Fractals Beyond Hierarchy—Analyzing the Temporal Patterns of Contact Networks in a French Public Sector Organization

Coming to a definition of what “fractal” entails in the social space is not particularly easy. The definition “fractal geometry is a workable geometric middle ground between the excessive geometric order of Euclid and the geometric chaos of general mathematics. It is based on [□] invariance under contraction or dilation” (Mandelbrot, 1989, p. 3) does introduce one of the basic concepts of fractal geometry, invariance of detail to scale, but does not give clear implications about fractal patterns in social interactions, organizations, and networks. The statement of invariance under contraction or dilation refers to the fact that fractals retain detailed structure at arbitrarily small scales. While this is often equated with self-similarity at arbitrarily small scales, like in the “Koch snowflake” (von Koch, 2004) or the “Sierpinski triangle” (Sierpinski, 1915), the fractal reality of nature is not typically self-similar. In a temporal context, for example, fluctuations over different time scales might exhibit different properties but are invariant to dilation in the sense that fluctuations remain regardless of the scale of the time series (Evertsz, 1995). Besides this, only mathematical fractals are infinite, thus the invariance of detail to scale in natural (physical) fractals only holds true for some orders of magnitude, leading to some scholars calling them “pseudo-fractals” (Kautz, 2011).

Early on, researchers also identified fractals in social and organizational spaces, which remained predominantly at the structural level. The first contribution on fractal organizations by Zimmerman and Hurst (1993) departed from the tension at and between different levels that underlies fractals in nature and generalized it to firms, attested self-similarity and fractal boundaries in organizations (Zimmerman & Hurst, 1993). The fractal structure of organizations comes as a little surprise, as many social groups have a discrete hierarchical organization that is fractal with a consistent scaling ratio of three (Zhou et al., 2005). Later organizational research on the “fractal organization” developed the concepts of self-similar “basic fractal units,” which are parts of a greater system but also a fully contained system with subsystems in them to arbitrarily small scales (Ryu & Jung, 2003; Tharumarajah, 1996). They notably included an interactionist component, pointing out that basic fractal units are networked via an efficient information and communication system. The concept of “fractal organization” was later extended to and formalized in the “fractal social organization” (FSO) (De Florio et al., 2012; De Florio et al., 2013a,b). In de Florio’s model, FSOs are characterized by a distributed, bio-inspired, and hierarchical architecture, in which a basic building block, similar to a basic fractal unit, is applied recursively at different levels. This provides a homogeneous way to model collective behaviors of different scales and complexity (De Florio et al., 2013a). In practice, FSOs have a higher performance, as the recursive application of the same structure, enabled by self-similarity, creates a high potential for parallel processing in the systems (De Florio et al., 2013b).

In the original definitions of both fractal organization and FSO, strong emphasis is placed on hierarchical federation in nested systems. While this applies to many firms, and based on evidence from hierarchical mammalian societies (Hill et al., 2008), it effectively excludes nonhierarchical forms of organization, such as networks and heterarchical federations from the discussion of fractal (social) organizations. Furthermore, the organization is only structurally fractal. No direct consideration is paid to the fractal patterns in interaction between individuals or units in the fractal system(s). Hence, the research into fractal patterns in organizations so far is predominantly atemporal, meaning that there are few insights into the fractal patterns of interaction or the development of organizational structure over time.

An important contribution to the analysis of temporal fractal patterns in the social and organizational space is the article of Aguilera et al. (2013) about political self-organization on social media. Building on work from the field of physics (Bak et al., 1987), they identify fractal scaling as a characteristic of self-organizing systems in the social space. They observe an “interesting mix between stability and instability creating complex structures of the variability of the systems activity” (Aguilera et al., 2013, p. 296). The temporal fractal patterns in a network can be described with a spectral density function S(f), or more specifically Sf∝fβ S\left( f \right) \propto {f^\beta } for fractal scaling, where f stands for the different frequencies of the interaction activity, and β defines a log-linear relationship between the spectral power content at different scales of f, delivering insights into the relative influence of each scale in the system. In cognitive neuropsychology, this approach is commonly used to describe interactivity among components in the system, as well as predict the emergence of new cognitive structures (Dixon et al., 2012). This approach is especially suitable for the analysis of organizational structures with distinct components, such as the departments in the data used for this article.

Three different types of processes can be identified from different values of the relative influence of each scale in the system (β):

- White noise (β=0) entails no correlations in time. They display fast changes in their activity, as well as high adaptation to rapid environmental changes, but are unable to maintain structured and coherent patterns.

