### Protected areas

We determined the structural connectivity of the global network of PAs by quantifying intact continuous pathways (areas largely devoid of high anthropogenic pressures) between PAs. Data on PA location and boundary were obtained from the May 2019 World Database of Protected Areas (WDPA)^{70}. We only considered PAs that had a land area of at least 10 km^{2}. As China removed most of its PAs from the public May 2019 WDPA version, we used the April 2018 WDPA for China only, which contained the full set of Chinese PAs at the time. It is important to note that our statistics may differ from those reported by countries and territories due to methodologies and dataset differences used to measure terrestrial area of a country or territory.

### Measure of human pressure

We used the latest global terrestrial human footprint (HFP) maps—a cumulative index of eight variables measuring human pressure on the global environment—to calculate the average human pressure between PAs^{35}. While there are other human pressure maps^{74,75,76}, the HFP is a well-accepted dataset that provided a validation analysis using scored pressures from 3114 × 1 km^{2} random sample plots. The root mean squared error for the 3114 validation plots was 0.125 on the normalized 0–1 scale, indicating an average error of approximately 13%. The Kappa statistic was 0.737, also indicating high concurrence between the HFP and the validation dataset. The HFP 2013 map uses the following variables: (1) the extent of built human environments, (2) population density, (3) electric infrastructure, (4) crop lands, (5) pasture lands, (6) roads, (7) railways, and (8) navigable waterways.

Navigable waterways such as rivers and lakes are included within HFP as they can act as conduits for people to access nature^{35}. In the latest HFP (2013), rivers and lakes are included based on size and visually identified shipping traffic and shore side settlements. Venter et al. treated the great lakes of North America, Lake Nicaragua, Lake Titicaca, Lake Onega, Lake Peipus, Lake Balkash, Lake Issyk Kul, Lake Victoria, Lake Tanganyika and Lake Malawi, as they did navigable marine coasts (i.e. only considered coasts as navigable for 80 km either direction of signs of a human settlement, which were mapped as a night lights signal with a Digital Number (DN) > 6 within 4 km of the coast^{35}). Rivers were included if their depth was >2 m and there were night-time lights (DN > = 6) within 4 km of their banks, or if contiguous with a navigable coast or large inland lake, and then for a distance of 80 km or until stream depth is too shallow for boats. To map rivers and their depth, Venter et al. used the hydrosheds (hydrological data and maps based on shuttle elevation derivatives at multiple scales) dataset on stream discharge, and the following formulae: stream width = 8.1× (discharge[m^{3}/s])0.58; and velocity = 4.0 × (discharge[m^{3}/s])0.6/(width[m]); and cross-sectional area = discharge/velocity; and depth = 1:5× area/width^{35}.

Each human pressure was scaled from 0–10, then weighted within that range according to estimates of their relative levels of human pressure following Sanderson et al.^{77}. The resulting standardized pressures were then summed together to create the HFP maps for all non-Antarctic land areas^{35}.

Within the main manuscript, we defined intact land as any 1 km^{2} pixel with a HFP value not higher than or equal to 4. Within this threshold, all areas with a HFP score higher than 4 are defined as nonintact. While previous analyses showed that a >4 score is a key threshold above which species extinction risk greatly increases^{39}, we recognized that there is no one true threshold, which impacts all species equally. Some species may require no human pressure to successfully disperse, while others might successfully navigate through more intensively modified landscapes. Therefore, we conducted our analyses for two additional HFP thresholds. The first used a HFP score <1 and the second incorporated all areas with a HFP < 10.

### Probability of connectivity

The probability of connectivity network-based metric underlies the analysis performed^{78}, with adaptations to account for structural connectivity provided by intact lands. Probability of Connectivity (PC) is given by the following formula:

$${mathrm{PC}} = frac{{mathop {sum}nolimits_{i = 1}^n {mathop {sum}nolimits_{j = 1}^n {a_ia_jp_{ij}^ ast } } }}{{A_L^2}},$$

(1)

where *n* is the total number of PAs in the study area (i.e. landmass of continent, country or territory), *a*_{i} and *a*_{j} are the total area of PAs *i* and *j*, *p**_{ij} is the maximum product probability between PAs *i* and *j*, and *A*_{L} is the total area of the study area. The maximum product probability (*p**_{ij}) considers both direct connections (movement from *i* to *j* without using any other intermediate PA in the network) and indirect connections (movement from *i* to *j* facilitated by one or several other intermediate PAs acting as stepping stones). The maximum product probability (*p**_{ij}) is calculated through network analysis using the values of the direct dispersal probabilities between nodes (*p*_{ij}). In this analysis, *p*_{ij} = 1 when PA *i* and *j* are connected (edge to edge) by a continuous pathway of intact land and *p*_{ij} = 0 if not. Both probabilities will be equal when the direct movement is the most favorable (probable) pathway between *i* and *j*. *p**_{ij} will be larger than *p*_{ij} when intermediate stepping stones increase the structural connectivity between *i* and *j* beyond what is possible by using only the direct connection between them^{78,79}. Therefore, two PAs may not be directly connected by intact lands (hence having *p*_{ij} = 0), but may be connected through an intermediate stepping-stone PAs, which would give *p**_{ij} = 1.

