  This paper considers distributed estimation of linear systems when the state
observations are corrupted with Gaussian noise of unbounded support and under
possible random adversarial attacks. We consider sensors equipped with single
time-scale estimators and local chi-square ($\chi^2$) detectors to
simultaneously opserve the states, share information, fuse the
noise/attack-corrupted data locally, and detect possible anomalies in their own
observations. While this scheme is applicable to a wide variety of systems
associated with full-rank (invertible) matrices, we discuss it within the
context of distributed inference in social networks. The proposed technique
outperforms existing results in the sense that: (i) we consider Gaussian noise
with no simplifying upper-bound assumption on the support; (ii) all existing
$\chi^2$-based techniques are centralized while our proposed technique is
distributed, where the sensors \textit{locally} detect attacks, with no central
coordinator, using specific probabilistic thresholds; and (iii) no
local-observability assumption at a sensor is made, which makes our method
feasible for large-scale social networks. Moreover, we consider a Linear Matrix
Inequalities (LMI) approach to design block-diagonal gain (estimator) matrices
under appropriate constraints for isolating the attacks.
