2007 Research Fair - Physics Abstracts
Phase Transitions of Entropy and Mutual Information
Abstract networks—Graphs—model complex systems. A graph consists of a set of nodes (called vertices) and a set of connections (called edges). Our networks are directed, weighted, and totally connected. Unlike static graphs, the networks are dynamic systems. These networks exhibit decay-to-fixed point, chaotic, and oscillatory behavior. We analyze network behavior in two ways: with the average joint entropy and the mutual information between pairs of nodes. These two measures offer a different perspective of network activity. High entropy and low mutual information indicates a chaotic network; low entropy and low mutual information indicates decay-to-fixed point behavior; and intermediate to low entropy and high mutual information indicates oscillatory networks. The networks demonstrate a sudden change in entropy and mutual information as we vary the weight bias. We posit that the abrupt change in the entropy or mutual information constitutes a phase transition with respect to the weight bias parameter.
Evolvability of Totally Connected Networks
Abstract networks model many complex systems. Metabolic pathways, the Internet and the nervous system are all examples readily modeled as networks. We study the evolvability of networks that simulate a variety of non-linear complex systems. Our networks are graphs that are directed, weighted, and totally connected. Moreover, they are dynamic systems. Signals propagate through them via modulating connections. The activity in these networks varies widely as we vary the mean of the connection weight. Nodes exhibit decay-to-fixed point, chaotic, and oscillatory activity. The transitions between these different activities are phase transitions in entropy. High entropy indicates a very active or chaotic network; low entropy indicates decay-to-fixed point behavior; intermediate entropy indicates oscillatory networks. Our hypothesis is that networks with a weight bias near a phase transition evolve readily. We evolve networks with different weight biases through twenty generations and rank them according to their performance on the test problem. Results confirm our hypothesis.