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Community landscapes: an integrative approach to determine overlapping network module hierarchy, identify key nodes and predict network dynamics

Istvan A. Kovacs, Robin Palotai, Mate S. Szalay, Peter Csermely

arXiv:0912.0161v4physics.comp-phcond-mat.dis-nncs.MSphysics.data-anphysics.soc-phq-bio.MN

TL;DR

Community detection lacks a method that is simultaneously fast, accurate, and broadly usable across heterogeneous networks. ModuLand identifies modules as hills in an influence-based community landscape, and its applications report overlapping functional assignments, hierarchical structure, and key-node analysis.

  • Problem

    Finding a fast, accurate, broadly usable method that can provide modular overlaps and partitions for heterogeneous networks remains difficult.

    Source: Introduction · p. 1

  • Method

    ModuLand is an integrative method family that identifies overlapping modules as hills of an influence-function-based, centrality-type community landscape.

    Source: Introduction · p. 2

  • Results

    The authors report that pervasive overlaps represent functional multiplicity at the single-node level; in yeast, NodeLand identified 10 highly overlapping modules in the cAMP-dependent kinase family.

    Source: Description of the ModuLand method… · p. 8

  • Takeaways & Limitations

    The authors identify potential uses including long-range-link identification, prediction of missing links or nodes, network classification, and efficient information transfer.

    Source: Discussion · p. 14

    On noisy community landscapes, local-maxima methods can produce a module at each local maximum, including highly overlapping modules; the authors merge groups with correlation above 90%.

    Source: Selecting the appropriate method of… · p. 13

Abstract

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Background: Network communities help the functional organization and evolution of complex networks. However, the development of a method, which is both fast and accurate, provides modular overlaps and partitions of a heterogeneous network, has proven to be rather difficult. Methodology/Principal Findings: Here we introduce the novel concept of ModuLand, an integrative method family determining overlapping network modules as hills of an influence function-based, centrality-type community landscape, and including several widely used modularization methods as special cases. As various adaptations of the method family, we developed several algorithms, which provide an efficient analysis of weighted and directed networks, and (1) determine pervasively overlapping modules with high resolution; (2) uncover a detailed hierarchical network structure allowing an efficient, zoom-in analysis of large networks; (3) allow the determination of key network nodes and (4) help to predict network dynamics. Conclusions/Significance: The concept opens a wide range of possibilities to develop new approaches and applications including network routing, classification, comparison and prediction.

Introduction

Community detection lacks a universally best method, despite many useful approaches. ModuLand offers an integrative framework that treats overlapping modules as hills in an influence-based community landscape.

Introduction

  • No universally best community-detection method is available, and widely used algorithms can have serious problems despite producing useful network dissections.Existing approaches draw on varied physical and algorithmic principles.Source: Introduction · p. 1
  • Community detection includes partitioning and overlapping approaches, alongside spectral, random-walk, spin-model, and synchronization methods.The paper identifies Girvan–Newman edge removal and modularity as influential developments and notes Clique Percolation as a popular overlapping method.Source: Introduction · p. 2
  • ModuLand frames overlapping modules as hills of an influence-function-based, centrality-type community landscape.Its method family provides a common framework for methods with different overlap levels, computational speeds, and accuracy levels.Source: Introduction · p. 2

Description of the ModuLand method family

ModuLand builds a link-based community landscape from influence functions, identifies modules as hills, and can extend them into overlapping modules and a hierarchy. Its applications include benchmark, protein-interaction, and word-association networks.

