Anton V. Proskurnikov

Opinion Dynamics in Social Networks With Hostile Camps: Consensus vs. Polarization

Most of the distributed protocols for multi-agent consensus assume that the agents are mutually cooperative and “trustful,” and so the couplings among the agents bring the values of their states closer. Opinion dynamics in social groups, however, require beyond these conventional models due to ubiquitous competition and distrust between some pairs of agents, which are usually characterized by repulsive couplings and may lead to clustering of the opinions. A simple yet insightful model of opinion dynamics with both attractive and repulsive couplings was proposed recently by C. Altafini, who examined first-order consensus algorithms over static signed graphs. This protocol establishes modulus consensus, where the opinions become the same in modulus but may differ in signs. In this paper, we extend the modulus consensus model to the case where the network topology is an arbitrary time-varying signed graph and prove reaching modulus consensus under mild sufficient conditions of uniform connectivity of the graph. For cut-balanced graphs, not only sufficient, but also necessary conditions for modulus consensus are given.

Network science on belief system dynamics under logic constraints

Belief system dynamics People tend to structure their beliefs in a way that appears consistent to them. But how do some beliefs within groups persist in the face of social pressure, whereas others change and, by changing, influence a cascade of other beliefs? Friedkin et al. developed a model that can describe complexes of attitudes in a group that interact and change (see the Perspective by Butts). Their model revealed how the changing views of the U.S. population on the existence of weapons of mass destruction in Iraq changed their views on whether the invasion by the United States was justified. Science, this issue p. 321; see also p. 286 An algorithmic approach shows how our belief systems change when facts and beliefs are in conflict. Breakthroughs have been made in algorithmic approaches to understanding how individuals in a group influence each other to reach a consensus. However, what happens to the group consensus if it depends on several statements, one of which is proven false? Here, we show how the existence of logical constraints on beliefs affect the collective convergence to a shared belief system and, in contrast, how an idiosyncratic set of arbitrarily linked beliefs held by a few may become held by many.

Consensus in switching networks with sectorial nonlinear couplings: Absolute stability approach

Consensus algorithms for multi-agent networks with high-order agent dynamics, time-varying topology, and uncertain symmetric nonlinear couplings are considered. Convergence conditions for these algorithms are obtained by means of the Kalman-Yakubovich-Popov lemma and absolute stability techniques. The conditions are similar in spirit and extend the celebrated circle criterion for the stability of Lurie systems.

Technical communique: Average consensus in networks with nonlinearly delayed couplings and switching topology

The paper addresses consensus under nonlinear couplings and bounded delays for multi-agent systems, where the agents have the single-integrator dynamics. The network topology is undirected and may alter as time progresses. The couplings are uncertain and satisfy a conventional sector condition with known sector slopes. The delays are uncertain, time-varying and obey known upper bounds. The network satisfies a symmetry condition that resembles the Newtons Third Law. Explicit analytical conditions for the robust consensus are offered that employ only the known upper bounds for the delays and the sector slopes.

Novel Multidimensional Models of Opinion Dynamics in Social Networks

Unlike many complex networks studied in the literature, social networks rarely exhibit unanimous behavior, or consensus. This requires a development of mathematical models that are sufficiently simple to be examined and capture, at the same time, the complex behavior of real social groups, where opinions and actions related to them may form clusters of different size. One such model, proposed by Friedkin and Johnsen, extends the idea of conventional consensus algorithm (also referred to as the iterative opinion pooling) to take into account the actors’ prejudices, caused by some exogenous factors and leading to disagreement in the final opinions. In this paper, we offer a novel multidimensional extension, describing the evolution of the agents’ opinions on several topics. Unlike the existing models, these topics are interdependent, and hence the opinions being formed on these topics are also mutually dependent. We rigorously examine stability properties of the proposed model, in particular, convergence of the agents’ opinions. Although our model assumes synchronous communication among the agents, we show that the same final opinions may be reached “on average” via asynchronous gossip-based protocols.

A tutorial on modeling and analysis of dynamic social networks. Part I

In recent years, we have observed a significant trend towards filling the gap between social network analysis and control. This trend was enabled by the introduction of new mathematical models describing dynamics of social groups, the advancement in complex networks theory and multi-agent systems, and the development of modern computational tools for big data analysis. The aim of this tutorial is to highlight a novel chapter of control theory, dealing with applications to social systems, to the attention of the broad research community. This paper is the first part of the tutorial, and it is focused on the most classical models of social dynamics and on their relations to the recent achievements in multi-agent systems.

Dissipativity of T-Periodic Linear Systems

It is proved that the existence of a positive-definite storage function is necessary and sufficient for strict dissipativity of linear systems with periodic coefficients. The connection between strict dissipativity of the system and a nonoscillatory property of an associated Hamiltonian system is established.

Popov-Type Criterion for Consensus in Nonlinearly Coupled Networks

This paper addresses consensus problems in nonlinearly coupled networks of dynamic agents described by a common and arbitrary linear model. Interagent interaction rules are uncertain but satisfy the standard sector condition with known sector bounds; both the agents model and interaction topology are time-invariant. A novel frequency-domain criterion for consensus is offered that is similar to and extends the classical Popovs absolute stability criterion.

CONSENSUS IN MULTI-AGENT SYSTEMS

Many cooperative behaviors of multi-agent teams emerge from local interactions among the agents, where an agent interacts with a few “adjacent” teammates, but has no information about the remaining agents. For instance, the selforganization of many biological populations – including swarms of insects, flocks of birds, and schools of fish – are based on such local interaction rules: the motion and decisions of an individual agent are determined by the behavior of its nearest neighbors in the population. A special case of multi-agent coordination is consensus, that is, the agreement of agents on some quantity of interest or, more generally, the full or partial synchronization of their state trajectories. Establishing consensus is a “benchmark” problem in multi-agent systems study, which allows to reveal the main principles of multi-agent coordination and, in particular, the role of the system’s interaction graph (or topology). Consensus lies in the heart of many natural phenomena (e.g., synchronous oscillation of neural cells, which maintains a stable heart rhythm) and engineering designs (e.g., attitude synchronization of satellites). In this article, we give a brief review of distributed consensus algorithms, focusing on the basic ideas and relevant mathematical theory, in particular, graph theoretic methods.

Problems and methods of network control

Control of network systems, or network control, is a rapidly developing field of modern automated control theory. Network control is characterized by a combination of the classical control theory toolbox (linear systems, nonlinear control, robust control and so on) and conceptually new mathematical ideas that come primarily from graph theory. Methods of network control let one solve analysis and synthesis problems for complex systems that arise in physics, biology, economics, sociology, and engineering sciences. In this survey, we present the main fields of application for modern theory of network control and formulate its key results obtained over the last decade.