White noise processes exhibit high fluctuations on a short time scale but normalize across a longer scale. On the other hand, Brown noise diffusion processes exhibit larger fluctuations on longer time scales, limited by the end of the diffusion process. On shorter time scales, the fluctuation per time becomes negligible. Pink noise processes are inherently fractal as the combination of white and brown noise leads to an invariance of fluctuation to scale. Insights from cognitive neuroscience (Dixon et al., 2012) as well as the human and nonhuman social space (Aguilera et al., 2013; Alados & Huffman, 2000) indicate that this particular mix of fluctuation size and frequency is a core component of functional and healthy systems of interactions. However, Aguilera et al. (2013) analyzed critically self-organizing systems. Other research on temporal fractal patterns such as fractal time series in the social and organizational space considered power-law distributions in the growth of the internet (Tadić, 2002) and the multifractal characteristics of warehouse-out behavioral sequences (Yao et al., 2017). Organizations with a clear structure despite the absence of a formal hierarchy among actors may present a different case, one has not yet investigated for its fractal scaling and temporal fractal properties. It remains unclear whether the three processes outlined above also replicate in environments that constrain flexibility and long-range structural patterns by the presence of structure external to the interactions. The remainder of this article will therefore investigate the (multi-)fractal properties of interaction sequences between humans in a structured, though not formally hierarchical organizational setting. Besides pink noise, higher levels of brown noise than in the context of Aguilera and colleagues are expected due to the more rigid structure of the organization.

The data used were collected by the SocioPatterns collaboration in one of the two office buildings of Institut de Veille Sanitaire (InVS), the French Institute for Public Health Surveillance, located near Paris and lasted 2 weeks in 2015 (Génois et al., 2015). The data collection was carried out using wearable sensors that can detect the proximity of individuals wearing them using radio signals. The body acts as a shield for the radio frequencies, resulting only in the detection of proximity if individuals face each other at a range of <1.5 m (Barrat et al. 2014; Cattuto et al., 2010). Furthermore, RFID readers located in the environment enabled the definition of a spatial location for each sensor (Génois & Barrat, 2018). As each RFID reader covered a range of approximately 30 m, two individuals within such a distance (assuming low overlap between sensors) were defined to have a co-presence, as their signal is picked up by the same sensor. This resulted in a temporal network of face-to-contacts and a temporal co-presence network, at a temporal resolution of 20 s each. Genois and Barrat (2018) provide a detailed description of the data collection process and data structure.

Furthermore, the data can be subdivided into groups corresponding to the five functional departments, with department sizes ranging from 5 to 60 people. A detailed description of the organizational structure can be found in Genois et al. (2015), which is the first fielding of the measurement at InVS in 2013, later repeated in 2015 (Génois & Barrat, 2018). The more recently collected data will be used in this analysis. Appendix 1 displays contact networks based on all contacts during the data collection period, both for the whole organization and the three most contact intensive departments separately.

A first visual inspection of the contact frequencies indicated that the time stamps (t) of the 20 s intervals were continuous and not limited to office hours. Hence, there were prolonged periods with no contacts during the evenings, nights, and weekends in the 2 weeks of measurement. These periods were excluded from the analysis, as they would bias the patterns of fluctuation. This yielded a data set of 18,488 with 20 s intervals. The analyses were performed on the face-to-face contact network.

DFA (Peng et al., 1992) method is widely used for discovering fractal properties of time series by determining the statistical self-affinity of a signal. As the name suggests, the time series is detrended globally by integrating it, before dividing it into “windows” of equal size t. This process is repeated for a range of different values of t, resulting in different number of windows. For each value of t a standard least squares line is fit to the data, which allows for computing the size of the fluctuation F(t) for every window individually. Usually, F(t) increases with t. While Aguilera et al. (2013) use the DFA results to calculate the Hurst exponent, an approximation of β in the spectral density function, which in turn is used to estimate β, as the relationship between the Hurst exponent and the spectral power content at different scales is β=2 * α-1, this article takes the more parsimonious approach of estimating β directly from the DFA results. As F(t) t^β, β can be estimated directly from the DFA results by regressing F(t) on t. The range of the regression is determined by the window sizes for which a linear relation in the log-log plot of t and F(t) holds true. The exponent of this regression is beta (Penzel et al., 2003).

For the interpretation of the DFA results, it is crucial to recognize the cited β values of white noise (0), brown noise (2), and pink noise (1) as ideal values, situated on a spectrum from β=0 to β=2. Hence the pink noise is observed as values tend toward 1 instead of only if they are exactly 1. This is coherent with the practically imperfect nature of fractals, both as “pseudo-fractals” (Kautz, 2011), limited in their dimensionality by the temporal resolution and length of observation, as well as the randomness of an observational context such as InVS. Previous research using DFA to detect fractal patterns in temporal data has not specified clear boundaries on the white-pink-brown noise spectrum (Aguilera et al., 2013; Alados & Huffman, 2000), but described values between s β=0.8 and β=1.2 as “quite close to pink noise” (Aguilera et al., 2013, 2). Also lados and Huffman (2000) confirm the presence of fractal scaling at β=1.247.

For the three biggest departments in terms of contacts (DMI, DSE, DISQ, with 14,945, 9,528, and 8,658 measured contacts, respectively), as well as for the whole organization, time series were created that were composed of number of contacts for each 20 s interval. The DFA procedure was performed on 200 window sizes of t, ranging from 30 (600 s) to 18,488 (369,760 s) intervals, resulting in a range of t=30 to t=18,488 for the time series, obtaining the F(t) values for each window size t across the different cases. Following this, the value of t was identified where the shape of F(t) began to have a log-linear relationship, indicating fractal scaling. This allows for an estimation of a range of scales across which the fractal relationship within the time series held true.