Description of the ModuLand method family

  • ModuLand sums node influence functions over each link to create a centrality-type landscape, then identifies modules as its hills.NodeLand approximates influence functions on weighted, undirected networks; the authors also describe LinkLand and PerturLand.Source: Description of the ModuLand method… · pp. 2–4
  • Threshold-based hills produce non-overlapping modules, whereas local-maxima hills identify modules simultaneously and can be extended into overlapping modules.Threshold choice trades off detection of small modules against merging large ones; ProportionalHill extends modules toward lower landscape regions.Source: Description of the ModuLand method… · pp. 3, 5
  • Higher-level networks represent modules as nodes and their overlaps as links, allowing repeated hierarchical analysis toward a single node and zoom-in analysis of large networks.The method also identified the karate-club split and a third previously identified module, and benchmark modules consistently corresponded to the originals across parameter settings.Source: Description of the ModuLand method… · pp. 6–7
  • On non-overlapping benchmark graphs, ModuLand compared well with InfoMap, Louvain, and CFinder, but was less accurate than InfoMap on less-defined modules.The authors note that these graphs were developed to compare non-overlapping methods, which may partly explain the overlapping methods’ performance.Source: Description of the ModuLand method… · p. 8
  • On the yeast cAMP-dependent kinase family, NodeLand identified 10 highly overlapping modules, while InfoMap, Louvain, and CFinder each assigned the family to one module.The 16 neighbors added 4, 11, 8, and 6 modules, respectively; CFinder found modules for 5 neighbors, while the other methods assigned all 16.Source: Description of the ModuLand method… · p. 8
  • The authors conclude that pervasive overlaps represent functional multiplicity at the single-node level, while related nodes’ differing module-membership strengths suggest functional specificity.They base this conclusion on comparisons with other methods and experimental data from the literature.Source: Description of the ModuLand method… · p. 8

Modular hierarchy of a social network

The Community-44 analysis found four main friendship modules and gender differences in multiracial and overlapping-community patterns. In the power grid, bridgeness-based removals affected network integrity more than degree-based attacks and performed comparably to or better than betweenness-based removals.

Modular hierarchy of a social network

  • Community-44 showed four distinguishable main modules and further sub-modules; girls were less likely to form multiracial friendship communities, while boys overlapped significantly more communities.The reported differences had chi-square p < 0.05 and p < 0.0001, respectively, and agreed with cited sociological observations.Source: Modular hierarchy of a social network · p. 9

Efficient determination of central, key nodes of power-grid network

  • In the USA Western Power Grid, bridgeness-based node removals reduced network integrity more than degree-based attacks and performed comparably to or better than betweenness-based removals.The authors note this comparison is notable because network integrity corresponds extremely well to global betweenness centrality.Source: Efficient determination of central,… · pp. 9–10

Discrimination between date- and party-hubs

ModuLand-derived bridgeness and centrality measures separated date- and party-hubs in a yeast protein-interaction network. The method family also supports overlapping membership, adaptable algorithms, and hierarchical analysis, with method choices trading detail against computational cost.

Discrimination between date- and party-hubs

  • 84 of 201 party-hubs and 307 of 318 date-hubs were classified correctly by the bridgeness-centrality separation.Of the incorrectly identified hubs, 91% of date-hubs and 76% of party-hubs were potentially misclassified relative to the classification consensus.Source: Discrimination between date- and… · p. 10
  • ModuLand measures also predicted node dynamics from network topology alone, and date-hubs were hypothesized to have higher bridgeness than party-hubs of similar centrality.The authors cite date-hubs' inter-modular position and similar efficiency of bridgeness-based and date-hub-based network disintegration as support for the hypothesis.Source: Discrimination between date- and… · p. 10

General characterization of the ModuLand method family

  • Fast, approximate treatments of influence-function calculation and landscape hill detection can yield sufficiently fine modularizations because construction and membership summations average out small details.The authors recommend more elaborate treatment when the application warrants it.Source: General characterization of the… · pp. 10–11
  • Earlier methods can supply ModuLand influence functions or landscape constructions; some yield binary containment rather than its fine-grained membership strengths.For example, local module finding can serve as a binary influence-function method, while random-walk methods can be interpreted as influence functions.Source: General characterization of the… · pp. 11–12
  • Changing influence functions, landscape generation, or hill detection generates variants, and local-maxima hills can upgrade threshold-based partitions to overlapping modules.The family also provides continuous link and node membership across modules, plus effective-number, centrality, and bridgeness measures.Source: General characterization of the… · p. 12

