The observed accelerated expansion of the Universe may be explained by dark energy or the breakdown of general relativity (GR) on cosmological scales. When the latter case, a modified gravity scenario, is considered, it is often assumed that the background evolution is the same as the $Lambda$CDM model but the density perturbation evolves differently. In this paper, we investigate more general classes of modified gravity, where both the background and perturbation evolutions are deviated from those in the $Lambda$CDM model. We introduce two phase diagrams, $alpha{rm-}fsigma _8$ and $H{rm-}fsigma _8$ diagrams; $H$ is the expansion rate, $fsigma_8$ is a combination of the growth rate of the Universe and the normalization of the density fluctuation which is directly constrained by redshift-space distortions, and $alpha$ is a parameter which characterizes the deviation of gravity from GR and can be probed by gravitational lensing. We consider several specific examples of Horndeskis theory, which is a general scalar-tensor theory, and demonstrate how deviations from the $Lambda$CDM model appears in the $alpha{rm-}fsigma _8$ and $H{rm-}fsigma _8$ diagrams. The predicted deviations will be useful for future large-scale structure observations to exclude some of the modified gravity models.
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