We have investigated a pair of connected networks of agents. Each node is an agent with opinion s_i=+1 or -1. Both networks A and B are Barabasi-Albert scale-free networks, and they are linked by E_AB internetwork connections. We used Ising dynamics to model a majority-driven behavior, such as social opinion formation.
Three stable states of the whole system are possible. First is fully ordered state. Second is two ordered subnetworks, but with opposite spins. Third is disordered, paramagnetic state.
If two networks are competing (are ordered with opposite spins) and we increase noise (temperature) or interaction strength (number of internetwork links) the system will swithch to full orderer state. Our main concern is which network will "win" such competition by imposing its initial spin on the whole system.
It is obvious, that for a symmetric case the winning network is "selected" at random. We investigate how the network size, density and the interlinking algorithms influence the winner. We also search for observable signals that could tell us in advance which network will win. Such signals can be extremely useful when given real interacting networks (for example social systems). Fluctuation-dissipation theorem predicts that the network with larger fluctuations will lose. We investigate if the theorem holds in this case. We also check, if unusual linking of both networks, or unusual network characteristics might change the outcome.

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