Response curves of excitable sensor networks for different spreading sources.

<p>The SIR (a), SIS (b) and Rumor (c) spreading models are applied on facebook social network. Four distinct nodes are selected as diffusion sources. The selected sources have degree <i>k</i> = 1089, 309, 82 and 10. The relationship between the response and spreading influence is presented. In (d) we plot the average degree of infected people versus the spreading influence for SIR model originating from different sources.</p

Illustrations of excitable sensor networks and the dynamics of sensors.

<p>(a) A schema of a sensor network. The lower layer is the underlying social network and the upper layer represents the sensor network. Both blue and red balls are individuals in the social network and red balls are selected to be sensors. The spreading dynamics and signal transmission occur in the lower and upper level separately. (b) The dynamics of excitable sensors. The number 0, 1 and 2 stand for the resting, excitation and refractory state, respectively. Each sensor in resting state can be activated either by means of infection in the spreading dynamics or by excited neighboring sensors with probability <i>s</i> independently. Once activated, the sensors will automatically turn into the refractory state in the next time step, where they cannot be activated again and activate other sensors. Then, these sensors change back to resting state.</p

Response of sensor networks to SIR epidemic spreading.

<p>Here we run SIR model on facebook social network and select 10% of nodes as sensors. The average degree of sensor network is set as ⟨<i>k</i>⟩ = 4, so the coupling strength of excitable sensors is <i>s</i> = 0.25. We set <i>μ</i> = 0.2 in simulations. The source is selected as a hub with degree <i>k</i> = 1089. (a) The random sensors fail to detect small-scale epidemic spreading and targeted sensors saturate only after the spreading occupies about 20% of population. (b) The excitable sensor network is capable of detecting small-scale epidemic spreading and distinguishing large-scale spreading. Straight lines indicate relevant parameters to calculate the dynamic range Δ. (c) and (d) display the fraction of active sensors <i>F</i>^<i>t</i> for different methods when we set <i>β</i> = 0.001 and <i>β</i> = 0.1 in SIR modeling respectively.</p

Validation of real diffusion instances in Twitter.

<p>The appearance frequency of selected 309 hashtags is shown in (a). Hashtag ids are ranked chronologically. In (b) we present the normalized response of different detecting strategies to these hashtags. We also display the detection rate <i>r</i> (the fraction of detected hashtags) if we consider a topic is detected only when the response is above a threshold <i>F</i>_<i>p</i> in (c).</p

Comparison of performances of different strategies in response to SIR spreading dynamics.

<p>We apply SIR model on facebook (a), coauthor (b) and email (c) social networks, and display the response curve for each strategy. 10% of nodes are selected as sensors. We construct an excitable sensor network with average degree ⟨<i>k</i>⟩ = 4, and set <i>μ</i> = 0.2 in simulations. The sources are selected as hubs with degree <i>k</i> = 1089, 343 and 1383 respectively. The response curves for all cases are normalized to the unit interval [0, 1]. The insets show the dynamic range for each case when we vary the calculation interval [<i>F</i>_<i>x</i>, <i>F</i>_1−<i>x</i>] from <i>x</i> = 0.01 to <i>x</i> = 0.15.</p

SIR modeling with links’ infection rate cannot reproduce the realistic viral spreading pattern.

<p><b>a</b>, distribution of the real-world infection rate for each social link <i>β</i> calculated from viral spreading instances. The ratio between the size distribution of SIR simulations and real viral spreading is displayed in <b>b</b>. Inset shows the ratio of depth distribution. In <b>c</b>, the proportion of diffusion trees with a given depth for both SIR simulations and real viral spreading is presented, and the ratio between real cases and SIR modeling is shown in the inset. <b>d</b> illustrates the proportion of spreading instances for diffusion with a given depth for both cases. The inset shows the ratio between real viral spreading and simulations. Same analyses are shown in <b>e</b> and <b>f</b> for real viral spreading without interactions with self-promotion and broadcast diffusion.</p

Analysis of social spreading.

<p><b>a</b> shows the probability distributions of the size of diffusion trees and viral spreading processes. The inset displays the distributions of spreading depth for both cases. The straight lines are fitted with the maximum likelihood method. In <b>b</b>, we present the proportion of diffusion instances in spreading processes with a given depth. The relation between the size of viral spreading and diffusion trees is displayed in <b>c</b>. Error bars indicate the 10% and 90% percentiles. The inset presents the diminishing ratio when mapping the diffusion trees to viral spreading. In <b>d</b>, we classify the nodes according to their depth in spreading processes and display their average branching number.</p

Properties of the broadcast.

<p>In <b>a</b>, we display the distributions of the size and depth for broadcast diffusion trees and broadcast spreading respectively. The proportion of broadcast links in diffusion processes with a certain depth is shown in <b>b</b>. The relation between the size of broadcast spreading and broadcast diffusion trees is displayed in <b>c</b>. Error bars indicate 10% and 90% percentiles. The inset presents the diminishing ratio when mapping the diffusion trees to broadcast spreading. We plot nodes’ average branching number versus their depth in diffusion in <b>d</b>.</p

Analysis of the self-promotion.

<p><b>a</b> shows the distributions of the size and depth of self-promotion diffusion trees. The fraction of self-promotion links in diffusion trees with a certain depth is displayed in <b>b</b>. In <b>c</b> we present the probability distribution of the total number of self-promotion for each user, which has a power-law shape with exponent <i>γ</i> = 1.62 ± 0.08. In <b>d</b> we plot the relationship between posts’ branching number and their depth in self-promotion diffusion trees.</p

Coupling of distinct mechanisms.

<p>We plot the fraction of the social spreading, self-promotion, and broadcast links in diffusion trees deeper than a given threshold in <b>a</b>. The x-axis value is the lower bound of selected trees’ depth. In <b>b</b>, the distribution of the route number for each type is displayed. We calculate the average route number of the diffusion links for each type in diffusion trees deeper than a certain depth, and present the results in <b>c</b>. After removing the leaves of diffusion trees, we obtain the information diffusion skeletons. We show the average route number and composition in diffusion skeletons whose depth exceeds certain values in the main panel and inset of <b>d</b> respectively. In the spreading processes among population, the fraction and average route number of social spreading and broadcast links are presented in <b>e</b> and <b>f</b>.</p