The other measure commonly taken of food web networks (in addition to connectance) is the link or degree distribution. Real-world networks, unlike random graphs, rarely have Poisson or normal link distributions, having instead scale-free or power law distributions. The terms scale-free and power law refer to the fact that these distributions lack characteristic scales (see below), and take the general form , where the probability of a value is a power function of the value itself. Scale-free distributions have been found in networks as diverse as the Internet, transportation networks, anatomical circulatory networks, social networks and food webs. There are two features of these distributions that are of importance to food web theory. First, being scale-free means that the distribution has no characteristic scale. Many distributions have a characteristic scale, often captured by a peak (or high density region) and measured as a mean or mode, for example Poisson or normal distributions. The form of a sample drawn from one of those distributions depends on the range from which it is drawn, whereas the shape of a power law distribution is invariant throughout its range. One part of the distribution may be used to predict another with a simple rescaling of the density (see figure). Therefore, a partial sampling of the range yields an overall view.
Second, power law distributions are long-tailed decay distributions. The decay of the distribution’s density with increasing X, dictated by the negative exponent , means that the distribution’s density is concentrated at low values of X. Nevertheless, the long tail means that there is measurable density high in the X range. Contrast this with the exponential distribution in the first figure, which is also a decay distribution, but with a rapidly decaying short tail. A long-tailed link distribution has nodes that are of considerably greater degree than others. These highly linked or hub nodes confer significant resistance to failure of network connectivity. The canonical example is the Internet. Random failure of any single server is unlikely to affect the network broadly because most servers are of low degree (drawn from the high density region of P(X), and hence of low degree), but there is a high probability that they are linked to high-degree hubs. The network is susceptible, however, to targeted attacks on hubs. This is the now classic work by Barabasi and others. It is not clear how long-tailed distributions arise in networks, but models of preferential growth, where new links have a greater probability of being added to already highly linked nodes (the “rich get richer” model) are reasonable hypotheses when applied to flow networks (for example, information, energy) or social networks (the blogosphere, personal relationships).
Food webs have been characterized most frequently by their in-link distributions, which are the frequency distributions of the number of prey per consumer species (species in-degree). In-link distributions therefore describe patterns of energy flow in the system, as well as the trophic habits or dietary breadths of the species. Most documented food webs have decay in-link distributions and those are either scale-free, power law distributions, or they have properties of exponential decay, or seem to be a mixture of the two types of distributions. Exponentially decaying distributions have greater concentrations of density at low degree. This latter group of distributions are best described as mixed exponential-power law distributions of the form
r is species in-degree, and M is the maximum number of prey species available.
Dunne et al. (2002) examined the link distributions of 16 published food webs, though the survey included both links to (in-links) and from (out-links) predators. They found significant variation among the networks, but distributions belonged mostly either to power law, exponential or uniform distributions. Camacho et al. (2002), in an analysis of six of those same food webs concluded that trophic in-link distributions in fact follow a universal functional form,
where z is the coordination number of the network and is the exponential integral function. The above is also a decay function, significantly related to a scaled number of prey, r/2z. The authors derived a value of z=7.5 from the pooled data of the six networks. The distribution itself was derived analytically from an interpretation of the niche model of Williams and Martinez, which has demonstrated some success in describing empirical trophic link distributions. The data are generally aggregated averages of species population distributions, however, and it is not clear to what extent, if any, the niche model actually predicts any underlying community mechanisms, rather than describing those specific parameterized and averaged representations.
The in-link distribution of the Greater Antillean coral reef raises again the issue of species aggregation. The distribution of the guild-level network, where 750 species are aggregated into 265 guilds on the basis of very precise trophic data, is a distinct power law distribution of the form
(The exponent is of particular importance in community resistance to secondary extinction [Roopnarine et al., 2007]; see below). The high resolution of this dataset allow us, however, to also examine the species-level network, for which the distribution is not a decay distribution, but instead has a distinct mode at 36 links (secondary consumers and higher; second figure). The most precise trophic data are available for the vertebrate species in the network and the vertebrate-only distribution is similar to the overall species-level distribution, though with a mode at 76 links. Clearly the discrepancy between the guild- and species-level distributions is caused by the omission of species richnesses from the aggregated guilds. Thus it remains to be resolved if communities in fact always have greater proportions of trophic specialists, or if this pattern is restricted to aggregated data, and if the pattern occurs naturally at all or is an artifact.