Detailed Balance Equation: X,Y∈[gV, u(X)P(X,Y)=u(Y)P(Y;X) o∈[glV：the proposals of all vertices C∈{0，IE：indicates whether each edge e∈passes its check 2x→y≌{(o,C)|X→Y when the random choice is(o,C)} P(X,Y) ）_∑o,e)EQx-y Pr(o)Pr(C|o,X) = (Y) P(Y,X) >(oc)Eyx Pr(o)Pr(ca,Y) (X) Bijection x,y:2x→y→2y→k is constructed as: 了C=C' (a,C)xy(a,C)）s.t if Ce =1 for all e incident with v,theno.,=X otherwise o=v > m头- (Yu) Π. Ae(Yu,Yo）=(Y) b.(X) v∈V Ae(Xu,X)u(X) e=uv∈EP(X, Y ) P(Y,X) = P (￾,C)2⌦X!Y Pr(￾)Pr(C | ￾, X) P (￾,C)2⌦Y !X Pr(￾)Pr(C | ￾, Y ) Detailed Balance Equation: 8X, Y 2 [q] µ(X)P(X, Y ) = µ(Y )P(Y,X) V , ￾ 2 [q] V : the proposals of all vertices C 2 {0, 1}E : indicates whether each edge e∈E passes its check ⌦X!Y , {(￾, C) | X ! Y when the random choice is (￾, C)} Bijection ￾X,Y : ⌦X!Y ! ⌦Y !X is constructed as: (￾, C) ￾X,Y 7￾! (￾ 0 , C0 ) C = C0 if for all Ce = 1 e incident with v, then ￾ 0 v = Xv otherwise ￾ 0 v = ￾v ⇢ s.t. = µ(Y ) µ(X) Pr(￾)Pr(C | ￾, X) Pr(￾0 )Pr(C0 | ￾0 , Y ) = Y v2V bv(Yv) bv(Xv) Y e=uv2E Ae(Yu, Yv) Ae(Xu, Xv) = µ(Y ) µ(X)