||Fumito Mori, Necessary Condition for Frequency Synchronization in Network Structures, PHYSICAL REVIEW LETTERS, 10.1103/PhysRevLett.104.108701, 104, 10, 2010.03, We present the necessary condition for complete frequency synchronization of phase-coupled oscillators in network structures. The surface area of a set of sites is defined as the number of links between the sites within the set and those outside the set. The necessary condition is that the surface area of any set of cN (0 < c < 1) oscillators in the N-oscillator system must exceed root N in the limit N -> infinity. We also provide the necessary condition for macroscopic frequency synchronization. Thus, we identify networks in which one or both of the above mentioned types of synchronization do not occur..
||Fumito Mori, Atsushi Mochizuki, Expected Number of Fixed Points in Boolean Networks with Arbitrary Topology, PHYSICAL REVIEW LETTERS, 10.1103/PhysRevLett.119.028301, 119, 2, 2017.07, Boolean network models describe genetic, neural, and social dynamics in complex networks, where the dynamics depend generally on network topology. Fixed points in a genetic regulatory network are typically considered to correspond to cell types in an organism. We prove that the expected number of fixed points in a Boolean network, with Boolean functions drawn from probability distributions that are not required to be uniform or identical, is one, and is independent of network topology if only a feedback arc set satisfies a stochastic neutrality condition. We also demonstrate that the expected number is increased by the predominance of positive feedback in a cycle..