Social diffusion and global drift on networks

We study a mathematical model of social diffusion on a symmetric weighted network where individual nodes' states gradually assimilate to local social norms made by their neighbors' average states. Unlike physical diffusion, this process is not state conservational and thus the global state of the network (i.e., sum of node states) will drift. The asymptotic average node state will be the average of initial node states weighted by their strengths. Here we show that, while the global state is not conserved in this process, the inner product of strength and state vectors is conserved instead, and perfect positive correlation between node states and local averages of their self-neighbor strength ratios always results in upward (or at least neutral) global drift. We also show that the strength assortativity negatively affects the speed of homogenization. Based on these findings, we propose an adaptive link weight adjustment method to achieve the highest upward global drift by increasing the strength-state correlation. The effectiveness of the method was confirmed through numerical simulations and implications for real-world social applications are discussed.

Figure 1

Visual illustration of the relationships between several vectors discussed ($h,w,g,\widehat{g},s_0,s_{\infty }$). This diagram is drawn in two dimensions for simplicity, but the actual vectors exist in an $n$-dimensional state space. Note that $h^{T}h=w^{T}h={\widehat{g}}^{T}h=n$. If the network is nonassortative, then $w\approx \widehat{g}$.

Figure 2

Time evolution of the global state $h^{T}s$ toward the asymptotic state $h^T{s}_{\infty }$ during social diffusion on networks. The absolute difference between $h^{T}s$ and $h^T{s}_{\infty }$ is plotted over time. Black solid, blue dashed, and pink dotted curves show results with nonassortative, disassortative, and assortative topologies, respectively. Each curve shows the mean of 100 independent numerical simulation runs, with shaded areas representing standard deviations. Each simulation was conducted using Eq. (3), starting with a randomly generated initial condition with $n=1000,c=1$, node state range $[-1,1]$ and link weight range $[0,10]$. Numerical integration was conducted for $t=0\sim 10$ using a simple Euler forward method with step size $\delta t=0.01$. (a) Results on random networks. (b) Results on scale-free networks. Network topologies were generated first as an unweighted network by using the Erdős-Rényi (for random) or Barabási-Albert (for scale-free) network generation algorithm. Disconnected networks were not used for the experiment. Self-loops were randomly added to nodes with 1% probability. Once the topology was generated, a random link weight was assigned to each link. For assortative or disassortative networks, a revised Xulvi-Brunet and Sokolov algorithm [20, 27] was additionally used to tune their strength assortativity. Assortative (or disassortative) network topologies were created by applying to a randomly generated nonassortative network $30000$ times of possible link rewirings that would enhance degree assortativity (or disassortativity) while preserving node degrees. Any link rewiring that would disconnect the network or create a multilink was forbidden. Self-loops were allowed in the above operations. Every time a new link was created between originally disconnected nodes, a random link weight was assigned to the new link.

Figure 3

Summary of systematic simulations of social diffusion with adaptive link weight adjustment with logarithmically varied $\alpha$ and $\beta$ (cases with $\alpha =0$ or $\beta =0$ are also added to the lower ends of the axes). The difference in the global state between before and after each simulation ($\Delta h^{T}s$) is plotted over the $\alpha -\beta$ parameter space. Ten independent simulation runs were conducted for each parameter setting. Each dot represents one simulation run, while the surface shows the trend of the average. The initial network topology was generated using the Erdős-Rényi (a) or Barabási-Albert (b) network generation algorithm, each with 200 nodes and 20% connection density. Node states and link weights were initially random in the same way as in the previous experiment (Fig. 2). $c=1$. Numerical integration was conducted for $t=0\sim 1$ using a simple Euler forward method with step size $\delta t=0.01$.