Graph_Sampler User’s Manual

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Table of Contents

1 Software and Documentation Licenses

1.1 Software license

Graph_Sampler is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

1.2 Documentation license

Version 1.2, November 2002

Copyright © 2000,2001,2002 Free Software Foundation, Inc.
51 Franklin St, Fifth Floor, Boston, MA  02110-1301, USA

Everyone is permitted to copy and distribute verbatim copies
of this license document, but changing it is not allowed.

1. PREAMBLE

   The purpose of this License is to make a manual, textbook, or other functional and useful document free in the sense of freedom: to assure everyone the effective freedom to copy and redistribute it, with or without modifying it, either commercially or noncommercially. Secondarily, this License preserves for the author and publisher a way to get credit for their work, while not being considered responsible for modifications made by others.

   This License is a kind of “copyleft”, which means that derivative works of the document must themselves be free in the same sense. It complements the GNU General Public License, which is a copyleft license designed for free software.

   We have designed this License in order to use it for manuals for free software, because free software needs free documentation: a free program should come with manuals providing the same freedoms that the software does. But this License is not limited to software manuals; it can be used for any textual work, regardless of subject matter or whether it is published as a printed book. We recommend this License principally for works whose purpose is instruction or reference.

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2 Overview

Graph_Sampler is an inference and simulation tool for networks (understood as graphs). It can simulate random graphs for general directed graphs (eventually cyclic) (see Bois & Gayraud 2013) or for directed acyclic graphs (Bayesian networks). The graphs are generated by Markov chain Monte Carlo simulations and their structure can be specified to follow probabilistic properties through the use of prior distributions. In the case of Bayesian networks, you can also infer about their probable structure through the joint use of priors and data about node values (via a likelihood function).

2.1 General procedure

You write an input files and run compiled graph_sampler program. The input files specifies the kind of graph to simulate, some simulation parameters and output options, the priors you want, the eventual data and their likelihood (see Running Graph_Sampler). The simulation output is written to standard ASCII files.

No knowledge of computer programming is required, unless you want to tailor the program to special needs (in which case you may want to contact us).

3 Installation

3.1 System requirements

Graph_Sampler is written in ANSI-standard C language. We are distributing the source code and you should be able to compile it for any system, provided you have an ANSI C compliant compiler.

On any system we recommend the GNU gcc compiler (freeware). An automated compilation script (called Makefile) is provided and can be used if the standard command make is available to you.

If you want to modify the input file parser you will need lex and yacc (that is for experienced C programmers).

3.2 Distribution

Graph_Sampler source code is available on Internet through:

https://sites.google.com/site/utcchairmmbsptp/graph_sampler.

3.3 Machine-specific installation

3.3.1 Unix and GNU/Linux operating systems

To install on a Unix or GNU/Linux machine, download (in binary mode) the distributed archive file to your machine. Place it in a directory where there is no existing graph_sampler subdirectory that could be erased (make sure you check that). Decompress the archive with GNU gunzip (gunzip <archive-name>.tar.gz). Untar the decompressed archive with tar (tar xf <archive-name>.tar) (do man tar for further help). Many other archiving tools can be used in place of gunzip and tar. Move to the graph_sampler directory just created and issue the following commands:

make

This command compiles the graph_sampler program.

You can also compiles this manual as an info file with the command make info or as an html file with make html.

3.3.2 Other operating systems

Under other operating systems (Windows, etc.) or if everything else fails you should be able to both uncompress and untar the archive with widely distributed archiving tools. Refer to the documentation of your C compiler to create an executable file from the source code files provided.

You are now ready to use Graph_Sampler.

4 Running Graph_Sampler

After having compiled graph_sampler, you are ready to run it. For this you need to write an input file. This chapter explains how to write such files with the proper syntax.

In Unix the command-line syntax to run that executable is simply:

graph_sampler [input-file [output-prefix]]

where the brackets indicate optional arguments. If no input file or/and output prefix are specified, the program will use defaults. The default input file name is script.txt, the output files created depend on your selection in the input file (see below) and their name is printed on exit. Default output file names are best_graph.out, graph_samples.out, degree_count.out, motifs_count.out, edge_p.out, results_mcmc.bin. If you just specify an input file name, output file names will still be the default ones. If you specify an input file name and an output prefix the standard output file names will be prefixed by it (i.e., with the prefix my the edge probabilities output file will be named my_edge_p.out).