Selecting the appropriate method of the ModuLand method family

  • For undirected networks the authors prefer LinkLand, and for directed networks PerturLand; threshold-based hills suit non-overlapping modules, while local maxima identify overlaps.ProportionalHill is suggested for practical use, TotalHill for the most detailed overlap information, and GradientHill when only important overlaps are needed; the hill-finding phase trades accuracy against computational cost.Source: Selecting the appropriate method of… · p. 13
  • On noisy landscapes, each local maximum can create a new, possibly highly overlapping module, so the authors merge groups with correlation above 90%.The local-maxima approaches are described as overcoming the giant-component problem and producing continuously overlapping modules, unlike the traditional threshold-based approach.Source: Selecting the appropriate method of… · p. 13

Discussion

ModuLand treats network modules as hills in a community landscape and combines existing partitioning approaches with local-maxima methods for overlapping modules. The resulting overlaps, hierarchy, and node measures are proposed for network analysis and applications.

Discussion

  • ModuLand frames modules as hills of an influence-function-based community landscape, integrating partitioning methods and local-maxima approaches that yield overlapping modules.Its examples include NodeLand, LinkLand, and PerturLand for influence functions, and ProportionaHill, GradientHill, and TotalHill for hill determination.Source: Discussion · p. 14
  • The framework's overlaps, hierarchy, centrality, and bridgeness measures are proposed for identifying stabilizing weak links and trend-setting nodes, and for predicting missing links or nodes.The passage also names network classification and efficient information transfer as possible applications.Source: Discussion · p. 14
  • The authors propose that module overlaps may play a role in disconnection and synchronization of complex-system modules and their re-assembly during and after crises.Source: Discussion · p. 14

Networks

The paper describes six networks used as examples, spanning collaboration, social, word-association, power-grid, and protein-interaction data.

Networks

  • The network examples include an undirected, unweighted science co-authorship giant component with 379 nodes and 914 links, and Zachary's weighted, undirected karate-club network with 34 nodes and 78 links.The karate-club network is described as a gold standard because its members later split into two factions.Source: Networks · p. 14
  • The word-association giant component has 10,167 nodes and 63,788 weighted links, while the high-school friendship component has 1,127 nodes and 5,096 weighted links.Directions were removed from both networks; weights represent association strength in the former and friendship strength in the latter.Source: Networks · p. 15
  • The examples also include an unweighted, undirected Western USA power grid with 4,941 nodes and 6,594 links, and a yeast protein-interaction giant component with 2,444 nodes and 6,271 links.The yeast network covers approximately half the genome and the most reliable roughly 3% of expected links.Source: Networks · p. 15

Electronic Supplementary Material S1

The supplement details ModuLand’s influence-function and module-assignment methods and illustrates their use in benchmarks and network examples. It also shows how modules form hierarchical levels and how assignment choices affect overlap and resolution.

Summary

  • The supplement describes NodeLand, LinkLand, and PerturLand, adapts earlier methods as BCLand and CliqueLand, and presents GradientHill, ProportionalHill, and TotalHill for minimal-to-detailed overlap.Its materials include 14 supplementary figures, five tables, and summaries of 18 module definitions, 129 modularization methods, and 13 module-comparison methods.Source: Summary · p. 26

Supplementary Figures

  • Threshold-based and SameHill detection select connected landscape regions, whereas local-maxima detection finds hilltops and assigns nodes or links to overlapping modules.GradientHill, ProportionalHill, and TotalHill yield minimal, moderate, and extensive overlap, respectively.Source: Figure S4 · p. 33, Figure S5 · p. 34
  • On the Community-44 school network, the four algorithms yielded rather consistent module counts, but NoLand failed to distinguish the four known main sections.With LinkLand, higher hierarchical levels showed fewer, more extended modules; with PerturLand, X=0.5 merged many smaller modules and two of the four major modules.Source: Figure S10 · p. 39, Figure S11 · p. 40, Figure S12 · p. 41
  • The co-authorship network hierarchy contained 206 level-1 modules (effective number 123.17) and 18 level-2 modules (effective number 9.34).The illustrated hierarchy continues until a single node at level 4; higher-level memberships can also be projected onto the original network.Source: Figure S6 · p. 35, Supplementary Figures · p. 39
  • Modules of the benchmark graph were consistently identified while they remained strongly defined (μ<0.5), using NodeLand with ProportionalHill and merging highly correlated modules.The benchmark averaged 100 representations; a correlation threshold that is too low merges distinct modules, while one that is too high can leave artifact modules unmerged.Source: Supplementary Figures · p. 42
  • In the yeast interaction network, functional labels were assigned from core modular proteins with at least 50% of their module’s local-maximum centrality, and those core proteins showed functional consensus.The figure presents the lowest hierarchy level, with node height representing centrality and color indicating maximum module membership.Source: Figure S14 · p. 43

