Leading indicators of instability based on different elements of the covariance matrix (S_y), including the maximum (in absolute value) element, <i>Max</i>[S_y], the difference between <i>Max</i>[S_y] and <i>Min</i>[S_y], the element of S_y corresponding to the most connected, least connected, or highest eigenvector centrality (24) network node.

<p>Random (l<i>eft</i>) and scale free (right) (30) network generated with <i>N</i> = 50 and <i>C</i> = 0.1 (main panels) and <i>N</i> = 0.1 and <i>C</i> = 0.5 (insets). Instability (i.e., decrease in <i>Max</i>[<i>Re</i>(λ)]) is attained by increasing the interaction strength <i>p</i> (mean field case). The figures represent average behavior over 100 realizations.</p

Distribution of the correlation, ρ_K, between <i>Max</i>[S_y] and the parameter <i>p</i>, after 1000 realizations for the full disordered (not mean-field) case.

<p>If ρ_K is significant (p-value<0.05) and ρ_K>0.5 the increase in <i>Max</i>[S_y] is interpreted as an early warning sign. We calculate these detection statistics for several realizations of each network structure and determine the probability of detecting the early warning sign of instability. We consider eleven different network architectures typical of ecological or social networks, including random (R), predator-prey (PP), cascade (Casc), compartmentalized (Comp), mutualistic (M), bipartite (Bip), nested (N), nested with competition (N+C), scale free (SF), and small world (SW). These networks have different structures for the adjacency matrix and different combination of interaction types, i.e (++) mutualistic, (+−) antagonistic, (−−) competitive or a combination of them (See <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0101851#pone.0101851.s019" target="_blank">Materials S1</a> for more details).</p

<i>Max</i> [S_y]as a leading indicator of instability in a “mean field” network with constant interaction intensity (in absolute value), <i>p</i>.

<p>Instability is attained by increasing <i>p</i> (main panel A, with <i>N</i> = 20, <i>C</i> = 0.2) or <i>C</i> (inset B, with <i>N</i> = 20, and <i>C</i> increasing from 0.1 to 1) with different network structures. The figures represent average behavior over 1000 realizations.</p

List of top 15 migrant source (sending) countries for 1960, 1980 and 2000 showing the number of people originating from that country (stock) and the percent of the total international migration stock for that census round.

<p>List of top 15 migrant source (sending) countries for 1960, 1980 and 2000 showing the number of people originating from that country (stock) and the percent of the total international migration stock for that census round.</p

Characteristics of human migration network.

<p>(A) Cumulative undirected degree distribution of 1960 (dashed) and 2000 (solid). Plots for all other decades (not shown) progressed from the 1960 line to the 2000 line with time. The number of countries considered remains constant with time and therefore sets an upper limit on <i>k</i>. (inset of A) Log of Source strength as a function of the log of source degree for the 2000 census round. Exponent values remained consistently ∼3 for all census rounds. (B) Total strength <i>s</i> and connectance (i.e. percentage of possible undirected connections) over time. (C) Degree of nearest neighbor, k_nn, as a function of undirected degree for each country in the 2000 census round with moving average line. (D) Network transitivity and average path length over time. (inset of D) Average source strength per source degree (thousands of people per degree) over time.</p

Normalized mutual information between the community structures in different decades.

<p>Normalized mutual information between the community structures in different decades.</p

Community maps and mutual information.

<p>(A–C) The color scale indicates the strength of modularity within a community decreasing from top to bottom. As another symptom of the ongoing globalization, the global modularity of the community structures slightly decreases with time: 0.62 in 1960; 0.61 in 1970 and 1980; 0.60 in 1990; and 0.57 in 2000. Similarly, the ratio between the internal and total fluxes slowly decreases in time: 0.80 in 1960; 0.81 in 1970; 0.76 in 1980; 0.75 in 1990 and 2000. (D) The agreement between migrant communities and communities defined on the basis of religion (•), language (+) or population-based gravity models (x) was evaluated using mutual information as a measure of non-linear correlation <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0053723#pone.0053723-DOdorico1" target="_blank">[29]</a>.</p

Major global migration stocks and net migration of the 2000 census.

<p>Any migrant stock of 600,000 people or more is shown. Units are in millions of migrants. Each country is designated as either a net immigration (blue) or net emigration (tan) country. The centroid of Malaysia is placed in the South China Sea between the two main halves of the country in order to make the connections from Indonesia to Malaysia and from Malaysia to Singapore visible. French Guyana is treated as a territory of France and so reflects the net migration of France.</p

Appendix C. Reliability in identification of increasing trends in variance and autocorrelation.

Reliability in identification of increasing trends in variance and autocorrelation

Appendix A. Examples from past climate transitions.

Examples from past climate transitions