Mathematical problems for complex networks

Copyright @ 2012 Zidong Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This article is made available through the Brunel Open Access Publishing Fund.Complex networks do exist in our lives. The brain is a neural network. The global economy is a network of national economies. Computer viruses routinely spread through the Internet. Food-webs, ecosystems, and metabolic pathways can be represented by networks. Energy is distributed through transportation networks in living organisms, man-made infrastructures, and other physical systems. Dynamic behaviors of complex networks, such as stability, periodic oscillation, bifurcation, or even chaos, are ubiquitous in the real world and often reconfigurable. Networks have been studied in the context of dynamical systems in a range of disciplines. However, until recently there has been relatively little work that treats dynamics as a function of network structure, where the states of both the nodes and the edges can change, and the topology of the network itself often evolves in time. Some major problems have not been fully investigated, such as the behavior of stability, synchronization and chaos control for complex networks, as well as their applications in, for example, communication and bioinformatics

Resilience and Controllability of Dynamic Collective Behaviors

The network paradigm is used to gain insight into the structural root causes of the resilience of consensus in dynamic collective behaviors, and to analyze the controllability of the swarm dynamics. Here we devise the dynamic signaling network which is the information transfer channel underpinning the swarm dynamics of the directed interagent connectivity based on a topological neighborhood of interactions. The study of the connectedness of the swarm signaling network reveals the profound relationship between group size and number of interacting neighbors, which is found to be in good agreement with field observations on flock of starlings [Ballerini et al. (2008) Proc. Natl. Acad. Sci. USA, 105: 1232]. Using a dynamical model, we generate dynamic collective behaviors enabling us to uncover that the swarm signaling network is a homogeneous clustered small-world network, thus facilitating emergent outcomes if connectedness is maintained. Resilience of the emergent consensus is tested by introducing exogenous environmental noise, which ultimately stresses how deeply intertwined are the swarm dynamics in the physical and network spaces. The availability of the signaling network allows us to analytically establish for the first time the number of driver agents necessary to fully control the swarm dynamics

Stability of Multi-Dimensional Switched Systems with an Application to Open Multi-Agent Systems

Extended from the classic switched system, themulti-dimensional switched system (MDSS) allows for subsystems(switching modes) with different state dimensions. In this work,we study the stability problem of the MDSS, whose state transi-tion at each switching instant is characterized by the dimensionvariation and the state jump, without extra constraint imposed.Based on the proposed transition-dependent average dwell time(TDADT) and the piecewise TDADT methods, along with the pro-posed parametric multiple Lyapunov functions (MLFs), sufficientconditions for the practical and the asymptotical stabilities of theMDSS are respectively derived for the MDSS in the presenceof unstable subsystems. The stability results for the MDSS areapplied to the consensus problem of the open multi-agent system(MAS) which exhibits dynamic circulation behaviors. It is shownthat the (practical) consensus of the open MAS with disconnectedswitching topologies can be ensured by (practically) stabilizingthe corresponding MDSS with unstable switching modes via theproposed TDADT and parametric MLF methods.Comment: 12 pages, 9 figure

Active-passive dynamic consensus filters: Theory and applications

”This dissertation presents a new method for distributively sensing dynamic environments utilizing integral action based system theoretic distributed information fusion methods. Specifically, the main contribution is a new class of dynamic consensus filters, termed active-passive dynamic consensus filters, in which agents are considered to be active, if they are able to sense an exogenous quantity of interest and are considered to be passive, otherwise, where the objective is to drive the states of all agents to the convex hull spanned by the exogenous inputs sensed by active agents. Additionally, we generalize these results to allow agents to locally set their value-of-information, characterizing an agents ability to sense a local quantity of interest, which may change with respect to time. The presented active-passive dynamic consensus filters utilize equations of motion in order to diffuse information across the network, requiring continuous information exchange and requiring agents to exchange their measurement and integral action states. Additionally, agents are assumed to be modeled as having single integrator dynamics. Motivated from this standpoint, we utilize the ideas and results from event-triggering control theory to develop a network of agents which only share their measurement state information as required based on errors exceeding a user-defined threshold. We also develop a static output-feedback controller which drives the outputs of a network of agents with general linear time-invariant dynamics to the average of a set of applied exogenous inputs. Finally, we also present a system state emulator based adaptive controller to guarantee that agents will reach a consensus even in the presence of input disturbances. For each proposed active-passive dynamic consensus filter, a rigorous analysis of the closed-loop system dynamics is performed to demonstrate stability. Finally, numerical examples and experimental studies are included to demonstrate the efficacy of the proposed information fusion filters”--Abstract, page iv