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i am implementing Link Communities community detection algorithm. I have trouble understanding explanation of partition density described in the paper

Here is only the part defining partition density: enter image description here

I cannot find the connection between definition of link density (equation 2) and definition of partition density (equation 3). Because of that, i dont understand why is partition density defined the way it is. And i especially dont see how is (equation 3) calculating average of (equation 2) - if there is average, i would expect number of partitions (c) to be below horizontal line (in the divisor)

I could not google any other definition of partition density.

Anyone who can shed some light into it would be appreciated.

hendrix
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2 Answers2

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It is not really an average, but rather a weighted sum: each community is weighted depending on the proportion of total links it contains (i.e. mc). So instead of having a uniform factor of 1/C, where C is the number of communities, you have a community-specific factor of mc/M.

Vincent Labatut
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(2) is the density of one community, (3) is the weighted density sum of all communities, the weight is $m_c / M$.

wa007
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