Optimization of the mean first passage time in complex networks

Complex, dynamical processes in nature can be represented by networks, such as the stochastic jumps between numerous metastable states in the energy landscape of proteins, or the intracellular transport of molecules in the endoplasmic reticulum in eukaryotic cells. The state of the system can be represented as a random walk on a complex network, whose structure frequently combines both hierarchy and modularity. The graph topology and the link weight distribution affect the dynamical transitions between the states of the random walkers and ultimetely the trapping efficiency. To study this, we consider a complete graph with weighted links in which selected nodes correspond to absorbing states. We use an electrical circuit analogue to calculate the mean first passage time of random walkers to reach a trap. We propose an optimization rule according to which the edge weights remodel in order to minimize the mean first passage time and the cost to maintain the graph. We investigate how time varying, correlated traps, which can be activated and deactivated in a collective manner but with different correlations, can give rise on hierarchy and modularity in the network.