164 research outputs found
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
Dynamics over Signed Networks
A signed network is a network with each link associated with a positive or
negative sign. Models for nodes interacting over such signed networks, where
two different types of interactions take place along the positive and negative
links, respectively, arise from various biological, social, political, and
economic systems. As modifications to the conventional DeGroot dynamics for
positive links, two basic types of negative interactions along negative links,
namely the opposing rule and the repelling rule, have been proposed and studied
in the literature. This paper reviews a few fundamental convergence results for
such dynamics over deterministic or random signed networks under a unified
algebraic-graphical method. We show that a systematic tool of studying node
state evolution over signed networks can be obtained utilizing generalized
Perron-Frobenius theory, graph theory, and elementary algebraic recursions.Comment: In press, SIAM Revie
A relative approach to opinion formation
Formal models of opinion formation commonly represent an individual’s opinion by a value on a fixed opinion interval. We propose an alternative modeling method wherein interpretation is only provided to the relative positions of opinions vis-à -vis each other. This method is then considered in a similar setting as the discrete-time Altafini model (an extension of the well-known DeGroot model), but with more general influence weights. Even in a linear framework, the model can describe, in the long run, polarization, dynamics with a periodic pattern, and (modulus) consensus formation. In addition, in our alternative approach key characteristics of the opinion dynamic can be derived from real-valued square matrices of influence weights, which immediately allows one to transfer matrix theory insights to the field of opinion formation dynamics under more relaxed conditions than in the DeGroot or discrete-time Altafini models. A few specific themes are covered: (i) We demonstrate how stable patterns in relative opinion dynamics are identified which are hidden when opinions are considered in an absolute opinion framework. (ii) For the two-agent case, we provide an exhaustive closed-form description of the relative opinion model’s dynamic in the long run. (iii) We explore group dynamics analytically, in particular providing a non-trivial condition under which a subgroup’s asymptotic behavior carries over to the entire population
A system-theoretic framework for privacy preservation in continuous-time multiagent dynamics
In multiagent dynamical systems, privacy protection corresponds to avoid
disclosing the initial states of the agents while accomplishing a distributed
task. The system-theoretic framework described in this paper for this scope,
denoted dynamical privacy, relies on introducing output maps which act as
masks, rendering the internal states of an agent indiscernible by the other
agents as well as by external agents monitoring all communications. Our output
masks are local (i.e., decided independently by each agent), time-varying
functions asymptotically converging to the true states. The resulting masked
system is also time-varying, and has the original unmasked system as its limit
system. When the unmasked system has a globally exponentially stable
equilibrium point, it is shown in the paper that the masked system has the same
point as a global attractor. It is also shown that existence of equilibrium
points in the masked system is not compatible with dynamical privacy.
Application of dynamical privacy to popular examples of multiagent dynamics,
such as models of social opinions, average consensus and synchronization, is
investigated in detail.Comment: 38 pages, 4 figures, extended version of arXiv preprint
arXiv:1808.0808
Switching Quantum Dynamics for Fast Stabilization
Control strategies for dissipative preparation of target quantum states, both
pure and mixed, and subspaces are obtained by switching between a set of
available semigroup generators. We show that the class of problems of interest
can be recast, from a control--theoretic perspective, into a
switched-stabilization problem for linear dynamics. This is attained by a
suitable affine transformation of the coherence-vector representation. In
particular, we propose and compare stabilizing time-based and state-based
switching rules for entangled state preparation, showing that the latter not
only ensure faster convergence with respect to non-switching methods, but can
designed so that they retain robustness with respect to initialization, as long
as the target is a pure state or a subspace.Comment: 15 pages, 4 figure
Recurrent Averaging Inequalities in Multi-Agent Control and Social Dynamics Modeling
Many multi-agent control algorithms and dynamic agent-based models arising in
natural and social sciences are based on the principle of iterative averaging.
Each agent is associated to a value of interest, which may represent, for
instance, the opinion of an individual in a social group, the velocity vector
of a mobile robot in a flock, or the measurement of a sensor within a sensor
network. This value is updated, at each iteration, to a weighted average of
itself and of the values of the adjacent agents. It is well known that, under
natural assumptions on the network's graph connectivity, this local averaging
procedure eventually leads to global consensus, or synchronization of the
values at all nodes. Applications of iterative averaging include, but are not
limited to, algorithms for distributed optimization, for solution of linear and
nonlinear equations, for multi-robot coordination and for opinion formation in
social groups. Although these algorithms have similar structures, the
mathematical techniques used for their analysis are diverse, and conditions for
their convergence and differ from case to case. In this paper, we review many
of these algorithms and we show that their properties can be analyzed in a
unified way by using a novel tool based on recurrent averaging inequalities
(RAIs). We develop a theory of RAIs and apply it to the analysis of several
important multi-agent algorithms recently proposed in the literature
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