Dealing with correlated choices: How a spin-glass model can help political parties select their policies

Starting from preferences on $N$ proposed policies obtained via questionnaires from a sample of the electorate, an Ising spin-glass model in a field can be constructed from which a political party could find the subset of the proposed policies which would maximize its appeal, form a coherent choice in the eyes of the electorate, and have maximum overlap with the party's existing policies. We illustrate the application of the procedure by simulations of a spin glass in a random field on scale-free networks.

Figure 1

Typical networks with $N=512$ spins like those which were simulated. The left panel is for $\lambda =4.5$, where the network resembles a random graph. The right panel is for $\lambda =2.5$, at which value the network has similarities to many scale-free networks in nature and sociology [15, 16]. The number of edges that each vertex (spin) has is encoded in the size of the dot. Large dots represent strongly connected spins, whereas small dots have few connections. Note that we choose a minimum connectivity of 3 to prevent dangling ends in the network. As can be clearly seen, smaller values of $\lambda$ make for higher connectivity on a few nodes.

Figure 2

Example distributions $P\left(q\right)$ for $N=512$ and individual different interactions $J_{ij}$, as well as random fields $h_i$ on scale-free networks with $\lambda =2.5$ (left) and $\lambda =4.5$ (right) and $T=0.2862\ll T_c$. Solid lines (red) are for $H=0$, whereas dashed (blue) lines are for $H=0.10$. Note that when $H=0$ (solid curves) the distributions are symmetric, i.e., $P\left(q\right)=P(-q)$. In panels (a), (b), and (c) there seem to be 4 peaks/shoulders for both $H=0$ and $H=0.1$, while for panels (d), (e), and (f) 12 $\left[1\right]$, 4 $\left[6\right]$, and 2 $\left[6\right]$ peaks are visible for $H=0$ $[H=0.10]$.