When the program starts, it announces which model description file was used to create it. While the input file is read or while simulations are running, some informations will be printed on your computer screen. They can help you check that the input file is correctly interpreted and that the program runs as it should. Graph_Sampler can also post error messages, which should be self-explanatory. Where appropriate, they show the line number in the input file where the error occurred.

The program ends (if everything is fine) by giving you the name of the output file generated. If you want to run the program in batch mode (in the background), you may want to redirect the screen output and error messages; refer for this to the man pages for your command shell.

4.1 Input file syntax

An input files specifies the kind of graph to simulate, some simulation parameters and output options, the priors you want, the eventual data and their likelihood. All that is done through the specification of predefined variables, using some keywords, user defined variables, numbers and operators.

A Graph_Sampler input file is a text (ASCII) file that obeys a relatively simple syntax:

• An input file can contain statements, matrix definitions and comments.
• Statements and matrix definitions must end with ’;’ and can span several lines. They can be placed in any order, except that matrix sizes (defined by specific predefined variables) must be defined before the corresponding matrix definition.
• Comments start with the pound sign # and go up to the end of line. They are ignored. Example:

  # this is a comment, comments are useful
  

• Variables are user defined symbols whose name must start with a letter, followed eventually by other letters, numbers and _. Letters can be upper-case or lower-case. Variable names are case sensitive. Example:

  Xa_2
  

  Note that unassigned variables have a default value of zero.

• Predefined variables are reserved names that Graph_Sampler understands. An exhaustive list of the predefined variables is given in the next section. Predefined variables not explicitly assigned have (hopefully useful) default values which may differ from zero (refer to their description, below). Example:

  nNodes
  

• Expressions are numbers (as in C) or formulae including numbers and/or user defined variables, operators (’+’, ’-’, ’*’, ’/’) or parentheses. Formulae are computed at they appear, with usual precedences. The division is always a real division (not an integer division). Example:

  (5 + 6) * (3.4 / 1.1E-8) + Xa_2;
  

• Statements are in the format:

  <variable> = <expression>;.

  Example:

  X_a2 = 5000;
  nNodes = 6 * Xa_2;
  

• White space consist of space, tab or carriage return. Several white space characers in a row are treated as just one white space. Example:

  Xa_2 =    (2   + 3)    /
            (25. - 5.76);
  

• Lists are comma separated lists of expressions. Example:

  1, 2, 2+1, 2*2, 5, Xa_2
  

• Array definitions can only be used with predefined variables at the lefthand side. They are in the forms: <variable> = array{<list of expressions>};.

  The term array is a reserved keyword (see the list of those keywords below). Example:

  n_data_levels = array{2, 2, 1+1};
  

• Matrix definitions can only be used with predefined variables at the lefthand side. They are in the forms:

  <variable> = matrix{<list of expressions>};.

  The term matrix is a reserved keyword (see the list of those keywords below). Example:

  data = matrix{
  1,   2,  2+1, 
  2*2, 5,  Xa_2};
  

  That is the general form, some matrices can accept keywords such as empty, full, or random instead of a list of expression inside the curly braces (see the specification of each predefined matrix, below).

4.2 Predefined variables

Here are the predefined variables that Graph_Sampler understands (they may have different synomyms, separed by commas):

bBN, bayesian_network

The predefined variable bBN indicate whether the graphs to sample are Bayesian networks (in that case it should be set to 1 or true) or general directed graphs (in which case it should set to 0 or false). General directed graphs can only be simulated on the basis of priors. For Bayesian networks both simulation and structural inference can be performed. The default value for bBN is false. Example:

bBN = true; # bBN = 1 would also work

dynamic_bayesian_network

If dynamic_bayesian_network is set to to 1 or true the graphs sampled are dynamic Bayesian networks. For such networks both simulation and structural inference can be performed. The default value for dynamic_bayesian_network is false.

nNodes, n_nodes

The number of nodes in the network considered is specified by setting nNodes to an integer (not long integer) value. nNodes must be set before the initial adjacency or prior on edges’ probability matrices are defined. The default value for nNodes is 0, which raises an error message, because nNodes should be set to a meaningful value.

