基于布朗粒子的随机运动度量网络中节点之间的距离

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基于布朗粒子的随机运动度量网络中节点之间的距离

算法思想

Consider a connected network of $N$ nodes and $M$ edges. Its node set is denoted by $V=\{1,\cdots,N\}$ and its connection pattern is specified by the generalized adjacency matrix $A$. If there is no edge between node $i$ and node $j$, $A_{ij}=0$; if there is an edge in between, $A_{ij}=A_{ji}=0$ and its value signifies the interaction strength (self-connection is allowed). The set of nearest neighbors of node $i$ is denoted by $E_i$ . A Brownian particle keeps moving on the network, and at each time step it jumps from its present position (say $i$) to a nearestneighboring position $j$. When no additional knowledge about the network is known, it is natural to assume the following jumping probability $P_{ij}=A_{ij}/\sum_{l=1}^NA_{il}$(the corresponding matrix $P$ is called the transfer matrix)

Define the node-node distance $d_{i,j}$ from $i$ to $j$ as the average number of steps needed for the Brownian particle to move from $i$ through the the network to $j$:

$$d_{i,j}=\sum_{l=1}^N(\frac{1}{I-B(j)})_{il}$$
where $I$ is the $N \times N$ identity matrix and matrix $B(j)$ equals to the transfer matrix $P$ except that $B_{lj}(j)$ for any $l\in V$.

算法实现

算法代码参考