## 4.4 Global Topological Organization of Complex Networks

Our objective here is to give a characterization of the global organization of complex networks. The first step for analyzing the global architecture of a network is to determine whether the network is homogeneous or modular. In a homogeneous network, what you see locally is what you get globally from a topological point of view. However, in a modular network, the organization of certain modules or clusters would be different from one to another and to the global characteristics of the network [27–29].

Formally, we consider a network homogeneous if it has good expansion (GE) properties. A network has GE if every subset *S* of nodes (*S* ≤ 50% of the nodes) has a neighborhood that is larger than some “expansion factor” Ω multiplied by the number of nodes in *S*. A neighborhood of *S* is the set of nodes that are linked to the nodes in *S* [30]. Formally, for each vertex *v* ∈ *V* (where *V* is the set of nodes in the network), the neighborhood of *v*, denoted as *Γ*(*v*), is defined as *Γ*(*v*) = {*u* ∈ *V*|(*u*, *v*)∈ *E*} (where *E* is the set of links in the network). Then, the neighborhood of a subset *S* ⊆ *V* is defined as the union of the neighborhoods of the nodes in *S*: *Γ*(*S*) = ∪_{v ∈S} *Γ*(*v*), and the network has GE if *Γ*(*v*) ≥ Ω |*S*| ∀ *S* ⊆ *V*.

Consequently, in a homogeneous network we should expect that some local topological properties scale as a power law of global topological properties. A power-law relationship between a two variables *x* and *y* of the network is ...

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