autocycle

The autocycle variable should be set to 1 (true) if edges from a node to itself are allowed, and to 0 (false) otherwise. Its default value is false. Setting it to true is incompatible with specifying bBN to true (loops are not allowed in Bayesian networks).

initial_adjacency

The starting value of the graph adjacency matrix is defined by setting initial_adjacency, a square matrix of dimension nNodes. Matrix elements should be either 0 or 1. Element [i,j] is set to 1 if an edge (link) goes from node i to node j. Setting it to 0 indicates no edge between the two nodes. Example:

nNodes = 3;
initial_adjacency = matrix {0, 0, 0, 
                            1, 0, 0,
                            1, 0, 0};  

initial_adjacency definition can also use an extended syntax:

initial_adjacency = matrix{empty | full | random};.

were “|” means “or”.

• If the keyword empty is used all elements will be set to zero.
• If full is used all elements will be set to 1 when bBN is false. If bBN is true the diagonal elements will be set to zero and the others to 1. If you want it to work with Bayesian networks, you should set bBN before defining initial_adjacency, because its default value is false.
• If random is used all elements will be set randomly to 0 or 1 (with equal probability) when bBN is false. If bBN is true the diagonal elements will be set to zero and the others to 0 or 1. If you want it to work with Bayesian networks, you should set bBN before defining initial_adjacency, because its default value is false.

hyper_pB

The matrix hyper_pB is a square matrix of dimension nNodes which specifies a prior distribution on edge probabilities. Each element [i,j] of hyper_pB is the parameter p (a real of double format) of a Bernoulli distribution for the presence of an edge from node i to node j. In the case of Bayesian networks, p values should be 0 on the first diagonal. Example:

bBN = true;
hyper_pB = matrix {0,   0.1, 0.1,
                   0.9, 0,   0.1,
                   0.9, 0.1, 0  };

Internally, hyper_pB is always used. If it is not defined by the user, p values will default to 0.5 (with zeroes on the diagonal if bBN is true), so that the prior is neutral (equal probability for the absence or presence of any edge).

In the case of a Bayesian network, nodes which have been assigned a zero probability of having parents (a column of zero in the hyper_pB matrix) are understood to be special “control” nodes for which the likelihood will not be computed. Such nodes will typically correspond to experimental design variables. Their likelihood is not computed. They condition the likelihood of their eventual children node and then take the values assigned to them in the input file (in which case the “data” are rather forcing values than actual observations).

bPriorConcordance, concordance_prior

The flag bPriorConcordance set to 1 or true indicates that a concordance prior should be used (in addition to the baseline Bernoulli prior on individual edges). By default it is false. A concordance prior is an unnormalized score of the edge-wise difference between a reference adjacency matrix and the matrix being examined (see below edge_requirements).

edge_requirements

The matrix edge_requirements is a square matrix of dimension nNodes which specifies the concordance between the edges of a reference adjacency matrix and the current one. Each element [i,j] of edge_requirements can take a value of 1, -1, or 0.

• The value 1 indicates that an edge from node i to node j is desired (rather than strictly required). The presence of that edge in the adjacency matrix evaluated raises its score by lambda_concordance (see below).
• The value -1 indicates that an edge from node i to node j is not desired. The presence of that edge in the adjacency matrix evaluated lowers its score. In the case of Bayesian networks, diagonal values should be -1, otherwise Hell might break loose.
• A value of 0 indicates no preference: the score is unaffected by the presence of absence of an edge from node i to node j. The presence of that edge in the adjacency matrix evaluated raises its score.

