Jacopo GrillipersName, Tim RogerspersName, Stefano AllesinapersName (2016date). “Modularity and Stability in Ecological Communities.” Nature Communicationstitle·j, ChicagoplaceName.
Object of study — Whether modular organisation of speciesterm·key interactionsterm·key makes an ecologicalterm·key communityterm·key more able to recover from small perturbations.
Research question — May suggested in 1972date that communities do better when interactionsterm·key are arranged in blocks; fifty years of simulations disagree with each other - what does the mathematics actually say?
Contribution — An analytic relation between modularityterm·dom Q and stabilityterm·key, showing modularityterm·dom stabilises only in narrow regimes while anti-modularity strongly destabilises.
phenomenon · claim-type · cross-domain-transfer · mechanism · claim-relation · replication-status · limitation — grounded in the full text| phenomenon | modularity and local stability of ecological communities — “Here we study the effect of modularity on the local stability of ecological dynamical systems, by presenting new results in random matrix theory, which are obtained using a quaternionic parameterization of the cavity method.” | |
| claim-type | mechanism — “The qualitative behaviour of these systems can be understood quite simply, when considering the distribution of the eigenvalues of the block-structured matrices in the complex plane.” | |
| method — evidence | — “For each parameter set, we produced 50 block-structured matrices M , and 50 unstructured matrices , obtained by setting Q =0.” | |
| object of study — evidence | — “In ecology, it has been posited that a modular organization of species interactions would benefit the dynamical stability of communities, even though evidence supporting this hypothesis is mixed.” | |
| research question — evidence | — “Though the relationship between block structure and dynamics on networks has since been investigated in many fields, including epidemiology 25 , neuroscience 26 and complex systems in general 27 , a systematic classification is still lacking.” | |
| contribution — evidence | — “Results show that modularity can have moderate stabilizing effects for particular parameter choices, while anti-modularity can greatly destabilize ecological networks.” | |
| transfer | from statistical physics / random matrix theory: random matrix theory and the cavity method applied to the stability of ecological communities — “presenting new results in random matrix theory, which are obtained using a quaternionic parameterization of the cavity method” | transfers[0] |
| transfer | from random matrix theory and statistical physics: the cavity method, extended with a quaternionic parameterisation, to obtain eigenvalue results for structured ecological community matrices — “Here we study the effect of modularity on the local stability of ecological dynamical systems, by presenting new results in random matrix theory, which are obtained using a quaternionic parameterization of the cavity method.” | transfers[1] |
| mechanism | modularity shifts the rightmost eigenvalue of the community interaction matrix, which governs local stability (May eigenvalue criterion) — “When μ ≈0, there are no outliers, and thus stability is determined in all cases by the rightmost eigenvalue” | mechanisms[0] |
| mechanism | structure and strength are separated by construction: the community matrix is an element-wise product of an interaction-strength matrix and an adjacency matrix, so the effect of topology can be isolated — “In summary, the parameterization M = W ○ K allows us to separate the effect on stability of a block structure (modelled by K through α, C and Q ) from those due to the distribution of interaction strengths (modelled by W ).” | mechanisms[1] |
| mechanism | modularity is a single dial: within- and between-subsystem connectance are re-expressed as overall connectance plus Q, so one number moves the structure from perfectly modular to perfectly bipartite — “For simplicity, we speak of a ‘modular' structure whenever C w > C b , and of an ‘anti-modular' structure when C w < C b .” | mechanisms[2] |
| mechanism | stability is read off the rightmost eigenvalue, interpretable as how much self-regulation the system would need to stay at equilibrium — “Re( λ M ,1 ), is a measure of stability, as it expresses the amount of self-regulation we would need to stabilize the equilibrium 20 32 .” | mechanisms[3] |
| mechanism | the unstructured random network is the reference point, so every result is a ratio against the same coefficients rearranged at random — “To calculate , we need to choose a reference model for network structure, and here we use the Erdős–Rényi random graph 34 , as we want to contrast our results with those found for an unstructured, random network.” | mechanisms[4] |
| builds-on | random matrix theory (from physics/mathematics) — “presenting new results in random matrix theory” | claim_relations[0] |
| extends | the hypothesis that modularity benefits ecological stability — “a modular organization of species interactions would benefit the dynamical stability of communities” | claim_relations[1] |
| contradicts | May's hypothesis (ref. 20) that model communities do better when interactions are arranged in blocks, and the subsequent challenges in refs 21-22 — “The picture emerging from these results is much more nuanced and complex than what was previously hypothesized 20 21 22 : modularity can have a moderate stabilizing effect when the two subsystem have about the same size, and the mean μ is negative or destabilizing, when μ ≥0 and the subsystems have different sizes.” | claim_relations[2] |
| builds-on | Allesina et al. (ref. 36), the authors' earlier decomposition of the community matrix — “Following the approach by Allesina et al . 36 , we write the matrix M as the sum of two matrices, M = A + B , where A is a matrix with block structure whose elements are and B is obtained by difference: B = M − A .” | claim_relations[3] |
| builds-on | O'Rourke & Renfrew (ref. 40) on outlier eigenvalues of low-rank-perturbed random matrices — “This problem has been recently studied by O'Rourke & Renfrew 40 , who considered the following case: B is a large, random matrix whose eigenvalues follow the elliptic law.” | claim_relations[4] |
| replicates | Aljadeff et al. (ref. 41) implicit formula for the limiting spectral density — “In this case, the distribution of the eigenvalues is not uniform, and Aljadeff et al . 41 provide an implicit formula for the density of the limiting spectral distribution, which is consistent to that found in the Supplementary Information using a different method.” | claim_relations[5] |
| contradicts | May's hypothesis that block-arranged interactions improve community stability — “This hypothesis was challenged by a number of authors 21 22 , who produced simulations showing the opposite effect.” | claim_relations[6] |
| extends | the random-community stability literature descending from May — “The case of C w = C b represents ‘unstructured' systems, such as those studied by May and other authors 20 33 .” | claim_relations[7] |
| sense | modular - the ecological sense: species grouped by where and when they are active, so interaction frequency tracks group membership — “Ecological communities are structured both in space, as in a stratified lake and in time, as for migratory birds.” | senses[0] |
| sense | modular - the formal sense: a value of Q above what an Erdos-Renyi reference model would give by chance — “Values of Q >0 ( Q <0) mean that we observe within-subsystem interactions more (less) frequently than expected by chance.” | senses[1] |
| sense | stability - local: recovery from a small perturbation around an equilibrium, as against persistence or robustness to species loss — “In ecology, the idea that a modular organization would be beneficial for the local stability of ecological communities (that is, their ability to recover from small perturbations) dates back to work on complexity and stability by May 20 , where he suggested ‘that our model multi-species communities [...] will do better if the interactions tend to be arranged in blocks'.” | senses[2] |
| sense | quaternionic parameterization - the parameterization of the cavity method used for the paper's random-matrix results — “Here we study the effect of modularity on the local stability of ecological dynamical systems, by presenting new results in random matrix theory, which are obtained using a quaternionic parameterization of the cavity method.” | senses[3] |
