View source for Leskovec et al., WWW 2010

== Citation ==

Jure Leskovec, Kevin J. Lang, and Michael Mahoney. 2010. Empirical comparison of algorithms for network community detection. In Proceedings of the 19th international conference on World wide web (WWW '10). ACM, New York, NY, USA, 631-640. DOI=10.1145/1772690.1772755 http://doi.acm.org/10.1145/1772690.1772755

== Abstract from the paper ==

Detecting clusters or communities in large real-world graphs such as large social or information networks is a problem of considerable interest. In practice, one typically chooses an objective function that captures the intuition of a network cluster as set of nodes with better internal connectivity than external connectivity, and then one applies approximation algorithms or heuristics to extract sets of nodes that are related to the objective function and that "look like" good communities for the application of interest.

In this paper, we explore a range of network community detection methods in order to compare them and to understand their relative performance and the systematic biases in the clusters they identify. We evaluate several common objective functions that are used to formalize the notion of a network community, and we examine several different classes of approximation algorithms that aim to optimize such objective functions. In addition, rather than simply fixing an objective and asking for an approximation to the best cluster of any size, we consider a size-resolved version of the optimization problem. Considering community quality as a function of its size provides a much finer lens with which to examine community detection algorithms, since objective functions and approximation algorithms often have non-obvious size-dependent behavior.

== Summary ==

=== Task Description ===
Detecting clusters or communities in graph network.

The authors try to compare and study performances of:
* Objective functions
* Heuristics / Approximation Algorithms that optimize the objectives.
This is because:
* Heuristics / Algorithms often find clusters that are systematically biased. 
* Certain methods tend to perform particularly well or particularly poorly on certain kinds of graphs
* In special cases, one might need to identify specific types of clusters.

=== Background ===
* Conductance (quality score for single cluster)
<math>\phi</math>(S) = (#edges outside S) / (#edges inside S)
(Small conductance means good clusters)
* Network Community Profile (NCP, size resolved score of clusters)
The score of best cluster of size k: <math>\Phi(k)=\min_{S\subset V, |S|=k}{\phi(S)}</math>

=== Comparison of algorithms===
We first compare two graph partitioning algorithms:"Local Spectral Partitioning" and "flow-based Metis+MQI"

Here are the findings:
* Metis+MQI generates sets with better conductance
* Local Spectral gives tighter and more well-rounded sets ("compact")
* At small size scales, Metis+MQI performs better on ratio of external-to-internal conductance, but Local Spectral performs better at larger clusters.
There are some other clustering methods and their properties:
* Leighton-Rao algorithm (based on multi-commodity flow) (works on mesh-like graphs)
* Graclus (prefer larger clusters, compact, conductance)
* Newman's modularity optimizing program (Dendrogram) (compact)
For detailed performance, please see the plots in the paper. (Figure 4)

=== Comparison of objective functions ===
（Notation: For set S, n means nodes, m means edges ,c means edges pointing ourside S)

==== Multi-criterion ====
(1) Conductance: c/(2m+c)
(2) Expansion: c/n
(3) Density: 1 - (2m)/{n(n-1)}
(4) Cut Ratio: c/{n(N-n)}
(5) Normalized Cut:  c/(2m+c) + c/{2(M-m)} + c
(6) Max ODF:  max frac. of edges of a node pointing outside S
(7) Average ODF: avg. frac. of edges of a node pointing outside S
(8) Flake ODF: frac. of nodes with more than 1/2 edges inside
Here are the findings:
* (1), (2), (4), (5) and (7) are similar
* (6) perfers smaller clusters, and (8) prefers larger clusters
* (3) performs bad, (4) has high variance

==== Single-criterion ====
* (9) Modularity : (m-E(m))/(4m)
* (10) Modularity Ratio: m/E(m)
* (11) Volume: 2m+c
* (12) Edges cut: c
Here are the findings:
* all measures are monotonic
* (9) prefers large clusters and ignore small ones