Example:

bPriorConcordance = true;
edge_requirements = matrix {-1, -1,  0,
                             1, -1, -1,
                             1, -1, -1};

By default, all elements of edge_requirements will be set to 0 when bBN is false. If bBN is true the diagonal elements will be set to -1 and the others to 0. If you want it to work with Bayesian networks, you should set bBN before defining edge_requirements, because its default value is false.

lambda_concord, lambda_concordance

The parameter lambda_concord is used to weight the differences between the reference adjacency matrix and the current adjacency matrix when bPriorConcordance is true. It should be set to a double (typically superior to zero). Its default value is 1.

bPriorDegreeNode, degree_prior

The flag degree_prior set to 1 or true indicates that an exponential prior is placed on the distribution of the nodes’ degrees (the number of incoming and outgoing edges for a given node) (see Bois & Gayraud 2013). It comes in addition to the baseline Bernoulli prior on individual edges. By default it is false.

gamma_degree

If bPriorDegreeNode is true, gamma_degree specifies the parameter of the exponential prior on degree counts. It should be set to a double (typically superior to zero). Its default value is 1.

bPriorMotif, motif_prior

The flag bPriorMotif set to 1 or true indicates that a beta-binomial prior is placed on the count of triangular feed-forward and feedback loops in the network (see Bois & Gayraud 2013). It comes in addition to the baseline Bernoulli prior on individual edges and is incompatible with Bayesian networks (an error message will be issued). By default it is false.

alpha_motif

If bPriorMotif is true, alpha_motif specifies the first parameter of the beta-binomial prior on loops’ counts. It should be set to an integer superior to zero. Its default value is 1.

beta_motif

If bPriorMotif is true, beta_motif specifies the first parameter of the beta-binomial prior on loops’ counts. It should be set to an integer superior to zero. Its default value is 1.

nData, n_data

If bBN is true, data can be input to infer on the probabiliy of the presence of edges on the basis of priors and data likelihood, in a fully Bayesian framework. The predefined variable nData should be set to an integer equal to the number of data points per node. Its default is zero. If no data are provided while bBN is true, simulations will proceed simply on the basis of priors distributions.

data

After nNodes, nData and bBN have been defined, a data matrix can also be defined (actually if nData is different from zero it must be defined). nData has no default value. It should have nNodes rows and each row should be a vector of nData values (of format doubles). Example:

nNodes = 3;
bBN = true;
nData = 4;
data = matrix {1.1, 1.3, 1.4, 1.35,
               2.1, 2.4, 2.5, 2.45,
               3.4, 3.6, 3.8, 3.85};

bDirichlet, dirichlet_score

The data likelihood is by default normal given a vague normal-gamma prior for the regression parameters. An alternative, for discrete data, is to use a Dirichlet-multinomial model (See Laskey and Myers 2003, Heckerman et al. 1994, Heckerman et al. 1995) . To that effect you just need to set dirichlet_score to true (its default value is false). In that case the data have to be coded by integers from zero to n. The number of levels for each node has to be specified using an n_data_levels declaration. The Dirichlet hyper-parameters are internally set to one, specify a uniform prior on configurations of parents for any node.

n_data_levels, nData_levels

If a Dirichlet-multinomial model is used, discrete data have to be specified for each node. Such data have to be coded as integers from zero to n, n being the number of levels for a given node. Those levels are specified using the n_data_levels array declaration.

n_data_levels = array{2, 2, 3, 2, 4};

bZellner, zellner_score

An alternative to the default data model is to use Zellner’s score. To that effect you just need to set zellner_score to true (its default value is false). The drawback is that any node cannot have more parents than it has data about it (arguably, that’s an artificial constraint).

nRuns, n_runs

The total number of iterations to be performed by the MCMC sampler is specified by setting nRuns to a long integer value. Its default value is 1000000000 (yes, a billion).

nBurnin, n_burnin

A certain number of “burn-in” iterations can be specified by setting nBurnin to a long integer value. In that case the MCMC chain recording, and computation of summary outputs (such as the edge probabilities) starts only after nBurnin iteration. Its default value is zero. This is typically used to discard the part of the MCMC chain that is not at equilibrium. However, checking that equilibrium is attained is best done, in our opinion, by running multiple independent chains and using Gelman and Rubin R^ diagnostic (see Gelman & Rubin 1992 and other relevant statistical literature).

seed, random_seed

The starting value of the pseudo-random generator seed can be explicitly set to any long integer number superior to zero. That allows repeating exactly the same sequence random numbers. That is required to generate different chains for the same problem in order to check the convergence of the MCMC simulations. If it is not set by the user, seed has a default value of 314159265.3589793.

