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Nestedness in complex networks: Observation, emergence, and implications

Manuel Sebastian Mariani, Zhuo-Ming Ren, Jordi Bascompte, Claudio Juan Tessone

arXiv:1905.07593v1physics.soc-phecon.THnlin.AOphysics.data-anq-bio.PE

TL;DR

Nestedness is a widespread non-random network pattern whose emergence and systemic implications require explanation. This review synthesizes methods, mechanisms, and consequences across network science, statistical physics, ecology, economics, and related fields. It finds nestedness across ecological, socio-economic, bipartite, and unipartite networks, while emphasizing that its assessment depends strongly on metrics and null models.

  • Problem

    The review addresses how nestedness emerges, how it affects system stability and feasibility, and how it relates to other network properties across ecological and socio-economic systems.

  • Method

    The paper synthesizes methodologies and findings from network science, statistical physics, ecology, economics, social sciences, graph theory, and dynamical systems.

  • Results

    Nestedness is reported across diverse systems, including product trade networks, contractor networks, and cultural-trait distributions, with more sophisticated products typically having more centralized and nested trade networks.

  • Takeaways & Limitations

    The review unifies usually disconnected bipartite and unipartite strands of nestedness research within a cross-disciplinary complex-systems perspective.

  • Takeaways & Limitations

    Understanding nestedness in temporal and multilayer networks remains incomplete, and proposed emergence mechanisms still require robust statistical validation.

Abstract

from arXiv · show

The observed architecture of ecological and socio-economic networks differs significantly from that of random networks. From a network science standpoint, non-random structural patterns observed in real networks call for an explanation of their emergence and an understanding of their potential systemic consequences. This article focuses on one of these patterns: nestedness. Given a network of interacting nodes, nestedness can be described as the tendency for nodes to interact with subsets of the interaction partners of better-connected nodes. Known since more than $80$ years in biogeography, nestedness has been found in systems as diverse as ecological mutualistic organizations, world trade, inter-organizational relations, among many others. This review article focuses on three main pillars: the existing methodologies to observe nestedness in networks; the main theoretical mechanisms conceived to explain the emergence of nestedness in ecological and socio-economic networks; the implications of a nested topology of interactions for the stability and feasibility of a given interacting system. We survey results from variegated disciplines, including statistical physics, graph theory, ecology, and theoretical economics. Nestedness was found to emerge both in bipartite networks and, more recently, in unipartite ones; this review is the first comprehensive attempt to unify both streams of studies, usually disconnected from each other. We believe that the truly interdisciplinary endeavour -- while rooted in a complex systems perspective -- may inspire new models and algorithms whose realm of application will undoubtedly transcend disciplinary boundaries.

1. Introduction

This introduction frames nestedness as a widespread non-random network pattern and reviews how it is observed, how it may emerge, and what implications it has for interacting systems. It unifies ecological and socio-economic perspectives with methods from several disciplines.

  • Motivation: Nestedness is presented as a widely observed structural pattern whose prevalence motivates questions about its emergence and implications for system functioning.The review asks how nestedness arises and relates to stability, persistence, modularity, core-periphery structure, and economic prediction.
  • Scope and contribution: The review provides a cross-disciplinary perspective spanning network science, statistical physics, ecology, economics, and social sciences.Its stated aim is to connect findings and methodologies from fields that have often studied nestedness separately.
  • Basic definition: Perfect nestedness requires lower-degree nodes’ neighborhoods to be contained within those of higher-degree nodes, with bipartite comparisons restricted to nodes in the same class.After degree-based reordering, the adjacency matrix has a monotonic separatrix separating ones from zeros.
  • Observation: Real networks are called nested when their measured nestedness cannot be explained by a reasonable null model, rather than only when they satisfy the stringent perfect topology.Metrics and null models are therefore central because different choices can produce different significance conclusions.
  • Empirical scope: The review covers nestedness in ecological and socio-economic networks, including evidence from plant-pollinator and plant-frugivore systems and changing patterns in country-product trade networks.Most of the ecological networks studied by Bascompte et al. showed nestedness beyond degree-preserving null models, while trade-network nestedness decreased before the 2007–2008 financial crisis.
  • Review structure: The article organizes its central discussion around observing nestedness, modeling its emergence, and evaluating its implications.Proposed emergence mechanisms include optimization of fitness or centrality and stochastic duplication with link randomization.