Furthermore, a network analysis was performed to gain descriptive insights into the underlying networks of the time series. Factors that were deemed to deliver insights into structural elements influencing the contact frequency and fluctuation were network size, density, average and maximum degree centrality in the network, the clustering coefficient (transitivity), and the size and percentage of the largest component.

The results of the network analysis summarized in Table 1 show a higher network density within departments, than across the whole organization. While for the whole organization, 9% of the possible contacts between individuals were realized over the 10-day period of data collection, this increases to 22%, 29%, and 45% in the respective departments. From this it is also evident that the percentage of other individuals a given individual has contact with over the time period increases as the department size decreases.

Significant differences are found in degree centrality at organizational and departmental levels. While the network density decreased with network size, the opposite seems to hold true for the degree centrality. This is predominantly because the smaller n of the departmental networks limits the maximum number of unique contacts an individual can have within the network. Comparing the average degree across the whole organization (39.39) to the average degree within departments, it is evident that the degree within the departments examined is consistently lower than across the whole organization. In all but the DISQ department where the maximum degree equals n-1, individuals also are not directly connected to all other members of their department.

Similarly to the network density, the clustering coefficient, as a measure of redundancy in the network, is also inversely related to network size. Finally, not only within departments but also across the whole organization all individuals are linked via some finite path, which entails that a message transmitted exclusively in face-to-face contacts could have reached all 217 individuals over the course of the 10-day data collection period.

The results of the network analysis analyzing both the structure of the three most contact intensive departments individually, as well as the organization as a whole, indicate that there are substantial differences between departments, mainly due to their differences in size. While departments as clusters can be analyzed as sub-networks, there is also a need for analyzing the organization as a whole. Hence, the article will proceed by first analyzing the fractal scaling for the DMI, DISQ, and DSE departments separately, before analyzing the aggregate results of the whole organization, also including the other, less contact-intensive departments.

Figure 4 displays the DFA organizations across all functional departments in the organization. For the range t=30-t=18,488, β=0.913, which is higher than in all previously examined departments separately. This is at least partly because of the limited range where the linear relationship in the log-log plot holds true (t=30-t=2,000), β=1.170. In contrast to the within-department data on interaction frequencies, the data for the whole organization yields a β value exceeding 1, indicating the presence of traces of brown noise in the pink noise, rather than white noise. Hence, the organization as a whole displays the highest β values out of all samples, both across all window sizes t and in the limited range.

Figure 4:

Research such as Aguilera et al. (2013) or Alados and Huffman (2000) shows the value of tracking temporal fractal patterns in various contexts, ranging from the online organization of social movements to the behavior of chimpanzees. This article substantiates their findings with insights from the organizational/institutional space, highlighting a certain universality of temporal fractal patterns in situations of social interaction. The theory also suggests however that they are only present in “functional” or “healthy” systems, and that systems without these patterns will not retain adaptability and stability over time (Dixon et al., 2012).

Based on this, it is possible to automatize the tracking and analysis of organizational contact data. Here, strategically crucial is the decision of “which interactions to track.” They should represent the primary form of functional interactions in the organization, which can present a considerable challenge in many organizations that heavily rely both on email communication as well as physical interactions. Practically, digital interactions such as emails can easily be tracked in organizations’ IT systems, while physical interactions can be tracked using either the GPS function in the employees’ smartphones or RFID badges in chipcards or company badges similar to the Sociopatterns collaboration (Génois & Barrat, 2018). To implement a continuous tracking system, it is possible to automatize and implement the creation of data sets and their analysis using DFA on a rolling basis (e.g., every week with the interaction data of the previous month).

To ensure compatibility with the standard DFA implementation in R, the data should have the format t, i, j where t refers to the time stamp of the interaction and i and j are the respective identifiers of the interacting individuals. The code below gives insight into the import of raw data in the specified format, including metadata on which departments’ individuals are in and the subsequent steps necessary to create a data set ready to use for DFA. In practice, this script could be automated to run for every wave of collected data.

Once the data concerning interaction frequencies is prepared, it is possible to perform DFA and produce both numerical and visual output. The code below shows a simple, computationally inexpensive implementation of DFA in R that can be automatized. Here, the user can specify the window size range. This corresponds to rows in the data, not necessarily seconds or minutes, so the relationship between the number and the actual window size is contingent on the temporal resolution in the data. The window size range depends on the practical setting and needs for information on fluctuation. In an organization with a high frequency of interaction and high adaptability, the lower bound of window size ranges should consequently be lower. If an organization is primarily concerned with long-term stability of interactions, the upper bound should be adjusted accordingly. Furthermore the user can specify the number of specific window sizes to perform DFA on in this range (i.e., n points). A lower number of window sizes to be integrated and detrended promise higher computational efficiency, whereas a higher number provides for a more fine-grained analysis.