AbstractIn this note, we study discrete time majority dynamics over an inhomogeneous random graph G obtained by including each edge e in the complete graph $$K_n$$
K
n
independently with probability $$p_n(e)$$
p
n
(
e
)
. Each vertex is independently assigned an initial state $$+1$$
+
1
(with probability $$p_+$$
p
+
) or $$-1$$
-
1
(with probability $$1-p_+$$
1
-
p
+
), updated at each time step following the majority of its neighbors’ states. Under some regularity and density conditions of the edge probability sequence, if $$p_+$$
p
+
is smaller than a threshold, then G will display a unanimous state $$-1$$
-
1
asymptotically almost surely, meaning that the probability of reaching consensus tends to one as $$n\rightarrow \infty $$
n
→
∞
. The consensus reaching process has a clear difference in terms of the initial state assignment probability: In a dense random graph $$p_+$$
p
+
can be near a half, while in a sparse random graph $$p_+$$
p
+
has to be vanishing. The size of a dynamic monopoly in G is also discussed.
A Note on the Majority Dynamics in Inhomogeneous Random Graphs
