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Lancichinetti–Fortunato–Radicchi benchmark is an algorithm that generates benchmark networks (artificial networks that resemble real-world networks). They have a priori known communities and are used to compare different community detection methods.  The advantage of the benchmark over other methods is that it accounts for the heterogeneity in the distributions of node degrees and of community sizes.

Lancichinetti–Fortunato–Radicchi benchmark is an algorithm that generates benchmark networks (artificial networks that resemble real-world networks). They have a priori known communities and are used to compare different community detection methods.  The advantage of the benchmark over other methods is that it accounts for the heterogeneity in the distributions of node degrees and of community sizes.

'''<font color="#ff8000">兰奇基内蒂-福图纳托-拉迪奇基准程序(Lancichinetti–Fortunato–Radicchi benchmark)</font>'''是一种生成基准网络(类似于真实世界网络的人工网络)的算法。他们有一个预先已知的社区，用于比较不同的社区检测方法。与其他方法相比，基准测试的优点在于它解释了'''<font color="#ff8000">节点度(node degree)</font>'''分布和社区规模分布的'''<font color="#ff8000">异质性(heterogeneity)</font>'''。

'''<font color="#ff8000">兰奇基内蒂-福图纳托-拉迪奇基准程序(Lancichinetti–Fortunato–Radicchi benchmark)</font>'''是一种生成'''<font color="#ff8000">基准网络（baseline network)</font>'''(类似于真实世界网络的人工网络)的算法。他们有一个预先已知的社区，用于比较不同的'''<font color="#ff8000">社区检测</font>'''方法。与其他方法相比，基准测试的优点在于它解释了'''<font color="#ff8000">节点度(node degree)</font>'''分布和'''<font color="#ff8000">社区规模(community sizes)</font>'''分布的异质性。

The node degrees and the community sizes are distributed according to a power law, with different exponents. The benchmark assumes that both the degree and the community size have power law distributions with different exponents, <math>\gamma</math> and <math>\beta</math>, respectively. <math>N</math> is the number of nodes and the average degree is <math>\langle k \rangle</math>. There is a mixing parameter <math>\mu</math>, which is the average fraction of neighboring nodes of a node that do not belong to any community that the benchmark node belongs to.  This parameter controls the fraction of edges that are between communities.

The node degrees and the community sizes are distributed according to a power law, with different exponents. The benchmark assumes that both the degree and the community size have power law distributions with different exponents, <math>\gamma</math> and <math>\beta</math>, respectively. <math>N</math> is the number of nodes and the average degree is <math>\langle k \rangle</math>. There is a mixing parameter <math>\mu</math>, which is the average fraction of neighboring nodes of a node that do not belong to any community that the benchmark node belongs to.  This parameter controls the fraction of edges that are between communities.

节点度和社区规模按幂律分布，但指数不同。基准测试假设度和社区规模都具有不同指数的'''<font color="#ff8000">幂律分布(power law distribution)</font>'''，分别为'''<font color="#32CD32">此处需插入公式</font>'''和'''<font color="#32CD32">此处需插入公式</font>'''。'''<font color="#32CD32">此处需插入公式</font>'''是节点的数量，平均度为'''<font color="#32CD32">此处需插入公式</font>'''。混合参数'''<font color="#32CD32">此处需插入公式</font>'''是一个节点的相邻节点的平均比例，这些相邻节点不属于基准节点所属的任何社区。这个参数控制着社区之间的边缘比例。

