Pure proportional voting rules can sometimes lead to highly suboptimal outcomes. We introduce the Method of Equal Shares with Bounded Overspending (BOS Equal Shares), a robust variant of the Method of Equal Shares that balances proportionality and efficiency. BOS Equal Shares addresses inefficiencies implied by strict proportionality axioms, yet still provides fairness guarantees, similar to the original Equal Shares. Our extensive empirical analysis shows excellent performance of BOS Equal Shares across several metrics. In the course of the analysis, we also study a fractional variant of the Method of Equal Shares.

Centrality measures are widely used to assign importance to graph-structured data. Recently, understanding the principles of such measures has attracted a lot of attention. Given that measures are diverse, this research has usually focused on classes of centrality measures. In this work, we provide a different approach by focusing on classes of graphs instead of classes of measures to understand the underlying principles among various measures. More precisely, we study the class of trees. We observe that even in \fix{the} case of trees, there is no consensus on which node should be selected as the most central. To analyze the behavior of centrality measures on trees, we introduce a property of \emph{tree rooting} that states a measure selects one or two adjacent nodes as the most important, and the importance decreases from them in all directions. This property is satisfied by closeness centrality but violated by PageRank. We show that, for several centrality measures that root trees, the comparison of adjacent nodes can be inferred by \emph{potential functions} that assess the quality of trees. We use these functions to give fundamental insights on rooting and derive a characterization explaining why some measure root trees. Moreover, we provide an almost liner-time algorithm to compute the root of a graph by using potential functions. Finally, using a family of potential functions, we show that many ways of tree rooting exist with desirable properties.

We study the complexity of computing the Shapley value in partition function form games. We focus on two representations based on marginal contribution nets (embedded MC-nets and weighted MC-nets) and five extensions of the Shapley value. Our results show that while weighted MC-nets are more concise than embedded MC-nets, they have slightly worse computational properties when it comes to computing the Shapley value: two out of five extensions can be computed in polynomial time for embedded MC-nets and only one for weighted MC-nets.

Edge centrality measures are functions that evaluate the importance of edges in a network. They can be used to assess the role of a backlink for the popularity of a website as well as the importance of a flight in virus spreading. Various node centralities have been translated to apply for edges, including Edge Betweenness, Eigenedge (edge version of eigenvector centrality), and Edge PageRank. With this paper, we initiate the discussion on the axiomatic properties of edge centrality measures. We do it by proposing an axiomatic characterization of Edge PageRank. Our characterization is the first characterization of any edge centrality measures in the literature.

We extend the well-known model of graph-restricted games due to Myerson to signed graphs. In our model, it is possible to explicitly define not only that some players are friends (as in Myerson's model) but also that some other players are enemies. As such our games can express a wider range of situations, e.g., animosities between political parties. We define the value for signed graph games using the axiomatic approach that closely follows the celebrated characterization of the Myerson value. Furthermore, we propose an algorithm for computing an arbitrary semivalue, including the extension of the Myerson value proposed by us. We also develop a pseudo-polynomial algorithm for power indices in weighted voting games for signed graphs with bounded treewidth. Moreover, we consider signed graph games with a priori defined alliances (unions) between players and propose algorithms to compute the extension of the Owen value to this setting.

Vitality indices form a class of centrality measures that assess the importance of a node based on the impact its removal has on the network. To date, theoretical analysis of this class is lacking. In this paper, we show that vitality indices can be characterized using the axiom of Balanced Contributions proposed by Myerson in the coalitional game theory literature. We explore the link between both fields and show an equivalence between vitality indices and induced game theoretic centralities based on the Shapley value. Our characterization allows us to easily determine which known centrality measures are vitality indices.

In recent years, the axiomatic approach to centrality measures has attracted attention in the literature. However, most papers propose a collection of axioms dedicated to one or two considered centrality measures. In result, it is hard to capture the differences and similarities between various measures. In this paper, we propose an axiom system for four classic feedback centralities: Eigenvector centrality, Katz centrality, Katz prestige and PageRank. We prove that each of these four centrality measures can be uniquely characterized with a subset of our axioms. Our system is the first one in the literature that considers all four feedback centralities.

