What Big Data tells: Sampling the social network by communication channels

Big Data has become the primary source of understanding the structure and dynamics of the society at large scale. The network of social interactions can be considered as a multiplex, where each layer corresponds to one communication channel and the aggregate of all of them constitutes the entire social network. However, usually one has information only about one of the channels or even a part of it, which should be considered as a subset or sample of the whole. Here we introduce a model based on a natural bilateral communication channel selection mechanism, which for one channel leads to consistent changes in the network properties. For example, while it is expected that the degree distribution of the whole social network has a maximum at a value larger than one, we get a monotonically decreasing distribution as observed in empirical studies of single-channel data. We also find that assortativity may occur or get strengthened due to the sampling method. We analyze the far-reaching consequences of our findings.

Figure 1

(a) Schematic representation of the different communication channels or layers (different colors) on a sample egocentric network (gray). (b) One of the communication layers on the top of the original egocentric network. For the selection of the users, we used the sampling method defined in Sec. 3 with $f_0=0.25$.

Figure 2

Empirical results of the mobile phone call (MPC) dataset (left) and the Hungarian social networking service iWiW dataset (right). Degree distributions for users with activity larger than some threshold values (top) and the average degree of neighbors of users with degree $k$ (bottom). For the degree distributions, frequencies are divided by the total number of nodes in each dataset, i.e., the curves for “all” are normalized while the others are not.

Figure 3

Degree distributions of the sampled networks with analytic curves obtained using Eq. (7). (a) The surrogate networks are regular random graphs (RR, $\times$) or Erdős-Rényi random graphs (ER, $\square$) with a network size of $N={10}^4$ and the degree of $k_0=150$ (RR) or the average degree of $\langle k\rangle =150$ (ER). Analytic solutions for several values of $f_0$ (curves) perfectly fit the simulation results (symbols). The simulation results are averaged over 50 independent runs. (b) The surrogate network is a weighted social network using the link deletion model of Ref. [29] with parameters $N={10}^4$ and $\langle k\rangle \approx 47.8$. Using $f_0=0.3$, we find that the size of the sampled network is 9287 and that its average degree is $\langle k\rangle \approx 7.7$. Since the surrogate network has a high average degree, most of the nodes remain in the sampled network even after majority of links are removed.

Figure 4

The degree distributions in the sampled networks from ER for nodes with affinity larger than the indicated values.

Figure 5

Average degrees of neighboring nodes $k_{nn}\left(k\right)$ as a function of the degree of the node in the sampled networks. (a) The surrogate networks are Erdős-Rényi random graphs used in Fig. 3. Blue squares show the result for all nodes, while others show the results only for nodes whose affinity is in the annotated ranges, respectively. Analytic solutions in Eqs. (11) and (12) are also depicted for comparison. (b) The assortativity $k_{nn}\left(k\right)$ for the network sampled from a weighted social network with link deletion used in Fig. 3. In the inset, $k_{nn}\left(k\right)$ for the surrogate network is plotted for comparison. In all cases, the simulation results are averaged over 50 independent runs.

Figure 6

The effects of sampling on (a) the node strength distribution and (b) the average node strength as a function of the degree when the WSN is used for the surrogate network. Parameter values for the WSN are the same as in Fig. 3.

Figure 7

(a) The fraction of nodes with degree 1 normalized by the peak value of $P\left(k\right)$ in the sampled networks. The horizontal axis indicates the exponent of the mean $\beta$. (b) Typical degree distributions for positive and negative $\beta$. Erdős-Rényi random graphs are used as the surrogate networks. The results are averaged over five independent runs.

Figure 8

(a) The assortativity coefficient of the sampled networks as a function of $\beta$. (b) Typical degree correlation $k_{nn}$ for positive and negative $\beta$. Erdős-Rényi random graphs are used as the surrogate networks. The results are averaged over five independent runs.

Figure 9

Degree distributions $P\left(k\right)$ in the sampled networks from the regular random graph of $N={10}^4$ and $k_0=150$, using different functional forms of $P\left(f\right)$. The definitions of $P\left(f\right)$ are given in Table 1. The results are averaged over 50 independent runs.