'''<font color="#ff8000">节点度(node degree)</font>'''和'''<font color="#ff8000">社区规模(community sizes)</font>'''按幂律分布，但指数不同。基准测试假设'''<font color="#ff8000">度(node degree)</font>'''和'''<font color="#ff8000">社区规模(community sizes)</font>'''都具有不同指数的'''<font color="#ff8000">幂律分布(power law distribution)</font>'''，分别为'''<font color="#32CD32">此处需插入公式</font>'''和'''<font color="#32CD32">此处需插入公式</font>'''。'''<font color="#32CD32">此处需插入公式</font>'''是节点的数量，平均度为'''<font color="#32CD32">此处需插入公式</font>'''。混合参数'''<font color="#32CD32">此处需插入公式</font>'''是一个节点的相邻节点的平均比例，这些相邻节点不属于基准节点所属的任何社区。这个参数控制着社区之间的边缘比例。

<big>Step 3:</big> Generate community sizes from a power law distribution with exponent <math>\beta</math>. The sum of all sizes must be equal to <math>N</math>. The minimal and maximal community sizes <math> s_{\min} </math> and <math> s_{\max} </math> must satisfy the definition of community so that every non-isolated node is in at least in one community:

<big>Step 3:</big> Generate community sizes from a power law distribution with exponent <math>\beta</math>. The sum of all sizes must be equal to <math>N</math>. The minimal and maximal community sizes <math> s_{\min} </math> and <math> s_{\max} </math> must satisfy the definition of community so that every non-isolated node is in at least in one community:

< big > 步骤3: </big > 根据指数为'''<font color="#32CD32">此处需插入公式</font>'''的幂律分布生成社区规模。所有规模大小的和必须等于'''<font color="#32CD32">此处需插入公式</font>'''。最小和最大的社区规模'''<font color="#32CD32">此处需插入公式</font>'''必须满足社区的定义，这样每个非孤立的节点至少存在于一个群落中:

< big > 步骤3: </big > 根据指数为'''<font color="#32CD32">此处需插入公式</font>'''的幂律分布生成'''<font color="#ff8000">社区规模(community sizes)</font>'''。所有规模大小的和必须等于'''<font color="#32CD32">此处需插入公式</font>'''。最小和最大的'''<font color="#ff8000">社区规模(community sizes)</font>''''''<font color="#32CD32">此处需插入公式</font>'''必须满足社区的定义，这样每个非孤立的节点至少存在于一个群落中:

<big>Step 4:</big> Initially, no nodes are assigned to communities. Then, each node is randomly assigned to a community. As long as the number of neighboring nodes within the community does not exceed the community size a new node is added to the community, otherwise stays out. In the following iterations the “homeless” node is randomly assigned to some community. If that community is complete, i.e. the size is exhausted, a randomly selected node of that community must be unlinked. Stop the iteration when all the communities are complete and all the nodes belong to at least one community.

<big>Step 4:</big> Initially, no nodes are assigned to communities. Then, each node is randomly assigned to a community. As long as the number of neighboring nodes within the community does not exceed the community size a new node is added to the community, otherwise stays out. In the following iterations the “homeless” node is randomly assigned to some community. If that community is complete, i.e. the size is exhausted, a randomly selected node of that community must be unlinked. Stop the iteration when all the communities are complete and all the nodes belong to at least one community.

< big > 步骤4: </big > 最初，没有为任何社区分配任何节点。然后，每个节点被随机分配到一个社区。只要社区内相邻节点的数量不超过社区规模，就会向社区添加一个新节点，否则就不会添加。在接下来的迭代中，无归属的节点被随机分配给某个社区。如果该社区是完备的，即，规模已经用尽，必须随机选择社区中的一个节点并断开其链接。当所有社区都完备且所有节点都至少属于一个社区时停止迭代。

< big > 步骤4: </big > 最初，没有为任何社区分配任何节点。然后，每个节点被随机分配到一个社区。只要社区内相邻节点的数量不超过'''<font color="#ff8000">社区规模(community sizes)</font>'''，就会向社区添加一个新节点，否则就不会添加。在接下来的迭代中，无归属的节点被随机分配给某个社区。如果该社区是完备的，即规模已经用尽，必须随机选择社区中的一个节点并断开其链接。当所有社区都完备且所有节点都至少属于一个社区时停止迭代。