论文标题
通过在无尺度的空间随机图中惩罚向枢纽的传输来停止爆炸
Stopping explosion by penalising transmission to hubs in scale-free spatial random graphs
论文作者
论文摘要
我们研究信息在有限和无限均匀的空间随机图中的传播。我们假设每个边缘的传输成本是I.I.D的产物。随机变量L和一个惩罚因素：预期度W_1和W_2的顶点之间的边缘受（W_1W_2）^μ> 0的（W_1W_2）^μ> 0的惩罚。我们研究了此过程，用于（有限和无限的）几何不均匀随机图，以及对于双曲线随机图，所有这些过程均具有带功率法律分布的指数τ>1。对于τ<3，我们发现阈值行为，取决于L衰减的累积分布功能，依赖于LED的速度。如果大多数多项式衰减，指数小于（3-τ）/（2μ），则爆炸发生，即，以积极的概率，我们可以以有限的成本（对于无限模型）到达无限的许多顶点，或达到以有界成本的所有顶点的线性分数（用于有限型模型）。另一方面，如果L的CDF在零时至少在多项式上，指数大于（3-τ）/（2μ），则不会发生爆炸。可以说，这种行为可以更好地表示社交网络中的信息传播过程比没有惩罚因素的情况，在这种情况下，爆炸总是发生，除非L的CDF在零周围呈指数呈指数呈指数呈指数呈指数呈呈指数成倍。最后，我们将结果扩展到其他惩罚功能，包括W_1和W_2中的任意多项式。在某些情况下，有趣的现象是，当我们扭转W_1和W_2的作用时，模型会改变行为（从爆炸性到保守性，反之亦然）。直觉上，这可能对应于逆转信息流：收集信息可能比发送信息更长。
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)^μfor all μ>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 τ> 1. For τ< 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-τ)/(2μ) 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-τ)/(2μ), 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.
