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This paper investigates the sparse <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> model with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>𝓁</mi><mn>1</mn></msub></semantics></math></inline-formula> penalty in the field of network data models, which is a hot topic in both statistical and social network research. We present a refined algorithm designed for parameter estimation in the proposed model. Its effectiveness is highlighted through its alignment with the proximal gradient descent method, stemming from the convexity of the loss function. We study the estimation consistency and establish an optimal bound for the proposed estimator. Empirical validations facilitated through meticulously designed simulation studies corroborate the efficacy of our methodology. These assessments highlight the prospective contributions of our methodology to the advanced field of network data analysis.