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The map equation

M. Rosvall, D. Axelsson, C. T. Bergstrom

arXiv:0906.1405v2physics.soc-ph

TL;DR

Large networks need simplification, but different community-detection methods expose different structural aspects and therefore suit different questions. The paper presents the map equation as a flow-based, information-theoretic method that compresses random-walk trajectories, compares it with modularity, and supplies visualization and search software. It concludes that flow-based methods are preferable for studying dynamics and system behavior when links induce flow.

  • Problem

    Large networks require coarse-grained descriptions, yet researchers need guidance on which community-detection methods fit the structures and questions they study.

  • Method

    The map equation uses information-theoretic compression of random-walk trajectories to simplify weighted and directed networks with respect to flow.

  • Results

    The map equation and modularity highlight different aspects of network structure because the former captures flow patterns while the latter captures topological or formation-related structure.

  • Takeaways & Limitations

    Method choice should follow the research question: flow-based approaches suit dynamics and system behavior, while modularity or topological methods suit formation and non-flow relationships.

Abstract

from arXiv · show

Many real-world networks are so large that we must simplify their structure before we can extract useful information about the systems they represent. As the tools for doing these simplifications proliferate within the network literature, researchers would benefit from some guidelines about which of the so-called community detection algorithms are most appropriate for the structures they are studying and the questions they are asking. Here we show that different methods highlight different aspects of a network's structure and that the the sort of information that we seek to extract about the system must guide us in our decision. For example, many community detection algorithms, including the popular modularity maximization approach, infer module assignments from an underlying model of the network formation process. However, we are not always as interested in how a system's network structure was formed, as we are in how a network's extant structure influences the system's behavior. To see how structure influences current behavior, we will recognize that links in a network induce movement across the network and result in system-wide interdependence. In doing so, we explicitly acknowledge that most networks carry flow. To highlight and simplify the network structure with respect to this flow, we use the map equation. We present an intuitive derivation of this flow-based and information-theoretic method and provide an interactive on-line application that anyone can use to explore the mechanics of the map equation. We also describe an algorithm and provide source code to efficiently decompose large weighted and directed networks based on the map equation.

I. INTRODUCTION

Large networks require coarse-grained descriptions, but the appropriate simplification depends on which structural aspects and system questions matter. The map equation focuses on how existing network structure constrains flow and dynamics.

  • Real-world networks are often too large to represent directly, making coarse-grained descriptions necessary for useful analysis.
  • Community detection must determine which aspects of a network should be highlighted before assigning nodes and links to modules.
  • Methods based on network-formation models address how a network was generated, whereas flow-based methods address dynamics on its existing structure.
  • The map equation simplifies network structure with respect to how links drive system-wide flow and interdependence.
  • For a given partition, minimizing the map equation over network partitions identifies structural features relevant to dynamics on the network.

II. MAPPING FLOW

The map equation uses information-theoretic compression to identify network regularities relevant to flow. It models likely trajectories as random walks and exploits persistent regional structure through module and index codebooks.

  • Minimum description length links data compression with detecting significant structures in the data being compressed.
  • When trajectories are unavailable, the method approximates likely movement as random walks guided by directed and weighted network links.
  • Efficient node codewords exploit movement regularities, so finding an optimal path encoding also identifies important flow-related network structure.
  • Huffman codeword lengths derive from ergodic node visit frequencies, with shorter codes assigned to more frequently visited nodes.
  • Modules are persistent regions encoded with separate module codebooks, while an index codebook signals which module codebook is active.
  • A two-level code using module and index codebooks yields an average 32% shorter description for the example network than a one-level description.
  • The full description length balances increasing between-module costs against decreasing within-module costs as the number of modules changes.
  • For the example in Figure 2, the full description length reaches its minimum at four modules.

III. THE MAP EQUATION

The map equation exploits the duality between compressing random-walk trajectories and detecting network structure relevant to flow. For any partition, it estimates the theoretical minimum description length by weighting index- and module-codebook entropies by their usage rates, then selects the partition with the shortest length.

  • III. THE MAP EQUATION: The map equation gives the theoretical limit for describing a network path under any partition without constructing the optimal code.It turns community detection into comparing the theoretical limits for alternative partitions.
  • III. THE MAP EQUATION: The total description length balances increasing module-transition costs against decreasing within-module costs as the number of modules grows.In the illustrated network, the Huffman description length reaches its minimum at four modules and averages 3.09 bits per step for that partition.
  • III. THE MAP EQUATION: The two-level code uses an index codebook for module transitions and separate module codebooks for within-module movements.Index-codeword lengths depend on module exit probabilities, while module-codeword lengths depend on node visit and exit probabilities within each module.
  • III. THE MAP EQUATION: For a partition M, Shannon’s source coding theorem supplies entropy lower bounds for each codebook, whose rates of use determine the map equation.The index codebook is used when the walker switches modules, while module codebook i is used during visits and exits associated with module i.
  • III. THE MAP EQUATION: Optimizing a partition requires tracking each module’s exit rate and aggregate random-walk visit probability, which can be updated efficiently during numerical searches.The node-level entropy term independent of partitioning need not be recomputed when comparing partitions.

