Differential Inequalities in Multi-Agent Coordination and Opinion Dynamics Modeling

Distributed algorithms of multi-agent coordination have attracted substantial attention from the research community; the simplest and most thoroughly studied of them are consensus protocols in the form of differential or difference equations over general time-varying weighted graphs. These graphs are usually characterized algebraically by their associated Laplacian matrices. Network algorithms with similar algebraic graph theoretic structures, called being of Laplacian-type in this paper, also arise in other related multi-agent control problems, such as aggregation and containment control, target surrounding, distributed optimization and modeling of opinion evolution in social groups. In spite of their similarities, each of such algorithms has often been studied using separate mathematical techniques. In this paper, a novel approach is offered, allowing a unified and elegant way to examine many Laplacian-type algorithms for multi-agent coordination. This approach is based on the analysis of some differential or difference inequalities that have to be satisfied by the some "outputs" of the agents (e.g. the distances to the desired set in aggregation problems). Although such inequalities may have many unbounded solutions, under natural graphic connectivity conditions all their bounded solutions converge (and even reach consensus), entailing the convergence of the corresponding distributed algorithms. In the theory of differential equations the absence of bounded non-convergent solutions is referred to as the equation's dichotomy. In this paper, we establish the dichotomy criteria of Laplacian-type differential and difference inequalities and show that these criteria enable one to extend a number of recent results, concerned with Laplacian-type algorithms for multi-agent coordination and modeling opinion formation in social groups.Comment: accepted to Automatic

Graph Theory and Networks in Biology

In this paper, we present a survey of the use of graph theoretical techniques in Biology. In particular, we discuss recent work on identifying and modelling the structure of bio-molecular networks, as well as the application of centrality measures to interaction networks and research on the hierarchical structure of such networks and network motifs. Work on the link between structural network properties and dynamics is also described, with emphasis on synchronization and disease propagation.Comment: 52 pages, 5 figures, Survey Pape

Analytical computation of the epidemic threshold on temporal networks

The time variation of contacts in a networked system may fundamentally alter the properties of spreading processes and affect the condition for large-scale propagation, as encoded in the epidemic threshold. Despite the great interest in the problem for the physics, applied mathematics, computer science and epidemiology communities, a full theoretical understanding is still missing and currently limited to the cases where the time-scale separation holds between spreading and network dynamics or to specific temporal network models. We consider a Markov chain description of the Susceptible-Infectious-Susceptible process on an arbitrary temporal network. By adopting a multilayer perspective, we develop a general analytical derivation of the epidemic threshold in terms of the spectral radius of a matrix that encodes both network structure and disease dynamics. The accuracy of the approach is confirmed on a set of temporal models and empirical networks and against numerical results. In addition, we explore how the threshold changes when varying the overall time of observation of the temporal network, so as to provide insights on the optimal time window for data collection of empirical temporal networked systems. Our framework is both of fundamental and practical interest, as it offers novel understanding of the interplay between temporal networks and spreading dynamics.Comment: 22 pages, 6 figure

Steering opinion dynamics via containment control

In this paper, we model the problem of influencing the opinions of groups of individuals as a containment control problem, as in many practical scenarios, the control goal is not full consensus among all the individual opinions, but rather their containment in a certain range, determined by a set of leaders. As in classical bounded confidence models, we consider individuals affected by the confirmation bias, thus tending to influence and to be influenced only if their opinions are sufficiently close. However, here we assume that the confidence level, modeled as a proximity threshold, is not constant and uniform across the individuals, as it depends on their opinions. Specifically, in an extremist society, the most radical agents (i.e., those with the most extreme opinions) have a higher appeal and are capable of influencing nodes with very diverse opinions. The opposite happens in a moderate society, where the more connected (i.e., influential) nodes are those with an average opinion. In three artificial societies, characterized by different levels of extremism, we test through extensive simulations the effectiveness of three alternative containment strategies, where leaders have to select the set of followers they try to directly influence. We found that, when the network size is small, a stochastic time-varying pinning strategy that does not rely on information on the network topology proves to be more effective than static strategies where this information is leveraged, while the opposite happens for large networks where the relevance of the topological information is prevalent