We propose a new centrality measure, called the Random Walk Decay centrality. While most centralities in the literature are based on the notion of shortest paths, this new centrality measure stems from the random walk on the network. We provide an axiomatic characterization and show that the new centrality is closely related to PageRank. More in detail, we show that replacing only one axiom, called Lack of Self-Impact, with another one, called Edge Swap, results in the new axiomatization of PageRank. Finally, we argue that Lack of Self-Impact is desirable in various settings and explain why violating Edge Swap may be beneficial and may contribute to promoting diversity in the centrality measure.

The intuitive notion of added value in groups represents a fundamental property of biological, physical, and economic systems: how the interaction or cooperation of multiple entities, substances, or other agents can produce synergistic effects. However, despite the ubiquity of group formation, a well-founded measure of added value has remained elusive. Here, we propose such a measure inspired by the Shapley value —a fundamental solution concept from Cooperative Game Theory. To this end, we start by developing a solution concept that measures the average impact of each player in a coalitional game and show how this measure uniquely satisfies a set of intuitive properties. Then, building upon our solution concept, we propose a measure of added value that not only analyzes the interactions of players inside their group, but also outside it, thereby reflecting otherwise-hidden information about how these individuals typically perform in various groups of the population.

At the heart of multi-agent systems is the ability to cooperate to improve the performance of individual agents and/or the system as a whole. While a widespread assumption in the literature is that such cooperation is essentially unrestricted, in many realistic settings this assumption does not hold. A highly influential approach for modelling such scenarios are graph-restricted games introduced by Myerson [36]. In this approach, agents are represented by nodes in a graph, edges represent communication channels, and a group can generate an arbitrary value only if there exists a direct or indirect communication channel between every pair of agents within the group. Two fundamental solution-concepts that were proposed for such games are the Myerson value and the Shapley value . While an algorithm has been developed to compute the Shapley value in arbitrary graph-restricted games, no such general-purpose algorithm has been developed for the Myerson value to date. With this in mind, we set out to develop for such games a general-purpose algorithm to compute the Myerson value, and a more efficient algorithm to compute the Shapley value. Since the computation of either value involves enumerating all connected induced subgraphs of the game’s underlying graph, we start by developing an algorithm dedicated to this enumeration, and then we show empirically that it is faster than the state of the art in the literature. Finally, we present a sample application of both algorithms, in which we test the Myerson value and the Shapley value as advanced measures of node centrality in networks.

We study the class of distance-based centralities that consists of centrality measures that depend solely on distances to other nodes in the graph. This class encompasses a number of centrality measures, including the classical Degree and Closeness Centralities, as well as their extensions: the Harmonic, Reach and Decay Centralities. We axiomatize the class of distance-based centralities and study what conditions are imposed by the axioms proposed in the literature. Building upon our analysis, we propose the class of additive distance-based centralities and pin-point properties which combined with the axiomatic characterization of the whole class uniquely characterize a number of centralities from the literature.

Feedback centralities are one of the key classes of centrality measures. They assess the importance of a vertex recursively, based on the importance of its neighbours. Feedback centralities includes the Eigenvector Centrality, as well as its variants, such as the Katz Centrality or the PageRank, and are used in various AI applications, such as ranking the importance of websites on the Internet and most influential users in the Twitter social network. In this paper, we study the theoretical underpinning of the feedback centralities. Specifically, we propose a novel axiomatization of the Eigenvector Centrality and the Katz Centrality based on six simple requirements. Our approach highlights the similarities and differences between both centralities which may help in choosing the right centrality for a specific application.

We study the problem of extending the classic centrality measures to weighted graphs. Unfortunately, in the existing extensions, paths in the graph are evaluated solely based on their weights, which is a restrictive and undesirable assumption for a variety of settings. Given this, we define a notion of the path evaluation function that assesses a path between two nodes by looking not only on the sum of edge weights, but also on the number of intermediaries. Using an axiomatic approach, we propose three classes of path evaluation functions. Building upon this analysis, we present the first systematic study how classic centrality measures can be extended to weighted graphs while taking into account an arbitrary path evaluation function. As an application, we use the newly-defined measures to identify the most well-linked districts in a sample public transport network.