A. Undirected weighted networks

For undirected weighted networks, the map equation derives node visit frequencies directly from link weights and expresses module quantities through relative weights of nodes and intermodule links.

  • A. Undirected weighted networks: In an undirected network, a node’s random-walk visit frequency equals the relative weight of its incident links.The relative weight is the total incident-link weight divided by twice the total network-link weight.
  • A. Undirected weighted networks: A module’s relative weight is the sum of its node weights, while its exit weight measures the relative weight of links leaving that module.The total intermodule relative weight is the sum of the module exit weights.
  • A. Undirected weighted networks: The undirected weighted-network form of the map equation combines module visit and exit weights within the entropy expression.These weights determine the codebook-use rates and the corresponding description length.

B. Directed weighted networks

For directed weighted networks, the method computes steady-state visit frequencies with a random surfer and incorporates teleportation into module-exit probabilities. In the illustrated comparison, flow direction changes the map-equation partition while modularity remains unchanged for structurally matched cases.

  • B. Directed weighted networks: Directed weighted networks use power iteration to calculate steady-state node visit frequencies, with teleportation ensuring a unique stationary distribution.The teleportation probability is set to τ = 0.15, and the results are described as relatively robust to this choice.
  • B. Directed weighted networks: The map equation assigns four modules to the flow network but leaves the source-sink network unpartitioned, with respective map-equation scores of 3.33 and 4.58 bits.The source-sink structure represents pairwise interactions rather than movement and induces no flow, so multiple codebooks provide no benefit.
  • B. Directed weighted networks: Modularity gives identical evaluations to the matched directed networks because their module weights and in- and out-weight patterns are the same.The comparison illustrates that modularity can miss differences caused by link direction and resulting flow.
  • B. Directed weighted networks: Nonuniform teleportation can reflect heterogeneous node importance, such as article counts for journals or flight counts for airports.This adjustment is intended for networks whose nodes represent inherently different objects.
  • B. Directed weighted networks: Teleportation contributes to a module’s exit probability because each node sends part of its weight outside the module.The contribution depends on the teleportation probability and the fraction of nodes outside the module.

IV. THE MAP EQUATION COMPARED WITH MODULARITY

The map equation and modularity assess different aspects of network structure: the map equation focuses on flow dynamics, while modularity focuses on link patterns associated with network formation.

  • Modularity assigns identical values to the compared four-module partitions because their within-module and total link weights are the same.
  • The map equation distinguishes networks that modularity treats as identical by detecting persistent within-module flow and limited movement between modules.
  • For networks without extended flow, random teleportation dominates, so the map equation minimizes description length with one single module.
  • The map equation captures small modules with long persistence times, whereas modularity captures modules with more than the expected number of incoming or outgoing link-ends.

V. FAST STOCHASTIC AND RECURSIVE SEARCH ALGORITHM

The authors develop a fast stochastic and recursive search algorithm to minimize the map equation while improving on the accuracy limitations of simpler hierarchical clustering.

  • The new algorithm balances speed and accuracy for minimizing the map equation on directed and undirected weighted networks.It is described as as fast as previous high-speed algorithms and more accurate than the authors’ previous simulated-annealing algorithm.
  • The core procedure repeatedly moves nodes to neighboring modules that most decrease the map equation, then rebuilds the network hierarchically until no further reduction occurs.
  • Because merged modules cannot later be separated, an early optimal move can have the opposite effect after network rebuilding.
  • Accuracy is improved through recursive submodule movements and single-node movements that reopen module assignments after the core search.
  • The two extensions are repeated while clustering improves, with submodule movements applied recursively and stochastic restarts used after improvement stops.

CONCLUSION

The paper presents the map equation as a flow-based alternative to formation-oriented methods such as modularity. The preferred approach depends on whether the research question concerns network formation or dynamics on an existing network.

  • The paper details the map equation’s mechanics and distinguishes flow-based methods from methods based on stochastic models of network formation.
  • The authors conclude that map equation and modularity highlight different aspects of network structure, so method choice should depend on the questions being asked.
  • Modularity or other topological methods may be preferred for analyzing network formation or simplifying networks whose links represent pairwise relationships.
  • Flow-based approaches such as the map equation are preferable for studying dynamics, system-wide flow, network interdependence, and relations between structure and behavior.
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