## Digraph 3 Plus VERIFIED

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The basic idea of the Digraph3 Python resources is to make easy python interactive sessions or write short Python3 scripts for computing all kind of results from a bipolar-valued digraph or graph. These include such features as maximal independent, maximal dominant or absorbent choices, rankings, outrankings, linear ordering, etc. Most of the available computing resources are meant to illustrate a Master Course on Algorithmic Decision Theory given at the University of Luxembourg in the context of its Master in Information and Computer Science (MICS).

Using the Digraph3 modules is easy. You only need to have installed on your system the Python programming language of version 3.+ (readily available under Linux and Mac OS). Notice that, from Version 3.3 on, the Python standard decimal module implements very efficiently its decimal.Decimal class in C. Now, Decimal objects are mainly used in the Digraph3 characteristic r-valuation functions, which makes the recent python-3.7+ versions much faster (more than twice as fast) when extensive digraph operations are performed.

In Listing 1.1 we import, for instance, from the randomDigraphs module the RandomDigraph class in order to generate a random digraph object dg of order 5 - number of nodes called (decision) actions - and arc probability of 50%. We may directly inspect the content of python object dg (Line 3).

We are starting this tutorial with generating a uniformly random [-1.0; +1.0]-valued digraph of order 7, denoted rdg and modelling, for instance, a binary relation (x S y) defined on the set of nodes of rdg. For this purpose, the Digraph3 collection contains a randomDigraphs module providing a specific RandomValuationDigraph constructor.

With the save() method (see Listing 1.4 Line 3) we may keep a backup version for future use of rdg which will be stored in a file called __tutRandValDigraph.py__ in the current working directory. The genuine Digraph class constructor may restore the rdg object from the stored file (Line 4). We may easily inspect the content of rdg (Lines 5). The digraph size 22 indicates the number of positively valued arcs. The valuation domain is uniformly distributed in the interval and the mean absolute arc valuation is (Line 12) .

All Digraph objects contain at least the list of attributes shown here: a name (string), a dictionary of actions (digraph nodes), an order (integer) attribute containing the number of actions, a valuationdomain dictionary, a double dictionary relation representing the adjency table of the digraph relation, a gamma and a notGamma dictionary containing the direct neighbourhood of each action.

We may also extract the border -the part of a digraph induced by the union of its initial and terminal prekernels (see tutorial On computing digraph kernels)- as well as, the inner part -the complement of the border- with the help of two corresponding class constructors: GraphBorder and GraphInner (see Listing 1.6).

The constructor of the partial digraphs bnf and inf (see Listing 1.6 Lines 3 and 6) puts to the indeterminate characteristic value all links not in the border, respectively not in the inner part (see Fig. 1.7).

We may as readily compute the dual (negated relation 14), the converse (transposed relation) and the codual (transposed and negated relation) of the digraph instance rdg.

The same closeTransitive() method with a Reverse = True flag may be readily used for eliminating all transitive arcs from a transitive digraph instance. We make usage of this feature when drawing Hasse diagrams of TransitiveDigraph objects.

As the original digraph rdg was connected (see above the result of the showShort() command), both the symmetric and the transitive closures operated together, will necessarily produce a single strong component, i.e. a complete digraph. We may sometimes wish to collapse all strong components in a given digraph and construct the so collapsed digraph. Using the StrongComponentsCollapsedDigraph constructor here will render a single hyper-node gathering all the original nodes (see Line 7 below).

Sometimes it is required to exchange the graph valuation data in CSV format with a statistical package like R. For this purpose it is possible to export the digraph data into a CSV file. The valuation domain is hereby normalized by default to the range [-1,1] and the diagonal put by default to the minimal value -1.

Let us finally mention some special universal classes of digraphs that are readily available in the digraphs module, like the CompleteDigraph, the EmptyDigraph and the IndeterminateDigraph classes, which put all characteristic values respectively to the maximum, the minimum or the median indeterminate characteristic value.