### Structural connectivity between protected areas

The Probability of Connectivity (PC) metric accounts for both intra-PA (*i* = *j*) and inter-PA area (*i* ≠ *j*) structural connectivity, which is, respectively, given by the intra-PA (PCintra) and inter-PA (PCinter) components of PC are

$${mathrm{PC}} = {mathrm{PCintra}} + {mathrm{PCinter}}{mathrm{.}}$$

(2)

PCintra is calculated using the formula

$${mathrm{PCintra}} = frac{{mathop {sum}nolimits_{i = 1}^n {mathop {sum}nolimits_{j = 1,i = j}^n {a_ia_jp_{ij}^ ast } } }}{{A_L^2}} = frac{{mathop {sum}nolimits_{i = 1}^n {a_i^2} }}{{A_L^2}}.$$

(3)

While PCinter is mathematically defined as

$${mathrm{PCinter}} = frac{{mathop {sum}nolimits_{i = 1}^n {mathop {sum}nolimits_{j = 1,i ne j}^n {a_ia_jp_{ij}^ ast } } }}{{A_L^2}}.$$

(4)

In this analysis, we investigated the connectivity between PAs (i.e. all PAs considered, regardless of how much intact land they contain) that is provided by intact land. For this reason, here the intra-node connectivity is removed and we focus only the inter-node (inter-PA) connectivity (PCinter) for both country/territory and continent level analyses. PCinter is defined as the probability that two points randomly located in two different PAs within the study area (therefore considering only the cases where *i* ≠ *j*) are connected to each other via intact habitat. We calculated PCinter using two scenarios: PCinter_intact and PCinter_all. PCinter_intact is the value when considering that only the intact lands provide structural connectivity between PAs. PCinter_all is the value when any land (all land, intact, or not) provides structural connectivity between PAs, (i.e. considering that two PAs are connected when they are located in the same landmass or island). This analysis provided us with the maximum terrestrial PA structural connectivity that could be theoretically achieved in a country/territory or continent if all of its land was intact. In both scenarios, an 8-neighbouhood rule between land cells was used when defining the continuity of land (using the 1 km^{2} resolution of the HFP layer).

### Structural connectivity provided by intact lands: ConnIntact

We combined PCinter_intact and PCinter_all, as defined above, to obtain ConnIntact, which quantifies the percentage of the PA system that is connected through intact pathways. It is calculated using the following ratio:

$${mathrm{ConnIntact}} = 100frac{{{mathrm{PCinter}}_{mathrm{intact}}}}{{{mathrm{PCinter}}_{mathrm{all}}}}.$$

(5)

Which, given the equation for PCinter above, can be expressed as:

$${mathrm{ConnIntact}} = 100frac{{mathop {sum}nolimits_{i = 1}^n {mathop {sum}nolimits_{j = 1,i ne j}^n {a_ia_jp_{{it{INTACT}}_{ij}}^ ast } } }}{{mathop {sum}nolimits_{i = 1}^n {mathop {sum}nolimits_{j = 1,i ne j}^n {a_ia_jp_{{mathrm{ALL}}_{ij}}^ ast } } }},$$

(6)

where *p*_{INTACT} refers to the maximum product probabilities when only the intact lands provide structural connectivity between PAs, *p*_{ALL} refers to the maximum product probabilities when all land would be intact and hence would provide the highest possible structural connectivity between PAs, *n* is the total number of PAs in the study area (e.g. a country, territory or continent), and *a*_{i} and *a*_{j} are the total area of PAs *i* and *j*. To calculate connectivity, we consider all possible land between two PAs if they are located in the same landmass or island (i.e. if there is a continuous land pathway between the PAs). An 8-neighbourhood rule between land cells is used when defining the continuity of land. This analysis provides insight into how well connected the PAs would be if all land was intact (i.e. the proportional connection based on the maximum terrestrial PA connectivity that could be theoretically achieved in a country, territory, or continent). It is important to note that not all the PAs in a given country, territory, or continent will be connected if they are located in different landmasses or islands, but calculated as an aggregation of the results at the country or territory level. ConnIntact provides the percentage of the PA network that is connected by intact lands. This metric is expressed as a percentage of the total area under protection (see Supplementary Figs. 1–3).

### Theoretical examples

The ConnIntact metric assumes that all PAs have the same area and that *n* is the number of PAs in a hypothetic country. In addition, we define *t* is the proportion of PAs that are located within the intact land. Therefore, *t*·*n* is the number of PAs within intact land.

If all the intact land is located in a single and continuous intact patch (so that all PAs within intact land are connected to each other), then *t***n* (*t***n* − 1) is the number of PA pairs that are connected (both directions) by intact land. The maximum number of PA pairs that would be connected (both directions) if all the land within the study area was intact would be *n*·(*n* − 1). As ConnIntact is expressed as a percentage, the value of ConnIntact in this case is equal to 100·*t*·*n*·(*t*·*n* − 1)/*n*·(*n* − 1) (a particular and simplified case of Eq. (5) in the main text).

