ SOURCE   │ TARGET   │   AMOUNT │
├──────────┼──────────┼──────────┤
│ A        │ B        │        5 │
│ B        │ C        │       15 │
└──────────┴──────────┴──────────┘

with compression efficiency and factor

Compression Efficiency CE = 0.667
Compression Factor CF(p=2) = 0.718

Non-conservative ED compression

...

Maximal non-conservative compression

...

The non-conservative maximally compressed graph can be obtained as

g_cc = g.compress(type='nc-max')
g_cc

which in our example above returns

compnet.Graph object:
┌──────────┬──────────┬──────────┐
│ SOURCE   │ TARGET   │   AMOUNT │
├──────────┼──────────┼──────────┤
│ B        │ C        │       10 │
│ A        │ C        │        5 │
└──────────┴──────────┴──────────┘

with compression efficiency and factor

Compression Efficiency CE = 1.0
Compression Factor CF(p=2) = 0.801

Grouping along additional dimensions

If an additional dimension exists...

Authors

Luca Mingarelli, 2022