The power of Connection is one of three powers needed to fight the 4D Beings, along with Destruction and Alteration. Unlike the other two genes, the Connection gene does not have any known practical use in the Eternal Sphere, but has an important purpose. It lets Third-Dimensional Beings connect to Fourth-Dimensional Space through special terminals.
It is never stated whether or not this power is related to Sophia's ability to use symbology.
The Connection gene was placed inside Sophia Esteed. Since Sophia was born after Fayt and Maria, the research team had less time to research the gene, and thus could not fully perfect it, meaning Sophia's powers aren't as stable as the other two's.
After the events of Moonbase, the party travels to Planet Styx, where they encounter the Time Gate. While Sophia knew her powers could open up a gate to Fourth-Dimensional Space, she had no idea of how to actually use her powers. However, when she approached it, her arms started reacting, and a powerful force was unleashed, opening a gate in the Time Gate.
A power given to Sophia for the fight against the 4D beings, that is capable of penetrating the barrier between 4D space and the Eternal Sphere. Since this ability was researched and created in a short period of time after the others were complete, it doesn't always work perfectly.