An "Opinion Reproduction Number" for Infodemics in a Bounded-Confidence Content-Spreading Process on Networks

TL;DR

The content-spreading model, which one can also interpret as an independent-cascade model, introduces a twist into BCMs by using bounded confidence for the content spread itself, and defines an analog of the basic reproduction number from disease dynamics that is called an opinion reproduction number.

Abstract

We study the spreading dynamics of content on networks. To do this, we use a model in which content spreads through a bounded-confidence mechanism. In a bounded-confidence model (BCM) of opinion dynamics, the agents of a network have continuous-valued opinions, which they adjust when they interact with agents whose opinions are sufficiently close to theirs. The employed content-spreading model introduces a twist into BCMs by using bounded confidence for the content spread itself. We define an analogue of the basic reproduction number from disease dynamics that we call an \emph{opinion reproduction number}. A critical value of the opinion reproduction number indicates whether or not there is an ``infodemic'' (i.e., a large content-spreading cascade) of content that reflects a particular opinion. By determining this critical value, one can determine whether or not an opinion dies off or propagates widely as a cascade in a population of agents. Using configuration-model networks, we quantify the size and shape of content dissemination by calculating a variety of summary statistics, and we illustrate how network structure and spreading-model parameters affect these statistics. We find that content spreads most widely when the agents have a large expected mean degree or a large receptiveness to content. When the spreading process slightly exceeds the infodemic threshold, there can be longer dissemination trees than when the expected mean degree or receptiveness are larger, even though the total number of content shares is smaller.

Figures (9)

• Figure 1: A schematic illustration of our content-spreading process (see \ref{['alg:update']}). In panel (a), we show several consecutive update steps of \ref{['alg:update']}. We initialize a graph at time $t = 0$ with one active node (in pink), which we label as node $0$. We suppose that the root node $0$ has content state $x_0 = 0.1$. We initialize all other nodes $j$ to have opinions $x_j \in (0,1)$. In this example, the receptiveness parameter is $c = 0.35$. At each time step, any neighbors $j$ of the previously active node that satisfy $\vert x_j - x_0\vert < c$ become active (i.e., turn pink). We thus add them to the dissemination tree $G_0$. In this example, \ref{['alg:update']} terminates after 4 steps. In panel (b), we show the resulting dissemination tree $G_0$ in orange, with the root node outlined in black.
• Figure 2: A comparison of numerical simulations of our content-spreading model with our analytical predictions in (a) an infodemic situation (when the receptiveness parameter is $c = 0.2$, which implies that $R > 1$) and (b) a situation without an infodemic (when the receptiveness parameter is $c = 0.05$, which implies that $R < 1$). For each numerical simulation, we generate a 10000-node configuration-model network with a degree sequence that we determine using a Poisson distribution with mean $\lambda = 5$. We draw the initial node opinions $x_i$ uniformly at random from $(0,1)$, and we set the content state to $x_0 = 0.5$. In each realization, we choose $5$ source nodes uniformly at random to seed with content state $x_0$. In our analytical calculations, we assume that the content spread from each source node is independent, as the number of source nodes is much smaller than the total number of nodes. The solid purple curve gives the total number of content shares (averaged over $100$ realizations) as a function of time. In each realization, we draw a new degree sequence and a new set of node opinions. The light purple shaded region indicates the standard deviation for these $100$ realizations. The dashed orange curve gives our analytical approximation.
• Figure 3: The $(x_0,c)$ phase diagram for 10000-node configuration-model networks with degree sequences that we generate using a Poisson distribution with mean $\lambda = 5$. We vary both the content state $x_0$ and the receptiveness parameter $c$. The shading of each square indicates the mean, across 100 trials, of the total number of content shares as a proportion of the total number of nodes, with lighter shades indicating that more nodes share the content. In each realization, we choose $5$ source nodes uniformly at random to seed with content state $x_0$. In each realization, we draw a new degree sequence and a new set of node opinions. The orange solid curve shows the critical receptiveness value $c^*$ for each $x_0$.
• Figure 4: Histograms of the total number of content shares in our content-spreading model across 1000 trials for 200-node, 2400-node, and 4800-node configuration-model networks. The dashed vertical lines indicate the numbers of content shares (i.e., the dissemination-tree sizes) that we predict from our analysis. In each realization, we generate an $N$-node configuration-model network with a degree sequence that we obtain from a Poisson distribution with mean $\lambda = 5$. We draw the node opinions $x_i$ uniformly at random from $(0,1)$, and we set the content state to $x_0 = 0.5$ and the receptiveness parameter to $c = 0.2$. In each realization, we draw a new degree sequence and a new set of node opinions.
• Figure 5: The effect of varying the network size $N$ on the total number of content shares, the width, the longest-path length, and the structural virality of dissemination trees of our content-spreading model on configuration-model networks. The solid curves give means across $1000$ realizations, and the shaded regions give the standard deviations. In each realization, we generate an $N$-node configuration-model network with a degree sequence from a Poisson distribution with mean $\lambda = 5$. We vary $N$ from $100$ to $5000$ in increments of $100$. Each realization has different initial node opinions, which we draw uniformly at random from $(0,1)$. The content state is $x_0 = 0.5$ and the receptiveness parameter is $c = 0.2$. In each realization, we draw a new degree sequence and a new set of node opinions. Both the total number of content shares (i.e., the dissemination-tree size) and the width grow linearly with $N$. The longest-path length grows quickly at first as we increase $N$, and then it grows much more slowly with $N$. The structural virality appears to saturate at a constant value for sufficiently large $N$.