In the first half of 2020 we kept on posting about the opinion dynamics. Though our focus has drifted towards direction I was working on during my postdoctoral project. One of the main results, compartmental voter model, was presented on Physics of Risk just before the summer holidays.

Obviously my life, as well as yours, was somewhat interrupted by the COVID-19, so I just had to have at least few epidemiological posts. Most of these came after summer holiday, as I tend to prepare lots of future posts in advance. The posts were mostly inspired by email discussions with my colleagues.

All in all, Physics Risk had 41 posts in 2020 (+2 posts in comparison to 2019). 22 of posts were filled under interactive models tag. I was not sure if I will be able to keep the tempo up, but I managed this year by shrugging of some of the extraneous responsibilities. Once again, I start new year with doubts about the future of Physics of Risk, because I still feel that I have too much responsibilities.

Fig. 1:The number of posts written in English and still available on this iteration of Physics of Risk. The wide bars represent total number of posts for each year since 2010, while the narrower bars represent a number of posts with 'Interactive models' tag.

Either way, we will continue with posts on anomalous diffusion by considering it in the voter models. Likely I'll make some new posts on opinion dynamics, too, as I have a few interesting models, which I thought to have already posted about. This will give me a chance to write about the last result from my postdoctoral project, which is related to the Latane's social impact theory.

Santa has a nice red hat, so for our Christmas and New Year holiday gap post we share a video about mathematical hat problems by Numberphile.

In a previous post we have used mean squared displacement to understand diffusion of Brownian and Levy's walks. Alternatively we could use a more familiar statistical measure: variance (or standard deviation).

A well-known example of "normal" diffusion is the Brownian walk. Brownian walk, and "normal" diffusion by consequence, is one of the basic concepts in statistical physics and stochastic modeling. Needless to say, it is one of the most important concepts in our work. But in the empirical we can find examples of different kind of diffusion, which notably faster (known as super-diffusion) or notably slower (known as sub-diffusion) than the "normal" diffusion.

In this post we will compare normal and anomalous diffusion from the point-of-view of mean square displacement (abbr. MSD).

Another group of people using statistical methods have shown that Biden has stolen Trump's votes! To be more precise, they have misused statistical methods to prove their point. Well as you know there are lies, damned lies and statistics.

The "proof" relies on a fact that multiple elections were held at the same time (at least in some states). The people who have conducted the analysis have calculated the difference between the fraction of votes for Trump (in presidential elections), \( v_t \) and the fraction of votes for Republican candidates in other elections, \( v_r \):

\begin{equation} \Delta = v_t - v_r . \end{equation}

They have found that \( \Delta \) decreases with \( v_r \). Their claim is that with larger \( v_r \) we should observe larger \( v_t \), therefore \( \Delta \approx 0 \). Which appears logical from the first glance, but is actually false.

Let us imagine that there is some non-zero probability of defecting (voting for president representing another party), \( p_d \). Assuming that there were \( 1 - v_r \) votes cast for Democrats (equivalent to assumption that there are no third parties), fraction of votes cast for Trump will be given by:

\begin{equation} v_t = v_r (1-p) + (1-v_r) p . \end{equation}

Fraction of votes cast for Trump is a sum of votes from non-defecting Republican voters and votes from defecting Democrat voters. Let us now obtain the expression for difference:

\begin{equation} \Delta = \left[ v_r (1-p) + (1-v_r) p \right] - v_r
= p ( 1 - 2 v_r ) . \end{equation}

So, it appears qualitatively the same as this simplistic model predicts. And we have no election rigging built into the model.

We invite you to watch a video by Matt Parker from Stand-up maths, which explores this topic.