Drawing uniformly graphs under partition constraints (Partition Adjacency Matrix). Commonly used for network testing.

# Uniform Graph Draw

This package implements random draw algorithm for networks. In particular it creates uniform samples of networks with a given degree-sequence and partition constraints (fixed number of crossing edges/arrows between node-groups in partition). The literature reefers to this set of constraint also as Partition Adjacency Matrix (PAM) restrictions.

It is implemented according to the paper:

## Get it Running

Install the paper via pip:

• pip install ugd

then run

``````#import modules
import ugd
import numpy

# create dictionary of node attributes
var_dict ={
0: {'gender': 'm'},
1: {'gender': 'm'},
2: {'gender': 'f'},
3: {'gender': 'f'},
}

# UNDIRECTED CASE, test whether there are abnormal many connection between groups:

# test undirected: note there are 3 possible networks, and only one has no male, female edges

# DIRECTED CASE, test a utility funktion with the local optimal test:

# testing whether there is taste for reciprocity:
# the utility is reciprocity -> i gets utility form an edge to j if j has an edge to i

# there are 9 graphs, for 3 the reciprocity is high, for 6 low, due to symmetry the optimal stat only takes on 2 values

print("is stat value of the original graph:")
print(out_dict["info_dict"]["original_value"])
print("the values of the simulated graphs:")
print(out_dict["stat_list"])
var_dict = var_dict, test_variable= ('gender','m','f'),mixing_time=1000, anz_sim=100, show_polt=True)
``````

### Working with ugd

The easiest way to use ugd is by simply passing in the adjacency matrix and set show_plot=True. This runs the simulation algorithm and plots a default statistic.

The statistic can be customized. Firstly by entering a dictionary with node characteristics and testing for one characteristic. Secondly by writing a costume test statistic and enter it into the function as 'stat_f'. How to write a "locally most powerful" test statistic for a specific network formation game is derived in Pelican, A. & Graham, B. S. (2019). The weights for the optimal test statistic is not calculated by this package, it has to be done with other statistical packages and feed in via 'stat_f' or directly applied to the list of graphs returned by the ugd package.

Node characteristic can be added as controls. The algorithm then generated uniformly graphs with also have the same number of edges between the node-groups induced by the controls. Note that the algorithm is slower if many controls are added. Hard constraints (where there are no edges within, or some the groups), such as the group constraint in a bipartite graph do not slow the algorithm.

The processing of the individual graphs can be easily customized by working directly with the simulated graphs.

An entry point of testing social and economic networks can be found here https://arxiv.org/abs/1908.00099.

## API

There are 3 functions provided.

1. graph_hyp_test
• generating a sequence of uniform sampled graphs under the desired set of constrains.
2. digraph_hyp_test
• generating a sequence of uniform sampled digraphs under the desired set of constrains.
3. make_lcl_mst_pwf_stat
• making a locally optimal test statistic from the edge utility and the observed network. The locally optimal test statistic can then be used in digraph_hpy_test.

For the API the first two functions only differs in that the interpretation of the adjacency matrix is once as digraph representation and once as graph representation.

``````INPUT:
:param adj_m:         A numpy array containing 0 and 1s as elements, representing
:param var_dict:      A dictionary with the integers 1..n as primary key (representing
the n nodes). The values are dictionaries containing the
Variable name as keys and the values can either be numbers or be
numbers or strings
:param stat_f:        A function which maps the adj_m and var_dict to a number "the
statistic of interest".
:param test_variable: Alternative to stat_f, creating a statistic which counts the
arrows form a node-subset into another. It is a triple with
first element variable name, second the value of the variable
for the set where the arrows leave and third the value of the
subset where the arrow go to.
:param controlls:     List of variable names, the number of arrows crossing the groups
induced by the controls is constant in all the simulation.
:param mixing_time:   Number of runs (steps in the markov graph) before a the graph
is considered random
:param anz_sim:       Number of simulations
:param show_polt:     Boolean whether a plot is desired

OUTPUT:
:return: out_dict     Dictionary with keys 'graph_list', 'stat_list', 'plot',
and 'info_dict'
graph_list:           List of random adjacency matrices with the given degree-sequence
and arrows between the controls
stat_list:            List of the statistics stat_f evaluated for the random graphs
plot:                 Plot with the illustration of the estimation output
info_dict:            Dictionary with the information about the simulation
``````

The API for make_lcl_mst_pwf_stat:

``````INPUT:
:param adj_m:                A numpy array containing 0 and 1s as elements, representing
:param var_dict:             A dictionary with the integers 1..n as primary key (representing
the n nodes). The values are dictionaries containing the
Variable name as keys and the values can either be numbers or be
numbers or strings
:param controlls:            List of variable names, the number of arrows crossing the groups
induced by the controls is constant in all the simulation.
:param edge_util_function:   A function mapping the an adjency matrix to a numpy matrix,
where the entries are the corresponding utility the agent
would get from forming the edge.

OUTPUT:
:param localy_optimal_stat:  A function which maps the adj_m and var_dict to a number "the
locally optimal statistic for the edge utility".
``````

## Architecture:

All the logic is implemented in the digraph_draw folder. it is divided into

• markov_walk

Implementation of algorithm 1 from the paper Markov Draw Algorithm

• schlaufen_construction

Implementation of algorithm 2 from the paper Schlaufen Detection Algorithm

• model

containing the data models (appropriate Graph representation and node representation for efficient construction of the altering paths in the Schlaufen)

• user_interface

Contains the all the logic used for input validation, parsing of input, estimation of runtime, transformation of the graph format, output processing.

• help_functions

### Comment

The current implementation, includes only controlling of a fixed number of crossing edges/arrows between node-groups as constraints. More complex complex can be implemented by writing a consum implementation of the no_violation function in constraint_violation_check. Note, that depending on the constraint the construction of the Schlaufensequence should not be stopped because a feasible one is found, but only due to the random stop. This in order to preserve correctness.

## Project details

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