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SUPPLEMENTARY INFORMATION
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For “The problem of pattern and scale in ecology: what have we learned in 20 years?”
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Jérôme Chave
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SI1. Measuring the modularity of a network
The search for modules in complex systems has often been equated to that of
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‘communities’, group of actors who interact more than expected by chance (Freemann 1977).
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Although theory has a long history (de Solla Price 1965, Freemann 1977), many methods to
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measure patterns of modularity have been developed in the wake of internet social networks
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in the early 2000s.
Modularity may be given a formal definition, as follows. In any network, biological or
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not, a module is defined as a relatively autonomous subset of components (nodes), such that
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any two components are highly connected, but such that a component within the module is
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loosely connected with any component outside the module. One measure of modularity,
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denoted by Q, measures the excess of links within modules in relative to the number expected
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by chance, summed over all modules (Girvan & Newman 2002). We study a graph composed
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of N nodes which are related by K undirected links. If we assign each node to one of M
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predefined communities, we may define the fraction 𝐸𝑖𝑗 of links connecting nodes in
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𝑀
𝑀
community i to nodes in community j (such that ∑𝑀
𝑖=1 ∑𝑗=1 𝐸𝑖𝑗 = 1). Then, 𝑎𝑖 = ∑𝑗=1 𝐸𝑖𝑗 is
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the fraction of links joining at least one node in community i. If the links are randomly placed
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between nodes, irrespective of the community, then 𝐸𝑖𝑗 = 𝑎𝑖 𝑎𝑗 . Modularity then is 𝑄 =
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∑𝑀
𝑖=1(𝐸𝑖𝑖 − 𝑎𝑖 ).
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The concept of betweenness centrality is another useful to quantify the notion of
modularity (Freeman 1977, Girvan & Newman 2002). It measures how important an edge is
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as a connection route between a priori defined modules. Formally, let 𝜎𝑖𝑗 be the number of
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shortest path between node i and j, with 𝜎𝑖𝑗 ≥ 1. We focus on the node k, and ask how many
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of these shortest paths connecting i and j go through an edge e. Let 𝜎𝑖𝑗𝑒 be the number of these
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paths connecting {𝑖, 𝑗} and going through edge e. The betweenness of edge e, 𝐵𝑒 , is defined as
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the sum, over all pairs {𝑖, 𝑗}, of the ratio 𝜎𝑖𝑗𝑒 /𝜎𝑖𝑗 (Girvan & Newman 2002, Newman & Girvan
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2002). Ma & Zeng (2003) used such an approach to understand the design principles and the
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patterns of organization in large metabolic networks, by unravelling the most central
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metabolites of the networks based on its topology. Salathé & Jones (2010) investigated the
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spread of disease in networks with community structure, and found that community structure
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has a major impact on disease dynamics. Using the ideas of betweenness centrality, they
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found that in networks with strong community structure, immunization interventions targeted
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at individuals bridging communities are more effective than those simply targeting highly
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connected individuals.
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With the rapid development of high-throughput DNA sequencing technologies, it is
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becoming obvious that species need to be defined statistically. In the past, microbiologists
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have used 16S prokaryotic ribosomal DNA sequences to infer the similarity among microbial
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strains, and this has resulted in astounding discoveries, including altogether new lineages
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(Pace 1997). Early such example is the discovery of the cyanobacteria Prochloroccocus
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marinus in the late 1980s, and that of picoplankton Ostreococcus tauri in 1994. Traditionally,
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techniques of species discovery have been predicated on the assumption that 16S DNA
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sequences with 97% similarity or more delimit species. However, the problem of sequence
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clustering is often far more complex than a simple 97% threshold (Meyer & Paulay 2005), as
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the rapid development of DNA-based identification of eukaryotic organisms, the large
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programs of DNA barcoding for a wide range of clades, have offered plenty of opportunities
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to verify (Moritz & Cicero 2004, Knowles & Carsten 2007).
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Instead, techniques for modularity discovery should be implemented, and these are not
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unrelated to the multivariate analyses invented by Fisher (1936) to analyse Anderson’s (1936)
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Iris flower dataset. One intriguing such technique to cluster data into natural groups (modules,
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operational taxonomic units, or others) is that developed based on the physics of diluted
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magnets (Wisemann et al. 1998). Another application has made use of the so-called Markov
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clustering (Enright et al. 2002), used for defining molecular taxonomic units on fungi using
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sequence data (Zinger et al. 2009). With the explosion of environmental genomics
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approaches, there is no doubt that these sequence clustering techniques will become crucial to
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avoid statistical artefacts in the process of diversity discovery and community detection.
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References
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Anderson, E. (1936). The species problem in Iris. Ann. Miss. Bot. Garden, 23, 457-509.
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de Solla Price, D.J. (1965). Networks of scientific papers. Science, 169, 510-515.
