We develop a network-based model to assign responsibility for direct and indirect emissions. We introduce a responsibility measure which is based on exponential discounting and takes into account the relative share of responsibility for both upstream and downstream emissions. The responsibility for emission is computed on the basis of the discounted sum of walks for a given node in a given network. It corresponds to a convex combination of the Bonacich centralities for the upstream and downstream weighted adjacency matrices. In view of legal disputes and international negotiations on assigning emissions responsibilities, we apply a normative approach based on the set of appropriate axioms and provide axiomatic foundations for designing liability of GHG-emissions. The responsibility measure is characterized by a set of natural and appealing axioms such as anonymity, monotonicity, additivity, tail-convergence, recursivity and consistency. We support our theoretical results by applying the model to the World Input-Output Database 2016 Release.

In this paper, we aim at incorporating externalities into centrality measures. We characterize axiomatically a class of measures based on the degree measure in weighted networks and modified for an externality. We give a foundation of these measures as utility functions over network positions, expressing preferences over positions in networks, and bringing externalities of connections between other positions. Other nodes being more connected can have positive as well as negative externalities. Our class of measures contains two parameters, in particular, an externality parameter. Depending on these parameters, we can recover some specific measures, e.g., the degree measure if the externality parameter is equal to zero. By considering some additional axioms on the preferences, we can specify the range for the parameters in the utility functions. Negative (respectively, positive) values of the externality parameter express negative (respectively, positive) externalities. Negative externalities of other nodes being connected to each other always occur when the sum of the values assigned to all nodes is a constant. This is the case, for instance, for the average degree externality measure which assigns to every node in a network its degree minus the average degree over all nodes, and therefore leads to the sum of the values equal to zero.

The paper presents a novel approach based on differential games to the study of criminal networks. We extend the static crime network game (Ballester et al., 2006, 2010) to a dynamic setting. First, we determine the relationship
between the Markov Perfect Equilibrium (MPE) and the vector of Bonacich centralities. The established proportionality between the Nash equilibrium
and the Bonacich centrality in the static game does not hold in general in the dynamic setting. Next, we provide an explicit characterization of equilibrium strategies, and conduct comparative dynamic analysis with respect to the network size, network density, and the marginal expected
punishment. Contrary to the static game, where aggregate equilibrium increases with network size and density, in the dynamic setting, more criminals or more connected criminals can lead to a decrease in total crime, both in the short run and at the steady state. We also examine another novel issue in the network theory literature, i.e., the existence of a voracity effect, occurring when an increase in the implicit growth rate of total wealth in the economy lowers economic growth. We do identify the presence of
such a voracity effect in our setting.