Introduction to the Modeling and Analysis of Complex Systems

346 CHAPTER 16. DYNAMICAL NETWORKS I: MODELING• random graph• barbell graph• ring-shaped graph (i.e., degree-2 regular graph)Discuss how the network topology affects the synchronization. Will it make synchronizationeasier or more difficult?Exercise 16.10 Conduct simulations of the Kuramoto model by systematically increasingthe amount of variations of ω i (currently set to 0.05 in the initializefunction) and see when/how the transition of the system behavior occurs.Here are some more exercises of dynamics on networks models. Have fun!Exercise 16.11 Hopfield network (a.k.a. attractor network) John Hopfield proposeda discrete state/time dynamical network model that can recover memorizedarrangements of states from incomplete initial conditions [20, 21]. This was one ofthe pioneering works of artificial neural network research, and its basic principlesare still actively used today in various computational intelligence applications.Here are the typical assumptions made in the Hopfield network model:• The nodes represent artificial neurons, which take either -1 or 1 as dynamicstates.• Their network is fully connected (i.e., a complete graph).• The edges are weighted and symmetric.• The node states will update synchronously in discrete time steps according tothe following rule:( )∑s i (t + 1) = sign w ij s j (t)(16.14)jHere, s i (t) is the state of node i at time t, w ij is the edge weight betweennodes i and j (w ij = w ji because of symmetry), and sign(x) is a function thatgives 1 if x > 0, -1 if x < 0, or 0 if x = 0.• There are no self-loops (w ii = 0 for all i).