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

M. Rosvall, C. T. Bergstrom

arXiv:1010.0431v2physics.soc-phcs.SIphysics.comp-ph

TL;DR

Large integrated networks often have organization spanning more than two levels, making richer hierarchical structure difficult to capture with standard modular descriptions. The paper generalizes the map equation to compress random-walker trajectories and optimize nested partitions, revealing multilevel organization in real networks. It finds intuitive country–continent and scientific-field hierarchies, with shallow structures in globally interconnected systems and richer ones in highly separated systems.

  • Problem

    Real-world networks frequently contain submodules within modules across multiple scales, while standard modular descriptions and some existing methods do not directly reveal all levels.

  • Method

    The hierarchical map equation uses random walks as proxies for flow and nested index codebooks to minimize the description length of movement over hierarchical network partitions.

  • Results

    The method reveals countries and continents in global air traffic, more than 100 scientific fields in four major disciplines, and generally shallow versus rich hierarchies in globally interconnected versus highly separated systems.

  • Takeaways & Limitations

    Compression-based flow analysis can expose both hierarchical structure and relationships among modules at multiple levels in large social and biological networks.

  • Takeaways & Limitations

    The study focuses on hard partitions and random-walker flow, postponing overlapping partitions and generalized flows to subsequent work.

Abstract

from arXiv · show

To comprehend the hierarchical organization of large integrated systems, we introduce the hierarchical map equation, which reveals multilevel structures in networks. In this information-theoretic approach, we exploit the duality between compression and pattern detection; by compressing a description of a random walker as a proxy for real flow on a network, we find regularities in the network that induce this system-wide flow. Finding the shortest multilevel description of the random walker therefore gives us the best hierarchical clustering of the network, the optimal number of levels and modular partition at each level, with respect to the dynamics on the network. With a novel search algorithm, we extract and illustrate the rich multilevel organization of several large social and biological networks. For example, from the global air traffic network we uncover countries and continents, and from the pattern of scientific communication we reveal more than 100 scientific fields organized in four major disciplines: life sciences, physical sciences, ecology and earth sciences, and social sciences. In general, we find shallow hierarchical structures in globally interconnected systems, such as neural networks, and rich multilevel organizations in systems with highly separated regions, such as road networks.

Introduction

The paper addresses the need to uncover hierarchical organization beyond two-level community structure. It introduces a flow-oriented method designed to reveal multiple interdependent levels without external resolution parameters.

  • Real-world networks often contain submodules within modules across multiple scales, so two-level modular descriptions capture only cross sections of their organization.
  • Existing hierarchical approaches often generate a dendrogram or grouping first, then use another method to choose cuts or resolution thresholds.
  • The paper focuses on network flows represented by directed and weighted links, including movement of people, passengers, ideas, and funds.
  • The proposed approach seeks multiple levels of interdependence with one method that inherently reveals natural system levels rather than requiring multiple external resolution parameters.
  • The hierarchical map equation generalizes the map equation to determine the number of levels, modules at each level, and node memberships.

The two-level map equation

The two-level map equation compresses random-walk dynamics by combining module and index codebooks. Its limitation is that it cannot exploit hierarchical structure at multiple scales simultaneously.

  • A random walk serves as a proxy for real flow, while compression identifies network structures that generate the flow.
  • The map equation minimizes the per-step average description length of a walker moving along weighted and directed network links.
  • Localized regions with long persistence times enable compression through reused short codewords in multiple module codebooks.
  • An index codebook identifies the active module codebook, making descriptions unambiguous across modules.
  • In the 27-node example, two-level clustering produces nine modules with a 3.57-bit description, versus 4.75 bits for one unpartitioned module.
  • The two-level code cannot simultaneously exploit module- and submodule-level structure, leaving additional module-level compression and organization unreported.

The hierarchical map equation

The hierarchical map equation extends coding across nested modules, allowing multilevel partitions to be optimized for network flow. It represents structure through nested index and module codebooks.

  • The hierarchical map equation: The hierarchical description adds index codebooks for each hierarchy level, reducing the example network’s description length from 3.57 to 3.48 bits.
  • The hierarchical map equation: The method permits an arbitrary number of nested index codebooks for movements between modules, submodules, and finer modular levels.
  • The hierarchical map equation: Figure 1’s description length combines average codeword lengths from index and module codebooks weighted by their rates of use.
  • The hierarchical map equation: For a hierarchical map, each module contains a submap whose submodules may themselves contain further submaps.
  • The hierarchical map equation: At each submodule level, the subindex codebook encodes entering submodules or exiting to a coarser level, weighted by its codeword-use rate.
  • The hierarchical map equation: The optimal hierarchical partition minimizes the hierarchical map equation over all possible hierarchical network partitions.
  • The hierarchical map equation: Figure 2 depicts module sizes by horizontal-line width and gives each module’s number of submodules beside braces, omitting submodules below 1 per mil of total flow.

