Leiden algorithm

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The Leiden algorithm is a community detection algorithm developed by Traag et al ^[1] at Leiden University. It was developed as a modification of the Louvain method to address the issues with disconnected communities.

Quality[edit]

Similar to modularity, the quality function is used to assess how well the communities have been allocated. The Leiden algorithm uses the Constant Potts Model (CPM):^[2]

${\displaystyle {\begin{aligned}{\mathcal {H}}(G,{\mathcal {P}})&=\sum _{C\in {\mathcal {P}}}|E(C,C)|-\gamma {\binom {||C||}{2}}\end{aligned}}}$

Algorithm[edit]

The Leiden algorithm is similar to that of the Louvain method, with some important modifications.

Step 1: First, each node in the network is assigned to its own community.

Step 2: Next, we decide which communities to move the nodes into and update the partition ${\displaystyle {\mathcal {P}}}$.

queue = V(G) # create a queue from the nodes  while queue != empty:   node = queue.next() # get the next node   delta_H = 0   for C in communities: # compute the change in quality for each community     if delta_H(node, C) > delta_H:       delta_H = delta_H(node, C)       community = C   if delta_H > 0:     move node to community     outside_nodes = { node_i | (node, node_i) are edges and node_i is not in community } # find the nodes which are connected to the node but not in the community     queue.add(outside_nodes not already in queue) 

Step 3: Assign each node in the graph to its own community in a new partition called ${\displaystyle {\mathcal {P}}_{\text{refined}}}$.

Step 4: The goal of this step is to separate poorly-connected communities:

for C in communities of P:   # find the nodes in the community which have lots of edges within the community   well_connected_nodes = { node | node is in C, |E(node, C\node)| >= gamma ||node||(||C|| - ||node||) }   for node in well_connected_nodes:     if node is singleton under P_refined:       well_connected_communities = { C_i | C_i is in P_refined, C_i is a subset of C, |E(C_i, C\C_i)| >= gamma*||C_i||(||C|| - ||C_i||)       for C_i in well_connected_communities:         compute probability P(C_i) # 0 if assignment of node to C_i decreases quality of P_refined, greater weights for greater quality increases       assign node to C_i by sampling P(C_i) distribution 

Step 5: Use the refined partition ${\displaystyle {\mathcal {P}}_{\text{refined}}}$ to aggregate the graph. Each community in ${\displaystyle {\mathcal {P}}_{\text{refined}}}$ becomes a node in the new graph ${\displaystyle G_{\text{agg}}}$.

Example: Suppose that we have:

${\displaystyle {\begin{aligned}V&=\{v_{1},v_{2},v_{3},v_{4},v_{5},v_{6},v_{7}\}\\C_{1}&=\{v_{1},v_{2},v_{3},v_{4}\}\\C_{2}&=\{v_{5},v_{6},v_{7}\}\\{\mathcal {P}}&=\{C_{1},C_{2}\}\\{\mathcal {P}}_{\text{refined}}&=\{C_{1a},C_{1b},C_{2}\}\end{aligned}}}$

Then our new set of nodes will be:

${\displaystyle {\begin{aligned}V_{agg}&=\{C_{1a}\mapsto w_{1a},C_{1b}\mapsto w_{1b},C_{2}\mapsto w_{2}\}\end{aligned}}}$

Step 6: Update the partition ${\displaystyle {\mathcal {P}}}$ using the aggregated graph. We keep the communities from partition ${\displaystyle {\mathcal {P}}}$, but the communities can be separated into multiple nodes from the refined partition ${\displaystyle {\mathcal {P}}_{\text{refined}}}$:

${\displaystyle {\begin{aligned}{\mathcal {P}}&=\{\{v~|~v\subseteq C,v\in V(G_{\text{agg}})\}~|~C\in {\mathcal {P}}\}\end{aligned}}}$

Example: Suppose that ${\displaystyle C}$ is a poorly-connected community from the partition ${\displaystyle {\mathcal {P}}}$:

${\displaystyle {\begin{aligned}C&=\{v_{1},v_{2},v_{3},v_{4},v_{5}\}\\{\mathcal {P}}&=\{C\}\end{aligned}}}$

Then suppose during the refinement step, it was separated into two communities, ${\displaystyle C_{1}}$ and ${\displaystyle C_{2}}$:

${\displaystyle {\begin{aligned}C_{1}&=\{v_{1},v_{2},v_{3}\}\\C_{2}&=\{v_{4},v_{5}\}\\{\mathcal {P}}_{\text{refined}}&=\{C_{1},C_{2}\}\end{aligned}}}$

When we aggregate the graph, the new nodes will be:

${\displaystyle {\begin{aligned}V(G_{\text{agg}})&=\{C_{1},C_{2}\}\end{aligned}}}$

but we will keep the old partition:

${\displaystyle {\begin{aligned}{\mathcal {P}}&=\{\{C_{1},C_{2}\}\}\end{aligned}}}$

Step 7: Repeat Steps 2 - 6 until each community consists of only one node.

References[edit]