Community-detection algorithm

source:
https://arxiv.org/pdf/0803.0476.pdf
https://sites.google.com/site/findcommunities/

• 有很多种 community detection algorithms：
  
  • divisive algorithms: detect inter-community links and remove them from the network
  • agglomerative algorithms: merge similar nodes/communities recursively
  • optimization methods: based on the maximisation of an objective function
• modularity show the quality of the partitions results
  
  • The modularity of a partition is a scalar value between -1 and 1
  • measures the density of links inside communities as compared to links between communities
  • can also be as an objective function to optimize

Louvain method:

Two phases (repeated iteratively):

• First phase
  
  • assign a different community to each node of the network. (这个时候，有多少个点就有多少个community)
  • for each node i, consider the neighbors j of i and evaluate the gain of modularity that would take place by removing i from its community and by placing it in the community of j. The node i is then placed in the community for which this gain is maximum
  • If no positive gain is possible, i stays in its original community
  • repeat and sequentially for all nodes until no further improvement can be achieved. The first phase is then complete.
• second phase (build a new network whose nodes
  are now the communities found during the first phase.)
  
  • weights between the new nodes are given by the sum of the weight of the links between nodes in the corresponding two communities
  • Links between nodes of the same community lead to self-loops for this community in the new network.
  • Once this second phase is completed, it is then possible to reapply the first phase of the algorithm to the resulting weighted network and to iterate.
• 这两个 phase 一起打包看作 “pass”的话，
  
  • the number of meta-communities decreases at each pass
  • most of the computing time is used in the first pass.
  • The passes are iterated until there are no more changes and a maximum of modularity is attained.
  • The first pass of the algorithm finds the natural partition of the network.
  • The second pass finds the global maximum of modularity where cliques are combined into groups of two.
• This first phase stops when a local maxima of
  the modularity is attained
• the output of the algorithm depends on the order in which the nodes are considered.

总结：
每个 pass 有两个阶段（phase）：
第一个阶段，modularity is optimized by allowing only local changes of communities。
第二个阶段，the found communities are aggregated in order to build a new network of communities.
The passes are repeated iteratively until no increase of modularity is possible