I. Networks used for the analysis

The analysis draws on six networks spanning collaboration, social ties, word associations, infrastructure, and protein interactions. Their constructions and sizes differ, including weighted and directed data that were sometimes consolidated for analysis.

1. Network science collaboration network

  • The network-science collaboration network contains 379 scientists and 914 weighted links, with link weights proportional to the number of joint publications.It is the giant component of an undirected network compiled by Mark J. Newman.Source: §1 · p. 44

2. Zachary karate club social network

  • The weighted, undirected Zachary karate-club network has 34 members and 78 links, and its recorded split into two factions makes it a community-detection test case.Link weights represent interaction strength.Source: §2 · p. 44

3. Word association network

  • The word-association analysis uses the largest component of an undirected network with 10,617 words and 63,788 links, formed by collapsing parallel forward and backward links and summing their weights.The source data included nearly three-quarters of a million responses from 6,000 participants to 5,019 stimulus words.Source: §3 · p. 44

4. School-friendship network

  • The Community-44 school network was reduced from 1,147 students and 6,189 directed links to a largest connected component of 1,127 nodes and 5,096 weighted undirected links.Parallel directed links were merged and their weights summed; resulting link weights ranged from 1 to 12.Source: §4 · p. 45

5. Electrical power-grid of the USA

  • The unweighted, undirected USA Western Power Grid network contains 4,941 nodes and 6,594 links.The paper presents it as an engineered-network example for studying error propagation and malicious attacks.Source: §5 · p. 45

6. Yeast protein-protein interaction network

  • The yeast protein-interaction network contains 2,444 nodes and 6,271 links in its largest connected component, covering approximately half the yeast genome’s proteins in the full network.The analyzed network is unweighted and undirected.Source: §6 · p. 45

II. Overview of the ModuLand network module determination method family

ModuLand represents network modules as hills in a community landscape built from influence functions. It assigns nodes and links to overlapping modules and recursively organizes those modules into a hierarchy for zoom-in analysis.

II. Overview of the ModuLand network module determination method family

  • ModuLand proceeds from influence functions to a community landscape, identifies hills and assigns nodes or links to overlapping modules, then builds higher levels by treating modules as nodes.The hierarchy can use original or higher-level modules as nodes at subsequent levels.Source: §II · p. 46
  • Influence-function choices, landscape construction, and hill assignment can vary; the methods are applicable to weighted directed networks as well as weighted undirected networks.The authors note that an optimal influence-function algorithm may differ across networks; self-loops are excluded, and non-existing links may be treated as zero-weight links.Source: §II · pp. 46–47, §III · p. 48
  • ModuLand versions provide fractional membership across modules, do not require a preset module count for local-maxima hill detection, and offer hierarchical views from bird’s-eye to detailed.Several algorithms explore multiple topology scales by giving progressively less weight to network segments farther from the examined node or link.Source: §II · p. 46

III. Starting considerations and definitions

  • Influence functions represent indirect impacts of starting nodes, links, or network segments on original links, while the community landscape height is the sum of influence-function values on each link.In this paper, influence values are assumed non-negative, and the implemented landscape construction sums them.Source: §II · p. 46, §III · p. 48

2. Input data of the ModuLand method family

This section describes suitable input representations and influence-function construction for ModuLand on weighted and directed networks, including density-based influence-zone growth and perturbation spreading. It also explains how network methods can be represented as community landscapes and covers landscape assignments and adaptations for overlapping modules and automatic module counts.

