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We investigate the spatial distribution of Lyman alpha emitting galaxies (LAEs) at $z \approx 2.67$, selected from the NOAO Deep Wide-Field Survey (NDWFS), using two-point statistics and topological diagnostics adopted from network science. We measure the clustering length, $r_0 \approx 4 h^{-1}$ Mpc, and the bias, $b_{LAE} = 2.2^{+0.2}_{-0.1}$. Fitting the clustering with halo occupation distribution (HOD) models results in two disparate possibilities: (1) where the fraction of central galaxies is $<$1% in halos of mass $>10^{12}$$M_\odot$; and (2) where the fraction is $\approx$20%. We refer to these two scenarios as the `Dusty Core Scenario' for Model#1 since most of central galaxies in massive halos are dead in Ly$\alpha$ emission, and the `Pristine Core Scenario' for Model#2 since the central galaxies are bright in Ly$\alpha$ emission. Traditional two-point statistics cannot distinguish between these disparate models given the current data sets. To overcome this degeneracy, we generate mock catalogs for each HOD model using a high resolution $N$-body simulation and adopt a network statistics approach, which provides excellent topological diagnostics for galaxy point distributions. We find three topological anomalies from the spatial distribution of observed LAEs, which are not reproduced by the HOD mocks. We find that Model#2 matches better all network statistics than Model#1, suggesting that the central galaxies in $> 10^{12} h^{-1} M_\odot$ halos at $z \approx 2.67$ need to be less dusty to be bright as LAEs, potentially implying some replenishing channels of pristine gas such as the cold mode accretion., Comment: 23 pages, 18 figures, accepted by MNRAS