Group sizes, durations and aggreation ===================================== The analysis of real-world face-to-face data is often based on the analysis of networks in terms of disconnected groups --- the group sizes, their lifetimes and the contact durations distributions. Furthermore, an aggregated static network is often used to analyze the social structure. In the following we show how to compute and analyze these observables easily. Base analysis function ---------------------- To do a complete measurement of the group size histograms, the group life time distributions, the contact and inter-contact distributions, and the aggregated weighted network use :func:tacoma.api.measure_group_sizes_and_durations .. code:: python result = tc.measure_group_sizes_and_durations(temporal_network) The returned object is an instance of :class:_tacoma.group_sizes_and_durations. Group sizes ----------- We define the group size distribution :math:N_g(t) describing the number of groups of size :math:g at time :math:t. Note that the group size distribution is restricted by .. math:: \sum_{g=1}^N gN_g(t) = N = \mathrm{const.} The time-dependent group size distribution is saved in two ways. For edge_lists ~~~~~~~~~~~~~~~~ For each edge list, a size histogram as a dictionary of (int, int)-pairs is saved in the attribute result.size_histograms. The key is the group size :math:g and the value is number of its occurences. For edge_changes ~~~~~~~~~~~~~~~~~~ A single group size histogram is computed for the initial edge list. Then, for each edge changing event, a dictionary is saved in result.size_histogram_differences, where keys are the group size :math:g and the corresponding value is the change of occurences in comparison to the network's previous state. .. note:: Since evaluating the size histograms might be heavy on memory, you can omit the detailed computation using .. code:: python result = tc.measure_group_sizes_and_durations( temporal_network, ignore_size_histogram=True ) Averaged size distribution ~~~~~~~~~~~~~~~~~~~~~~~~~~ The average group size distribution .. math:: \overline{N_g} = \frac{1}{t_\mathrm{max}-t_0} \int\limits_{t_0}^{t_\mathrm{max}}dt\, N_g(t) is saved in result.aggregated_size_histogram, a list of floats where the g-th entry contains the average number of groups of size g. Note that this implies that the 0-th entry is always equal to 0. You can easily get a numpy.ndarray of group sizes g and corresponding average number of groups of this size N_g using the function :func:tacoma.tools.group_size_histogram. .. code:: g, N_g = tc.group_size_histogram(result) Note that this function returns values for non-zero occurences only. If you are just interested in the mean group size, use :func:tacoma.tools.mean_group_size .. code:: mean_g = tc.mean_group_size(result) Number of groups ~~~~~~~~~~~~~~~~ At all times, the number of seperated components (groups) is given by .. math:: c(t) = \sum_{g=1}^N N_g(t) and thus the mean number of components over time is .. math:: \overline{c} = \sum_{g=1}^N \overline{N_g}. It can be computed with :func:tacoma.tools.mean_number_of_groups. .. code:: python mean_c = tc.mean_number_of_groups(result) Coordination number ~~~~~~~~~~~~~~~~~~~ The coordination number :math:n of a single node is defined as the group size this node is part of, as was first done in a study by Zhao, StehlĂ©, Bianconi, and Barrat_. The probability distribution of the coordination number is given by the group size distribution :math:N_g as .. math:: P_n(t) = \frac{nN_n}{N}. It can also be interpreted as the probability of a single node to be in a group of size :math:n. The mean coordination number :math:\left\langle n \right\rangle can be computed as .. code:: python mean_n = tc.mean_coordination_number(result) Analysis ~~~~~~~~ Plot a distribution of group sizes as .. code:: python from tacoma.analysis import plot_group_size_histogram import matplotlib.pyplot as pl ht09 = tc.load_sociopatterns_hypertext_2009() result = tc.measure_group_sizes_and_durations(ht09) fig, ax = pl.subplots(1,1) plot_group_size_histogram(result, ax) pl.show() .. figure:: img/N_g.png :alt: group size distribution Average group size histogram for the SocioPatterns Hypertext 09 dataset. Durations --------- Group durations ~~~~~~~~~~~~~~~ A group is initiated as soon as an event leads to a change in members. The duration of this group is defined as the time it takes until the next event at which the constituents of the group changes. All durations of groups of size :math:g are saved in result.group_durations[g] (hence, result.group_durations[0] is always empty). Groups which were active at :math:t_0 are not considered in the measurement since we do not know when they were initiated, so including them in the analysis would skew the distribution. Likewise, groups which are still active at :math:t_\mathrm{max} are omitted for the same reason. The durations can be analyzed using :func:tacoma.analysis.plot_group_durations. .. code:: python from tacoma.analysis import plot_group_durations import matplotlib.pyplot as pl ht09 = tc.load_sociopatterns_hypertext_2009() result = tc.measure_group_sizes_and_durations(ht09) fig, ax = pl.subplots(1, 1) plot_group_durations(result, ax, max_group=4, time_unit='s') pl.show() .. figure:: img/tau_g.png :alt: group duration distribution Duration distribution of different group sizes for the SocioPatterns Hypertext 09 dataset. (Inter-) Contact durations ~~~~~~~~~~~~~~~~~~~~~~~~~~ The durations of all contacts is saved in result.contact_durations. However, contacts which were active at :math:t_0 are omitted since we do not know when they started, so including them in the analysis would skew the distribution. Likewise, contacts which are still active at :math:t_\mathrm{max} are omitted for the same reason. The inter-contact duration of a node is defined as the time a node spends alone (i.e. in a group of size :math:g=1). Hence, all inter-contact durations are saved in result.group_durations[1]. The contact and inter-contact durations can be analyzed using :func:tacoma.analysis.plot_group_durations. .. code:: python from tacoma.analysis import plot_contact_durations import matplotlib.pyplot as pl ht09 = tc.load_sociopatterns_hypertext_2009() result = tc.measure_group_sizes_and_durations(ht09) fig, ax = pl.subplots(1, 1, figsize=(4,3)) plot_contact_durations(result, ax, time_unit='s') pl.show() .. figure:: img/cont.png :alt: contact and inter-contact distributions Contact and inter-contact distributions for the SocioPatterns Hypertext 09 dataset. Aggregated network ------------------ The aggregated network .. math:: W_{ij} = \int\limits_{t_0}^{t_\mathrm{max}}dt\,A_{ij}(t) is given as a dictionary in result.aggregated_network. Each key is a pair of ints, representing the edge :math:(i, j), the corresponding value is :math:W_{ij}. If you just want the aggregated network without the other results, use :func:tacoma.api.aggregated_network. .. _Zhao, StehlĂ©, Bianconi, and Barrat: https://arxiv.org/abs/1102.2423