8.
Two properties of diagraph nodes <ul><li>Out-degree: Number of connections going out from a node </li></ul><ul><li>In-degree: Number of connections going in to a node </li></ul><ul><li>Edge: Connection between any two nodes </li></ul>11/07/11 ARRI, UTA
9.
Important types of Diagraphs Balanced Strongly Connected Tree 11/07/11 ARRI, UTA
10.
What is Consensus among nodes Consensus in the English language is defined firstly as unanimous or general agreement Before Consensus After Consensus 11/07/11 ARRI, UTA
11.
Graph Dynamics (Diagraph) Adjacency Matrix or Diagonal Matrix Laplacian matrix Note that is row stochastic 11/07/11 ARRI, UTA
12.
Continuous Time System <ul><li>Each node if assumed to have simple integrator dynamics, for -th node, </li></ul><ul><li>Input </li></ul><ul><li>Resultant Dynamics of the graph with all node </li></ul>11/07/11 ARRI, UTA
13.
Comment As is row stochastic The first eigenvalue of will be 0 The right eigenvector corresponding to 0 eigenvalue will be At steady state all state values will be equal 11/07/11 ARRI, UTA
14.
State solution Eigen decomposition and Left and right eigenvector Right eigenvector Left eigenvector 11/07/11 ARRI, UTA
20.
What if there is one leader in the graph Assuming rest of the graph is connected The Laplacian matrix of a graph with a leader with may be anything Left eigenvector 11/07/11 ARRI, UTA
21.
Consensus value for one leader graph Note that if there is more than one leaders then no single solution is possible 11/07/11 ARRI, UTA
22.
Simulation result (one leader case) For tree network the result will be equivalent 11/07/11 ARRI, UTA
23.
Graph contains a spanning tree How the value of can be determined ? 11/07/11 ARRI, UTA
24.
Eigenvalue properties <ul><li>For stability all the eigenvalues should be in the left half of the plane </li></ul><ul><li>The second largest eigenvalue is of a standard laplacian matrix is known as Fiedler eigenvalue </li></ul><ul><li>Fiedler eigenvalue determines the speed of the whole network, thus it is important to maximize its value </li></ul><ul><li>Note that Fiedler eigenvalue in general can not be determined from the dominant eigenvalue of the inverse of laplacian matrix </li></ul>11/07/11 ARRI, UTA
25.
Gershgorin disk of a network 11/07/11 ARRI, UTA
26.
More comments <ul><li>Fiedler eigenvalue is also known as algebraic connectivity or spectral gap of a graph </li></ul><ul><li>Algebraic connectivity is different from connectivity or vertex-connectivity </li></ul><ul><li>Network synchronization speed does NOT depend on vertex-connectivity </li></ul><ul><li>Number of zero eigenvalues in a laplacian matrix reveals, number of connected components in a graph </li></ul>11/07/11 ARRI, UTA
27.
Reducibility Consider a matrix with . If is reducible, there exist an integer and a Permutation matrix such that 11/07/11 ARRI, UTA
28.
Irreducibility 11/07/11 ARRI, UTA Consider a matrix . Then, is irreducible if and only if For any scalar .
29.
Comment on reducibility <ul><li>A connected graph (strongly/balanced) is generally have irreducible adjacency and laplacian matrix </li></ul><ul><li>A tree network generally posses a reducible adjacency and laplacian matrix </li></ul>11/07/11 ARRI, UTA
30.
Discrete time system Murray-Saber, 2004 Continuous time system Max out-degree Discretized Perron matrix 11/07/11 ARRI, UTA
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