Contents -- Preface -- 1. Introduction -- 2. Graphs, Networks, Laplacian Matrices and Algebraic Connectivity -- 2.1 Graphs and digraphs -- 2.2 Matrices and graphs -- 2.3 Algebraic connectivity -- 2.3.1 Undirected graphs -- 2.3.2 Directed graphs -- 2.3.3 Basic properties of a1 and b1 -- 2.3.4 Basic properties of a2 -- 2.4 Locally connected graphs -- 2.4.1 Basic properties of a3 and a4 -- 2.5 Examples -- 2.6 Hypergraphs -- 2.7 Further reading -- 3. Graph Models -- 3.1 Examples of complex networks -- 3.1.1 Neural networks -- 3.1.2 Transportation networks -- 3.1.3 World Wide Web -- 3.1.4 Erdos number and scientific collaboration network -- 3.1.5 Film actor network -- 3.1.6 Citation network -- 3.2 Classical random graph models -- 3.2.1 Algebraic connectivity of random graphs -- 3.3 Small-world networks -- 3.3.1 Watts-Strogatz model -- 3.3.2 Newman-Watts model -- 3.3.3 Algebraic connectivity of small-world networks -- 3.4 Scale-free networks -- 3.5 Random geometric graphs -- 3.6 Graphs with a prescribed degree sequence -- 3.7 Algebraic connectivity and degree sequence -- 3.7.1 Regular graphs -- 3.7.1.1 Construction 1: graph with low 2 and r -- 3.7.1.2 Construction 2: graph with high 2 and r -- 3.7.2 Graphs with prescribed degree sequence -- 3.7.2.1 Construction 1: graph with low 2 and r -- 3.7.2.2 Construction 2: graph with high 2 and r -- 3.8 Further reading -- 4. Synchronization in Networks of Nonlinear Continuous-time Dynamical Systems -- 4.1 Static coupling topology -- 4.1.1 Properties of (G) -- 4.1.2 Computing (G) -- 4.1.3 Zero row sums matrices -- 4.1.4 Matrices in W -- 4.1.5 Synchronization and algebraic connectivity -- 4.2 Coupling topology with a spanning directed tree -- 4.3 Time-varying coupling topology -- 4.4 Coupling between delayed state variables -- 4.4.1 Choosing the factorization B1B2 = UG V D.

4.4.2 Choosing the matrix U 2 Ws -- 4.4.3 Choosing the matrix K -- 4.4.4 The case D = 0 -- 4.5 Synchronization criteria based on algebraic connectivity -- 4.6 Further reading -- 5. Synchronization in Networks of Coupled Discrete-time Systems -- 5.1 Synchronization of coupled scalar maps via contractivity of operators -- 6. Synchronization in Network of Systems with Linear Dynamics -- 6.1 Autonomous coupling -- 6.2 Nonautonomous coupling: continuous-time case -- 6.2.1 Slowly varying coupling -- 6.3 Nonautonomous coupling: discrete-time case -- 6.3.1 A discrete-time consensus problem -- 6.4 Ergodicity of inhomogeneous Markov chains -- 6.5 Contractive matrices -- 6.5.1 Pseudocontractivity and scrambling stochastic matrices -- 6.5.2 Set-nonexpansive and set-contractive operators -- 6.5.3 Set-contractivity under the max-norm -- 6.5.4 Set-contractivity under the Euclidean norm -- 6.5.5 Set-contractivity under a weighted Euclidean norm -- 6.5.6 Set-contractivity and coefficient of ergodicity -- 6.6 Further reading -- 7. Agreement and Consensus Problems in Groups of Interacting Agents -- 7.1 Continuous-time models -- 7.1.1 Rate of exponential convergence -- 7.1.2 Dynamic coupling topology -- 7.2 Discrete-time models -- 7.2.1 Follow the leader dynamics and leadership in coordinated agents -- 7.3 A nonlinear model for consensus -- 7.4 Agreement in random networks -- 7.4.1 Synchronization in random networks without the scrambling condition -- 7.5 Further reading -- Appendix A Algebraic Connectivity and Combinatorial Properties of a Graph -- A.1 Properties of algebraic connectivity -- A.2 Vertex and edge connectivity -- A.3 Graph partitions -- A.3.1 Maximum directed cut -- A.3.2 Edge-forwarding index -- A.3.3 Bisection width -- A.3.4 Isoperimetric number -- A.3.5 Minimum ratio cut -- A.3.6 Independence number -- A.4 Semibalanced graphs -- Bibliography -- Index.