We propose an axiomatization of PageRank. Specifically, we introduce five simple axioms—Foreseeability, Outgoing Homogeneity, Monotonicity, Merging, and Dummy Node—and show that PageRank is the only centrality measure that satisfies all of them. Our axioms give a new conceptual and theoretical underpinnings of PageRank and show how it differs from other centralities.

One of the fundamental research challenges in network science is centrality analysis, i.e., identifying the nodes that play the most important roles in the network. In this article, we focus on the game-theoretic approach to centrality analysis. While various centrality indices have been recently proposed based on this approach, it is still unknown how general is the game-theoretic approach to centrality and what distinguishes some game-theoretic centralities from others. In this article, we attempt to answer this question by providing the first axiomatic characterization of game-theoretic centralities. Specifically, we show that every possible centrality measure can be obtained following the game-theoretic approach. Furthermore, we study three natural classes of game-theoretic centrality, and prove that they can be characterized by certain intuitive properties pertaining to the well-known notion of Fairness due to Myerson.

One of the fundamental research challenges in network science is the centrality analysis, i.e., identifying the nodes that play the most important roles in the network. In this paper, we focus on the game-theoretic approach to centrality analysis. While various centrality indices have been proposed based on this approach, it is still unknown what distinguishes this family of indices from the more classical ones. In this paper, we answer this question by providing the first axiomatic characterization of game-theoretic centralities. Specifically, we show that every centrality can be obtained following the game-theoretic approach, and show that two natural classes of game-theoretic centrality can be characterized by two intuitive properties pertaining to Myerson's notion of Fairness.

Measuring how central nodes are in terms of connecting a network has recently received increasing attention in the literature. While a few dedicated centrality measures have been proposed, Skibski et al. [2016] showed that the Attachment Centrality is the only one that satisfies certain natural axioms desirable for connectivity. Unfortunately, the Attachment Centrality is defined only for unweighted graphs which makes this measure ill-fitted for various applications. For instance, covert networks are typically weighted, where the weights carry additional intelligence available about criminals or terrorists and the links between them. To analyse such settings, in this paper we extend the Attachment Centrality to node-weighted and edge-weighted graphs. By an axiomatic analysis, we show that the Attachment Centrality is closely related to the Degree Centrality in weighted graphs.

The coalition structure generation problem is one of the key challenges in multi-agent coalition formation. It involves partitioning a set of agents into coalitions so that system performance is optimized. To date, the multi-agent systems literature has focused exclusively on the utilitarian version of this problem which seeks to maximize the sum of the values of the coalitions involved. However, there are many examples of situations in which other performance metrics are of interest. In particular, in games with non-transferable utility, we may be more interested in an egalitarian optimal coalition structure, or in minimizing the difference between the utilities of the most affluent and poorest agents. In this paper, we present a number of exact algorithms to solve such non-utilitarian formulations of the coalition structure generation problem.

We propose a novel representation for coalitional games with externalities, called Partition Decision Trees. This representation is based on rooted directed trees, where non-leaf nodes are labelled with agents' names, leaf nodes are labelled with payoff vectors, and edges indicate membership of agents in coalitions. We show that this representation is fully expressive, and for certain classes of games significantly more concise than an extensive representation. Most importantly, Partition Decision Trees are the first formalism in the literature under which most of the direct extensions of the Shapley value to games with externalities can be computed in polynomial time.

Inspired by emerging applications of social networks, we introduce in this paper a new centrality measure termed gate-keeper centrality. The new centrality is based on the well-known game-theoretic concept of Shapley value and, as we demonstrate, possesses unique qualities compared to the existing metrics. Furthermore, we present a dedicated approximate algorithm, based on the Monte Carlo sampling method, to compute the gatekeeper centrality. We also consider two well known applications in social network analysis, namely community detection and limiting the spread of mis-information; and show the merit of using the proposed framework to solve these two problems in comparison with the respective benchmark algorithms.