La Folie, Quo Vadis, Evo-Devo?

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Network models are crucial for shaping our understanding of complex networks and help to explain the origin of observed network characteristics. There are three models that had a direct impact on our understanding of biological networks.

Random networks
The Erdös–Rényi (ER) model of a random network (see figure, part A) starts with N nodes and connects each pair of nodes with probability p, which creates a graph with approximately pN(N–1)/2 randomly placed links (see figure, part Aa). The node degrees follow a Poisson distribution (see figure, part Ab), which indicates that most nodes have approximately the same number of links (close to the average degree ). The tail (high k region) of the degree distribution P(k) decreases exponentially, which indicates that nodes that significantly deviate from the average are extremely rare. The clustering coefficient is independent of a node’s degree, so C(k) appears as a horizontal line if plotted as a function of k (see figure, part Ac). The mean path length is proportional to the logarithm of the network size, l ~ log N, which indicates that it is characterized by the small-world property.

Scale-free networks
Scale-free networks (see figure, part B) are characterized by a power-law degree distribution; the probability that a node has k links follows P(k) ~ k exp(–γ), where γ is the degree exponent. The probability that a node is highly connected is statistically more significant than in a random graph, the network’s properties often being determined by a relatively small number of highly connected nodes that are known as hubs (see figure, part Ba; blue nodes). In the Barabási–Albert model of a scale-free network15, at each time point a node with M links is added to the network, which connects to an already existing node I with probability ΠI= kI/ΣJkJ, where kI is the degree of node I and J is the index denoting the sum over network nodes. The network that is generated by this growth process has a power-law degree distribution that is characterized by the degree exponent γ = 3. Such distributions are seen as a straight line on a log–log plot (see figure, part Bb). The network that is created by the Barabási–Albert model does not have an inherent modularity, so C(k) is independent of k (see figure, part Bc). Scale-free networks with degree exponents 2<γ<3, a range that is observed in most biological and non-biological networks, are ultra-small, with the average path length following L~ log log N, which is significantly shorter than log N that characterizes random small-world networks.

Hierarchical networks
To account for the coexistence of modularity, local clustering and scale-free topology in many real systems it has to be assumed that clusters combine in an iterative manner, generating a hierarchical network47,53 (see figure, part C). The starting point of this construction is a small cluster of four densely linked nodes (see the four central nodes in figure, part Ca).Next, three replicas of this module are generated and the three external nodes of the replicated clusters connected to the central node of the old cluster, which produces a large 16-node module. Three replicas of this 16-node module are then generated and the 16 peripheral nodes connected to the central node of the old module, which produces a new module of 64 nodes. The hierarchical network model seamlessly integrates a scale-free topology with an inherent modular structure by generating a network that has a power-law degree distribution with degree exponent γ = 1 + n4/n3 = 2.26 (see figure, part Cb) and a large, system-size independent average clustering coefficient ~ 0.6. The most important signature of hierarchical modularity is the scaling of the clustering coefficient, which follows C(k) ~ exp(–k) a straight line of slope –1 on a log–log plot (see figure, part Cc). A hierarchical architecture implies that sparsely connected nodes are part of highly clustered areas, with communication between the different highly clustered neighbourhoods being maintained by a few hubs (see figure, part Ca).