Like a spider's web

“It is anticipated that the whole of the populous parts of the United States will, within two or three years, be covered with net-work like a spider's web.”

— Illustration and quote from The London Anecdotes, 1848
Quoted in The Victorian Internet

Google+ vs. Facebook engagement

Here's a little statistically-insignificant self-experimentation, based on 51 near-simultaneous posts to both Google+ and Facebook, from the beginning of September 2011 to the present.

"Engagement" is the number of unique people (excluding me) who responded, either by commenting, liking or +1'ing the post, or liking or +1'ing a comment on the post. (The scatterplot points are perturbed slightly from their true integral values so they don't completely overlap.) What's remarkable here is how coincidentally similar the engagement is on the two networks — the difference is under 2 percent (!) despite the fact that my social network on Facebook is currently 2.25 times as large as on Google+.

Google+ has very slightly higher engagement on STEM-related posts (science, technology, engineering, and mathematics), while Facebook is slightly higher for other posts, but the differences are well within 95% confidence intervals.

It's possible my social network has somewhat shifted to Google+. Here is the post set split into five chronological partitions with 10 or 11 posts in each.

Engagement as defined above excludes re-sharing posts, because I wasn't confident the two social networks are reporting these in the same way (e.g., do they both report recursive shares?). But there is some interestingly significant difference in sharing behavior on Google+ with STEM posts seeing nearly seven times as much sharing as non-STEM posts in this very small data set, an effect which didn't appear on Facebook.

Of course, all of this is specific to my social network, and really, the sample size is too small to draw any conclusions at all. Now, if someone were to compare posts for a large number of people that cross-post publicly to Facebook and Google+, that could start to get interesting...

Matrix multiplication algorithms over time

The asymptotically fastest algorithm for matrix multiplication takes time O(n^ω) for some value of ω. Here are the best known upper bounds on ω over time.

The latest improvements, the first in over 20 years, are due to Andrew Stothers and Virginia Vassilevska Williams. The latter gave an O(n^2.3727)-time algorithm for multiplying matrices.

When will the sometimes-conjectured ω = 2 be reached? Certainly nothing wrong with taking a linear fit of this data, right?

So that would be around the year 2043. Unfortunately, the pessimist's exponential fit asymptotes to ω = 2.30041...