It is a longstanding debate on the absence of threshold for SIS model on networks with finite second order moment of degree distribution. The eigenvector localization of the adjacency matrix for a network gives rise to the inactive Griffiths phase featuring slow decay of the activity localized around highly connected nodes due to the dynamical fluctuation. We show how it dramatically changes our understanding of SIS model, opening up new possibilities for the debate. We derive the critical condition for Griffiths to active phase transition: on average, an infected node can further infect another one in the characteristic lifespan of the star subgraph composed of the node and its nearest neighbors. The system approaches the critical point of avoiding the irreversible dynamical fluctuation and the trap of absorbing state. As a signature of the phase transition, the infection density of a node is not only proportional to its degree, but also proportional to the exponentially growing lifespan of the star. And the divergence of the average lifespan of the stars is responsible for the vanishing threshold in the thermodynamic limit. The eigenvector localization exponentially reinforces the infection of highly connected nodes, while inversely suppresses the infection of small-degree nodes.

中文翻译：

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特征向量定位网络上的敏感感染敏感模型

对于具有有限二阶矩分布的网络上的SIS模型没有阈值问题，这是一个长期存在的争论。网络邻接矩阵的特征向量局部化会导致非活动Griffiths相位，该相位的特征是由于动态波动而导致局部活动在高连接节点周围缓慢衰减。我们展示了它如何极大地改变了我们对SIS模型的理解，为辩论开辟了新的可能性。我们得出格里菲斯向活动相变的临界条件：平均而言，受感染的节点可以在由该节点及其最近的邻居组成的星形子图的特征寿命中进一步感染另一个节点。该系统接近避免不可逆的动态波动和吸收状态陷阱的临界点。作为相变的标志，一个节点的感染密度不仅与其度成正比，而且还与恒星的指数增长寿命成正比。恒星平均寿命的差异是导致热力学极限消失的原因。特征向量定位以指数方式增强了高度连接节点的感染，而相反地则抑制了小角度节点的感染。