2. Input data of the ModuLand method family

  • ModuLand accepts weighted and directed links, but requires a single link type with non-negative weights where larger values indicate stronger connections.When inputs instead encode distances, negative weights, multiple link types, or hyperedges, conversion is needed; there is no universal recipe, though endpoint-similarity weights are advised. Node features can also be converted to a network using a non-negative linear similarity metric.Source: §2 · pp. 48–49

IV. Determination of the community landscape

  • When the community landscape is already known, ModuLand can start from it directly and omit influence-function calculation.The described applications produce static module predictions from static or time-averaged network descriptions, although dynamic factors may also be incorporated.Source: §IV · p. 50

1. The NodeLand and LinkLand influence function calculation algorithms for weighted and undirected networks

  • For weighted undirected networks, NodeLand and LinkLand approximate influence functions through density-based influence-zone growth, with LinkLand accepting extensions that do not decrease density.NodeLand instead requires a density increase and favors maximal growth; it usually yields smaller modules and faster results. Growing influence zones favor connection-rich regions, which become landscape hills, while connection-poor regions form valleys.Source: §1 · p. 50, a. The NodeLand influence function… · p. 50, b. The LinkLand influence function… · p. 52

2. The PerturLand influence function calculation algorithm for weighted and directed networks

  • PerturLand models directed-network influence as information spreading from a starting node, with link weights setting transfer ratios and parameter X controlling attenuation.Link weights must be normalized to [0,1] when outside that range; node activity can set the initial information content, and the process tracks maximum incoming and outgoing quantities rather than sums.Source: §2 · pp. 53–54
  • PerturLand propagates perturbation through links until no further nonzero flow reaches nodes, then defines node influence by maximum outflow and link influence by flow through that link.Community landscape height is obtained by summing each link’s influence-function values across influence zones.Source: §2 · p. 55
  • PerturLand’s stated runtime is O(ne) per influence function; modified Dijkstra or Floyd–Warshall alternatives are described, and undirected links can be represented as opposing directed links.The alternatives have complexities O(e + n log n) per influence function and O(n^3) for the whole landscape, respectively; undirected-link landscape height is the maximum of the two directed heights.Source: §2 · pp. 55–56

3. Two extreme cases of centrality landscapes for weighted and directed networks

  • The strict extreme assigns a node’s outgoing links their weights, whereas the loose extreme assigns equal values across its connected subgraph, producing plateaus sized by node or link count.These extremes may be unsuitable for community detection, but can support topology analysis or suit particular networks.Source: §3 · p. 57
  • Former module-detection methods can be recast as ModuLand landscapes, enabling overlap detection and automatic module counts; some implementations omit influence-function calculation.Partition-based methods can use landscape thresholds, while local-maxima methods can yield overlapping modules and determine the number automatically.Source: §3 · p. 57, c, Methods yielding partitions of the… · p. 58
  • CliqueLand adapts clique percolation for unweighted, undirected networks, assigning each link the largest k for which it belongs to a k-clique community, or 1 if none exists.Clique communities unite adjacent k-cliques and can overlap.Source: §3 · p. 57
  • BCLand represents Girvan–Newman link removal as a landscape, setting each link’s height to the iteration in which it is removed.The underlying method repeatedly removes the link with highest betweenness centrality and recalculates centralities.Source: §3 · p. 58
  • For methods yielding multiple partitions or stochastic runs, link height can count how often its endpoints share a module, with structures optionally weighted by fitness or probability.Source: d, Stochastic module detection methods · p. 58

5. Summary of the community landscape determination methods

ModuLand incorporates existing modularization methods and offers algorithms that adapt to different network structures. Its landscape can be partitioned by horizontal cuts or used to identify overlapping modules around local maxima.