2. Classes of (potentially) nested networks

Nestedness appears across diverse socio-economic networks, including trade, interbank, contractor, spatial, communication, and social-media systems. These examples connect network topology with economic complexity, firm failure, financial crises, organizational efficiency, and critical events.

  • Significant nestedness occurs in both unipartite and bipartite socio-economic networks, including interbank, trade, country-product, contractor, and firm-location systems.
  • Country-level trade networks: More sophisticated product trade networks are typically more centralized and nested than networks for less complex products.The positive correlation between product complexity and centralization is imperfect, suggesting that complexity and nestedness provide complementary information.
  • Country-level trade networks: Country diversification is strongly correlated with economic fitness, and countries whose fitness exceeds that of similarly developed countries tend to grow economically in the future.
  • Contractor networks: In apparel-firm networks, generalist firms are less likely to fail, while firms with neither overly specialist nor overly generalist partners have lower failure probabilities.
  • Contractor networks: Manufacturer-contractor nestedness affects firm failure probability, while bipartite cooperation has been proposed as a parsimonious mechanism for generating the observed structure.
  • Interbank networks: Interbank networks combine a densely connected core with a low-degree periphery, and core banks’ reduced outgoing links contributed to declining interbank lending during the 2008 crisis.
  • Interbank networks: Core-periphery structure is a special case of nestedness, and Austrian and global banking networks have been found to be significantly nested.
  • Social-media networks: A sharp transition from modular to nested topology occurred near a critical event in a Twitter user-hashtag network during Spain’s 2011 civil protests.

3. Observing nestedness: metrics and null models

Because imperfect nestedness is common, measuring it requires comparing alternative metrics and null models whose choices can change significance conclusions. The review surveys gap-counting, distance, overlap, and eigenvalue-based metrics, alongside degree-preserving randomization and matrix-packing methods.

  • Motivation: Imperfect nestedness is widespread, but its measured significance depends on network size, density, degree distribution, metric choice, and null model.The review therefore asks how nestedness should be measured and compared across heterogeneous datasets.
  • Metrics: Nestedness metrics are grouped into gap-counting, distance, overlap, and eigenvalue-based categories applicable to binary matrices.Examples surveyed include unexpected absences, nestedness temperature, NODF, and spectral radius.
  • Overlap metrics: S-NODF extends NODF to include equal-degree node pairs, reducing abrupt changes caused by small degree perturbations.In the example, a contribution changes from 0.75 to 0.72 rather than dropping to zero after one interaction is added.
  • Eigenvalue-based metrics: The spectral-radius approach is supported by theorems stating that perfectly nested networks maximize spectral radius among connected bipartite networks with fixed node and edge counts.The result is stated both for total node count S and edge count E, and for separate row and column counts N and M.
  • Null models: Null-model selection is critical because loosely constrained equiprobable-interaction models can produce false claims of significant nestedness.Preserving exactly row and column degrees is one stricter alternative, while randomized-network generation can be computationally slow and depends on the number of realizations.
  • Temperature minimization: Matrix-packing algorithms alter row and column order to minimize temperature, and BINMATNEST generally achieves lower temperature than NTC, with the gap larger in pollination networks.Fitness-complexity and BINMATNEST outperform degree ordering; fitness-complexity is usually better except in small, dense networks.

4. Relation between nestedness and other systemic properties

Nestedness is closely related to degree sequence, disassortativity, core-periphery structure, and modularity, but the strength and interpretation of these relationships depend on metrics, null models, and network connectivity.

