Social dynamics models may present discontinuities in the right-hand side of the dynamics for multiple reasons, including topology changes and quantization. Several concepts of generalized solutions for discontinuous equations are available in the li
terature and are useful to analyze these models. In this chapter, we study Caratheodory and Krasovsky generalized solutions for discontinuous models of opinion dynamics with state dependent interactions. We consider two definitions of bounded confidence interactions, which we respectively call metric and topological: in the former, individuals interact if their opinions are closer than a threshold; in the latter, individuals interact with a fixed number of nearest neighbors. We compare the dynamics produced by the two kinds of interactions, in terms of existence, uniqueness and asymptotic behavior of different types of solutions.
In this paper we present a continuation method which transforms spatially distributed ODE systems into continuous PDE. We show that this continuation can be performed both for linear and nonlinear systems, including multidimensional, space- and time-
varying systems. When applied to a large-scale network, the continuation provides a PDE describing evolution of continuous state approximation that respects the spatial structure of the original ODE. Our method is illustrated by multiple examples including transport equations, Kuramoto equations and heat diffusion equations. As a main example, we perform the continuation of a Newtonian system of interacting particles and obtain the Euler equations for compressible fluids, thereby providing an original alternative solution to Hilberts 6th problem. Finally, we leverage our derivation of the Euler equations to control multiagent systems, designing a nonlinear control algorithm for robot formation based on its continuous approximation.
Context. The lockdown orders established in multiple countries in response to the Covid-19 pandemics are arguably one of the most widespread and deepest shock experienced by societies in recent years. Studying their impact trough the lens of social m
edia offers an unprecedented opportunity to understand the susceptibility and the resilience of human activity patterns to large-scale exogenous shocks. Firstly, we investigate the changes that this upheaval has caused in online activity in terms of time spent online, themes and emotion shared on the platforms, and rhythms of content consumption. Secondly, we examine the resilience of certain platform characteristics, such as the daily rhythms of emotion expression. Data. Two independent datasets about the French cyberspace: a fine-grained temporal record of almost 100 thousand YouTube videos and a collection of 8 million Tweets between February 17 and April 14, 2020. Findings. In both datasets we observe a reshaping of the circadian rhythms with an increase of night activity during the lockdown. The analysis of the videos and tweets published during lockdown shows a general decrease in emotional contents and a shift from themes like work and money to themes like death and safety. However, the daily patterns of emotions remain mostly unchanged, thereby suggesting that emotional cycles are resilient to exogenous shocks.
In this paper we study the influence of additive noise in randomized consensus algorithms. Assuming that the update matrices are symmetric, we derive a closed form expression for the mean square error induced by the noise, together with upper and low
er bounds that are simpler to evaluate. Motivated by the study of Open Multi-Agent Systems, we concentrate on Randomly Induced Discretized Laplacians, a family of update matrices that are generated by sampling subgraphs of a large undirected graph. For these matrices, we express the bounds by using the eigenvalues of the Laplacian matrix of the underlying graph or the graphs average effective resistance, thereby proving their tightness. Finally, we derive expressions for the bounds on some examples of graphs and numerically evaluate them.
This paper studies the Laplacian spectrum and the average effective resistance of (large) graphs that are sampled from graphons. Broadly speaking, our main finding is that the Laplacian eigenvalues of a large dense graph can be effectively approximat
ed by using the degree function of the corresponding graphon. More specifically, we show how to approximate the distribution of the Laplacian eigenvalues and the average effective resistance (Kirchhoff index) of the graph. For all cases, we provide explicit bounds on the approximation errors and derive the asymptotic rates at which the errors go to zero when the number of nodes goes to infinity. Our main results are proved under the conditions that the graphon is piecewise Lipschitz and bounded away from zero.
In this paper, we present a type of media disorder which we call `junk news bubbles and which derives from the effort invested by online platforms and their users to identify and share contents with rising popularity. Such emphasis on trending matter
s, we claim, can have two detrimental effects on public debates: first, it shortens the amount of time available to discuss each matter; second it increases the ephemeral concentration of media attention. We provide a formal description of the dynamic of junk news bubbles, through a mathematical exploration the famous public arenas model developed by Hilgartner and Bosk in 1988. Our objective is to describe the dynamics of the junk news bubbles as precisely as possible to facilitate its further investigation with empirical data.
This work explores models of opinion dynamics with opinion-dependent connectivity. Our starting point is that individuals have limited capabilities to engage in interactions with their peers. Motivated by this observation, we propose a continuous-tim
e opinion dynamics model such that interactions take place with a limited number of peers: we refer to these interactions as topological, as opposed to metric interactions that are postulated in classical bounded-confidence models. We observe that topological interactions produce equilibria that are very robust to perturbations.
In this work, we use the spectral properties of graphons to study stability and sensitivity to noise of deterministic SIS epidemics over large networks. We consider the presence of additive noise in a linearized SIS model and we derive a noise index
to quantify the deviation from the disease-free state due to noise. For finite networks, we show that the index depends on the adjacency eigenvalues of its graph. We then assume that the graph is a random sample from a piecewise Lipschitz graphon with finite rank and, using the eigenvalues of the associated graphon operator, we find an approximation of the index that is tight when the network size goes to infinity. A numerical example is included to illustrate the results.
A digital security breach, by which confidential information is leaked, does not only affect the agent whose system is infiltrated, but is also detrimental to other agents socially connected to the infiltrated system. Although it has been argued that
these externalities create incentives to under-invest in security, this presumption is challenged by the possibility of strategic adversaries that attack the least protected agents. In this paper we study a new model of security games in which agents share tokens of sensitive information in a network of contacts. The agents have the opportunity to invest in security to protect against an attack that can be either strategically or randomly targeted. We show that, in the presence of random attack, under-investments always prevail at the Nash equilibrium in comparison with the social optimum. Instead, when the attack is strategic, either under-investments or over-investments are possible, depending on the network topology and on the characteristics of the process of the spreading of information. Actually, agents invest more in security than socially optimal when dependencies among agents are low (which can happen because the information network is sparsely connected or because the probability that information tokens are shared is small). These over-investments pass on to under-investments when information sharing is more likely (and therefore, when the risk brought by the attack is higher).
This paper is about obtaining stable vehicle platooning by using Cooperative Adaptive Cruise Control when the communication is unreliable and suffers from message losses. We model communication losses as independent random events and we propose an or
iginal design for the cooperative controller, which mitigates the effect of losses. This objective is obtained by a switching controller that has a twofold objective: on the one hand, it promotes both plant stability and string stability of the average error dynamics by an $H_infty$ approach, and on the other hand it minimizes the variance around the average. We show by simulations that the proposed controller is able to compensate even for high probability of losses.
