To be able to use Theorem
1 to make predictions, we first need to estimate the proportion of potential stubborn nodes in the population, that is the percentage of votes which are guaranteed either for Conservatives or other parties. Let
\(s_0\) denote the number of stubborn state-0 (non-Conservative) nodes and
\(s_1\) that of state-1 (Conservative) ones. We look for the values
\((s_0^\star ,s_1^\star )\) that maximise the log-likelihood of the observed data
\((x_1, \ldots , x_m)\) under the assumption that those were generated via a realisation of the voter model. Let’s say we want to predict results for the
ith election. Because we need at least two datapoints to make an estimation, we require
\(3\leqslant i \leqslant m+1\). Following the notations introduced in Sect.
3 we let
\(p_{k,l}^{(s_0,s_1)}(t)\) denote the theoretical probability for
\(N_1(t)\) to go from
k to
l in
t units of time when there are respectively
\(s_0\) and
\(s_1\) state-0 and state-1 stubborn nodes. We seek to solve
$${\mathop{\arg \max }\limits_{{s_{0} ,s_{1} }}} \; \sum _{j=1}^{i-2} \log \left( p_{x_j,x_{j+1}}^{(s_0,s_1)}(t_{j+1}-t_j) \right).$$
(6)
Indeed,
\(p_{x_j,x_{j+1}}^{(s_0,s_1)}(t_{j+1}-t_j)\) is by definition the probability for Conservatives to win
\(x_{j+1}\) percent of the votes in the
\((j+1)\)th election knowing they won
\(x_j\) percent in the
jth one. Thus we seek to simultaneously maximise the likelihood of all past elections results. Let
\(Q^{(s_0,s_1)}\) be the matrix with entries calculated via (
2). By Theorem
1, we have that (
6) is equivalent to
$${\mathop{\arg \max }\limits_{{s_{0} ,s_{1} }}} \; \sum _{j=1}^{i-2} \log \left[ e^{(t_{j+1}-t_j)Q^{(s_0,s_1)}} \right] _{x_j,x_{j+1}}$$
(7)
The computation of matrix exponential is typically done in cubic time and quickly becomes intractable as the size of the matrix increases. Here however, because we have
\(n=100\), the number of possible couples
\((s_0,s_1)\) is small enough here that (
7) can be solved by directly computing the sum for each of these couples individually. The optimal value
\(s_1^\star\) for
\(s_1\) then gives us an estimation of the percentage of votes “locked” by the Conservative party, proportion of the population that will always root for them. The optimal value
\(s_0^\star\) for
\(s_0\) is an estimate of the quantity of such votes for all other parties aggregated.
To make a forecast for the
ith election, we just have to apply Theorem
1 with
\(Q=Q^{(s_0^\star ,s_1^\star )}\),
\(n_1=x_{i-1}\) and
\(t=t_i-t_{i-1}\). Equation
4 then gives us the expected percentage
\({\tilde{x}}_i\) of votes gathered by Conservatives on that occasion. This can then be compared to the actual value
\(x_i\) to assess the efficacy of our approach.