  Standard multiparty Bell experiments involve a single source shared by a set
of observers. In contrast, network Bell experiments feature multiple
independent sources, and each of them may distribute physical systems to a set
of observers who perform randomly chosen measurements. The $n$-locality
scenario in star-network configuration involves $n$ number of edge observers
(Alices), a central observer (Bob), and $n$ number of independent sources
having no prior correlation. Each Alice shares an independent state with the
central observer Bob. Usually, in network Bell experiments, one considers that
each party measures only two observables. In this work, we propose a
non-trivial generalization of $n$-locality scenario in star-network
configuration, where each Alice performs some integer $m$ number of
binary-outcome measurements, and the central party Bob performs $2^{m-1}$
binary-outcome measurements. We derive a family of generalized $n$-locality
inequalities for any arbitrary $m$. Using {blue}{an elegant} sum-of-squares
approach, we derive the optimal quantum violation of the aforementioned
inequalities can be attained when each and every Alice measures $m$ number of
mutually anticommuting observables. For $m=2$ and $3$, one obtains the optimal
quantum value {blue}{for qubit system local to each Alice, and it is sufficient
to consider the sharing of} a two-qubit entangled state between each Alice and
Bob. We further demonstrate that the optimal quantum violation of $n$-locality
inequality for any arbitrary $m$ can be obtained when every Alice shares
$\lfloor m/2\rfloor$ copies of two-qubit maximally entangled state with the
central party Bob. We also argue that for $m>3$, a single copy of a two-qubit
entangled state may not be enough to exhibit the violation of $n$-locality
inequality but multiple copies of it can activate the quantum violation.