bsave_the_chain, save_chain

The MCMC sampling chain can be saved in binary format to a file (named results_mcmc.bin) by setting bsave_the_chain to 1 or true. By default, the chain is not saved. Beware, MCMC chains can be very large, even though the recording format is very compact: results_mcmc.bin starts with the number of nodes in the graph (as a binary integer, i.e. a byte), followed by the value of the adjacency matrix (nNodes by nNodes bytes) at the end of burn-in period, followed by a one-byte encoding of the difference between successive adjacency matrices. The difference d between adjacency matrices (equal to -1 for removing an edge, +1 for adding an edge) and its location [i,j] are encoded as: (i + j * nNodes + 1) * d. No difference is encoded as zero. The results_mcmc.bin file can be used to recreate the successive adjacency matrices sampled.

nSaved_adjacency, n_saved_adjacency

The user can request the output of a number n_saved_adjacency (integer) of randomly generated adjacency matrices. Those matrices are saved at regularly spaced iterations along the MCMC chain (after the burn-in period) in the file graph_samples.out in text format, along with the logarithmes of the prior probability, data likelihood (if data were specified) and posterior probability. By default n_saved_adjacency is zero and no matrices are recorded.

bsave_best_graph, save_best_graph

By setting bsave_best_graph to true, the user can request the output of the adjacency matrix of the graph having the highest posterior probablity among all random graphs generated after the burn-in period. That matrice is saved in the file best_graph.out in text format, along with the logarithmes of its prior probability, data likelihood (if data were specified) and posterior probability. By default bsaved_best_graph is false.

bsave_the_edge_probabilies, save_the_edge_probabilies

Setting bsave_the_edge_probabilies to true, forces the output of a matrix of the individual edge probabilities in the file edge_p.out, in text format. By default bsaved_the_edge_probabilies is false.

bsave_the_degree_counts, save_the_degree_counts

Setting bsave_the_degree_counts to true, forces the output of a count of the nodes’ degrees in the graphs sampled after the burn-in period to the file degree_count.out, in text format. By default bsaved_the_degree_counts is false.

bsave_the_motifs_probabilies, save_the_motifs_probabilies

Setting bsave_the_motifs_probabilies to true, forces the output (to the file motifs_count.out, in text format) of a count of triangular feed-forward and feedback loops in the graphs sampled after the burn-in period. By default bsaved_the_motifs_probabilies is false.

4.3 Reserved keywords

The following keywords can be used in Graph_Sampler input files:

false (or FALSE) keyword

This keyword is equivalent to zero and can be used when assigning variables.

true (or TRUE) keyword

This keyword is equivalent to 1 and can be used when assigning variables.

array keyword

This keyword is used for vector definition. Example:

n_data_levels = array {2, 2, 1+1};

matrix keyword

This keyword is used for matrix definition. Example:

data = matrix {1.1, 1.3, 1.4, 1.35,
               2.1, 2.4, 2.5, 2.45,
               3.4, 3.6, 3.8, 3.85};

empty keyword

This keyword can be used to create an empty initial adjacency matrix (syntax: matrix{empty};).

full keyword

This keyword can be used to create a full initial adjacency matrix (all elements at 1, except a zeroed diagonal in Bayesian networks) (syntax: matrix{full};).

random keyword

This keyword can be used to create a random initial adjacency matrix (all elements 0 or 1 at random, except a zeroed diagonal in Bayesian networks) (syntax: matrix{random};).

Bibliographic References

Barry T.M. (1996). Recommendations on the testing and use of pseudo-random number generators used in Monte Carlo analysis for risk assessment. Risk Analysis 16:93-105.

Bernardo J.M. and Smith A.F.M. (1994). Bayesian Theory. Wiley, New York.

Bois F. and Gayraud G. (2013). Probabilistic generation of random networks taking into account information on motifs occurrence, arXiv:1311.6443 [q-bio.QM].

Gelman A. and Rubin D.B. (1992). Inference from iterative simulation using multiple sequences (with discussion). Statistical Science 7:457-511.

Heckerman et al. (1994). in Proceedings of Tenth Conference on Uncertainty in Artificial Intelligence, Seattle, WA, p. 293-301. Morgan Kanfmann.

Heckerman et al. (1995). Machine Learning, 20, 197-243.

Laskey and Myers (2003). Machine Learning, 50:175-196.

Concept Index