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The first return time (FRT) is the time it takes a random walker to firstreturn to its original site, and the global first passage time (GFPT) is thefirst passage time for a random walker to move from a randomly selected site toa given site. We find that in finite networks the variance of FRT, Var(FRT),can be expressed Var(FRT)~$=2\langle$FRT$ \rangle \langle $GFPT$ \rangle-\langle $FRT$ \rangle^2-\langle $FRT$ \rangle$, where $\langle \cdot \rangle$is the mean of the random variable. Therefore a method of calculating thevariance of FRT on general finite networks is presented. We then calculateVar(FRT) and analyze the fluctuation of FRT on regular branched networks (i.e.,Cayley tree) by using Var(FRT) and its variant as the metric. We find that theresults differ from those in such other networks as Sierpinski gaskets, Vicsekfractals, T-graphs, pseudofractal scale-free webs, ($u,v$) flowers, and fractaland non-fractal scale-free trees.

Analysis of fluctuations in the first return times of random walks on regular branched networks