【DM论文阅读杂记】复杂社区网络

Paper Title

Community Structure in Time-Dependent, Multiscale, and Multiplex Networks

Basic algorithm and main steps

Basic ideas

The paper generalizes the determination of community structure via quality functions to multislice networks, and derive a null model in terms of stability of communities under Laplacian dynamics.

Derivation of the quality function

Restricted our attention to unipartite, undirected network slices \((A_{ijs}= A_{jis})\) and couplings $(C_{jrs} = C_{jsr}) $ .

$ \omega $: Slice coupling strengths.

$ A_{ijs} $ : at slice \(s\), the connection node \(i\) and node \(j\)

$ C_{jrs} $： the connection between slice \(r\) and slice \(s\)

$ k_{js} = \sum_i A_{ijs} $ : the degree / strength of the node $ j $ on slice $ s $

$ C_{js} = \sum_r C_{jsr} $ : the strength across slice $ s $

multiple strength : $ \kappa {js} = k + C_{js} $

The expected weight of the edge between $ i $ and $ j $ under Laplacian dynamics:

\[\dot{P_{is}} = \sum_{jr} \frac{(A_{ijs}\delta_{sr}+\delta_{ij}C_{jsr})p_{jjr}}{\kappa_{jr}} - p_{is}
\]

Using the steady-state probability distribution

$ p^*{jr} = \kappa / 2\mu , ( 2\mu = \sum_{jr} \kappa_{jr} ) $

$ \gamma_s $: revolution parameter

Conditional propability:

\[\rho_{is|js}P^*_{jr} = (\frac{k_{is}}{2m_s}\frac{k_jr}{\kappa_jr}\delta_{sr} + \frac{C_{jsr}}{C_{jr}} \frac{C_{jr}}{\kappa_{jr}} \delta_{ij}) \frac{\kappa_{jr}}{2 \mu}
\]

$ m_s = \sum_j k_{js} $

Quality function:

\[Q = \frac{1}{2\mu}\sum_{ijsr} \bigg[\bigg( A_{ijs} - \gamma_s \frac{k_{is} k_{js}}{2m_s}\bigg)\delta_{sr} + \delta_{ij}C_{jsr} \bigg]
\]

Recover null model

Recovered the standard null model for directed networks (with a resolution parameter) by generalizing the Laplacian dynamics to include motion along different kinds of connections, giving multiple resolution parameters and spreading weights.

Motivation

• In terms of community detection, departed null models have not been available for time-dependent networks.
• One solution: piece together the structures obtained at different times or have abandoned quality functions in favor of such alternatives as the Minimum
  
  Description Length principle.
• Another solution: tensor decomposition, without qualtiy-function.

Contribution

• Generalize the determination of community structure via quality functions to multislice networks, removing the limits.
• Formulate a null model in terms of stability of communities under Laplacian dynamics.

My own idea

Some analysis

• Fig 2 is the experiment result on the dataset of the Zachary Karate Club network. There is 34 nodes and 16 slices (with resolution parameters $\gamma_s $= { 0 . 25, 0 . 5 , …, 4 } and $\omega $= {0,0.1,1}). Other things being equal, the larger \(\gamma\) is, the more communities is. The $ \omega $ means tighter connections among time slices. The horizontal axis is $ \gamma $, and the vertical axis is the 34 members. For any one of the three pictures, the number of communities increases as the $\gamma $ increases. With $\omega $ = 0.1,1, with \(\gamma\) increasing, nodes assigned to the same may keep in the same communities or be partitioned to different communities. However, comparing to the ones with larger slice coupling strengths( the second and the third picture ), the one ignoring slice coupling ( the first picture, with $ \omega $ = 0 ) will lead to messy clustering results (eg. both the \(\gamma\) = 0.25 and the \(\gamma\) have two communities, but they are not the same two communities) . Therefore, taking slice coupling strengths into consideration can improve the performance of the community detection.

Confuse

• What confuses me is the details of derivating the quality function.

Shortcoming

• The paper lacks comparing the performance of their novel algorithm with others.

Others

• I have learnt the null model and quality function of community detection in one dimesion, which is in the monority and restricted greatly. Through this paper, I know the methology in mutiscale and mutiplex networks.

  \[Q = \frac{1}{2m}\sum_{s \in S}\sum_{i, j \in s}(A_{ij} - \frac{k_i k_j}{2m}) =\\
  = \frac{1}{2m}\sum_{i, j}(A_{ij} - \frac{k_i k_j}{2m}) \delta(g_i,g_j)
  \]

  $ \delta(g_i, g_j )$ = 1 if nodes \(i\) and \(j\) are in the same communities and 0 otherwise.

• Unfinished: reproduct the code and results.