| sense | cavity method - the method used with quaternions to obtain the new random-matrix-theory results — “Here we study the effect of modularity on the local stability of ecological dynamical systems, by presenting new results in random matrix theory, which are obtained using a quaternionic parameterization of the cavity method.” | senses[4] |
| sense | Hadamard (element-by-element) product - the multiplication of W and K that yields M — “In both cases, the matrix M is obtained via the Hadamard (element-by-element) product of W and the adjacency matrix K .” | senses[5] |
| sense | adjacency matrix - K, combined element by element with W to obtain M — “In both cases, the matrix M is obtained via the Hadamard (element-by-element) product of W and the adjacency matrix K .” | senses[6] |
| sense | spectrum of M - a bulk plus up to two real outliers in the two-subsystem case — “when there are two subsystems the spectrum of M is composed of a ‘bulk' of eigenvalues, and up to two ‘outlier', real eigenvalues.” | senses[7] |
| sense | bulk - the main body of eigenvalues in M's spectrum — “when there are two subsystems the spectrum of M is composed of a ‘bulk' of eigenvalues, and up to two ‘outlier', real eigenvalues.” | senses[8] |
| sense | outlier eigenvalue - one of up to two real outliers outside the spectral bulk — “when there are two subsystems the spectrum of M is composed of a ‘bulk' of eigenvalues, and up to two ‘outlier', real eigenvalues.” | senses[9] |
| sense | cascade model - a size-ordered food web in which species consume only smaller species — “We therefore analysed matrices constructed using a variation of the cascade model, where we assign a ‘size' to each species and each species can only consume smaller species, and has a preference for those in the same subsystem ( Q >0), or for those in the other subsystem ( Q <0).” | senses[10] |
| sense | universality property - distribution choice does not alter the large-system result once moments are fixed and fourth moments bounded — “We do not need to specify the exact form of the distributions and , given that, as for many results in random matrix theory, our findings are consistent with the ‘universality' property 38 39 : once fixed the mean and covariance matrices of and , and provided that the fourth moment of each is bounded, any choice of distributions yields the same result, for S →∞.” | senses[11] |
| sense | elliptic law - B's large-system eigenvalues are approximately uniform in an ellipse — “In this case, the eigenvalues of B follow the ‘elliptic law' 39 , and for S large, are approximately uniformly distributed in an ellipse, centred at (− μ w , 0), and with horizontal semi-axis and vertical semi-axis .” | senses[12] |
| sense | feasible equilibrium point - the point around which M represents population dynamics — “M is the S × S community matrix, representing the population dynamics of an unknown dynamical system around a feasible equilibrium point.” | senses[13] |
| sense | local asymptotic stability - the system property on which block structure can have an effect — “Block structure can have an effect on the local asymptotic stability of the underlying system.” | senses[14] |
| differs in | May's hypothesis (ref. 20) that model communities do better when interactions are arranged in blocks, and the subsequent challenges in refs 21-22 — THE PARAMETER IS SUBSYSTEM SIZE, AND THE CONTRADICTION IS CONDITIONAL ON IT. The span: 'The picture emerging from these results is much more nuanced and complex than what was previously hypothesized: MODULARITY CAN HAVE A moderate stabilizing effect when the two subsystem have about the same size.' A 'contradicts' edge without this reads as a flat denial of May's hypothesis; the paper's own word is 'nuanced'. — “The picture emerging from these results is much more nuanced and complex than what was previously hypothesized 20 21 22 : modularity can have a moderate stabilizing effect when the two subsystem have about the same size, and the mean μ is negative or destabilizing, when μ ≥0 and the subsystems have different sizes.” — “modularity can have a moderate stabilizing effect when the two subsystem have about the same size” | claim_relations[2] |
| edge | bipartite network —is-a→ block structure — “If, on the other hand, interactions occur exclusively between groups, we obtain a bipartite network—another type of block structure with many ecological applications 5 .” | concept_edges[1] |
| edge | bulk —part-of→ spectrum of M — “when there are two subsystems the spectrum of M is composed of a ‘bulk' of eigenvalues, and up to two ‘outlier', real eigenvalues.” | concept_edges[6] |
| audit | outlier eigenvalue —part-of→ spectrum of M — Audit 2026-09-28: DOUBTFUL - read against its full span and kept, but the span writes 'outlier', real eigenvalues; the endpoint is a variant. — “when there are two subsystems the spectrum of M is composed of a ‘bulk' of eigenvalues, and up to two ‘outlier', real eigenvalues.” | concept_edges[7] |
| evidence status note | limitation — QUOTE RECOVERED FROM THE PAPER'S OWN CHARACTERS. The recorded quote was LETTERS-ONLY IDENTICAL to this stretch of the staged text but not an exact substring of it - spaces, hyphens, ligature glyphs or mathematical operators differed, so it could never anchor. This is the file's own text between the same two letters: not produced by a rule about the damage, copied out of the file. Round-trip checked (the recovered span is letters-only equal to the original) and length-checked. The text as originally recorded is kept in evidence as extracted. Update, 15 September 2026: CLEARED evidence status='quote-not-in-current-staged-text' - the quote now locates in the current staged text, so the condition the marker recorded no longer holds. — “our findings are consistent with the ‘universality' property” | limitations[0] |
| for which use | limitation — the unequal-subsystem case: WE CANNOT PREDICT THE RATIO IN FULL GENERALITY FOR THE CASE OF alpha < 1/2, so only the extreme cases are treated — “Though we cannot predict the ratio in full generality for the case of α <1/2, we can treat the extreme cases of a perfectly modular and perfectly bipartite structure, and this is sufficient to understand the qualitative behaviour of all cases (Methods).” | limitations[1] |
| for which use | limitation — the bipartite results: WE DO NOT HAVE A FORMULA FOR CORRECTING THE EIGENVALUES OF BIPARTITE MATRICES — “We do not have a formula for correcting the eigenvalues of bipartite matrices, but, as for the other cases, the correction is negligible when | μ | is large enough.” | limitations[2] |
| for which use | limitation — the transfer to real ecosystems: CLEARLY, ECOLOGICAL SYSTEMS DO NOT FOLLOW A RANDOM GRAPH STRUCTURE ... DISPLAYING A DIRECTIONALITY IN THE FLOW OF ENERGY FROM PRODUCERS TO CONSUMERS — “Clearly, ecological systems do not follow a random graph structure, for example, displaying a directionality in the flow of energy from producers to consumers.” | limitations[3] |
| for which use | limitation — every result computed from the matrix: WE REMOVE SELF-INTERACTIONS ... ADDING INTRA-SPECIFIC EFFECTS WOULD NOT QUALITATIVELY ALTER THE RESULTS - asserted, not shown — “We remove self-interactions from the matrix (setting M ii =0), so that we can concentrate on inter-specific effects (adding intra-specific effects would not qualitatively alter the results 32 ).” | limitations[4] |
| for which use | limitation — the scope of the modularity treated: the community is assumed to be COMPOSED OF TWO SUBSYSTEMS, the simplest case rather than modularity in general — “Here we study the case of a block-structured K : we assume that the community is composed of two subsystems, of sizes αS and (1− α ) S , respectively (with α ≤1/2), and that species in the same subsystem interact with probability C w (within-subsystem connectance), while species in different subsystems with probability C b (between-subsystem connectance).” | limitations[5] |
| for which use | limitation — every modularity value quoted: TO CALCULATE [it] WE NEED TO CHOOSE A REFERENCE MODEL FOR NETWORK STRUCTURE, AND HERE WE USE THE ERDOS-RENYI RANDOM GRAPH - the number belongs to the network and the chosen null model together — “To calculate , we need to choose a reference model for network structure, and here we use the Erdős–Rényi random graph 34 , as we want to contrast our results with those found for an unstructured, random network.” | limitations[6] |
| replication | untested | |
| limitation | results rely on the random-matrix universality assumption — “our findings are consistent with the ‘universality' property” | limitations[0] |
