The functional architecture undergoes alterations during the preclinical phase of Alzheimer's disease. Consequently, the primary research focus has shifted towards identifying Alzheimer's disease and its early stages by constructing a functional connectivity network based on resting-state fMRI data. Recent investigations show that as Alzheimer's Disease (AD) progresses, modular tissue and connections in the core brain areas of AD patients diminish. Sparse learning methods are powerful tools for understanding Functional Brain Networks (FBNs) with Regions of Interest (ROIs) and a connectivity matrix measuring functional coherence between them. However, these tools often focus exclusively on functional connectivity measures, neglecting the brain network's modularity. Modularity orchestrates dynamic activities within the FBN to execute intricate cognitive tasks. To provide a comprehensive delineation of the FBN, we propose a local similarity-constrained low-rank sparse representation (LSLRSR) method that encodes modularity information under a manifold-regularized network learning framework and further formulate it as a low-rank sparse graph learning problem, which can be solved by an efficient optimization algorithm. Specifically, for each modularity structure, the Schatten p-norm regularizer reduces the reconstruction error and provides a better approximation of the low-rank constraint. Furthermore, we adopt a manifold-regularized local similarity prior to infer the intricate relationship between subnetwork similarity and modularity, guiding the modeling of FBN. Additionally, the proximal average method approximates the joint solution's proximal map, and the resulting nonconvex optimization problems are solved using the alternating direction multiplier method (ADMM). Compared to state-of-the-art methods for constructing FBNs, our algorithm generates a more modular FBN. This lays the groundwork for further research into alterations in brain network modularity resulting from diseases.