We consider a model of influence with a set of non-strategic agents and two strategic agents. The non-strategic agents have initial opinions and are linked through a simply connected network. They update their opinions as in the DeGroot model. The two strategic agents have fixed and opposed opinions. They each form a link with a non-strategic agent in order to influence the average opinion that emerges due to interactions in the network. This procedure defines a zero-sum game whose players are the two strategic agents and whose strategy set is the set of nonstrategic agents. We focus on the existence and the characterization of pure strategy equilibria in this setting. Simple examples show that the existence of a pure strategy equilibrium does depend on the structure of the network. The characterization of equilibrium we obtain emphasizes on the one hand the influenceability of target agents and on the other hand their centrality whose characterization in our context induces a new notion that we call intermediacy. We also show that in the case where the two strategic agents have the same impact, symmetric equilibria emerge as natural solutions whereas in the case where the impacts are uneven, the strategic players generally have differentiated equilibrium strategies, the high-impact agent focusing on central targets and the low-impact agent on influenceable ones.

We develop a modification of the connections model by Jackson and Wolinsky that takes into account negative externalities arising from the connectivity of direct and indirect neighbors, thus combining aspects of the connections model and the coauthor model. We consider a general functional form for agents' utility that incorporates both the effects of distance and of neighbors' degree. Consequently, we introduce a framework that can be seen as a degree‐distance‐based connections model with both negative and positive externalities. Our analysis shows how the introduction of negative externalities modifies certain results about stability and efficiency compared to the original connections model. In particular, we see the emergence of new stable structures, such as a star with links between peripheral nodes. We also identify structures, for example, certain disconnected networks, that are efficient in our model but which could not be efficient in the original connections model. While our results are proved for the general utility function, some of them are illustrated by using a specific functional form of the degree‐distance‐based utility.

Small world graphs are examples of random graphs which mimic empirically observed features of social networks. We propose an intrinsic definition of small world graphs, based on a probabilistic formulation of scaling properties of the graph, which does not rely on any particular construction. Our definition is shown to encompass existing models of small world graphs, proposed by Watts and studied by Barbour & Reinert, which are based on random perturbations of a regular lattice. We also propose alternative constructions of small world graphs which are not based on lattices and study their scaling properties.

International audienceThis paper presents a model of influence where agents' beliefs are based on an objective reality, such as the properties of an environment. The perception of the objective reality is not direct: all agents know is that the more correct a belief, the more successful the actions that are deduced from this belief. (A pair of agents can influence each other when )Agents can influence eachother by pair when they perform a joint action. They are not only defined by individual beliefs, but also idyosynchratic confidence in their belief - this means that they are not all willing to (engage in action with) act with agents with a different belief and to be influenced by them. We show here that the distribution of confidence in the group has a huge impact on the speed and quality of collective learning and in particular that a small number of overconfident agents can prevent the whole group frow learning properly