Close and ordinary social contacts: How important are they in promoting large-scale contagion?

An outstanding problem of interdisciplinary interest is to understand quantitatively the role of social contacts in contagion dynamics. In general, there are two types of contacts: close ones among friends, colleagues and family members, etc., and ordinary contacts from encounters with strangers. Typically, social reinforcement occurs for close contacts. Taking into account both types of contacts, we develop a contact-based model for social contagion. We find that, associated with the spreading dynamics, for random networks there is coexistence of continuous and discontinuous phase transitions, but for heterogeneous networks the transition is continuous. We also find that ordinary contacts play a crucial role in promoting large-scale spreading, and the number of close contacts determines not only the nature of the phase transitions but also the value of the outbreak threshold in random networks. For heterogeneous networks from the real world, the abundance of close contacts affects the epidemic threshold, while its role in facilitating the spreading depends on the adoption threshold assigned to it. We uncover two striking phenomena. First, a strong interplay between ordinary and close contacts is necessary for generating prevalent spreading. In fact, only when there are propagation paths of reasonable length which involve both close and ordinary contacts are large-scale outbreaks of social contagions possible. Second, abundant close contacts in heterogeneous networks promote both outbreak and spreading of the contagion through the transmission channels among the hubs, when both values of the threshold and transmission rate among ordinary contacts are small. We develop a theoretical framework to obtain an analytic understanding of the main findings on random networks, with support from extensive numerical computations. Overall, ordinary contacts facilitate spreading over the entire network, while close contacts determine the way by which outbreaks occur, i.e., through a second- or first-order phase transition. These results provide quantitative insights into how certain social behaviors can emerge and become prevalent, which has potential implications not only to social science but also to economics and political science.

Figure 1

Numerically obtained and theoretically predicted final fraction of the recovered population for different parameter values. Simulation results (red circles) and theoretical prediction of Eq. (12) (blue solid lines) for the final recovered population versus transmission rate $p$ for three different values of $T$: (a) $T=3$, (b) $T=4$, and (c) $T=5$, where for each value of $T$, the results for four different values of $\mu$ are presented. Each data point is obtained from a single realization. The black vertical dashed lines indicating the position of $p^{\prime }=0.3$ are included in (a) for corresponding the results to the dynamical behaviors of the system in Fig. 2.

Figure 2

Understanding the effects of ordinary contacts on spreading dynamics. Four key statistical characterizing quantities are shown for $p^{\prime }=0.3,T=3,$ and $q=1.0$: the fraction of recovered population (the first column), the numbers of three types of transmission events (the second column), the distribution $P\left(L\right)$ of diffusion path lengths $L$ (the third column), and the frequency spectrum of the occurrence of the ordinary transmission events in the various diffusion paths (the fourth column). Three different values of the parameter $\mu$ are used: (a) $\mu =0.3$, (b) $\mu =0.5,$ and (c) $\mu =0.7$. The red solid lines in the first column are the analytical predictions, the blue circles represent the fractions of the recovered populations obtained by averaging $N_r$ independent realizations, and each light gray solid curve in each panel in the left column corresponds to one independent realization. The position of $p^{\prime }=0.3$ has also been specified in Fig. 1.

Figure 3

Existence of an optimal fraction of close transmission events for global spreading. Shown is the final recovered population as a function of $\mu$ for three different values of $T$ for $q=1.0$: (a) $T=3$, (b) $T=4,$ and (c) $T=5$. For each value of $T$, simulation results (red circles) and analytical predictions (blue solid lines) are shown for four different values of $p$. Two groups of vertical solid lines (${\mu }_1,{\mu }_2,{\mu }_3$) and (${\mu }_4,{\mu }_5,{\mu }_6$) are indicated in (a), corresponding to Figs. 4 and 5, respectively.

Figure 4

Illustrations of the role of ordinary contacts in promoting spreading. Shown are the behaviors of four statistical characterizing quantities for different values of ${\mu }_o$: (a) ${\mu }_o>{\mu }_1=0.18$, (b) ${\mu }_o\approx {\mu }_2=0.31,$ and (c) ${\mu }_o<{\mu }_3=0.34$, the positions of which are also labeled in Fig. 3. Other parameter values are $p=0.15$ and $q=1.0$.

Figure 5

Further illustrations of the role of ordinary contacts in promoting spreading. Shown are the behaviors of the four statistical characterizing quantities for (a) ${\mu }_4=0.15$, (b) ${\mu }_5=0.48$, and (c) ${\mu }_6=0.53$, corresponding to the three points of $\mu$ labeled in the second panel of Fig. 3. Other parameters are $p=0.2$ and $q=1.0$.