| limitation | The stability ratio cannot be predicted analytically for unequal subsystem sizes; only extreme cases are treated. — “Though we cannot predict the ratio in full generality for the case of α <1/2, we can treat the extreme cases of a perfectly modular and perfectly bipartite structure, and this is sufficient to understand the qualitative behaviour of all cases (Methods).” | limitations[1] |
| limitation | No correction formula exists for the outlier eigenvalues of bipartite matrices. — “We do not have a formula for correcting the eigenvalues of bipartite matrices, but, as for the other cases, the correction is negligible when | μ | is large enough.” | limitations[2] |
| limitation | Real ecological systems are not random graphs, which is why a cascade food-web variant had to be added. — “Clearly, ecological systems do not follow a random graph structure, for example, displaying a directionality in the flow of energy from producers to consumers.” | limitations[3] |
| limitation | Self-interactions are removed from the matrix by construction, with the claim that including them would not qualitatively change the result. — “We remove self-interactions from the matrix (setting M ii =0), so that we can concentrate on inter-specific effects (adding intra-specific effects would not qualitatively alter the results 32 ).” | limitations[4] |
| limitation | The whole analysis is of a two-subsystem block structure, which is the simplest case of modularity rather than the general one. — “Here we study the case of a block-structured K : we assume that the community is composed of two subsystems, of sizes αS and (1− α ) S , respectively (with α ≤1/2), and that species in the same subsystem interact with probability C w (within-subsystem connectance), while species in different subsystems with probability C b (between-subsystem connectance).” | limitations[5] |
| limitation | The measure of modularity depends on a chosen reference model, and the achievable range of Q depends on that choice. — “To calculate , we need to choose a reference model for network structure, and here we use the Erdős–Rényi random graph 34 , as we want to contrast our results with those found for an unstructured, random network.” | limitations[6] |
finding · quantity · variable · definition · key-term — grounded in the full text| finding | Modularity can have moderate stabilizing effects for particular parameter choices, while anti-modularity can greatly destabilize ecological networks. — “Results show that modularity can have moderate stabilizing effects for particular parameter choices, while anti-modularity can greatly destabilize ecological networks.” | content_tags.findings[0] |
| finding | It has been posited that a modular organization of species interactions benefits the stability of communities, though evidence for this hypothesis is mixed. — “it has been posited that a modular organization of species interactions would benefit the dynamical stability of communities, even though evidence supporting this hypothesis is mixed” | content_tags.findings[1] |
| finding | Modularity stabilises only in particular parameter regimes; anti-modularity destabilises strongly. — “Results show that modularity can have moderate stabilizing effects for particular parameter choices, while anti-modularity can greatly destabilize ecological networks.” | content_tags.findings[2] |
| finding | The same level of modularity can help or hurt depending on regime - so no structure is stabilising in itself. — “The rich range of possible effects associated with the same level of modularity stresses the fact that a given network structure is not ‘stabilizing' or ‘destabilizing' per se , but only for particular regimes.” | content_tags.findings[3] |
| finding | The hypothesis was challenged by authors whose simulations showed the opposite effect. — “This hypothesis was challenged by a number of authors 21 22 , who produced simulations showing the opposite effect.” | content_tags.findings[4] |
| finding | Despite decades of work across epidemiology, neuroscience and complex systems, no systematic classification of structure-to-dynamics relations existed. — “Though the relationship between block structure and dynamics on networks has since been investigated in many fields, including epidemiology 25 , neuroscience 26 and complex systems in general 27 , a systematic classification is still lacking.” | content_tags.findings[5] |
| quantity | Q > 0 modular, Q < 0 anti-modular — sign convention of the modularity measure against a chance expectation — “Values of Q >0 ( Q <0) mean that we observe within-subsystem interactions more (less) frequently than expected by chance.” | content_tags.quantities[0] |
| variable | modularity (independent) — “the effect of modularity on the local stability of ecological dynamical systems” | content_tags.variables[0] |
| variable | connectance (independent) — “In summary, the parameterization M = W ○ K allows us to separate the effect on stability of a block structure (modelled by K through α, C and Q ) from those due to the distribution of interaction strengths (modelled by W ).” | content_tags.variables[2] |
| definition | community matrix — “We study the stability of a community matrix M , modelling a continuous-time, dynamical ecological system” | content_tags.definitions[1] |
| evidence status note | definition — QUOTE RECOVERED FROM THE PAPER'S OWN CHARACTERS. The recorded quote was LETTERS-ONLY IDENTICAL to this stretch of the staged text but not an exact substring of it - spaces, hyphens, ligature glyphs or mathematical operators differed, so it could never anchor. This is the file's own text between the same two letters: not produced by a rule about the damage, copied out of the file. Round-trip checked (the recovered span is letters-only equal to the original) and length-checked. The text as originally recorded is kept in evidence as extracted. | content_tags.definitions[1] |
| definition | block structure — “Networks with this property are said to be ‘block-structured': a modular structure is a particular block structure, in which a network is divided into subsystems, and within-subsystem interactions are much more frequent than those between subsystems 3 4 .” | content_tags.definitions[3] |
| definition | modularity — “The appealingly simple idea of a block structure has been formalized in the measure of modularity, Q 3 4 , calculated as the difference between observed and expected within-group interactions, divided by the total number of interactions.” | content_tags.definitions[4] |
| definition | bipartite network — “If, on the other hand, interactions occur exclusively between groups, we obtain a bipartite network—another type of block structure with many ecological applications 5 .” | content_tags.definitions[5] |
| key-terms | modularity, stability, ecological, community, species, network, interactions, eigenvalue, matrix, anti-modular, block structure, bipartite network, community matrix, local stability, connectance, random matrix theory, cavity method, Erdős–Rényi random graph, subsystems |
| concept | modularity — “The appealingly simple idea of a block structure has been formalized in the measure of modularity, Q 3 4 , calculated as the difference between observed and expected within-group interactions, divided by the total number of interactions.” | concepts[1] |
| concept | block structure — not defined in this paper's own words; any partition of a network into groups where interaction frequency depends on group membership - the genus of which modularity and bipartiteness are species — “Networks with this property are said to be ‘block-structured': a modular structure is a particular block structure, in which a network is divided into subsystems, and within-subsystem interactions are much more frequent than those between subsystems 3 4 .” | concepts[4] |
| concept | bipartite network — not defined in this paper's own words; the extreme case where interactions occur only between groups and never within them — “If, on the other hand, interactions occur exclusively between groups, we obtain a bipartite network—another type of block structure with many ecological applications 5 .” | concepts[5] |
| concept | community matrix — not defined in this paper's own words; the matrix of pairwise effects at equilibrium, built as an element-wise product of interaction strengths and an adjacency matrix — “M is obtained by multiplying element-by-element two matrices, a matrix of interaction strengths, W , where W ij expresses the effect of species j on species i around equilibrium, and the adjacency matrix of an undirected graph, K . The community matrix is therefore M = W ○ K ( Fig. 1 ).” | concepts[6] |