  • Nestedness and degree distribution: Degree-preserving null models can substantially alter whether empirical networks are classified as significantly nested.Across 286 empirical networks, 113 were significantly nested under at least one metric, but only 11 under all four metrics considered.
  • Nestedness and degree distribution: In mutualistic networks, studies disagree on whether nestedness is explained by degree sequences alone.One maximum-entropy analysis found z-scores above two in only a tiny fraction of 167 networks, whereas earlier work often reported excess nestedness relative to degree-based null models.
  • Nestedness and degree distribution: Null-model constraints create opposing statistical risks: unconstrained models can inflate Type-I errors, whereas exact degree-preserving models can inflate Type-II errors.Fixing the degree sequence can even classify a perfectly nested matrix as maximally non-nested when it is the only matrix with that sequence.
  • Nestedness and assortativity: Nestedness and disassortativity are typically linked: perfectly nested networks have negative assortativity, and the association appears in synthetic and empirical networks.The relationship was reported across 60 unipartite and bipartite empirical networks, with nestedness increasing as Pearson degree correlation becomes more negative.
  • Nestedness and core-periphery structure: Nestedness correlates strongly with core-periphery structure, although both measures can also track edge density and degree heterogeneity.The correlation persisted after randomization procedures preserving edge density and then the exact degree sequence, suggesting related but non-equivalent structural properties.
  • Nestedness and modularity: The relationship between nestedness and modularity depends on network connectivity and can change over time.Low-connectivity networks tend to show positive correlations, high-connectivity networks negative correlations, and a Twitter protest network shifted from increasing modularity to increasing nestedness as protests peaked.

5. Emergence of nestedness: rewiring and formation mechanisms

The review surveys mechanisms proposed to explain nestedness across ecological and socio-economic networks, including graph-theoretic threshold rules, optimization, cooperation, and social processes. It emphasizes that theoretical ability to generate nestedness does not establish empirical relevance.

  • 5. Emergence of nestedness: rewiring and formation mechanisms: Network-growth explanations matter because microscopic mechanisms can produce distinct macroscopic topologies that should match observed data.Candidate mechanisms may need to be ruled out when their generated structures mismatch real networks.
  • 5. Emergence of nestedness: rewiring and formation mechanisms: Ecological proposals include competition-load minimization, fitness optimization, trait matching, speciation and divergence, and invasion dynamics.These mechanisms may optimize species-level or community-level fitness, or involve no optimization.
  • 5. Emergence of nestedness: rewiring and formation mechanisms: Socio-economic nestedness can emerge when actors maximize the centrality, or social status, of their interaction partners through social climbing dynamics.This mechanism has been studied in unipartite socio-economic networks.
  • 5.1. Graph-theoretic mechanisms: Threshold fitness models generate perfectly nested networks, and different fitness distributions can produce different degree distributions.With an exponential fitness distribution, the threshold model generates a power-law degree distribution without growth or preferential attachment.
  • 5.1.2. Growing perfectly nested networks with arbitrary degree distribution: Creation-sequence mechanisms can generate perfectly nested networks with approximately arbitrary degree distributions for sufficiently large networks.The construction is equivalent to the threshold model, linking node fitness and threshold values to a unique creation sequence.
  • 5.2. Ecological mechanisms: The review cautions that a mechanism’s theoretical production of nestedness does not show that it shaped observed empirical networks.This applies both to optimization mechanisms and to mechanisms without species-level or community-level optimization.
  • 5.2.1. Self-organizing network model: Self-organizing rewiring reaches perfect nestedness after many iterations and can be stopped at the nestedness level observed in a target real network.The resulting networks have truncated power-law degree distributions similar to those observed in real networks, regardless of the fraction of forbidden links.
  • 5.2.2. Bipartite cooperation: specialization and interaction: A bipartite cooperation model reproduces more than 70% of empirical degree-distribution, modularity, and nestedness metrics in ecological and manufacturer-contractor networks.Its nestedness levels are compatible with empirical pollination and New York garment-industry networks.

