community detection algorithms

Community Detection Algorithms

DIRECTED BY : ALIREZA ANDALIB

Member-Based Community Detection

1-Similarity characteristics are more often in the same community

Important Node Feature : node similarity - node degree(familiarity) - node reachability

similarity is based on overlap between the neighborhood

Two Methods to find similarity:

The similarity values between nodes v2 and v5 are :


2- sub graphs based on node degrees is a clique

We can cut graph to complete sub graphs -> NP harduse brute force-polynomial solvable - use cliques as core of community

Brute-force clique identification Method -> can find all maximal cliques in a graph

Clique percolation method -> CMP

Though sharing no neighborhood overlap, the social circles of these players (coach, players, fans, etc.) might look quite similar due to their social status. In other words, nodes are regularly equivalent when they are connected to nodes that are themselves similar (a self-referential definition).


3-The two extremes of reachability

(1) there is a path between them (regardless of the distance)

BFS & DFS Methods ->is not useful in large community

(2) so close to be immediate neighbors

we can find shortest paths between their nodes in Clique

but There are predefined sub graphs, with roots in community

Group-Based Community DetectionIn graph-based clustering, we cut the graph into several partitions

Cut size = how many cut edge and the summation of weights

12 4

Minimum Cut

Are not perfect coz often find singleton

nodsBalance Cut More Balance Cut

Group-Based Community Detection1-balance partitioning mod :

Graph G = (V,E) (Vertices, Edge) to K partition that have Pi vertices

P = (P1, P2, P3, ....... , Pk) , Pi ∩ Pj = 0 , =V , ¯Pi=V-Pi

Group-Based Community Detection

1-balance partitioning mod in matrix format :

Let matrix X Xi,j= 1 if node i is in community j , otherwise Xi,j= 0

Let D = diag(d1, d2, …. ,dn)

X’AX -> edge inside i community

Graph(G) Adjacency matrix(A)

1

7

4

2 6

10

53

8 9

Graph(G) with 3 community

1

3

2

Community matrix(X)Degree matrix(D)


Robust Communities:

goal is to find sub graphs robust enough such that removing edges or nodes does not disconnect the sub graph

K-vertex connected graph method -> we must find minimum number of nodes that must be removed to disconnect the graph =K

minimum degree for any node in the graph should not be less than k


Modular Communities:

How community structure found is at random(structures must far from random)

G(V, E) , |E| = m , we have degrees but don’t have Edges , v

Consider vi , vj nodes with di , dj degrees P(connect vi to vj ) = =

SO number of edges between vi and vj ->

Group-Based Community DetectionModular Communities:

modularity maximization try to maximize this distance

Consider Graph G = (V,E) (Vertices, Edge) to K partition that have Pi vertices P = (P1, P2, P3, ....... , Pk)

For partition Px this distance can be defined

generalize by partitioning P with k partitions


Modular Communities:

In all graph this distance is defined

And in matrix form


Dense Communities:

Cliques , clubs, and clans are examples of connected dense

we focus on sub graphs that should be disconnected

We can utilize the brute-force clique identification algorithm

Density


Hierarchical Communities:

community can have sub/super communities. Girvan-Newman algorithm designed for divisive hierarchical clustering

Girvan-Newman have measure called “edge between ness” removes edges with higher edge between ness.

For an edge E, edge between ness is defined as the number Edge of shortest paths between node pairs (Vi , Vj) such that the shortest path Between ness between Vi and Vj passes through E.


Hierarchical Communities (Girvan-Newman Algorithm):

1. Calculate edge between ness for all edges in the graph.

2. Remove the edge with the highest between ness

3. Recalculate between ness for all edges a edged by the edge removal

4. Repeat until all edges are removed

Group-Based Community DetectionHierarchical Communities:

community detection algorithms

Engineering