726 research outputs found
Analysis and applications of spectral properties of grounded Laplacian matrices for directed networks
In-depth understanding of the spectral properties of grounded Laplacian matrices is critical for the analysis of convergence speeds of dynamical processes over complex networks, such as opinion dynamics in social networks with stubborn agents. We focus on grounded Laplacian matrices for directed graphs and show that their eigenvalues with the smallest real part must be real. Power and upper bounds for such eigenvalues are provided utilizing tools from nonnegative matrix theory. For those eigenvectors corresponding to such eigenvalues, we discuss two cases when we can identify the vertex that corresponds to the smallest eigenvector component. We then discuss an application in leader-follower social networks where the grounded Laplacian matrices arise naturally. With the knowledge of the vertex corresponding to the smallest eigenvector component for the smallest eigenvalue, we prove that by removing or weakening specic directed couplings pointing to the vertex having the smallest eigenvector component, all the states of the other vertices converge faster to that of the leading vertex. This result is in sharp contrast to the well-known fact that when the vertices are connected together through undirected links, removing or weakening links does not accelerate and in general decelerates the converging process
Synchronization of Nonlinear Circuits in Dynamic Electrical Networks with General Topologies
Sufficient conditions are derived for global asymptotic synchronization in a
system of identical nonlinear electrical circuits coupled through linear
time-invariant (LTI) electrical networks. In particular, the conditions we
derive apply to settings where: i) the nonlinear circuits are composed of a
parallel combination of passive LTI circuit elements and a nonlinear
voltage-dependent current source with finite gain; and ii) a collection of
these circuits are coupled through either uniform or homogeneous LTI electrical
networks. Uniform electrical networks have identical per-unit-length
impedances. Homogeneous electrical networks are characterized by having the
same effective impedance between any two terminals with the others open
circuited. Synchronization in these networks is guaranteed by ensuring the
stability of an equivalent coordinate-transformed differential system that
emphasizes signal differences. The applicability of the synchronization
conditions to this broad class of networks follows from leveraging recent
results on structural and spectral properties of Kron reduction---a
model-reduction procedure that isolates the interactions of the nonlinear
circuits in the network. The validity of the analytical results is demonstrated
with simulations in networks of coupled Chua's circuits
On Spectral Properties of the Grounded Laplacian Matrix
Linear consensus and opinion dynamics in networks that contain stubborn agents are
studied in this thesis. Previous works have shown that the convergence rate of such dynam-
ics is given by the smallest eigenvalue of the grounded Laplacian induced by the stubborn
agents. Building on those works, we study the smallest eigenvalue of grounded Laplacian
matrices, and provide bounds on this eigenvalue in terms of the number of edges between
the grounded nodes and the rest of the network, bottlenecks in the network, and the small-
est component of the eigenvector for the smallest eigenvalue. We show that these bounds
are tight when the smallest eigenvector component is close to the largest component, and
provide graph-theoretic conditions that cause the smallest component to converge to the
largest component. An outcome of our analysis is a tight bound for Erdos-Renyi random
graphs and d-regular random graphs. Moreover, we de ne a new notion of centrality for
each node in the network based upon the smallest eigenvalue obtained by removing that
node from the network. We show that this centrality can deviate from other well known
centralities. Finally we interpret this centrality via the notion of absorption time in a
random walk on the graph
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