| concept | local stability — not defined in this paper's own words; a community's ability to recover from small perturbations, read off the rightmost eigenvalue of the community matrix — “Re( λ M ,1 ), is a measure of stability, as it expresses the amount of self-regulation we would need to stabilize the equilibrium 20 32 .” | concepts[7] |
| concept | Erdős–Rényi random graph — not defined in this paper's own words; the unstructured reference model against which modularity and stability are both measured — “To calculate , we need to choose a reference model for network structure, and here we use the Erdős–Rényi random graph 34 , as we want to contrast our results with those found for an unstructured, random network.” | concepts[8] |
| concept | connectance — not defined in this paper's own words; the density of realised interactions, split here into within-subsystem and between-subsystem components — “Here we study the case of a block-structured K : we assume that the community is composed of two subsystems, of sizes αS and (1− α ) S , respectively (with α ≤1/2), and that species in the same subsystem interact with probability C w (within-subsystem connectance), while species in different subsystems with probability C b (between-subsystem connectance).” | concepts[9] |
| concept | quaternionic parameterization — not defined in this paper's own words; the parameterization used with the cavity method to obtain the paper's random-matrix-theory results — “Here we study the effect of modularity on the local stability of ecological dynamical systems, by presenting new results in random matrix theory, which are obtained using a quaternionic parameterization of the cavity method.” | concepts[10] |
| concept | cavity method — not defined in this paper's own words; the method used in a quaternionic parameterization to derive the paper's random-matrix-theory results — “Here we study the effect of modularity on the local stability of ecological dynamical systems, by presenting new results in random matrix theory, which are obtained using a quaternionic parameterization of the cavity method.” | concepts[11] |
| concept | Hadamard (element-by-element) product — not defined in this paper's own words; the element-by-element product of the interaction-strength and adjacency matrices used to construct the community matrix — “In both cases, the matrix M is obtained via the Hadamard (element-by-element) product of W and the adjacency matrix K .” | concepts[12] |
| concept | adjacency matrix — not defined in this paper's own words; matrix K, combined element by element with W to obtain community matrix M — “In both cases, the matrix M is obtained via the Hadamard (element-by-element) product of W and the adjacency matrix K .” | concepts[13] |
| concept | spectrum of M — not defined in this paper's own words; the spectrum of M, decomposed for two subsystems into a bulk and as many as two real outliers — “when there are two subsystems the spectrum of M is composed of a ‘bulk' of eigenvalues, and up to two ‘outlier', real eigenvalues.” | concepts[14] |
| concept | bulk — not defined in this paper's own words; the bulk component of the community matrix's eigenvalue spectrum — “when there are two subsystems the spectrum of M is composed of a ‘bulk' of eigenvalues, and up to two ‘outlier', real eigenvalues.” | concepts[15] |
| concept | outlier eigenvalue — not defined in this paper's own words; one of up to two real eigenvalues lying outside the spectral bulk in the two-subsystem case — “when there are two subsystems the spectrum of M is composed of a ‘bulk' of eigenvalues, and up to two ‘outlier', real eigenvalues.” | concepts[16] |
| concept | cascade model — not defined in this paper's own words; a size-ordered food-web model in which each species consumes only smaller species, modified here to favor interactions within or between subsystems — “We therefore analysed matrices constructed using a variation of the cascade model, where we assign a ‘size' to each species and each species can only consume smaller species, and has a preference for those in the same subsystem ( Q >0), or for those in the other subsystem ( Q <0).” | concepts[17] |
| concept | universality property — not defined in this paper's own words; the random-matrix property under which fixed mean and covariance matrices plus bounded fourth moments yield the same large-system result across distribution choices — “We do not need to specify the exact form of the distributions and , given that, as for many results in random matrix theory, our findings are consistent with the ‘universality' property 38 39 : once fixed the mean and covariance matrices of and , and provided that the fourth moment of each is bounded, any choice of distributions yields the same result, for S →∞.” | concepts[18] |
| concept | elliptic law — not defined in this paper's own words; the large-S result under which B's eigenvalues are approximately uniform in an ellipse — “In this case, the eigenvalues of B follow the ‘elliptic law' 39 , and for S large, are approximately uniformly distributed in an ellipse, centred at (− μ w , 0), and with horizontal semi-axis and vertical semi-axis .” | concepts[19] |
| concept | feasible equilibrium point — not defined in this paper's own words; the equilibrium around which the community matrix represents the unknown system's population dynamics — “M is the S × S community matrix, representing the population dynamics of an unknown dynamical system around a feasible equilibrium point.” | concepts[20] |
| concept | local asymptotic stability — not defined in this paper's own words; the stability property of the underlying system that block structure can affect — “Block structure can have an effect on the local asymptotic stability of the underlying system.” | concepts[21] |
| random-matrix-theory Re(λ) | — “Re( λ M ,1 ), is a measure of stability, as it expresses the amount of self-regulation we would need to stabilize the equilibrium 20 32 .” | content_tags.statistics[0] |
| correlation ρ | — “Initially, we independently sample the coefficients in W in pairs, drawing ( W ij , W ji ) from a bivariate distribution with identical marginals, defined by the mean , the variance and the correlation .” | content_tags.statistics[1] |
| claim type | null-result — a second reading DIFFERING from the recorded value; recorded: mechanism — “The rich range of possible effects associated with the same level of modularity stresses the fact that a given network structure is not ‘stabilizing' or ‘destabilizing' per se , but only for particular regimes.” | alternatives.claim_type |
| method | mathematical-proof — a second reading DIFFERING from the recorded value; recorded: formal-model — “Here we study the effect of modularity on the local stability of ecological dynamical systems, by presenting new results in random matrix theory, which are obtained using a quaternionic parameterization of the cavity method.” | alternatives.method |
| phenomenon | the relation between modular network structure and the dynamical stability of the system it describes — a second reading DIFFERING from the recorded value; recorded: modularity and local stability of ecological communities — “Networks composed of distinct, densely connected subsystems are called modular.” | alternatives.phenomenon |
Full text — tagged
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Abstract. Networksterm·key composed of distinct, densely connected subsystemsterm·key are called modular. In ecology, it has been posited that a modular organization of speciesterm·key interactionsterm·key would benefit the dynamical stabilityterm·key of communities, even though evidence supporting this hypothesis is mixed. Here we study the effect of modularityterm·dom on the local stabilityterm·dom of ecologicalterm·key dynamical systems, by presenting new results in random matrix theoryterm·mth, which are obtained using a quaternionic parameterizationterm of the cavity methodterm. Results show that modularityterm·dom can have moderate stabilizing effects for particular parameter choices, while anti-modularity can greatly destabilize ecologicalterm·key networksterm·key. Modularityterm·dom in food webs can be caused by spatial and temporal mismatches in interactionsterm·key. Here, Jacopo GrillipersName, Tim RogerspersName and Stefano AllesinapersName show that modularityterm·dom, contrary to expectations, does not generally help stabilizing ecologicalterm·key communities. Ecologicalterm·key communities are structured both in space, as in a stratified lake and in time, as for migratory birds. This temporal and spatial organization is reflected in the strength of speciesterm·key interactionsterm·key, where we expect populations dwelling in the same location, or being active in the same season, to interact more frequently