Kleinberg小世界网络中的复杂传染路由
The Routing of Complex Contagion in Kleinberg's Small-World Networks
摘要：
在Kleinberg的小世界网络模型中，强联系被建模为底层基础网格中的确定性边缘，弱联系被建模为连接远程节点的随机边缘。通过弱连接将节点$u$与节点$v$连接的概率与$1/|uv|^α$成比例，其中$|uv|$是$u$与$v$之间的网格距离，而$α\ge 0$是模型的参数。复杂传染是指网络中的传播机制，其中每个节点仅在节点的$k \ge 2$邻居被激活后才被激活。本文提出了复杂传染路由（或复杂路由）的概念，即在一个时间步内激活一个节点，最终激活目标节点。我们考虑分散式路由方案，只有弱关系，从激活的节点被发现。我们研究了复杂传染的路由时间，并将结果与简单路由和复杂扩散（复杂传染的扩散，所有可以激活的节点都在同一步中立即激活，目标是最终激活所有节点）进行比较。我们发现，对于分散的复杂路由，路由时间是一个多项式在$n$（网络中的节点数）的所有范围内的$α$的期望和高概率（特别是，$Ω(n^{\frac{1}{α+2}})$的$α\le 2$和$Ω(n^{\fracα{2(α+2)}})$的$α> 2$的期望），而简单的传染的路由时间有多对数上限时，$α= 2$。我们的研究结果表明，复杂的路由比复杂的扩散和路由时间的复杂传染相比，简单的传染在甜蜜点指数不同。
英文摘要：
In Kleinberg's small-world network model, strong ties are modeled as deterministic edges in the underlying base grid and weak ties are modeled as random edges connecting remote nodes. The probability of connecting a node $u$ with node $v$ through a weak tie is proportional to $1/|uv|^α$, where $|uv|$ is the grid distance between $u$ and $v$ and $α\ge 0$ is the parameter of the model. Complex contagion refers to the propagation mechanism in a network where each node is activated only after $k \ge 2$ neighbors of the node are activated. In this paper, we propose the concept of routing of complex contagion (or complex routing), where we can activate one node at one time step with the goal of activating the targeted node in the end. We consider decentralized routing scheme where only the weak ties from the activated nodes are revealed. We study the routing time of complex contagion and compare the result with simple routing and complex diffusion (the diffusion of complex contagion, where all nodes that could be activated are activated immediately in the same step with the goal of activating all nodes in the end). We show that for decentralized complex routing, the routing time is lower bounded by a polynomial in $n$ (the number of nodes in the network) for all range of $α$ both in expectation and with high probability (in particular, $Ω(n^{\frac{1}{α+2}})$ for $α\le 2$ and $Ω(n^{\fracα{2(α+2)}})$ for $α> 2$ in expectation), while the routing time of simple contagion has polylogarithmic upper bound when $α= 2$. Our results indicate that complex routing is harder than complex diffusion and the routing time of complex contagion differs exponentially compared to simple contagion at sweetspot.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