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Enright, A. J., van Dongen, S. & Ouzounis, C.A. (2002). An efficient algorithm for large-
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scale detection of protein families. Nucleic Acids Res., 30, 1575-1584.
Fisher, R.A. (1936). The use of multiple measurements in taxonomic problems. Ann.
Eugenics, 7, 179-188.
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Freeman, L.C. (1977). A measure of centrality based on betweenness. Sociometry, 40, 35-41.
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Girvan, M. & Newman, M.E.J. (2002). Community structure in social and biological
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networks. Proc. Natl. Acad. Sci. USA, 99, 7821–7826
Ma, H.-W. & Zeng, A.-P. (2003). The connectivity structure, giant strong component and
centrality of metabolic networks. Bioinformatics, 19, 1423-1430.
Meyer, C.P. & Paulay, G. (2005). DNA barcoding: error rates based on comprehensive
sampling. PLoS Biol., 3, e422.
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Newman, M.E.J. & Girvan, M. (2004). Finding and evaluating community structure in
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Pace, N.R. (1997). A molecular view of microbial diversity and the biosphere. Science, 276,
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Salathé, M. & Jones, J.H. (2010). Dynamics and control of diseases in networks with
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Wiseman, S., Blatt, M. & Domany, E. (1998). Superparamagnetic clustering of data. Phys.
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Zinger, L., Coissac, E., Choler, P. & Geremia, R.A. (2009). Assessment of microbial
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communities by graph partitioning in a study of soil fungi in two alpine meadows. Appl.
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Environm. Microbiol., 75, 5863-5870.
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SI2 Coarsening dynamics in discrete spatial models.
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Durrett & Levin (1994) famously illustrated the importance of being spatial and
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discrete by modelling three classical biological problems using four models: individual-based
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versus mean-field models, and spatial versus non-spatial models. Here I offer a simplified
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version alng the same lines of reasoning with spatial models which both exhibit domain
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growth dynamics. The spatial dynamics of these models is illustrated in Fig. 7 of the main
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text.
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The voter model is the first model. It is usually defined using electors regularly placed
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on a grid and who may take one of two possible opinions (Clifford & Sudbury 1973), but here
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I translate the same model in terms of a species coexistence model. In a square lattice, every
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site is occupied with an individual of two possible species. The total number of individuals is
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N. The dynamics follows how each site changes its species occupancy as a result of local
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interactions. During an infinitesimal time step, one randomly chosen individual dies, and it is
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replaced by the offspring of a randomly chosen neighbour (of four possible neighbours). A
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macroscopic time step consists of N such draws (so that each individual is chosen once on
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average). As time t goes by, clusters occupied by the same species grow in size, and the
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number of neighbouring pairs with different species – henceforth, zones of tension – declines
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proportionally to 1/ln⁡(𝑡) (Fig 7), as may be shown based on the duality of the voter model
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with a system of coalescing random walks. Now assume that, with some probability 𝑚 > 0,
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some vacated sites can be occupied by offspring produced anywhere in the lattice. Adding
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even a small amount of long-distance dispersal, the domain growth depicted in Fig 7 is
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destroyed, and the model no more shows any coarsening.
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A second toy model is the majority model (known as the zero-temperature Ising model
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in physics, Krapivski et al. 2010). It is defined similarly that the voter model, but the local
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update rules are slightly different: all four neighbours send an offspring into the vacated cell,
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and the majority always wins. In case of a tie, the winner is chosen at random. In this model,
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isolated clusters are at a disadvantage and the number of zones of tension declines much faster
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than in the voter model, as 1/√𝑡. Importantly, the domain growth behaviour of this model is
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not altered by the addition of a probability 𝑚 > 0 of long-distance dispersal (up to a point
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𝑚 = 𝑚𝑐 where a critical transition occurs).
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These two models are lattice-based, and they could be approximated by non-spatial
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models such as in the example offered by Durrett & Levin (1994) or following methods
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developed in Bolker & Pacala (1997). In fact, for these particular cases, much is known about
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the macroscopic behaviour of the models (for overviews, see Hinrichsen 2000 & Krapivski et
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al. 2010).
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Bolker, B. & Pacala S.W. (1997). Using moment equations to understand stochastically
driven spatial pattern formation in ecological systems. Theor. Pop. Biol., 52, 179-197.
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Clifford, P. & Sudbury, A.W. (1973). A model for spatial conflict. Biometrika, 60, 581-588.
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Durrett, R. & Levin, S.A. (1994). The importance of being discrete (and spatial). Theor. Pop,
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Biol., 46, 363-394.
Hinrichsen, H. (2000). Non-equilibrium critical phenomena and phase transitions into
absorbing states. Adv. Phy., 49, 815-958
Krapivski, P.R., Redner, S. & Ben-Naim, E. (2010). A Kinetic View of Statistical Physics.
Cambridge University Press, 2010.