5. Summary of the community landscape determination methods

  • NodeLand, LinkLand, and PerturLand use progressively broader topological scales and detect communities in very low- and very high-density networks, including hierarchies with dense and sparse segments.The framework also represents Girvan and Newman’s distinct-community method (2002; 2004) and Palla et al.’s overlapping-community method (2005), along with other previously described methods.Source: §5 · p. 58
  • ModuLand can use influence functions based on other community definitions, including k-clan, k-club, and k-plex, or on assemblies such as cliques or motifs.Source: §5 · p. 58
  • Sales-Pardo et al. (2007) also presented this idea, though they did not describe their method as a community landscape.Source: §5 · p. 58
  • Further variants can change influence-function summation and landscape exploration; influence-zone borders should be chosen according to the meaning of links.The authors expect additional methods to suit particular networks or community structures.Source: §5 · p. 59

V. Determining modules based on the community landscape

  • Local-maxima methods identify module cores, then assign links and nodes to them using GradientHill, ProportionalHill, or TotalHill membership methods.Source: §V · p. 60
  • Horizontal cuts produce distinct modules, while local maxima and membership assignment can yield overlapping modules.Source: §V · p. 60

1. Horizontal cut of the centrality landscape as a detection threshold of previous community detection methods

Horizontal cuts represent many partition-based community detection methods as thresholds on a centrality landscape. The threshold choice affects which communities are detected, while local-maxima methods can identify large and small modules simultaneously.

1. Horizontal cut of the centrality landscape as a detection threshold of previous community detection methods

  • Horizontal cuts treat connected subgraphs above or, occasionally, below a selected landscape height as distinct modules, representing partition-based methods such as Girvan and Newman (2002; 2004).Source: §1 · p. 60
  • Setting the detection limit too high can miss small communities, while setting it too low can merge modules into a giant component; local-maxima methods find small and large modules simultaneously.Source: §1 · p. 60
  • Strictly increasing or decreasing landscape-height transformations leave horizontal-cut modules unchanged but can significantly alter overlaps found by local-maxima methods.Source: §1 · p. 60

2. Local maxima-based methods

These methods identify community-landscape hill-tops as module cores and assign network elements to modules, supporting overlapping assignments and deriving node memberships from link memberships.

2. Local maxima-based methods

  • The number of local-maxima-based modules equals the number of hill-tops, so these methods require no prior assumption about module count.Source: §2 · p. 61

Algorithm 4: Algorithm to find the hill-tops or highlands of community landscapes

3. Other definitions of hills on a centrality landscape (the SameHill and NumberHill methods)

  • SameHill groups connected components with similar node or link values; NumberHill grows modules from unassigned hill-tops to a fixed size, potentially creating overlaps and leaving nodes unassigned.When landscape hills are flat enough, SameHill can provide fast, mostly non-overlapping modularization.Source: §3 · pp. 68–69

4. Finding the nodes of the overlapping communities

  • Local-maxima-based methods assign all links and nodes, determine the total number of modules automatically, and avoid giant components for a detailed community-structure picture.Detailed node membership reflects the extent to which a node’s links belong to each module; isolated nodes may be assigned one-node modules for visualization.Source: §4 · p. 69

5. Summary of the hill finder methods

  • Hill-finder membership assignments can be varied, for example by weighting links according to landscape height differences or using nonlinear functions of neighboring links’ heights.The authors invite further exploration of these possible assignment procedures.Source: §5 · p. 69

6. Metrics based on the ModuLand modularization methods

The section defines weighted measures for effective group size, node overlap, bridgeness, and similarity, then describes merging highly correlated communities as a post-processing step.

