# Null models

EcologicalNetwork offers a number of ways to draw random binary networks from a template of probabilities. This is useful to generate networks under a null model, for example. All these functions will respect the fact that the network in bipartite or unipartite.

## Creating a deterministic network from a probabilistic network

There are a number of ways to generate a deterministic network from a probabilistic one. All of these functions take a network on a class belonging to ProbabilisticNetwork, and return a network of a class belonging to DeterministicNetwork.

### Convert to deterministic

The first is simply to assing true to all interactions with a non-0 probability, and false to the others. This is done with the make_binary function:

N = UnipartiteProbaNetwork(eye(3))
B = make_binary(N)
B.A

3×3 Array{Bool,2}:
true  false  false
false   true  false
false  false   true


# EcologicalNetwork.make_binaryFunction.

Returns a matrix B of the same size as A, in which each element B(i,j) is 1 if A(i,j) is greater than 0.

### Using a threshold

The second way is to determine a cutoff for probabilities, below which they will be assigned false. This is done through make_threshold:

N = UnipartiteProbaNetwork(rand((4, 4)))
B = make_threshold(N, 0.5)
B.A

4×4 Array{Bool,2}:
false  false  false  false
true  false   true   true
false  false  false  false
false  false   true   true


# EcologicalNetwork.make_thresholdFunction.

Returns a matrix B of the same size as A, in which each element B(i,j) is 1 if A(i,j) is > k. This is probably unwise to use this function since this practice is of questionnable relevance, but it is included for the sake of exhaustivity.

k must be in [0;1[.

### Random draws

The last way to convert a probabilistic network to a deterministic one is to perform one random draw for each interaction. In this scenario, true is assigned with a probability $P_{ij}$. This is done with the make_bernoulli function:

N = BipartiteProbaNetwork(rand((4, 4)))
B = make_bernoulli(N)
B.A

4×4 Array{Bool,2}:
false  false  false  false
false   true   true  false
false   true   true   true
false   true  false   true


# EcologicalNetwork.make_bernoulliFunction.

Returns a matrix B of the same size as A, in which each element B(i,j) is 1 with probability A(i,j).

## Creating a probabilistic network from a deterministic network

The inverse operation can be done using the nullX functions. These functions use informations about the degree distribution to generate probabilistic networks:

# EcologicalNetwork.null1Function.

Type I null model

null1(N::DeterministicNetwork)


Given a matrix A, null1(A) returns a matrix with the same dimensions, where every interaction happens with a probability equal to the connectance of A.

# EcologicalNetwork.null2Function.

Type II null model

null2(N::DeterministicNetwork)


Given a matrix A, null2(A) returns a matrix with the same dimensions, where every interaction happens with a probability equal to the degree of each species.

# EcologicalNetwork.null3inFunction.

Type IIIin null model

null3in(N::DeterministicNetwork)


Given a matrix A, null3in(A) returns a matrix with the same dimensions, where every interaction happens with a probability equal to the in-degree (number of predecessors) of each species, divided by the total number of possible predecessors.

# EcologicalNetwork.null3outFunction.

Type IIIout null model

null3out(N::DeterministicNetwork)


Given a matrix A, null3out(A) returns a matrix with the same dimensions, where every interaction happens with a probability equal to the out-degree (number of successors) of each species, divided by the total number of possible successors.

For an example:

N = make_bernoulli(BipartiteProbaNetwork(rand(3, 5)))
null2(N).A

3×5 Array{Float64,2}:
0.4  0.566667  0.733333  0.733333  0.9
0.3  0.466667  0.633333  0.633333  0.8
0.1  0.266667  0.433333  0.433333  0.6


## Null model wrapper

EcologicalNetwork has a wrapper to generate an arbitrary number of Bernoulli networks from a probability matrix. This approach is encourage over simply generating your own networks, because the wrapper will make sure that all networks have no species without any interactions. This ensures that the networks have the same size.

For example, we can generate a hundred replicates from the stony food web dataset, using the type 2 model:

template = null2(stony())

# Generate up to 100 networks
N = nullmodel(template, n=100, max=1000)

# Average connectance
mean(map(connectance, N))

0.0665125425170068


It must be noted that the number of networks returned by nullmodel may be lower than the requested number of networks. This is because of the constraint on the fact that no species can end up without interactions. When this constrained is enforced, some networks have very low success rates. This can be measured using the species_is_free function:

template = null2(mcmullen())

# Probability that every species has at least one interaction
at_least_one = 1.-species_is_free(make_unipartite(template))

# Probability that a randomized network has no unconnected species
prod(at_least_one)

1.1674941263628132e-10


# EcologicalNetwork.nullmodelFunction.

Generation of random matrices from a null model

nullmodel(N::ProbabilisticNetwork; n=1000, max=10000)


This function is a wrapper to generate replicated binary matrices from a template probability matrix A.

If you use julia on more than one CPU, i.e. if you started it with julia -p k where k is more than 1, this function will distribute each trial to one worker. Which means that it's fast.

• n (def. 1000), number of replicates to generate
• max (def. 10000), number of trials to make