than those that are not. When drawing ecologicalterm·key interaction networksterm·dom—where speciesterm·key are the nodes and the edges connecting the speciesterm·key stand for interactionsterm·key (for example, consumption, pollination and competition)—we therefore expect to find that speciesterm·key can be partitioned into distinct ‘groups', such that the frequency of interactionterm·key largely depends on group-membership 1 2 . Networksterm·key with this property are said to be ‘block-structured': a modular structure is a particular block structureterm·dom, in which a networkterm·key is divided into subsystemsterm·key, and within-subsystem interactionsterm·key are much more frequent than those between subsystemsterm·key 3 4 . If, on the other hand, interactionsterm·key occur exclusively between groups, we obtain a bipartite networkterm·dom—another type of block structureterm·dom with many ecologicalterm·key applications 5 . The appealingly simple idea of a block structureterm·dom has been formalized in the measure of modularityterm·dom, Q 3 4 , calculated as the difference between observed and expected within-group interactionsterm·key, divided by the total number of interactionsterm·key. Positive values indicate that interactionsterm·key occur predominantly within-groups, while negative values that interactionsterm·key are more frequent betweenthan within-group. Modularityterm·dom has become one of the most investigated networkterm·key metrics, with applications spanning biological, social and technological networksterm·key 6 7 8 9 10 11 . Unveiling the relationship between the networkterm·key structure of biological systems and their dynamical properties has been a long-sought goal of the discipline, and many authors have hypothesized that biological networksterm·key must be shaped by evolution 12 13 14 15 and co-evolution 16 , favouring configurations yielding controlled dynamics 17 18 19 . In ecology, the idea that a modular organization would be beneficial for the local stabilityterm·dom of ecologicalterm·key communities (that is, their ability to recover from small perturbations) dates back to work on complexity and stabilityterm·key by May 20 , where he suggested ‘that our model multi-species communities [...] will do better if the interactionsterm·key tend to be arranged in blocks'. This hypothesis was challenged by a number of authors 21 22 , who produced simulations showing the opposite effect. However, recent studies found that modularityterm·dom can indeed enhance speciesterm·key persistence 23 24 , further complicating the picture. Though the relationship between block structureterm·dom and dynamics on networksterm·key has since been investigated in many fields, including epidemiology 25 , neuroscience 26 and complex systems in general 27 , a systematic classification is still lacking. Here we study how the local stabilityterm·dom of ecologicalterm·key systems is influenced by modularityterm·dom, providing new results on the theory of random matrices that can be used to draw a direct relationship between modularityterm·dom Q and local stabilityterm·dom. The mathematical results are briefly stated in the Methods section, while a calculation based on the cavity methodterm 28 29 30 31 with quaternions is carried out in the Supplementary Notes . We show that, with respect to the corresponding unstructured case, modularityterm·dom can have moderate stabilizing effects for particular parameter choices, while anti-modularity can greatly destabilize networksterm·key. The rich range of possible effects associated with the same level of modularityterm·dom stresses the fact that a given networkterm·key structure is not ‘stabilizing' or ‘destabilizing' per se , but only for particular regimes. Results Building communityterm·key matrices We study the stabilityterm·key of a community matrixterm·mth M , modelling a continuous-time, dynamical ecologicalterm·key system composed of S populations, resting at a feasible equilibrium pointterm. We remove self-interactions from the matrixterm·key (setting M ii =0), so that we can concentrate on inter-specific effects (adding intra-specific effects would not qualitatively alter the results 32 ). M is obtained by multiplying element-by-element two matrices, a matrixterm·key of interactionterm·key strengths, W , where W ij expresses the effect of speciesterm·key j on speciesterm·key i around equilibrium, and the adjacency matrixterm of an undirected graph, K . The community matrixterm·mth is therefore M = W ○ K ( Fig. 1 ). Initially, we independently sample the coefficients in W in pairs, drawing ( W ij , W ji ) from a bivariate distribution with identical marginals, defined by the mean , the variance and the correlation . By varying these parameters, we can model different types of interactionsterm·key between the speciesterm·key from preponderantly predator–prey to dominated by competition or mutualism 33 . The binary matrixterm·key K dictates ‘who interacts with whom', and is symmetric because we assume pairwise interactionsterm·key. Thus, K determines which interactionsterm·key in W are activated, and which are suppressed. Here we study the case of a block-structured K : we assume that the communityterm·key is composed of two subsystemsterm·key, of sizes αS and (1− α ) S , respectively (with α ≤1/2), and that speciesterm·key in the same subsystemterm·key interact with probability C w (within-subsystem connectanceterm·mth), while speciesterm·key in different subsystemsterm·key with probability C b (between-subsystem connectanceterm·mth). This parameterization is especially intuitive, because for C b = C w , we recover the well-studied case of a random ecologicalterm·key communityterm·key 20 33 . Hence, by varying C b and C w , we can isolate the effect of having a modular or anti-modularterm·key structure ( Fig. 2 ), with the two extreme cases being networksterm·key composed of two separate subsystemsterm·key (perfectly modular), and those in which interactionsterm·key occur exclusively between subsystemsterm·key (perfectly anti-modularterm·key or bipartite). For simplicity, we speak of a ‘modular' structure whenever C w > C b , and of an ‘anti-modularterm·key' structure when C w < C b . The case of C w = C b represents ‘unstructured' systems, such as those studied by May and other authors 20 33 . Equivalently (Methods), we can express C w and C b in terms of the overall connectanceterm·mth C (that is, the overall density of interactionsterm·key in K ) and Q , the modularityterm·dom of the networkterm·key 3 4 , defined as: where L w is the observed number of interactionsterm·key within the subsystemsterm·key, L b the number of between-subsystem interactionsterm·key and is the number of within-subsystem interactionsterm·key we would expect by chance. Values of Q >0 ( Q <0) mean that we observe within-subsystem interactionsterm·key more (less) frequently than expected by chance. To calculate , we need to choose a reference model for networkterm·key structure, and here we use the Erdős–Rényi random graphterm·mth 34 , as we want to contrast our results with those found for an unstructured, random networkterm·key. Though Q is bounded by −1 and 1, only a smaller interval of values might be achievable for a given choice of a reference model, α and C (Methods; for example, in our case, when α =1/2, then Q ∈[−1/2, 1/2], provided that C <1/2). In summary, the parameterization M = W ○ K allows us to separate the effect on stabilityterm·key of a block structureterm·dom (modelled by K through α, C and Q ) from those due to the distribution of interactionterm·key strengths (modelled by W ). Given that the case of Q =0 (unstructured networkterm·key) has been studied intensively, and that the calculation of the stabilityterm·key of these matrices can be achieved analytically 20 32 33 35 , we use it as a reference point to determine the effect of Q on stabilityterm·key. Effect of modularityterm·dom on stabilityterm·key We want to study the effect of modularityterm·dom on stabilityterm·key. Therefore, we contrast the value for the real part of the ‘rightmost' eigenvalueterm·key of M , Re( λ M ,1 ), with , the value found for , a matrixterm·key with exactly the same coefficients, but re-arranged according to a random networkterm·key structure ( Q =0). Re( λ M ,1 ), is a measure of stabilityterm·key, as it expresses the amount of self-regulation we would need to stabilize the equilibrium 20 32 . Our analysis ( Fig. 3 ; Methods) highlights that there are three main parameterizations we need to consider: (a) mean interactionterm·key strength close to zero ( μ ≈0); (b) strongly negative mean interactionterm·key strength; and (c) strongly