Graphs

A.1 Construction of a D-EquiStatic graph A practical method to construct a D-EquiStatic weight matrix W is provided in Alg. 3. We should mention that the "while" loop in the algorithm is adopted to guarantee kΠWk2 ρ. Construct W by (3); end Output: The D-EquiStatic weight matrix W and its associated basis indices {ut}Mt=1 A.2 Proof of Theorem 1 Before showing properties of W defined by (3), we provide two lemmas as follows. Referring to Theorem 1.6 of [32], we have the following result for a sequence of random matrices. Lemma 1 (Matrix Bernstein) Consider a sequence of K independent random n n matrices {Mi}Ki=1. Assume that each random matrix satisfies E[Mi] = 0, and kMik2 R almost surely. Theorem 1 (Formal restatement of Theorem 1) Let A(u) be defined by (2) for any u [n 1]and the D-EquiStatic weight matrix W be constructed by (3) with {ui}Mi=1 following an independent and identical uniform distribution from [n 1].