MAXIMIZATION OF SPECIES POPULATION ABUNDANCES

This section examines species-abundance maximization and related formation mechanisms as routes to nested interaction structures. It connects optimization, evolutionary assembly, centrality-seeking, and export diversification with changes in nestedness and network organization.

  • MAXIMIZATION OF SPECIES POPULATION ABUNDANCES: Species-abundance maximization rewires a random interaction matrix so that the interaction structure approaches nestedness while plant and pollinator populations increase.Figure 20 presents these changes as optimization steps increase.
  • MAXIMIZATION OF SPECIES POPULATION ABUNDANCES: Community-level optimization rewires interactions by accepting swaps that increase the system’s total population at equilibrium.The dynamics begins with equal abundances and a randomly generated interaction matrix.
  • MAXIMIZATION OF SPECIES POPULATION ABUNDANCES: Both species-level and community-level optimization lead from random initial networks to highly nested interaction matrices.In the mean-field scenario, total population depends linearly on total interaction overlap.
  • MAXIMIZATION OF SPECIES POPULATION ABUNDANCES: As NODF increases with total interaction overlap, nestedness increases with the total population of the system.This relation connects a structural nestedness metric to population abundance.
  • MAXIMIZATION OF SPECIES POPULATION ABUNDANCES: Speciation and divergence generate networks with low to high nestedness depending on input parameters, suggesting nestedness can arise as an evolutionary spandrel.The model also produces heterogeneous degree and weight distributions and link asymmetry without optimization rules.
  • MAXIMIZATION OF SPECIES POPULATION ABUNDANCES: Immigration-generated networks have randomized nestedness, whereas radiation-generated networks are significantly more nested than randomized counterparts.Thus, different assemblage mechanisms can produce different topologies without optimization mechanisms.
  • MAXIMIZATION OF SPECIES POPULATION ABUNDANCES: Centrality-seeking dynamics reinforces nestedness by making already central agents increasingly attractive connection targets and leaving peripheral agents peripheral.In nested networks, agents can identify the best connection target using local information.

6. Implications of nestedness for systemic stability and feasibility

Nestedness affects systemic robustness, feasibility, and stability in ways that depend on the perturbation strategy and dynamical setting. More nested ecological networks can resist specialist-first losses and support larger feasibility domains, while stability also depends on interaction parameters and topology.

  • Topological robustness: Understanding co-extinction cascades helps identify species whose loss threatens ecosystem persistence and analogous failures in economic and financial systems.The review connects node-removal consequences to conservation and systemic-risk questions.
  • Topological robustness: The attack-tolerance curve measures surviving passive species against deleted active species, with systemic robustness defined as its area R ∈(0, 1).Values near one indicate persistence after substantial active-node deletion; small values indicate rapid collapse.
  • Topological robustness: More nested networks are more robust under specialist-first removal but more fragile under generalist-first removal.The sensitivity follows from which species are removed first in a nested structure.
  • Topological robustness: In mutualistic and country-product bipartite networks, rankings producing more nested adjacency matrices better identify structurally important or vulnerable nodes.Fitness-complexity produces larger extinction areas than several standard centrality and ranking methods in mutualistic networks and outperforms degree centrality and method of reflections in trade networks.
  • Stability and feasibility: Larger nestedness tends to accompany larger feasibility domains, alongside high mutualistic interaction strength and small mutualistic trade-off.For the analyzed empirical network, maximal nestedness with these parameter conditions characterizes the most structurally stable systems, subject to global-stability constraints.
  • Stability and feasibility: Feasibility depends critically on dynamical parameters, so structural stability evaluates the size of the parameter region supporting feasible stable coexistence.A feasible equilibrium may occur for some intrinsic growth rates but not others, and nestedness affects the resulting feasibility domain.

7. Observing nestedness at the mesoscopic scale

Mesoscopic nestedness studies nested structure within network subgraphs, including geographic regions, modules, and core-periphery blocks. New methods jointly detect nested components or in-block structure because sequential community and nestedness detection can fail.