In the Supplementary Fig. 1 example, *n* = 20; therefore, the maximum possible number of connections (pairs of PAs connected) is 380. Example 1a illustrates when the entire country is covered by one continuous patch of intact land, all PAs pairs are connected by intact land and ConnIntact is 100% (Supplementary Fig. 1a). This means that the 380 potential connections among PAs are all possible through intact land. Example 1b illustrates when we divide the country in two patches of intact land disconnected by nonintact land (Supplementary Fig. 1b) the protected areas on the right side become disconnected to the ones in the left side. However, inside each patch, PAs are still connected to each other by intact land. We can consider this case as the combination of two sets of PAs, *n*_{1} = 10 and *n*_{2} = 10, within which all PAs are connected. The number of PA pairs connected within each of the two intact land patches is therefore 90, which gives a total of 180 connections for the two intact land patches. This gives a value for ConnIntact of 47.4%, since 180 of the 380 potential connections are facilitated by intact pathways. Example 1c shows only ten of the PAs are located within a single patch of intact land, while the other 10 PAs are found in nonintact land (as in Supplementary Fig. 1c), then there are only 90 pairs of PAs connected via intact land and hence ConnIntact = 23.7%.

Within our analysis, connectivity is calculated from edge to edge and considers both direct and indirect connections. This means that two PAs are considered connected if they have a direct connection (a single patch of intact land connecting them), as is the case of all PAs in Supplementary Fig. 2b and 2c, but also if they have an indirect connection between them, facilitated by intermediate stepping-stone PAs. The latter is the case of the PAs labelled as X and Z in Supplementary Fig. 2a. PA X and PA Z are not directly connected (it is not possible to move from X to Z through a single continuous pathway of intact land). It is, however, possible to move from X to Y (edge to edge) through a continuous intact land patch, and from Y to Z (edge to edge) through another continuous intact land patch. Therefore, X is connected to Z as quantified by the connectivity analyses and ConnIntact metric here considered. In Supplementary Fig. 2, all the PAs are, in each of these three cases, connected by intact lands and have therefore the same value of the connectivity metric here considered (ConnIntact attains the maximum value of 100% in all these three cases).

We note that *PCinter* and ConnIntact can be used to evaluate the percentage of the PA pairs that are connected by intact lands, but cannot be used to state in general which PA network is ‘best’ or ‘best’ designed. Because *PCinter* and ConnIntact only consider inter-PA connections, it would be theoretically possible to have lower *PCinter* and ConnIntact for PA systems that are not more ‘poorly connected’ than others in certain comparisons. This is illustrated in Supplementary Fig. 3a; the ConnIntact metric, which quantifies how many pairs of PAs are connected (under the simplified case, as this one, in which all PAs have the same area), is equal to zero because the two PAs are isolated (not connected by intact lands). In Supplementary Fig. 3b, the ConnIntact metric is close to 50% (47.4%) because many of the pairs of smaller PAs are ‘locally’ (within each of the individual intact land patches) connected to each other. This comparison illustrates that, because the ConnIntact metric does not consider the intra-PA connectivity but focuses only in the inter-PA connectivity, it cannot be used to make a judgement about which of the PA systems is best. While ConnIntact is higher in Supplementary Fig. 3b than in Supplementary Fig. 3a, there are no reasons to think that, in general, Supplementary Fig. 3b can be regarded as a better PA system than Supplementary Fig. 3a.

### Three conditions analysis

We converted the three conditions dataset to a raster, and snapped to the same resolution (1 km^{2}) and projection (Mollweide) as the HFP dataset. We then used the tabulate area tool in ArcGIS 10.6 to calculate the area of each condition per country or territory.

### Sensitivity analysis

We tested the sensitivity of our results to the data obtained from the HFP score <4, using two additional HFP thresholds. With a HFP score <1, globally the proportion of connected PAs results in 8.2% rather than 9.7% for a HFP threshold of <4 (1.5% when considering the absolute difference in the proportion of connected PAs globally). When a HFP threshold of <10 is considered instead of <4, the proportion of the area under protection that is connected increases from 9.7% to 43.5% globally. This result does not alter our conclusions appreciably because the percentage of total land considered intact also varies in a similar fashion: 25% of all terrestrial land under <1 HFP threshold, 41.6% of all terrestrial land under <4 HFP threshold, and 74% of all terrestrial land under <10 HFP threshold.

Certain areas change considerably under different HFP thresholds. For example, the increase in the PA structural connectivity for HFP threshold <10 is particularly noticeable in Oceania (from 15.6 to 94.9%) and Russia (from 1.9 to 79%), and increases to more than 30% of PAs connected in Africa, Americas and Asia. This occurs because for this HFP threshold, most of the roads are no longer considered as a barrier, which can influence the movement of some particular species or group of species^{17}. This suggests that the structural connectivity depends also on how species respond to the permeability of the landscape. Therefore, the HFP threshold of <1 might be better to assess the structural connectivity for sensitive species to human activities such as the boreal woodland caribou, yet for species that can move through more human-modified landscapes, such as the American black bear, the HFP threshold of <10 might be more acceptable.

### Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.