Multilevel compression of random walks on networks reveals ...

Multilevel compression of random walks on networks reveals hierarchical organization in large integrated systems

M. Rosvall and C. T. Bergstrom Presented by Thang N. Dinh CISE Department, University of Florida Introduction Community structure gives 2-level map

of the network Real-world networks, the organization rarely is limited to two levels Flows of commuters in UK

Hierarchical Community structure How many hierarchical levels? How many modules at each level? What are members of each

module? Hierarchical Community structure Hierarchical clustering:

Agglomerative algorithms Divisive algorithms Sales-Pardo, 2007: Deriving hierarchical from node similarities. Clauset et al. 2007, 2008: Divide the

dendogram using Maximum Likelihood Lancichinetti et al. 2009: Local optimization of a fitness function Encoding Network Flows Optimal compression of network flows

Network structure (M. Rosvall, C. T. Bergstrom, PNAS 2008) Top three performers (LFR benchmark, Fortunato et al. 2009) Can be extended to reveal hierarchical structure of network

Encoding network flows Encoding Network Flows Compressing an infinite flow (random walk)

Minimize the expected code length Shannons source coding theorem: Two-level map encoding Each module has a code Within module:

Each node has a code Special exit code Two-level map encoding http://www.mapequation.org/mapde

mo/index.html Two-level map encoding Community structure M, with modules Pi The rate in which code for modules used

Probability of entering modules i Entropy of code for modules The entropy of code for nodes in module i Two-level encoding

Hierarchical map encoding Hierarchical map encoding Algorithm Fast stochastic and recursive search

algorithm Join neighboring nodes -> modules -> super-modules (Brondel. 2008) Recursive operate on modules at any levels Local improvement Submodules movement

Single-node movement Experiments Experiments

Experiments Experiments