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We introduce the notion of the $i$th homological strong persistence property for monomial ideals $I$, providing an algebraic characterization that ensures the chain of inclusions $\\text{Ass}\\,\\text{HS}_i(I)\\subseteq\\text{Ass}\\,\\text{HS}_i(I^2)\\subseteq\\text{Ass}\\,\\text{HS}_i(I^3) \\subseteq\\cdots$. 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In this paper, we investigate the asymptotic behavior of the syzygies of powers of edge ideals through the lens of homological shift ideals $\\text{HS}_i(I(G)^k)$. We introduce the notion of the $i$th homological strong persistence property for monomial ideals $I$, providing an algebraic characterization that ensures the chain of inclusions $\\text{Ass}\\,\\text{HS}_i(I)\\subseteq\\text{Ass}\\,\\text{HS}_i(I^2)\\subseteq\\text{Ass}\\,\\text{HS}_i(I^3) \\subseteq\\cdots$. 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