Analysis of visitors’ mobility patterns through random walk in the Louvre Museum

Abstract

This paper examines visitors’ sequential movements and their patterns in a large-scale art museum. Visitors’ available time makes their visiting styles different, resulting in dissimilarity in the order and number of visited places and in path sequence length. Since the probability of the appearance of short combinations of nodes is higher than that of long combinations of nodes, shorter path sequences tend to appear more frequently than longer path sequences. This prevents us from evaluating the strength of visitors’ mobility patterns, independent of their path sequence length. In order to solve this problem, we propose the random walk simulation model and compare the results with observed data. A random walk is a minimalistic model providing a reference line for the frequency of sequences as induced by the graph structure of the museum. The random walk simulations can therefore provide us with the probability of transitions between nodes and hence with the probability of each path of a given length. Thus, it enables us to compare the frequency of different path sequence lengths in the same framework. Our results indicate that short-stay visitors exhibit stronger patterns than long-stay visitors, confirming that short-stay visitors are more selective than long-stay visitors in terms of their visiting style. This is suggestive of the informal learning settings in which visitors shape their experiences through exploration in space.

Appendix (or “materials and methods”)

Algorithm for sequences extraction. Let’s define S_i = {s_1i, s_2i… s_pi … s_Pi} the set of all the P possible sequences of length i, where i = {k, k + 1, k + 2 … I}, where i is the length of the longest trajectory in the dataset, and k is the minimum meaningful trajectory in the dataset. Let’s also define f(s_pi) as the number of visitors in the dataset that used the sequence s_pi during their visit to the museum. Finally, lets define T as a table containing the resulting patterns and their frequencies.

Given that N = {1, 2 … n}, the set of nodes in the museum, the high-level steps of the algorithm are as follow:

being that s ⁿ_pi+1 is a sequence of length i + 1 that results from adding a node n belonging to the neighbors of the last node of the sequence s_pi (i.e. s_pi.[i]), to the sequence s_pi.

This algorithm starts from the basis that any sequence including a subsequence cannot be present if such subsequence is not present in the dataset. It then iteratively finds all sequences of a minimum length k that are actually used by at least one visitor. Then builds all the possible sequences of length k + 1 based on the existing sequences of length k and discarding the inexistent ones. By discarding the shortest-length sequences, the algorithm converges faster than if every possible sequence was tested.

We used k = 3 as a starting sequence length, based on the fact that the shortest possible trajectory, (e.g. 0-8-0 or 0-3-0) has a length of three. The resulting table T includes all sequences that appear in the dataset, along with their respective frequencies. Not every possible sequence is going to appear in T, since there are paths that are impossible for visitors to follow due to the physical distribution of the museum.

Spearman’s correlation coefficient The Spearman’s correlation coefficient ρ (Corder and Foreman 2009) is a non-parametric measure of statistical dependence between two variables. The coefficient evaluates how well the relationship between two variables (x, y) can be described by a monotonic function. The coefficient assumes values between − 1 (where $\frac{dy}{dx} < 0 x \in {\mathbb{R}}$) and + 1 (where $\frac{dy}{dx} > 0 x \in {\mathbb{R}}$), the extremes reached when one of the variable is a perfect monotone of the other. A correlation coefficient of zero indicates there is no tendency for y to increase or decrease as x increases. So, if x and y are the variables to correlate, x_i and y_i are the ranked values, then the Spearman’s correlation coefficient can be calculated from:

$$\rho = \frac{{ \mathop \sum \nolimits_{i} \left( {x_{i} - \bar{x}} \right)\left( {y_{i} - \bar{y}} \right)}}{{\sqrt {\mathop \sum \nolimits_{i} (x_{i} - x)^{2} \mathop \sum \nolimits_{i} (y_{i} - \bar{y})^{2} } }}$$