  In this paper, we propose co-evolution models for both dynamics of opinions
(people's view on a particular topic) and dynamics of social appraisals (the
approval or disapproval towards each other). Opinion dynamics and dynamics of
signed networks, respectively, have been extensively studied. We propose a
co-evolution model, where each vertex $i$ in the network has a current opinion
vector $v_i$ and each edge $(i,j)$ has a weight $w_{ij}$ that models the
relationship between $i,j$. The system evolves as the opinions and edge weights
are updated over time by the following rules, Opinion Dynamics and Appraisal
Dynamics. We are interested in characterizing the long-time behavior of the
dynamic model -- i.e., whether edge weights evolve to have stable signs
(positive or negative) and structural balance (the multiplication of weights on
any triangle is non-negative).
  Our main theoretical result solves the above dynamic system with
time-evolving opinions $V(t)=[v1(t),\cdots,vn(t)]$ and social tie weights
$W(t)=[w_{ij}(t)]_{n\times n}$. For a generic initial opinion vector $V(0)$ and
weight matrix $W(0)$, one of the two phenomena must occur at the limit. The
first one is that both sign stability and structural balance (for any triangle
with individual $i,j,k$, $w_{ij}w_{jk}w_{ki}\geq0$) occur. In the special case
that $V(0)$ is an eigenvector of $W(0)$, we are able to obtain the explicit
solution to the co-evolution equation and give exact estimates on the blowup
time and rate convergence. The second one is that all the opinions converge to
$0$, i.e., $\lim_{t\rightarrow \infty}|V(t)|=0$.
