Multifractality in individual honeybee behavior hints at colony-specific social cascades: Reanalysis of radio-frequency identification data from five different colonies

Honeybees (Apis mellifera) exhibit complex coordination and interaction across multiple behaviors such as swarming. This coordination among honeybees in the same colony is remarkably similar to the concept of informational cascades. The multifractal geometry of cascades suggests that multifractal measures of individual honeybee activity might carry signatures of these colony-wide coordinations. The present work reanalyzes time stamps of entrances to and exits from the hive captured by radio-frequency identification (RFID) sensors reading RFID tags on individual bees. Indeed, both multifractal spectrum width for individual bees’ inter-reading interval series and differences of those widths from surrogates significantly predicted not just whether the individual bee's hive had a mesh enclosure but also predicted the specific membership of individual bees in one of five colonies. The significant effects of multifractality in matching honeybee activity to type of colony and, further, matching individual honeybees to their exact home colony suggests that multifractality quantifies key features of the colony-wide interactions across many scales. This relevance of multifractality to predicting colony type or colony membership adds additional credence to the cascade metaphor for colony organization. Perhaps, multifractality provides a new tool for exploring the relationship between individual organisms and larger, more complex social behaviors.

Figure 1

Four example plots of RFID-reading interval series from four example honeybees. Original series lengths varied, but these plots depict only 100 successive intervals from each so as to show examples of the same length. All series depicted passed the augmented Dickey-Fuller test for stationarity, with nonstationarity rejected at $p<0.05$.

Figure 2

Example plot of RFID-reading time stamp series for an individual honeybee (left panel) and of inter-reading intervals series for the same honeybee (right panel). Arrows indicate the location on the right panel's vertical axis of the median, mean, and maximum inter-reading interval $(I_{\mathrm{Median}},I_{\mathrm{Mean}}$, and $I_{\mathrm{Max}}$, respectively), as well as the median, mean, and maximum “long-trip” inter-reading interval $(I_{\mathrm{LTMedian}},I_{\mathrm{LTMean}}$, and $I_{\mathrm{LTMax}}$, respectively) describing the distribution of intervals above the 30 000-s cutoff shown by the dashed horizontal line. The left panel also includes brackets to indicate the intervals $B$ between these long-trip intervals, specifically the median, the maximum, and, in the gray-shaded box inset, the mean $(B_{\mathrm{Median}},B_{\mathrm{Mean}}$, and $B_{\mathrm{Max}}$, respectively). To aid comparison, both panels show the 22nd and 23rd values of $B$ (i.e., 85 708 and 83 884 seconds apart) on the RFID-reading domain (left panel) and on the inter-reading interval domain (right panel).

Figure 3

Example plot of relations between negative Shannon entropy of bin masses μ(q,L) and logarithmic bin size $L$ in Eq. (3) for an example interval series of length 150. This depicts all values of $q$ that passed our $r>0.995$ criterion for inclusion into the multifractal spectrum we used for calculating $w_{\mathrm{MF}}$. Plots for different values of $q$ appear separated by vertical constant only to aid visibility. All relations in Eq. (1) for which either Eq. (1) or Eq. (2) for corresponding $q$ did not exhibit a $r>0.995$ relationship were omitted from the calculation of $w_{\mathrm{MF}}$.

Figure 4

Example plot of relations between mass μ(q,L)-weighted bin proportion and logarithmic bin size $L$ in Eq. (2) for an example interval series of length 150. This depicts all values of $q$ that passed our $r>0.995$ criterion for inclusion into the multifractal spectrum we used for calculating $w_{\mathrm{MF}}$. Plots for different values of $q$ appear separated by vertical constant only to aid visibility. All relations in Eq. (2) for which either Eq. (1) or Eq. (2) for corresponding $q$ did not exhibit a $r>0.995$ relationship were omitted from the calculation of $w_{\mathrm{MF}}$.

Figure 5

Example plot of multifractal spectra showing the multifractal spectrum for the original inter-reading interval series shown in the right panel of Fig. 2 (solid black) as well as for five IAAFT surrogates preserving the same values as the original and its linear autocorrelation (gray lines, dashed or solid or light to distinguish them from one another when overlapping). The width of the original series’ multifractal spectrum $w_{\mathrm{MF}}$ is 0.5134 indicated by the dashed black arrow at the bottom of the plot. Surrogates generally had narrower multifractal spectra, with surrogates 1–4 having widths 0.1505, 0.1224, 0.1153, and 0.3465. Surrogate 5 had the closest spectrum width (0.5026) to that for the original.