Measuring centrality by a generalization of degree

Network analysis has emerged as a key technique in communication studies, economics, geography, history and sociology, among others. A fundamental issue is how to identify key nodes, for which purpose a number of centrality measures have been developed. This paper proposes a new parametric family of centrality measures called generalized degree. It is based on the idea that a relationship to a more interconnected node contributes to centrality in a greater extent than a connection to a less central one. Generalized degree improves on degree by redistributing its sum over the network with the consideration of the global structure. Application of the measure is supported by a set of basic properties. A sufficient condition is given for generalized degree to be rank monotonic, excluding counter-intuitive changes in the centrality ranking after certain modifications of the network. The measure has a graph interpretation and can be calculated iteratively. Generalized degree is recommended to apply besides degree since it preserves most favourable attributes of degree, but better reflects the role of the nodes in the network and has an increased ability to distinguish among their importance.Comment: 20 pages, 8 figure

A graph interpretation of the least squares ranking method

The paper aims at analyzing the least squares ranking method for generalized tournaments with possible missing and multiple paired comparisons. The bilateral relationships may reflect the outcomes of a sport competition, product comparisons, or evaluation of political candidates and policies. It is shown that the rating vector can be obtained as a limit point of an iterative process based on the scores in almost all cases. The calculation is interpreted on an undirected graph with loops attached to some nodes, revealing that the procedure takes into account not only the given object's results but also the strength of objects compared with it. We explore the connection between this method and another procedure defined for ranking the nodes in a digraph, the positional power measure. The decomposition of the least squares solution offers a number of ways to modify the method

On the additivity of preference aggregation methods

The paper reviews some axioms of additivity concerning ranking methods used for generalized tournaments with possible missing values and multiple comparisons. It is shown that one of the most natural properties, called consistency, has strong links to independence of irrelevant comparisons, an axiom judged unfavourable when players have different opponents. Therefore some directions of weakening consistency are suggested, and several ranking methods, the score, generalized row sum and least squares as well as fair bets and its two variants (one of them entirely new) are analysed whether they satisfy the properties discussed. It turns out that least squares and generalized row sum with an appropriate parameter choice preserve the relative ranking of two objects if the ranking problems added have the same comparison structure.Comment: 24 pages, 9 figure

How to avoid ordinal violations in incomplete pairwise comparisons

Assume that some ordinal preferences can be represented by a weakly connected directed acyclic graph. The data are collected into an incomplete pairwise comparison matrix, the missing entries are estimated, and the priorities are derived from the optimally filled pairwise comparison matrix. Our paper studies whether these weights are consistent with the partial order given by the underlying graph. According to previous results from the literature, two popular procedures, the incomplete eigenvector and the incomplete logarithmic least squares methods fail to satisfy the required property. Here, it is shown that the recently introduced lexicographically optimal completion combined with any of these weighting methods avoids ordinal violation in the above setting. This finding provides a powerful argument for using the lexicographically optimal completion to determine the missing elements in an incomplete pairwise comparison matrix.Comment: 11 pages, 2 figure

Improving the fairness of group draw in sports tournaments

Several sports tournaments establish restrictions on the draw of the group stage. In this case, the usual mechanism to assign the teams into groups is unevenly distributed: the valid allocations are not equally likely. We show by two illustrative examples how the lack of randomisation in the size of the groups and the inflexible treatment of a sophisticated draw constraint can increase unfairness. Based on these ideas, the draw of the European Qualifiers for the 2022 FIFA World Cup could have been closer to uniform distribution without any negative effects. Our recommendation for a careful relabelling of the pots from which the teams are drawn has a decent chance to be implemented in practice.Comment: 10 pages, 1 figure, 3 table