  We study the problem of allocating indivisible resources under the
connectivity constraints of a graph $G$. This model, initially introduced by
Bouveret et al. (published in IJCAI, 2017), effectively encompasses a diverse
array of scenarios characterized by spatial or temporal limitations, including
the division of land plots and the allocation of time plots. In this paper, we
introduce a novel fairness concept that integrates local comparisons within the
social network formed by a connected allocation of the item graph. Our
particular focus is to achieve pairwise-maximin fair share (PMMS) among the
"neighbors" within this network. For any underlying graph structure, we show
that a connected allocation that maximizes Nash welfare guarantees a
$(1/2)$-PMMS fairness. Moreover, for two agents, we establish that a
$(3/4)$-PMMS allocation can be efficiently computed. Additionally, we
demonstrate that for three agents and the items aligned on a path, a PMMS
allocation is always attainable and can be computed in polynomial time. Lastly,
when agents have identical additive utilities, we present a
pseudo-polynomial-time algorithm for a $(3/4)$-PMMS allocation, irrespective of
the underlying graph $G$. Furthermore, we provide a polynomial-time algorithm
for obtaining a PMMS allocation when $G$ is a tree.
