We study the spread of information in finite and infinite inhomogeneous spatial random graphs. We assume that each edge has a transmission cost that is a product of an i.i.d. random variable L and a penalty factor: edges between vertices of expected degrees w_1 and w_2 are penalised by a factor of (w_1w_2)^\mu for all \mu >0. We study this process for scale-free percolation, for (finite and infinite) Geometric Inhomogeneous Random Graphs, and for Hyperbolic Random Graphs, all with power law degree distributions with exponent \tau > 1. For \tau < 3, we find a threshold behaviour, depending on how fast the cumulative distribution function of L decays at zero. If it decays at most polynomially with exponent smaller than (3-\tau)/(2\mu) then explosion happens, i.e., with positive probability we can reach infinitely many vertices with finite cost (for the infinite models), or reach a linear fraction of all vertices with bounded costs (for the finite models). On the other hand, if the cdf of L decays at zero at least polynomially with exponent larger than (3-\tau)/(2\mu), then no explosion happens. This behaviour is arguably a better representation of information spreading processes in social networks than the case without penalising factor, in which explosion always happens unless the cdf of L is doubly exponentially flat around zero. Finally, we extend the results to other penalty functions, including arbitrary polynomials in w_1 and w_2. In some cases the interesting phenomenon occurs that the model changes behaviour (from explosive to conservative and vice versa) when we reverse the role of w_1 and w_2. Intuitively, this could corresponds to reversing the flow of information: gathering information might take much longer than sending it out.

中文翻译：

---

通过在无标度空间随机图中惩罚到集线器的传输来阻止爆炸

我们研究有限和无限非均匀空间随机图中的信息传播。我们假设每条边都有一个传输成本，它是一个 iid 随机变量 L 和一个惩罚因子的乘积：期望度 w_1 和 w_2 的顶点之间的边被一个因子 (w_1w_2)^\mu 惩罚为所有 \mu > 0. 我们研究了无标度渗透、（有限和无限）几何非齐次随机图和双曲随机图的这个过程，所有这些都具有指数 \tau > 1 的幂律度分布。对于 \tau < 3，我们找到了一个阈值行为，取决于 L 的累积分布函数在零处衰减的速度。如果它最多以多项式衰减，指数小于 (3-\tau)/(2\mu) 那么就会发生爆炸，即，以正概率，我们可以达到无限多个具有有限成本的顶点（对于无限模型），或者达到所有具有有限成本的顶点的线性分数（对于有限模型）。另一方面，如果 L 的 cdf 至少以多项式衰减为零，且指数大于 (3-\tau)/(2\mu)，则不会发生爆炸。与没有惩罚因子的情况相比，这种行为可以说是社交网络中信息传播过程的更好表示，在这种情况下，除非 L 的 cdf 在零附近成倍指数平坦，否则总是会发生爆炸。最后，我们将结果扩展到其他惩罚函数，包括 w_1 和 w_2 中的任意多项式。在某些情况下，当我们反转 w_1 和 w_2 的角色时，会出现有趣的现象，即模型会改变行为（从爆炸性变为保守性，反之亦然）。