Mind the subtle difference between the neighborhoods of an empty and the neighborhoods of an indeterminate digraph instance. In the first kind, the neighborhoods are known to be completely empty (see Listing 1.10 Lines 22-27) whereas, in the latter, nothing is known about the actual neighborhoods of the nodes (see Listing 1.10 Lines 45-50). These two cases illustrate why in the case of bipolar-valued digraphs, we may need both a gamma and a notGamma attribute.

In this Digraph3 module, the BipolarOutrankingDigraph class from the outrankingDigraphs module provides our standard outranking digraph constructor. Such an instance represents a hybrid object of both, the PerformanceTableau type and the OutrankingDigraph type. A given object consists hence in:

the digraph valuationdomain, a dictionary with three entries: the minimum (-1.0, certainly outranked), the median (0.0, indeterminate) and the maximum characteristic value (+1.0, certainly outranking),

The resulting digraph contains 20 positive (valid) outranking realtions. And, the mean majority criteria significance support of all the pairwise outranking situations is 63.3% (see Listing 1.11 Lines 8-9). We may inspect the complete [-1.0,+1.0]-valued adjacency table as follows.

All outranking digraphs, being of root type Digraph, inherit the methods available under this latter class. The characteristic valuation domain of a digraph may, for instance, be recoded with the recodeValutaion() method below to the integer range [-7,+7], i.e. plus or minus the global significance of the family of criteria considered in this example instance.

From the theory (see [BIS-2013], [ADT-L7] ) we know that a bipolar-valued outranking digraph is weakly complete, i.e. if then . A bipolar-valued outranking relation verifies furthermore the coduality principle: the dual (strict negation - 14) of the converse (inverse ) of the outranking relation corresponds to its strict outranking part.

In this tutorial we have constructed a random outranking digraph with the help of a random performance tableau instance. The next Digraph3 tutorial presents now different models of random performance tableaux illustrating various types of decision problems.

In order to study aggregation of election results (see the tutorial onComputing the winner of an election with the votingProfiles module) in the contextof bipolar-valued outranking digraphs, we provide furthermore aspecific performance tableau model called RandomRankPerformanceTableauwhich provides ranks (linearly ordered performances without ties) of a given number of election candidates (decision actions) for a given number of weighted voters (performance criteria).

The three partial digraphs: geco, gsoc and genv, hence model the preferences represented in each one of the partial performance tableaux. And, we may aggregate these three outranking digraphs with an epistemic fusion operator.

We may notice in the outranking relation table above (see Listing 1.23) that decision alternative a2 positively outranks all the other four alternatives (Line 6). Similarly, alternative a5 is positively outranked by all the other alternatives (see Line 9). We may orient this way the graphviz drawing of the template outranking digraph.

For instance, candidate a1 is ranked four times before and once behind candidate a2. Hence the corresponding majority margin M(a1,a2) is 4 - 1 = +3. These majority margins define on the set of candidates what we call the majority margins digraph. The MajorityMarginsDigraph class (a specialization of the Digraph class) is available for handling such kind of digraphs.

Notice that in the case of linear voting profiles, majority margins always verify a zero sum property: M(x,y) + M(y,x) = 0 for all candidates x and y (see Listing 1.28 Lines 26-28). This is not true in general for arbitrary voting profiles. The majority margins digraph of linear voting profiles defines in fact a weak tournament and belongs, hence, to the class of self-codual bipolar-valued digraphs (13).

Now, a candidate x, showing a positive majority margin M(x,y), is beating candidate y with an absolute majority in a pairwise voting. Hence, a candidate showing only positive terms in her row in the majority margins digraph relation table, beats all other candidates with absolute majority of votes. Condorcet recommends to declare this candidate (is always unique, why) the winner of the election. Here we are lucky, it is again candidate a1 who is hence the Condorcet winner (see Listing 1.28 Line 26).

Now, we cannot find any completely positive row in the relation table (see Listing 1.29 Lines 17 - ). No one of the five candidates is beating all the others with an absolute majority of votes. There is no Condorcet winner anymore. In fact, when looking at a graphviz drawing of this majority margins digraph, we may observe cyclic preferences, like (a1 > a2 > a3 > a1) for instance (see Fig. 1.17). 153554b96e

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