Introduction to the Modeling and Analysis of Complex Systems

16.2. SIMULATING DYNAMICS ON NETWORKS 347What Hopfield showed is that one can “imprint” a finite number of pre-determinedpatterns into this network by carefully designing the edge weights using the followingsimple encoding formula:w ij = ∑ ks i,k s j,k (16.15)Here s i,k is the state of node i in the k-th pattern to be imprinted in the network. Implementa simulator of the Hopfield network model, and construct the edge weightsw ij from a few state patterns of your choice. Then simulate the dynamics of thenetwork from a random initial condition and see how the network behaves. Onceyour model successfully demonstrates the recovery of imprinted patterns, try increasingthe number of patterns to be imprinted, and see when/how the networkloses the capability to memorize all of those patterns.Exercise 16.12 Cascading failure This model is a continuous-state, discretetimedynamical network model that represents how a functional failure of a componentin an infrastructure network can trigger subsequent failures and cause alarge-scale systemic failure of the whole network. It is often used as a stylizedmodel of massive power blackouts, financial catastrophe, and other (undesirable)socio-technological and socio-economical events.Here are the typical assumptions made in the cascading failure model:• The nodes represent a component of an infrastructure network, such aspower transmitters, or financial institutions. The nodes take non-negative realnumbers as their dynamic states, which represent the amount of load or burdenthey are handling. The nodes can also take a specially designated “dead”state.• Each node also has its own capacity as a static property.• Their network can be in any topology.• The edges can be directed or undirected.• The node states will update either synchronously or asynchronously in discretetime steps, according to the following simple rules:– If the node is dead, nothing happens.– If the node is not dead but its load exceeds its capacity, it will turn to adead state, and the load it was handling will be evenly distributed to its