Efficient Computation of Semivalues for Game-Theoretic Network Centrality

Abstract


 
 
 Some game-theoretic solution concepts such as the Shapley value and the Banzhaf index have recently gained popularity as measures of node centrality in networks. While this direction of research is promising, the computational problems that surround it are challenging and have largely been left open. To date there are only a few positive results in the literature, which show that some game-theoretic extensions of degree-, closeness- and betweenness-centrality measures are computable in polynomial time, i.e., without the need to enumerate the exponential number of all possible coalitions. In this article, we show that these results can be extended to a much larger class of centrality measures that are based on a family of solution concepts known as semivalues. The family of semivalues includes, among others, the Shapley value and the Banzhaf index. To this end, we present a generic framework for defining game-theoretic network centralities and prove that all centrality measures that can be expressed in this framework are computable in polynomial time. Using our framework, we present a number of new and polynomial-time computable game-theoretic centrality measures.
 
 

A Game-Theoretic Algorithm for Link Prediction

“…A number of such studies have been recently proposed in the literature [38,41,42,7]. Interestingly, in a similar line of research on evading detection by centrality measures [37,36], it has been shown that game-theoretic centrality measures [34,29,25] are more difficult to evade than the conventional ones [3]. We believe the same will be the case…”

Calculation of Centrality in the Analysis of Congestion of City Roads on the Example of Petrozavodsk

“…Mathematical properties of centrality of arbitrary subgraphs are studied in detail in [10,11,13]. In mathematical game theory, the characteristics of arbitrary subsets of graph vertices as coalitions of players are analyzed, so the centrality of vertex groups [32,33] is of great importance. In [34], the concept of centrality is extended to subsets of vertices using fuzzy measures that can be defined for both numerical and categorical characteristics of a graph.…”

El control coalicional en el marco de la teoría de juegos cooperativos

“…Asimismo, sea un juego de enlaces (E , r v ) donde E = {I, II, III, IV}, y cuyo valor de la función de coste r v (Λ) evaluada en cada topología Λ viene dado por el vector r v = [0 1 2 2 1 4 0 2 1 3 5 4 3 2 5 6] T . En la Tabla 2 se detallan los valores específicos de todos los conceptos de solución presentados, donde como medidas de poder α y ω se han considerado respectivamente la medida de grado d i = |E i | (Sun and Tang, 2011;van den Brink et al, 2008), y la análoga medida de grado para enlaces d l = |E i | n |E j | n (Muros et al, 2017b), con l = {i, j} ∈ E y tomando n = 1. Asimismo, los pesos de los valores de Shapley ponderados se han escogido coincidentes con ω l (N , E ), ∀l ∈ E .…”

“…Porúltimo, otra solución a los problemas de explosión combinacional reside en los llamados métodos randomizados (Castro et al, 2009(Castro et al, , 2017Ishii and Tempo, 2010;Tarkowski et al, 2018). En particular, en (Castro et al, 2009(Castro et al, , 2017 se computa en tiempo polinomial una estimación del valor de Shapley de un enlace l, i.e., φ l (E , r v ), mediante la toma, con reemplazo e igual probabilidad, de un número q de muestras aleatorias del conjunto completo de permutaciones de las que puede formar parte cualquier enlace, dado por |E |!.…”