positive mean interactionterm·key strength. Without loss of generality, we can set σ 2 =1, and study the effect of the modularityterm·dom Q on the stabilityterm·key of the communityterm·key, measured as the ratio , for a given choice of α (controlling the size of the smaller subsystemterm·key), ρ (correlation of interactionterm·key strengths) and C (overall connectanceterm·mth of the system). Values Γ<1 are found when imposing the block structureterm·dom helps stabilizing the communityterm·key (for example, in Fig. 3 , for Q <0 and μ =0), while ratios Γ >1 stand for destabilizing effects (for example, any Q ≠0 for positive mean). For an unstructured matrixterm·key ( Q =0), the ratio is exactly 1. In Fig. 4 , we show the effect of modularityterm·dom on stabilityterm·key in a communityterm·key composed of 1,000 speciesterm·key, when we set C =0.2. Take the case of two equally sized subsystemsterm·key ( α =1/2), for which we derive new results allowing us to express the ratio analytically (Methods; Supplementary Information ): when μ ≥0, we have no effect of modularityterm·dom on stabilityterm·key; when μ <0, on the other hand, a bipartite structure is highly destabilizing, while a modular structure is moderately stabilizing. Both effects are stronger in the case of a negative correlation. When the two subsystemsterm·key have different sizes ( α <1/2), the stabilizing effect of modularityterm·dom found for the case of μ <0 is greatly diminished, and eventually also modularityterm·dom can become destabilizing (especially for positive ρ and ). For μ >0, any Q ≠0 is destabilizing, while for μ ≈0, a modular structure is always destabilizing, and a anti-modularterm·key structure can be stabilizing, provided that ρ is sufficiently negative. These effects hold qualitatively for different levels of C (Methods; Supplementary Figs 2–3 ), with higher connectancesterm·mth leading to more marked effects. Though we cannot predict the ratio in full generality for the case of α <1/2, we can treat the extreme cases of a perfectly modular and perfectly bipartite structure, and this is sufficient to understand the qualitative behaviour of all cases (Methods). The picture emerging from these results is much more nuanced and complex than what was previously hypothesized 20 21 22 : modularityterm·dom can have a moderate stabilizing effect when the two subsystemterm·key have about the same size, and the mean μ is negative or destabilizing, when μ ≥0 and the subsystemsterm·key have different sizes. Similarly, anti-modularity is highly destabilizing for μ ≠0, but can be stabilizing for μ =0. The qualitative behaviour of these systems can be understood quite simply, when considering the distribution of the eigenvaluesterm·key of the block-structured matrices in the complex plane. As shown in Fig. 5 , when there are two subsystemsterm·key the spectrum of Mterm is composed of a ‘bulkterm' of eigenvaluesterm·key, and up to two ‘outlier', real eigenvaluesterm·key. When μ ≈0, there are no outliers, and thus stabilityterm·key is determined in all cases by the rightmost eigenvalueterm·key(s) in the bulkterm. When μ ≠0, we have only one outlier in the case of unstructured networksterm·key: if μ <0, then the outlier lies to the left of the bulkterm and thus has limited effects on stabilityterm·key; if μ >0, on the other hand, the outlier lies to the right of the bulkterm and therefore solely determines stabilityterm·key. The modular case is similar to the unstructured one, though we now have two outilers, in that both lie either to the right ( μ >0) or the left ( μ <0) of the bulkterm. In the bipartite case, however, for any μ ≠0 the spectrum presents an outlier to the right (determining stabilityterm·key) and one to the left of the bulkterm. These simple observations are sufficient to understand the very strong destabilizing effect of a bipartite structure when μ <0: in this case, the stabilityterm·key of the unstructured networkterm·key, , is determined by the bulkterm of the eigenvaluesterm·key, while that of the block-structured networkterm·key, Re( λ M ,1 ), by the outlier to the right of the bulkterm ( Fig. 5 , red). When both Re( λ M ,1 ) and are determined by the bulkterm (for example, modular case with μ <0 or any structure with μ ≈0), the either stabilizing or destabilizing effect is going to be moderate. Moderate effects are also observed when both Re( λ M ,1 ) and are associated with an outlier lying to the right of the bulkterm ( μ >0). When both Re( λ M ,1 ) and are determined by the same type (bulkterm, outlier) of eigenvalueterm·key, the precise stabilizing or destabilizing effect depends nonlinearly on the parameters α , C , Q , μ , σ and ρ (Methods). To summarize, a block structureterm·dom for an otherwise random ecologicalterm·key system can help stabilization in only two cases: (a) when the structure is modular, and μ <0 (though small α or a positive ρ could reverse this effect); and (b) when the structure is bipartite, μ ≈0, and the correlation is negative. For all the other cases, the effect of a block structureterm·dom ranges from neutral to highly destabilizing. Food-web structure Clearly, ecologicalterm·key systems do not follow a random graph structure, for example, displaying a directionality in the flow of energy from producers to consumers. This directionality proved important in our previous study 36 , where we showed that when the mean of the negative effects dominates that of the positive effects, systems built according to the cascade 37 model (in which ‘larger' speciesterm·key consume ‘smaller' ones) are more likely to be stable than their random counterparts. We therefore analysed matrices constructed using a variation of the cascade modelterm, where we assign a ‘size' to each speciesterm·key and each speciesterm·key can only consume smaller speciesterm·key, and has a preference for those in the same subsystemterm·key ( Q >0), or for those in the other subsystemterm·key ( Q <0). Note that in this case, we need to set a mean for the positive effects and another one for the negative effects (Methods). Figure 6 shows that the stabilizing effect of modularityterm·dom found before for the case of μ <0 is practically negligible, while the other results are qualitatively the same. Discussion We have studied the effect of a modular or anti-modularterm·key networkterm·key structure on the stabilityterm·key of an otherwise random ecologicalterm·key system. Our parameterization makes it easy to compare the effect of the networkterm·key structure with that one would obtain for unstructured, random systems such as those studied in the past 20 33 : a ratio Γ<1 stand for stabilizing and Γ>1 for destabilizing effect of networkterm·key structure. For block-structured matrices, we showed that modularityterm·dom can have a positive effect on stabilityterm·key only when (a) the system is composed of two subsystemsterm·key of about the same size and (b) the overall mean interactionterm·key strength is negative. The stabilizing effect is stronger for negative correlations. Anti-modularity, on the other hand, is typically strongly destabilizing, besides the case in which the average interactionterm·key strength is close to 0 (a well-studied case 20 33 , though of limited biological realism). When the mean interactionterm·key strength is positive, both modularityterm·dom and anti-modularity are destabilizing. Through numerical simulations, we have investigated the more complex case of an interactionterm·key between modularityterm·dom and food-web structure, and found that the results are qualitatively unchanged. The picture emerging from both simulations and mathematical analysis is much more complex than previously hypothesized. Block structureterm·dom can have an effect on the local asymptotic stabilityterm of the underlying system. However, unless we are in particular areas of the parameter space, the effect tends to be destabilizing. Our results stress the fact that, when discussing the relationship between networkterm·key structure and local stabilityterm·dom, we need to qualify our statements, as a given structure is not stabilizing or destabilizing per se , but is only so under certain specific conditions. Though we have illustrated this point by studying the modular structures, we believe this phenomenon to hold generally: any networkterm·key structure can have different effects on stabilityterm·key, depending on the choice of parameters. To reinforce this message, in Fig. 7 , we show three cases in which an empirical networkterm·key structure makes the system more or less stable than its random counterpart, depending on the parameterization of the coefficients. Practically, this means