Conformity in numbers – does criticality in social responses exist? Piotr Nyczka, Katarzyna Byrka, Paul R. Nail, Katarzyna Sznajd-Weron Supporting information: S1 Appendix. 1 Derivation of an effective potential and Landau’s approach. 2 3 The goal is to determine the critical point p ∗ , below the system is ordered and the tricritical 4 point z ∗ , i.e. such a value of probability of independence that for z<z ∗ transition is continuous, 5 but for z>z ∗ discontinuous. We will use Landau’s approach and therefore first we need to derive 6 an effective potential in term of the order parameter m. To do this we first need to rewrite the 7 conditional probabilities of conformity α ± , anticonformity β ± and independence γ ± in term of the 8 order parameter m: 9 α + = 1 2 q+1 q  i=r i  k=0 q−i+1  l=0  q i  i k  q − i +1 l  (−1) l m k+l , (1) α − = 1 2 q+1 q  i=r i  k=0 q−i+1  l=0  q i  i k  q − i +1 l  (−1) k m k+l , β + = 1 2 q+1 q  i=r i+1  k=0 q−i  l=0  q i  i +1 k  q − i l  (−1) k m k+l , β − = 1 2 q+1 q  i=r i+1  k=0 q−i  l=0  q i  i +1 k  q − i l  (−1) l m k+l , γ + = 1 − m 4 , γ − = 1+ m 4 . Furthermore we define: 10 α ≡ α + − α − , β ≡ β + − β − , γ ≡ γ + − γ − . (2) (3) By combining (2) and (3) we obtain: 11 α = 1 2 q+1 q  i=r i  k=0 q−i+1  l=0  q i  i k  q − i +1 l  ((−1) l − (−1) k )m k+l , (4) β = 1 2 q+1 q  i=r i+1  k=0 q−i  l=0  q i  i +1 k  q − i l  ((−1) k − (−1) l )m k+l , γ = − m 2 . 018 1/3