  • Mesoscopic nestedness: Mesoscopic nestedness examines one or more network subgraphs rather than connectivity patterns across the entire network.Recent methods address subgraph quantification, nested-component detection, and nestedness within communities.
  • Geographical subgraphs: Geographical boundaries can produce heterogeneous nestedness, with Cyclades islands more nested and Anatolian islands less nested than comparable random subgraphs.The comparison uses randomly extracted subgraphs of equal size.
  • Largest nested subgraph: NESTLON sequentially builds a nested component from high-degree nodes by testing whether neighborhoods of lower-degree nodes are sufficiently contained.The confirmation ratio controls tolerance for deviations from perfect nestedness.
  • Largest nested subgraph: The metric µ_nest = |Γ_nest|/N ranges from 1 for perfect nestedness to 0 when no node pair satisfies the nestedness condition.Synthetic results suggest it can outperform NODF and BINMATNEST for sparse and high-density matrices, while local information supports scalability.
  • In-block nestedness: Modularity maximization can miss planted in-block nested communities because sparse nested structures cluster high-degree nodes together.This motivates methods that detect community partitioning and internal nested topology simultaneously.
  • In-block nestedness: The in-block nestedness quality function reconstructs planted blocks more accurately than modularity across broad low-p and low-µ regions, although performance deteriorates as both increase.In real networks, nestedness and in-block nestedness are somewhat independent, with small correlation between nestedness and modularity.
  • Core-periphery structure: The QMCP-CM maximization algorithm outperforms divisive and QMCP-ER methods in reconstructing planted core-periphery pairs, including cases where hubs occur in both core and periphery.In empirical networks, it uniquely produced substantially lower periphery-periphery edge density than expected for a random graph.

8. Outlook

The review synthesizes observation, emergence mechanisms, and dynamical implications of nestedness across ecological and socio-economic networks. It closes by identifying unresolved methodological and empirical gaps, especially for temporal, multilayer, and mesoscopic networks.

  • Outlook: The review organizes nestedness research around detection methods, mechanisms of emergence, and implications for network dynamics.Its interdisciplinary scope spans ecological and socio-economic systems.
  • Open questions: Universal answers remain unavailable for several questions, including how to measure nestedness and its statistical significance.The review presents these as continuing methodological challenges.
  • Temporal emergence: No statistical studies using time-stamped data have validated proposed mechanisms that generate nestedness.Because final network structure cannot identify the dynamical generative process, the review calls for thorough temporal analysis.
  • Temporal and multilayer networks: The review focuses on unipartite and bipartite networks, while nestedness in temporal and multilayer networks remains largely unexplored.The causes and implications of multilayer nestedness are still incompletely understood.
  • Mesoscopic nestedness: Mesoscopic detection methods are recent, and their performance, causes, and implications remain at an early stage of understanding.This limits current knowledge of nested structures within network subgraphs.
  • Scope: The review is not exhaustive and emphasizes physical mechanisms of nestedness and its effects on dynamical network processes.It aims to serve researchers across physics, ecology, and economics.

A. Software and relevant datasets

The appendix notes that the listed URLs were active when the article was published, but their future availability is not guaranteed.

  • Software and relevant datasets: The software and dataset-related URLs may no longer be active when the article is read.Their activity was confirmed only at publication time.

A.1. Software for nestedness and network analysis

The appendix identifies general-purpose and ecology-focused software packages that support network and nestedness analysis.

  • Software: NetworkX, igraph, and graph_tool provide general network-analysis functionality, while R packages bipartite and vegan implement nestedness metrics.bipartite targets bipartite networks; vegan targets ecological communities.

A.1.1. Metrics and null models

Nestedness analysis uses dedicated software packages alongside general network-analysis tools. These packages support metric computation, null models, and statistical significance testing.