that the challenge of proving that biological networkterm·key structure emerges because of a selective process, removing configurations yielding unfavourable dynamics 17 18 19 is much harder than expected: networkterm·key structure, without estimates of the distribution of the coefficients, cannot be used to determine the effect on dynamical properties. Methods Constructing the community matrixterm·mth M is the S × S community matrixterm·mth, representing the population dynamics of an unknown dynamical system around a feasible equilibrium pointterm. We consider two cases: (a) random ecologicalterm·key networksterm·key with block structureterm·dom and (b) food webs with block structureterm·dom. For case (a), we sampled the pairs ( W ij , W ji ) independently from a bivariate normal distribution with means ( μ , μ ) T , and covariance matrixterm·key For case (b), we first assigned a ‘size' to each speciesterm·key (randomly sampling it from a uniform distribution between 0 and 1), and then sampled the pairs ( W ij , W ji ) independently from a normal bivariate distribution with means ((1+ ξ ) μ , (1− ξ ) μ ) T (ensuring that , for Fig. 6 , ξ =3), and covariance matrixterm·key Σ, whenever i was larger than j . This means that speciesterm·key can only consume ‘smaller' prey, such as in the cascade modelterm. In particular, in this case we could order the rows and columns of matrixterm·key W so that all the positive effects would be confined to the upper-triangular part and the negative effects to the lower-triangular part. In both cases, the matrixterm·key M is obtained via the Hadamard (element-by-element) productterm of W and the adjacency matrixterm K . The matrixterm·key K is characterized by four parameters: S , α , C and Q . S is the size, α is the proportion of speciesterm·key belonging to the first subsystemterm·key, Q is the modularityterm·dom and C the overall connectanceterm·mth of K (density of the nonzero elements). The first αS speciesterm·key are assigned to the first subsystemterm·key, and the remaining (1− α ) S speciesterm·key to the second subsystemterm·key. The vector encodes the subsystem-membership of each speciesterm·key. Then, we set ( K ij , K ji ) to (1, 1) with probability C w , when γ i = γ j and with probability C b when γ i ≠ γ j . The ‘within-subsystem connectanceterm·mth' C w is: and the ‘between-subsystem connectanceterm·mth' C b is: Note that, given the Erdős–Rényi reference model, the values of Q that are attainable depend on both α and C : Numerical simulations For each α and ρ , we set S =1,000, C =0.2, σ =1 and μ =0 (green), μ =−1 (red) or μ =1 (blue), and varied Q from its minimum to its maximum possible value in twenty equally sized steps. For each parameter set, we produced 50 block-structured matrices M , and 50 unstructured matrices , obtained by setting Q =0. The ratio was computed by averaging over the replicates. In many cases, one can obtain the expectation for the ratio analytically (below; Supplementary Information ). The simulations were repeated for both random ecologicalterm·key networksterm·key ( Fig. 4 ) and cascade-based food webs ( Fig. 6 ). The spectrum of block-structured matrices For our derivations, we adopt a slightly more general notation, which includes that discussed above as a special case. We consider the matrixterm·key M , with M ii =0 and the off-diagonal coefficients independently sampled in pairs from either of two distributions: Hence, the pairs come from a certain bivariate distribution , when i and j belong to the same subsystemterm·key, while from a different distribution , when i and j belong to different subsystemsterm·key. We do not need to specify the exact form of the distributions and , given that, as for many results in random matrix theoryterm·mth, our findings are consistent with the ‘universality' property 38 39 : once fixed the mean and covariance matrices of and , and provided that the fourth moment of each is bounded, any choice of distributions yields the same result, for S →∞. When considering the case examined in the main text, where the elements M ij = W ij K ij , the universality propertyterm helps us in two ways. First, consider that the pairs ( M ij , M ji ) are zero with probability 1− C w (case γ i = γ j ) or probability 1− C b (case γ i ≠ γ j ), and that the nonzero pairs are sampled from a bivariate distribution (for the elements of W ), defined by the parameters μ , σ 2 and ρ . This is sufficient to calculate 32 the relevant parameters for the distributions and : This means that the effect of the connectancesterm·mth is somewhat trivial: we can ‘absorb' the connectancesterm·mth in the parameters μ w , μ b , σ w , σ b , ρ w and ρ b , which are our ‘effective' parameters, dictating the shape of the distribution of the eigenvaluesterm·key of M . The second advantage of having universal results is that we are free to choose any distribution for the pairs ( W ij , W ji ). In all our examples, these are sampled from a bivariate normal distribution. Decomposition We want to study the limiting (that is, when S is large) distribution of the eigenvaluesterm·key of M for the case of a random communityterm·key (for the food webs following the cascade modelterm, we rely exclusively on simulations). Following the approach by AllesinapersName et al . 36 , we write the matrixterm·key M as the sum of two matrices, M = A + B , where A is a matrixterm·key with block structureterm·dom whose elements are and B is obtained by difference: B = M − A . Thus, the diagonal elements of B ii =− μ w , while for the off-diagonal , and (when γ i = γ j ), or (when γ i ≠ γ j ). This parameterization is very convenient, as the spectrum of matrixterm·key B describes the bulkterm of the eigenvaluesterm·key of M , while the outlier eigenvaluesterm of M are given by the nonzero eigenvaluesterm·key of A , modified by a small correction 40 . The eigenvaluesterm·key of A The eigenvaluesterm·key of A are easy to obtain for any choice of S , α , μ w and μ b , with all being zero besides which can be both zero as well ( μ w = μ b =0), both different from zero ( α ≠1/2, μ w ≠ μ b ), or one zero and one nonzero ( μ w = μ b ≠0). Thus, there are going to be up to two outlier eigenvaluesterm. These are the approximate locations of the two outlier eigenvaluesterm of M (only one outlier when μ b = μ w ≠0, as found for example in the ‘unstructured' case). The exact location of the outliers depends also on B , as explained below. The eigenvaluesterm·key of B The spectrum of B has never been studied in full generality. We start by discussing the known cases, and then introduce new results that allows us to understand the qualitative behaviour of our simulations. These results can be derived by a calculation making use of the cavity methodterm ( Supplementary Information ). Known case: σ w = σ b , ρ w = ρ b In this case, the eigenvaluesterm·key of B follow the ‘elliptic lawterm' 39 , and for S large, are approximately uniformly distributed in an ellipse, centred at (− μ w , 0), and with horizontal semi-axis and vertical semi-axis . Known case: σ b =0 (perfectly modular) When there are no connections between-subsystem, we have two independent subsystemsterm·key. Hence, the eigenvaluesterm·key of B are simply the union of the eigenvaluesterm·key of the two squared block matrices found on the diagonal. The eigenvaluesterm·key of each diagonal block follow the elliptic lawterm, so that the distribution of the eigenvaluesterm·key of B is a combination of two uniform ellipses, both centred at (− μ w , 0), and with horizontal semi-axes and , and vertical semi-axes and , respectively. Known case: ρ b = ρ w =0 New results 41 can be applied to this case, showing that the eigenvaluesterm·key of B are contained in a circle in the complex plane, with centre (− μ w , 0) and radius In this case, the distribution of the eigenvaluesterm·key is not uniform, and Aljadeff et al . 