6. Metrics based on the ModuLand modularization methods

  • Effective number weights members by importance, so a low-weight member contributes only a fraction rather than one full member.It measures the number of important or frequent members continuously.Source: b. Effective number · p. 70
  • Unlike discrete counting, effective number needs no threshold and varies continuously with member weights.The effective size of support is defined as exp(H), where H is Shannon entropy.Source: b. Effective number · p. 70, a. Effective size of support · p. 70
  • A node’s modular overlap is the effective number of modules to which it is assigned.Source: c. Modular overlap · p. 70
  • Bridgeness measures a node’s or link’s overlap across two or more modules relative to the overlap of other nodes or links.Source: c. Modular overlap · p. 70
  • Node similarity is based on module-membership vectors and is maximal when those vectors are identical.Source: c. Modular overlap · p. 71

1. Merging the overlapping communities

  • Modules with highly correlated node-membership rankings are merged, and the resulting membership strengths are summed.The paper uses a correlation threshold of 0.9; for TotalHill, merging instead uses ProportionalHill-based module information because TotalHill overlaps are too extensive for that threshold.Source: §1 · p. 72

2. Robustness of the ModuLand method for the measurement, sampling and computational errors

ModuLand addresses measurement and computational error while building hierarchical network representations from modules and their overlaps. Higher levels, projection across levels, and ModuLand-based visualization support analysis of complex networks at a more manageable scale.

2. Robustness of the ModuLand method for the measurement, sampling and computational errors

  • Across parameter settings, ModuLand modules consistently corresponded to Lancichinetti et al. (2008) benchmark modules under strong-sense membership; comparison with an overlapping benchmark was beyond this paper's scope.Strong sense requires at least half of a node's neighbors to be assigned to the same module; discretization can test error sensitivity, while cumulative errors can create artificial hill-tops and unreal modules.Source: §2 · pp. 72–73

VII. Construction of the higher hierarchical level representations of the network

  • Higher hierarchical levels reduce detailed community structure to a few comprehensible modules, especially in networks with more than 10,000 nodes, and can eliminate occasional artificial hill-tops.The hierarchy is built recursively by treating modules as nodes in a higher-level network.Source: §VII · p. 74

1. The strength of the link between two modules

  • Higher-level link strengths encode direct and indirect interactions among original nodes, while self-loops are discarded because they add no information about inter-node connections.Source: §1 · pp. 74–75

2. The modularization of a higher hierarchical level

  • Higher-level networks are recursively modularized until no links remain; the resulting top-level node count depends on the influence-function algorithm.If interactions between modules differ from those between original nodes, the authors recommend selecting or adjusting an influence-function model for the higher level.Source: §2 · p. 75

3. Projection of the modules of higher hierarchical levels

  • Projection composes assignment matrices across levels to map lower-level nodes into higher-level modules, whose projected sizes generally grow with hierarchy depth.This allows analysis and visualization of nodes from earlier levels, including the original network, from the perspective of a selected higher level.Source: §3 · pp. 75–76

VIII. ModuLand-based network visualization methods

  • ModuLand-based visualization uses community centers and node-to-community contributions to position bottom-level nodes, with hierarchical zoom-in for large networks.The authors note that ModuLand's expected polynomial complexity is O(n^3), while conventional visualization tools may be slower on extremely large networks.Source: §VIII · p. 77

Supplementary Tables

The supplementary tables define network modules and compare modularization methods, including their use on yeast protein-interaction data. They document ModuLand’s analysis of overlapping and hierarchical structure, hub classes, central nodes, and module-size distributions, alongside limits in benchmarks and comparison measures.