Figure 6

Illustrations of graphical solutions of the bifurcation analysis for three cases: (a) continuous (second order) transition for $\mu =0.1$; (b) transition crossing the triple point for ${\mu }_C=0.245$; (c) discontinuous (first order) transition for $\mu =0.5$. Other parameters are $T=3$ and $q=1.0$.

Figure 7

Numerically obtained final stationary distributions of the recovered population. Shown are the distributions versus $p$ and $\mu$ for different values of $T$ and $q=1.0$. The large yellow inverted triangle represents the theoretically estimated triple point, while other markers denote the numerically estimated phase boundaries in terms of the relative variance $\nu \left(\infty \right)$. The white circles (pink inverted triangles) represent the boundaries across which continuous (discontinuous) transitions occur.

Figure 8

Theoretically predicted final stationary distributions of recovered population. Legends are the same as those in Fig. 7. There is a good agreement between the numerical results in Fig. 7 and the theoretical predictions here.

Figure 9

Effect of average degree on the contagion dynamics. Shown are the color coded, numerically calculated (top panels) and theoretical predicted (bottom panels) final stationary distributions of recovered population for ER random networks in the parameter plane $(p,\mu )$. The values of the average degree tested are $\langle k\rangle =6$ (left column), $\langle k\rangle =7$ (middle column), and $\langle k\rangle =8$ (right column), with the corresponding values of $p_e$ being $p_e=0.0006,p_e=0.0007,$ and $p_e=0.0008$, respectively. The main feature is that a relatively high average degree tends to facilitate contagion spreading. Other parameters are the same as those in Fig. 7.

Figure 10

Degree distributions of the two empirical networks. (a) The network pretty good privacy (PGP). The structural parameters are $N=10680,\langle k\rangle =4.554,\langle k^2\rangle =85.976$, maximum degree $k_{\text{max}}=205$, degree-degree correlation $r=0.238$, and clustering coefficient $c=0.378$. (b) The network Brightkite. The structural parameters are $N=56739,\langle k\rangle =7.506, \langle k^2\rangle =480.61$, maximum degree $k_{\text{max}}=1134$, degree-degree correlation $r=0.01$, and clustering coefficient $c=0.111$.

Figure 11

Contagion dynamics and phase transition on the empirical network pretty good privacy (PGP). (a) Numerically obtained final stationary distributions of the recovered population. (b) The corresponding result from randomly-reconnected networks with the degree of each node unchanged (first-order null network model). In detail, a new first-order null network with the same degree sequence of PGP is built after every 25 independent realizations of the spreading dynamics. The solid white circles indicate the numerically estimated phase boundaries with respect to the relative variance $\nu \left(\infty \right)$. The dashed lines indicate the position of $p_h$, which are absent for $T>6$ in the top panels and $T=10$ in the bottom panels because closed contacts tend to block the contagion when the value of $T$ becomes large.

Figure 12

Contagion dynamics and phase transition on the empirical network Brightkite. Legends are the same as in Fig. 11. Due to the weak degree correlation ($r=0.01$) of this network, testing the first-order null network model is not essential.

Figure 13

Statistical behaviors of close contacts in the PGP network. Three key statistical characterizing quantities of close contacts are shown: (a) the expected number of close contacts that each node in different degree class can own for two different values of $\mu$, (b) the distribution of the edges between two nodes (node 1 and node 2, where $k_1$ and $k_2$ are the degrees of the two end nodes of an edge), normalized by the total number $E$ of edges in the network, and (c) the distribution of the close contacts between two nodes for $\mu =0.1$, normalized by $E$. (d) The distribution of the close contacts between two nodes for $\mu =0.8$. Because of the symmetry in the distribution, only half of the matrix is given in (b, c), where each color point indicates the value of $d(k_1,k_2)$ or $d(k_2,k_1)$, and $d(k_1,k_2)$ is the fraction of edges with two nodes of degree $k_1$ and $k_2$, respectively. The pink squares in (a) indicate that those nodes of high degree have considerable numbers of close contacts—a great advantage for transmission. The contagion spreads readily among these hubs due to the sufficient close contacts among them. The adopted areas are localized for small values of $p$. Similar phenomena are also found for the first-order null network model of PGP.

Figure 14

The roles of two types of contacts in spreading dynamics in the PGP network. Shown are the behaviors of four statistical characterizing quantities for different values of $\mu$: (a) $\mu =0.1$, (b) ${\mu }_2=0.4,$ and (c) $\mu =0.8$, the positions of which are also labeled in Fig. 2. Other parameter values are $p_h>p=0.16, q=1.0,$ and $T=2$.

Figure 15

The roles of two types of contacts on spreading dynamics in the PGP network—additional support. Legends are the same as in Fig. 14 except for $p_h<p=0.9$.