Supplemental Material - Analyzing urban scaling laws in the United States over 115 years

Supplemental Material for Analyzing urban scaling laws in the United States over 115 years by Keith Burghardt, Johannes H Uhl, Kristina Lerman, Stefan Leyk in Environment and Planning B: Urban Analytics and City Science.</p

Supplemental Material - Analyzing urban scaling laws in the United States over 115 years

Supplemental Material for Analyzing urban scaling laws in the United States over 115 years by Keith Burghardt, Johannes H Uhl, Kristina Lerman, Stefan Leyk in Environment and Planning B: Urban Analytics and City Science.</p

Supplemental Material - Analyzing urban scaling laws in the United States over 115 years

Supplemental Material for Analyzing urban scaling laws in the United States over 115 years by Keith Burghardt, Johannes H Uhl, Kristina Lerman, Stefan Leyk in Environment and Planning B: Urban Analytics and City Science.</p

Distribution of the final number of answers to each question and question popularity versus the final number of answers.

<p>(Top row) Complementary cumulative distribution of the final number of answers posted in reply to a question as of September, 2014, on (a) technical, (b) non-technical, and (c) meta sites. Shaded areas correspond to the standard deviation in the distributions made for each board. (Bottom row) Number of views per question as a function of the number of answers on (d) technical, (e) non-technical, and (f) meta sites. Boxes indicate 50% confidence intervals, with a red line to indicate the median view count, and a red dot to represent the mean viewcount.</p

Word share regression coefficients increase with the number of answers users see.

<p>Word share regression coefficients for voting before (red triangles) and after (blue squares) an answer is accepted, as well as accepting an answer (green circles) for (a) technical, (b) non-technical, and (c) meta boards, with 2 to 20 answers. The shaded region represents the uncertainty in our values (see Section 2). Across all boards, voters appear increasingly likely to choose answers that take up a relatively large amount of web page space as the number of answers grows.</p

Answer acceptance increases the probability an answer will be voted on.

<p>Regression coefficients for voting on an (eventually) accepted answer before (red triangles) and after (blue squares) that answer is accepted for (a) technical, (b) non-technical, and (c) meta boards, with 2 to 20 answers. The shaded region represents the uncertainty in our values (see Section 2). There is a large and increasing vote dependence on the accepted answer once the asker accepts it, compared to before the answer is accepted, meaning the signal that this answer is accepted appears to have a statistically significant effect on voter behavior.</p

Relative size of regression parameters for answer acceptance and voting.

<p>Regression coefficients for answerers to accept (green circles) and voters to vote for an answer both before (red triangles) and after (blue squares) an answer is accepted on (a) technical, (b) non-technical, and (c) meta boards, averaged over the number of available answers from two to twenty. Higher values indicate a stronger relationship between attributes and user behavior (voting or accepting an answer). Error bars indicate the variance of the best-fit values values across two to twenty answers.</p

AUC versus number of answers for the full model, position null model, and social influence null model for technical boards.

<p>AUC for (a) voting before an answer is accepted, (b) accepting an answer, and (c) voting after an answer is accepted versus the number of answers in technical boards (See <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0173610#pone.0173610.s004" target="_blank">S4 Fig</a> for similar plots with non-technical and meta boards). Solid lines correspond to the full models, while the lighter dashed lines correspond to the position null model, in which the probability of picking answers decreases monotonically with the web page order. Finally, the dark dashed lines correspond to the “social influence” null model, in which social signals are the only attributes used in the model. Shaded regions correspond to standard deviations in values based on bootstrapping the test data.</p

Mean AUC versus models with various attributes removed.

<p>The AUC for voting before (red lines) and after (blue lines) an answer is accepted, as well as accepting an answer (green lines) averaged over two through twenty answers, weighted by the amount of test data, in (a) technical, (b) non-technical, and (c) meta boards. See main text for descriptions of models. Error bars are standard deviations of the mean values. We remove attributes and measure the drop in mean AUC to determine the relative importance of various attributes. The statistical significance of the drop in AUC values compared to the full model is as follows: “***”, p<0.001; “**”, p<0.0028 (Bonferroni correction for 18 variables with <i>α</i> = 0.05 [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0173610#pone.0173610.ref042" target="_blank">42</a>]); “–”, not significant.</p