Supplementary Tables

  • Table S1 orders local-topology-based module definitions by approximate stringency and notes the LS-set or modified LS-set as the most widely used definition.Other definitions can reflect communities through stronger links among members or their greater closeness or reachability relative to non-members.Source: Table S1 · pp. 78–79, Supplementary Tables · p. 79
  • Most local-topology definitions exclude loops and multiple links, and do not account for link weights or directedness.The supplement also notes a paucity of mathematically explicit definitions incorporating dynamic network properties.Source: Supplementary Tables · p. 79
  • Table S2 compares network module-determination methods and includes network-walk-based agglomerative methods and methods using alternative modularity definitions.The supplied table text identifies these method categories but does not provide enough readable cell data to support specific comparative rankings.Source: Table S2 · pp. 80, 81, 82, 83, 86, Supplementary Tables · p. 83
  • The method inventory was closed on July 29, 2010, and the authors cautioned that it was far from complete.The table’s test-module scores refer to a Girvan–Newman model network, while its Zachary-network column counts misplaced members.Source: Supplementary Tables · pp. 94–95
  • The comparison measures were designed to assess efficiency, reproducibility, and usefulness, but most did not accommodate link weights or directedness.These properties can be significant for assigning module membership in real-world networks.Source: Table S3 · pp. 96–97, Supplementary Tables · p. 97
  • On the yeast cAMP-kinase network, ModuLand used NodeLand and ProportionalHill with a 0.9 merge threshold, alongside InfoMap, Louvain, and CFinder with k=4.ModuLand’s landscape-based centrality let authors infer module function from core proteins; other methods required assessing all members, a tedious task for large Louvain modules.Source: Table S4 · p. 98, Supplementary Tables · p. 100
  • The module-alignment table treats two modules as aligned when most members of the smaller module also occur in the larger one.For ModuLand, reported module size is an effective member count; otherwise its modules would list all 2,444 nodes in the network’s giant component.Source: Table S5 · p. 101, Supplementary Tables · pp. 102–103
  • ModuLand combines varied influence-function calculations with landscape-hill module detection, letting users tune overlap, hierarchy, speed, and accuracy to a network.The hill-finding step provides flexibility to tailor simpler, faster but less accurate methods or more accurate but computationally costly approaches.Source: Supplementary Discussion · pp. 104–105
  • ModuLand’s word-association network distributions showed heterogeneous patterns but deviated from log-log linearity at both low and high values.The authors note that earlier studies also observed substantial nonlinearity in several module-parameter distributions.Source: Supplementary Discussion · p. 106
  • Local-maxima hill methods cannot distinguish landscapes built from small versus large influence zones, while noise-induced maxima can fragment modules; these effects shift module sizes toward intermediate values.Deviations at higher hierarchy levels also reflect link weights defined by overlaps of primary modules.Source: Supplementary Discussion · p. 106

Robustness of the results obtained by the ModuLand method

  • On Lancichinetti et al. (2008)’s non-overlapping benchmark, identified modules corresponded consistently across parameter settings under the strong-sense criterion; comparison with the overlapping benchmark was beyond scope.Here, strong sense means that at least half of a node’s neighbors are assigned to the same module.Source: Supplementary Discussion · pp. 106–107

Overlapping word-association modules

  • In the word-association network, homophones “bright” and “focus” occupied densely interconnected modules; a 13% similarity threshold excluded intellect-related neighbors of “bright.”NodeLand ranked money, cold, car, water, food, tree, book, church, and dinner as the highest community-landscape nodes, in that order.Source: Supplementary Discussion · p. 107

Modular hierarchy of a school friendship network

  • ModuLand builds hierarchy by treating primary modules as nodes at the next layer; school-network analyses recovered four major modules at higher levels and up to 232 modules at lower levels.Lowering PerturLand’s X value improved module separation; the authors also describe hierarchical representations as useful for fast visualization and packet delivery.Source: Supplementary Discussion · pp. 107–108

Central nodes of power-grid a network

  • ModuLand bridgeness identified more important nodes in the Western Power Grid than either degree or betweenness centrality.The comparison removed nodes in descending order of their initial centrality without recalculating centralities after each removal.Source: Supplementary Discussion · p. 109

Discrimination between date- and party-hubs

  • Yeast date-hubs had greater bridgeness than party-hubs at similar centrality and greater overlap at the same degree; classification misidentified 1/201 date-hubs and 28/318 party-hubs.The analysis also recovered functionally related, overlapping modules and placed superoxide dismutase in the mitochondrial ribosome, a relationship previously reported by Zielinski et al. (2002).Source: Supplementary Discussion · p. 109

Conclusions and perspectives

  • The authors characterize ModuLand as recovering earlier findings, exposing network hierarchy, providing node-characterization measures, and predicting node dynamics, with potential applications in routing and network analysis.They also propose possible uses including predicting missing or hidden nodes or links and assessing network dynamics and evolution.Source: Supplementary Discussion · p. 110
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