41 provide an implicit formula for the density of the limiting spectral distribution, which is consistent to that found in the Supplementary Information using a different method. New case: α =1/2 (equally sized subsystemsterm·key) When the two subsystemsterm·key have the same size ( α =1/2), we find ( Supplementary Information ) that the eigenvaluesterm·key of B are approximately uniformly distributed in the ellipse in the complex plane with centre in (− μ w , 0), horizontal semi-axis and vertical semi-axis Note that this would also be the limiting distribution for the eigenvaluesterm·key of the unstructured matrixterm·key with − μ w on the diagonal, and off-diagonal elements sampled independently in pairs from the bivariate normal distribution with means (0, 0) T , a correlation that is a weighted average of the correlations in B , and a variance that is the arithmetic mean of the variances in B , In this case, it is convenient to express these values in terms of μ , σ 2 , ρ , C and Q , because this makes the role of modularityterm·dom in modulating the stabilityterm·key much clearer: From the two equations above, it is clear that (a) for μ =0, modularityterm·dom has no effect on the spectrum, while for μ ≠0 the sign of μ does not affect the spectrum; (b) modular and bipartite structures have the same effect: the eigenvaluesterm·key of B will be approximately the same, when we have Q = q or Q =− q ; (c) the effect of modularityterm·dom is going to be more marked for large C or | μ |; and (d) the radius of B is always lower or equal than that we would find by setting Q =0. Summarizing, for α =1/2, the eigenvaluesterm·key of B are contained in an ellipse whose horizontal semi-axis is always smaller or equal than that found for the corresponding unstructured matrixterm·key. This explains the stabilizing effect of a modular structure ( Q <0) we observed for μ <0, as in that case the rightmost eigenvalueterm·key of M is the rightmost eigenvalueterm·key of B . New case: σ w =0 (perfectly bipartite) When σ w =0 (that is, C w =0), the nonzero coefficients of B are exclusively contained in the two blocks, representing the interactionsterm·key between subsystemsterm·key, as in a bipartite networkterm·dom. Hence, we can write the matrixterm·key B in block form: where X is a αS × (1− α ) S matrixterm·key and Y is a (1− α ) S × αS matrixterm·key. The two matrices on the diagonal contain all zeros, and have size αS × αS and (1− α ) S × (1− α ) S , respectively. The eigenvaluesterm·key of B 2 , are the eigenvaluesterm·key of B , squared: if λ i is an eigenvalueterm·key of B , then is an eigenvalueterm·key of B 2 . Squaring B , we obtain: XY has αS eigenvaluesterm·key, while YX has (1− α ) S eigenvaluesterm·key. The eigenvaluesterm·key of YX are the same as those of XY , with the exception of (1− α ) S − αS eigenvaluesterm·key which are exactly 0 (take v to be an eigenvector of XY . Then XYv=λv . Let w = Yv , hence, XYv = Xw = λv . Finally, consider YXw = Y ( λv )= λYv=λw . Thus, if λ ≠0, it is an eigenvalueterm·key of both XY and YX ). Hence, we can study the eigenvaluesterm·key of XY (the smaller matrixterm·key) without loss of generality. In the Supplementary Information , we show that the eigenvaluesterm·key of XY are contained in an ellipse in the complex plane, with centre in , horizontal semi-axis and vertical semi-axis ( Supplementary Fig. 7 , top). If the support of the distribution of the eigenvaluesterm·key of XY is an ellipse in the complex plane, then the support of the eigenvaluesterm·key of B for perfectly bipartite interactionterm·key matrices is obtained via a square-root transformation of the ellipse in the complex plane ( Supplementary Fig. 7 , bottom), with the addition of the point (0, 0), stemming from the extra eigenvaluesterm·key of YX . To find the real part of the rightmost eigenvalueterm·key of B , we thus need to consider the square-root transformation of the ellipse found for XY . The eigenvaluesterm·key of XY , which are the squared eigenvaluesterm·key of B , are contained in the ellipse: where x is the real part and y the imaginary part of the point z = x + iy (we can consider the case y >0 without loss of generality, given that the spectrum of B is symmetric about the real and the imaginary axis). Then, the square root of z has real part a : To approximate the maximum real part for the eigenvaluesterm·key of B , we need to find the x that maximizes a . First, we can rewrite the equation for a , exploiting the fact that we know that all the points z we want to consider are on the curve describing the ellipse: where we have substituted the value of y 2 by constraining the point z to be on the curve describing the ellipse. Substituting the values for x c , r x and r y , we can write: Where we maximize the function for a by taking values x in [ x c , x c + r x ], which is sufficient because of the symmetry discussed above. Maximizing, we find two cases: Combining the eigenvaluesterm·key of A and B Having derived the position of the eigenvaluesterm·key of A , and, for particular cases, the support of the distribution of those of B , we want to combine the results to obtain an approximation for Re( λ M ,1 ), the real part of the rightmost eigenvalueterm·key of M = A + B . This problem has been recently studied by O'Rourke & Renfrew 40 , who considered the following case: B is a large, random matrixterm·key whose eigenvaluesterm·key follow the elliptic lawterm. It is defined by its size, S and the distribution of the coefficients, which are independently sampled in pairs from a bivariate distribution with mean zero, unitary variance and correlation ρ . A is a matrixterm·key with low rank (that is, few nonzero eigenvaluesterm·key), and nonzero eigenvaluesterm·key that are sufficiently larger than those of B . Then (Theorem 2.4), we can order the eigenvaluesterm·key of such that: where the term o (1) goes to zero as S →∞. This means (ref. 40 ; Theorem 2.8) that a random matrixterm·key with a nonzero mean μ will have a single outlier located approximately at μS , exactly as found for the unstructured case 32 33 studied above. Clearly, the correction above is well suited for the unstructured case, and for the perfectly modular one (which is the combination of two unstructured cases). We also corrected in the same way the eigenvaluesterm·key for matrices with α =1/2, reasoning that the correction would have the same form, given that the spectra of these matrices converge to those of equivalent unstructured cases. We do not have a formula for correcting the eigenvaluesterm·key of bipartite matrices, but, as for the other cases, the correction is negligible when | μ | is large enough. Supplementary Fig. 8 shows that our approximation is indeed excellent for all the cases considered here. Simulating empirical networkterm·key structures We parameterized three empirical networksterm·key, and studied the effect of networkterm·key structure by measuring the ratio , when varying a critical parameter θ . For simplicity, we always consider the case of matrices with zero on the diagonal. Contact networkterm·key We took a symmetric adjacency matrixterm, A , specifying whether two members of an high school were in contact (see ref. 42 ), and built the matrixterm·key M by sampling the coefficients M ij = M ji from a normal distribution with mean θ and variance 0.0025, whenever A ij = A ji =1. We sampled θ independently from a uniform distribution and for each M , we obtained by shuffling the interactionsterm·key while maintaining the pairs (so that both matrices are symmetric), following ref. 35 . In Fig. 7 , we show 250 realizations. Food web We took the adjacency matrixterm A , specifying trophic interactionsterm·key in the Little Rock lake 43 , and built M by sampling M ij from the half-normal distribution , whenever A ij =1. These are the (negative) effects of predators on prey. For each M ij <0, we chose M ij by multiplying − M ij by a random value drawn from . Thus, for θ ≈1 the positive coefficients have about the same strength as the negative ones; when θ >1 positive effects dominate; and for θ <1 negative effects are stronger. Again, for each of the realizations, we built both M and , obtained by shuffling the interactionsterm·key. In Fig. 7 , we show results obtained by sampling θ 250 times from the distribution . Pollinator networkterm·key We took the pollination networkterm·key compiled by Robertson 44 45 , and we used the rectangular adjacency matrixterm to determine the position of the nonzero, mutualistic effects between plants and pollinators: M ij and M ji were sampled independently from the uniform distribution , whenever plant i and pollinator j interacted. We then sampled competitive effects by sampling uniformly the coefficients for k and l being both plants or both pollinators. Thus, M has a block structureterm·dom, very similar to that studied here. In Fig. 7 , we show the effects of the strength of competition ( θ ) on stabilityterm·key. Data availability The data and code needed to reproduce all results presented in the article can be downloaded from https://github.com/StefanoAllesina/blockstructure Additional information How to cite this article: GrillipersName, J. et al . Modularityterm·dom and stabilityterm·key in ecologicalterm·key communities. Nat. Commun. 7:12031 doi: 10.1038/ncomms12031 (2016). Supplementary Material Supplementary Information Supplementary Figures 1-8, Supplementary Notes and Supplementary