  • Dedicated ecology-oriented packages compute nestedness metrics and assess their statistical significance.
  • FALCON and Nestedness for Dummies are examples of packages developed specifically for nestedness analysis.
  • Both named packages implement multiple nestedness metrics, null models, and significance tests.

A.1.2. Nested network generation

Perfectly or highly nested networks can be generated either by specifying an adjacency-matrix separatrix or by using generative mechanisms. Threshold graphs provide one mechanism for producing perfectly nested networks.

  • Network generation offers two options: define a separatrix shape in the adjacency or incidence matrix, or use a generative mechanism.
  • Adding links above a chosen separatrix can produce perfectly nested or highly nested networks.
  • Threshold graphs are identified as perfectly nested networks and can be generated through mechanisms described in the review.

A.1.3. Community detection and nestedness at the mesoscopic scale

Community detection provides algorithms for studying mesoscopic network structure. Among modularity-maximization methods, the passage identifies Combo as more effective than other existing methods.

  • General network-analysis packages implement several algorithms for detecting communities.
  • The Combo algorithm is reported as more effective than other existing modularity-maximization methods.
  • The review points readers to an exhaustive list of available community-detection software in a separate review article.

A.2.1. World Trade data

World Trade data come from multiple sources, including the United Nations–curated COMTRADE dataset. Reported import and export values can disagree, requiring sanitation procedures that this review does not discuss in detail.

  • The United Nations Statistics Division curates COMTRADE, which contains bilateral trade-flow declarations spanning several decades.
  • World Trade datasets are available from several sources, including the Observatory of Economic Complexity.
  • Declared yearly import and export volumes may not match for the same exchange, making data-sanitation procedures necessary.
  • Detailed discussion of World Trade data sanitation lies beyond the review’s scope, with further discussions cited elsewhere.

A.2.2. Ecological spatial networks

The section identifies online sources for nestedness-related ecological network data, including the original NTC matrices and repositories for several network types.

  • Data sources: The original Nestedness Temperature Calculator includes 294 presence-absence matrices from the ecology literature.The matrices and their original references are available through the cited null-model data resource.
  • Data sources: The original NTC software can be downloaded from the cited repository link.
  • Data sources: Mutualistic and host-parasite network datasets are available through the Interaction Web DataBase.
  • Data sources: Food-web network datasets are available through Web of Life and the Interaction Web DataBase.

B. Network-based metrics of node importance in bipartite networks

The review compares centrality and ranking metrics for assessing node importance in bipartite networks, alongside methods for nestedness contribution and fitness-complexity ranking. These metrics range from degree-based rankings to path-, eigenvector-, random-walk-, and overlap-based measures.

  • Comparison framework: The review compares bipartite centrality metrics according to their ability to rank nodes by structural importance.
  • Centrality metrics: Degree ranks row- and column-nodes by decreasing numbers of connections, treating highly connected nodes as central.The row- and column-node degrees are denoted k_i and k_α.
  • Centrality metrics: Closeness centrality uses average shortest-path distance from a node to the other nodes, with a normalization factor discussed in the cited reference.
  • Centrality metrics: Betweenness centrality sums the share of shortest paths passing through each row- or column-node.Its normalization in bipartite networks requires separate discussion.
  • Centrality metrics: Eigenvector centrality assigns scores through the leading eigenvector of the adjacency matrix, making each node’s score proportional to its neighbors’ scores.
  • Centrality metrics: Google’s PageRank is defined through the leading eigenvector of an S × S matrix, with α described as a teleportation or damping parameter and Fig. 26 using c = 0.999.
  • Nestedness and ranking methods: Node contributions to nestedness can be estimated using overlap-based metrics such as JDM-NODF, including a contribution formula for individual nodes.
  • Nestedness and ranking methods: The fitness-complexity algorithm ranks row-nodes by decreasing fitness and column-nodes by decreasing complexity, while its reversed variant exchanges these roles.The fitness-complexity and MusRank algorithms are mathematically equivalent; Fig. 26 labels them MUS and MUSrev.
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