Multilevel organization in real-world networks

The hierarchical map equation reveals multilevel organization and relationships among modules across real-world networks. It finds rich hierarchies in regionally separated systems and shallow structures in globally interconnected systems, with examples from science, air traffic, disease, web, and road networks.

  • The hierarchical map equation reveals multilevel structures and relationships among modules in several large real-world networks.It captures both hierarchical structure and flow between modules at different levels.
  • Science: More than 100 scientific fields are organized into four major disciplines, with physical sciences further divided into physics and chemistry and mathematics.The citation network contains more than nine million citations among nearly 8,000 journals.
  • Global air traffic: 3,883 cities connected by 14,142 links are organized into countries grouped within continents, including 55 and 75 submodules in the two largest modules.The largest modules comprise European and African cities, and North and South American cities, respectively.
  • Human disease network: The disease-network partition groups class-related diseases, but only its largest cancer module matches a natural disease classification.The authors interpret the difference as an effect of missing data and bias toward cancer-related studies.
  • Cross-network pattern: Shallow hierarchies occur in globally interconnected systems, whereas rich multilevel organizations occur in systems with highly separated regions.The California road network has the highest compression gain, while C. Elegans and US air-travel networks are best compressed by two-level descriptions.
  • Cross-network pattern: The road-network representation and geographic range affect apparent hierarchical depth by changing connectivity, distances, and the number of nested regions represented.The dual Stockholm road network suppresses distances and makes the network more interconnected, while California includes multiple cities beyond neighborhood streets.

Materials and Methods

The hierarchical map equation measures the information needed to describe random-walker movements through a hierarchical partition, summing codebook contributions across levels. The method includes a recursive search objective and adjustments for directed-network teleportation.

  • Optimization target: The search minimizes description length to determine the number of hierarchy levels, modules at each level, and node assignments.These are the three structural decisions posed by the optimization problem.
  • Directed-network flow: For directed networks, node visit frequencies are adjusted to exclude random teleportation, so in-flow and out-flow need not remain equal.The procedure first calculates ergodic frequencies with teleportation and then removes teleportation contributions.
  • Hierarchical map equation: It measures the per-step average minimal information needed to track a random walker moving along network links under a given hierarchical partition.The walker may remain within a finest-level submodule or move across one or more hierarchy levels.
  • Hierarchical map equation: The hierarchical map equation sums description-length contributions from every module across all hierarchy levels.Intermediate levels use submap descriptions, while the finest level describes movements to nodes.

Fast stochastic and recursive search algorithm

The algorithm recursively searches for multilevel partitions by adding codebooks when they shorten the description, combining stochastic local moves with module refinement. Its design targets a balance between speed and accuracy.

  • Recursive multilevel search: The recursive search tests additional coarser and finer codebooks whenever they further compress movements between or within modules.It operates on modules at any hierarchy level and stops descending when no shorter description is found.
  • Speed and accuracy: The C++ stochastic recursive algorithm is as fast as earlier high-speed algorithms and more accurate than the previous simulated-annealing method.It is presented as a compromise between fast but inaccurate greedy search and accurate but slow Monte Carlo approaches.
  • Core search: The core algorithm repeatedly moves nodes to neighboring modules that decrease the map equation, then rebuilds the network hierarchically until no further reduction occurs.Nodes begin in separate modules and are processed in random sequential order.
  • Refinement moves: Submodule movements and single-node movements refine a final clustering by freeing larger groups or individual nodes to move between existing modules.Both procedures reapply the main algorithm after reassignment.

Performance test of the hierarchical map equation

The performance test evaluates whether the hierarchical map equation recovers known coarse and fine partitions in synthetic benchmark networks. Performance is excellent when the hierarchy is topologically well defined and degrades when within-level separation is insufficient.

  • Hierarchy definition: A three-level hierarchy is topologically well defined when p3 > p2 > p1, ordering links within fine modules above links across fine but within coarse modules and links across coarse modules.The probabilities correspond respectively to same-fine-module, same-coarse/different-fine-module, and different-coarse-module links.
  • Evaluation metric: Figure 4 maps mixing-parameter ranges for well-defined three-level structures using 10,000-node networks, coarse modules of 400–4,000 nodes, and fine modules of 10–100 nodes.Connected points indicate the mixing-parameter sets used in the paper.
  • Evaluation metric: Relative mutual information quantifies how much the inferred partitions reveal about the true benchmark partitions at the coarse and fine levels.The normalized measure ranges from 0 for independent partitions to 1 for identical partitions.
  • Benchmark design: The benchmark uses scale-free networks with 10,000 nodes, average degree 20, maximum degree 100, and module sizes of 400–4,000 and 10–100 nodes.Both module-size distributions are scale-free, with exponents -1 for sizes and -2 for network degree.
  • Performance outcome: Performance is excellent while the hierarchical organization is well defined and nodes have more links within than between both fine- and coarse-level modules.Outside this range, the well-defined region becomes too narrow.
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