Entropy analysis of Boolean network reduction according to the determinative power of nodes

Abstract

Boolean networks are utilized to model systems in a variety of disciplines. The complexity of the systems under exploration often necessitates the construction of model networks with large numbers of nodes and unwieldy state spaces. A recently developed, entropy-based method for measuring the determinative power of each node offers a new method for identifying the most relevant nodes to include in subnetworks that may facilitate analysis of the parent network. We develop a determinative-power-based reduction algorithm and deploy it on 36 network types constructed through various combinations of settings with regards to the connectivity, topology, and functionality of networks. We construct subnetworks by eliminating nodes one-by-one beginning with the least determinative node. We compare entropy ratios between these subnetworks and the parent network and find that, for all network types, the change in network entropies (sums of conditional node entropies) follows a concave down decreasing curve, and the slightest reductions in network entropy occur with the initial reductions which eliminate the nodes with the least determinative power. Comparing across the three network characteristics, we find trends in the rates of decrease in the entropy ratios. In general, the decline occurs more slowly in networks with degree values assigned from a power-law distribution and canalyzing functions of higher canalization depth. We compare results of the determinative-power-based reduction with those of a randomized reduction and find that, in forming subnetworks with maximal network entropy, the determinative-power-based method performs as well as or better than the random method in all cases. Lastly, we compare findings based on this conditional-entropy-based calculation of network entropy with those of an alternative calculation using simple sums of (independent) node entropies to demonstrate the vast differences resulting from the two approaches.