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Having seen this example, it should not be surprising that several evaluation measures can be extended in the same manner Because these extensions are usually very easy to find, we do not consider them explicitly In the following we state all evaluation measures wrt two attributes A and C, where C is the child attribute if we have a directed edge and A is the parent After an assessment of all components of a graphical model has been computed, the components have to be aggregated Usually this is done by simply summing them Such a simple aggregation is, obviously, satisfactory if the graph is a directed acyclic graph, because in this case the components of the decomposition are clearly separated from each other by the conditioning Hence there can be no interaction between the evaluations of these components For undirected graphs, however, the situation is different.

Information given includes survivorship and inundation length (days) with depth ooded above root crown (cm). The ooding depth above the root crown is not known unless given in parentheses.

Enlargement of external CSF spaces (enlargement of cortical sulci, narrowing of the gyri) Enlargement of internal CSF spaces (ventricular enlargement) Decreased total gray matter volume Decreased total white matter volume Mild atrophy of the thalamus and midbrain area Reduced pituitary size

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If two cliques overlap on more than one attribute, then the dependence of the attributes in the overlap is, in a way, counted twice , because it influences the assessment of both cliques As a consequence the quality of the graph may be overrated As an example consider the simple undirected graph shown in Figure 711 The maximal cliques are induced by the node sets {A, B, C} and {B, C, D}, which overlap on {B, C} Therefore the mutual dependence of B and C is counted twice As a consequence, the graph may be overrated compared to one in which the cliques overlap only on single attributes In order to cope with this problem one may consider deducing the assessments of overlaps from the summed assessments of the maximal cliques.

28. V PP or 29. V PP Since, according to (10),

EAs can achieve starting from randomly generated solution (from more than 110 million to several thousand cost units) This means that the strongest interference in the network has been avoided The second fact concerns the best solutions found so far by the solvers, which are included in the column best of Table 95 They point out that all the con gurations of the (1 + 10) EA are able to compute very accurate frequency assignments As a consequence, we can conclude that these algorithms are very suitable for solving this optimization problem We now turn to further analyze how the different strategies proposed for initializing and perturbing work within the (1 + 10) EA framework With this goal in mind, Figure 96 displays the average costs of the con gurations using 1 The Random Init strategy versus those using the Advanced Init method 2.

OneHopNeighborhood versus TwoHopNeighborhood strategies for determining the number of TRXs to be reallocated a channel 3 The random scheme versus interference-based one for selecting the TRXs 4 The random versus interference-based channel selection strategies Concerning the initialization method, Figure 96 shows that the (1 + 10) EAs using the Advanced Init scheme reach, on average, better frequency assignments than the con gurations with Random Init It is clear from these results that our proposed EAs can pro t from good initial plannings that guide the search toward promising regions of the search space If we compare the different strategies used in the perturbation method, several conclusions can be drawn First of all, con gurations of the (1 + 10) EA that reallocate the channel to a smaller number of TRXs, that is, OneHopNeighborhood strategy, against using the TwoHopNeighborhood scheme report a small improvement in the AFP cost.

(Note that this may require fairly complicated computations in case the overlap of several pairs of cliques has a nonempty intersection, since we may have to re-add the assessment of the intersection of overlaps etc) Of course, this is only a heuristic approach, since it may not be justifiable for all evaluation measures (It can be justified, though, for mutual (Shannon) information; see below) An alternative way to cope with the evaluation problem for undirected graphs is to direct the edges and then to apply the directed version of the chosen evaluation measure Unfortunately, not all undirected graphs can be turned into equivalent directed acyclic graphs (cf Section 414) This can be achieved, though, if the undirected graphs have hypertree structure,6 which may be seen as another argument in favor of such graphs.

However, it is clear that randomly choosing the TRX (and its corresponding.

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