In this part of the thesis, we consider unweighted and undirected networks. Let’s denote the complex network as G = (V, E), with V and E being the set of nodes and links, respectively. The network can be represented by its adjacency matrix A = [aij], where aij = 1 if an edge exists
between nodes vi and vj, and aij = 0 otherwise. We consider networks without any self-loops,
i.e., aii = 0. The centrality of a node (or edge) in a network determines its importance in a
particular functionality of the network. A centrality index assigns a score to all nodes indicating their vitality in the network. Due to the implicit meaning of ‘importance’, disparate indices have been introduced to cover the concept.
Degree Centrality
The most trivial method to evaluate the centrality of nodes accounts for the number of immediate neighbours of a node which is presented as degree centrality. The degree ki of node
vi is the number of edges incidents on that node:
𝑘𝑖 = ∑ 𝑎𝑗 𝑖𝑗 (2-3)
Because the degree centrality is computationally a low complex method compared to more complicated centralities like betweenness, closeness and eigenvector centralities; it is applied in many applications like controlling the animal disease epidemics (Candeloro et al., 2016). In a directed network, the edges have direction on them. This property provides two different degrees for each node; in-degree and out-degree. For example, in a citation network, if paper A cites paper B, there is a link from node A to node B, and the in-degree of a node shows its popularity and importance.
The Average Node Degree is calculated as the average of degrees of all nodes as 𝑘̅ =𝑁1∑𝑁 𝑘𝑖
𝑖=1 (2-4)
This parameter is an indication of how densely the nodes in the network are connected.
While the average node degree represents a particular structural property of a network, it happens that the networks with the same average node degree have entirely distinct topologies.
2.2 Structural Centrality Measures 17
To grasp another feature of the network using its topological properties, the Node Degree
Distribution, P(k), is applied that is defined as the probability that a randomly chosen node has
the degree k. It is described as equation 2-5.
𝑃(𝑘) =𝑁(𝑘)𝑁 (2-5) where N(k) represents the number of the nodes with degree k.
Fig. 2. 7 Node degree distributions (a) power-law distribution, (b) Poisson distribution.
Two popular node degree distributions in real networks are the power-law distribution and the Poisson distribution. In power-law distribution, P(k) relates to the degree parameter k, with equation 2-6.
𝑃(𝑘)∿𝑘−𝛾 (2-6)
In this equation, γ is the power-law exponent. In a network with power-law degree distribution, a significant portion of nodes have a small number of connections whereas a small part of nodes has a large number of edges connected; i.e. the network is heterogeneous.
In the Poisson distribution, the probability P(k) is changed with k as equation 2-7. 𝑃(𝑘) =𝑒−𝜆𝑘!𝜆−𝑘 (2-7)
18 Complex Network Models with 𝜆 = 𝑘̅, being the average node degree of the network. In networks with a Poisson degree distribution, no hub node can be formed and most of the nodes have the same degree; i.e. the network is homogeneous.
Poisson and power-law degree distributions are shown in Fig. 2.7.
It is essential to know if the nodes with the same degrees are connected (Degree Assortativity). One way to quantify this is by using the degree-degree correlation displayed in equation 2-8
𝑟 = 1 𝐸∑𝑗>𝑖𝑘𝑖𝑘𝑗𝑎𝑖𝑗−[ 1 𝐸∑ 1 2(𝑘𝑖+𝑘𝑗)𝑎𝑖𝑗 𝑗>𝑖 ] 2 1 𝐸∑ (𝑘𝑖 2+𝑘 𝑗2)𝑎𝑖𝑗 𝑗>𝑖 −[1𝐸∑𝑗>𝑖12(𝑘𝑖+𝑘𝑗)𝑎𝑖𝑗] 2 (2-8)
where E is the total number of edges, ki is the degree of node vi and aij is the entry (i,j) in
the adjacency matrix. According to the value of r, there are three types of networks:
• r > 0: The network is called assortative; i.e. the nodes with higher degree intend to connect and the nodes with lower degrees are connected (rich with rich, poor with poor)
• r < 0: The network is called disassortative; i.e. the nodes with higher degree intend to connect the nodes with lower degrees and the nodes with low degrees intend to connect to the nodes with higher degrees (rich with poor)
• r = 0: The is no intention for the connection between the nodes regarding their degree Node degree is a simple centrality measure that needs only local information on the nodes. The degree is indeed the most natural centrality measure to compute and has been shown to control many of the network functions.
Local Rank
To obtain an effective ranking parameter to overcome computationally complex calculations in large-scale networks, Chen et al. (Chen et al., 2012) proposed another local centrality measure, called Local Rank, which considers information on nodes’ fourth-order neighbours. Local rank is a compromise between the degree centrality and other time- consuming measures. The Local Rank of each node vi is computed as follows:
2.2 Structural Centrality Measures 19
𝑄(𝑗) = ∑𝑘∈𝛤𝑗𝑅(𝑘) (2-10) where Γi is the set of immediate neighbours of vi and R(k) accounts for the number of immediate
and the next immediate neighbours of vk.
Clustering Coefficient
Another measure which quantifies the interconnection in the network is the clustering coefficient. It measures the local connectivity in the network (i.e., indicating to what extent the neighbours of a node are interconnected). For node vi with degree ki, the maximum number of
possible edges among its neighbours is ki(ki-1). The clustering coefficient is the portion of these
possible edges that indeed exist and is computed as: 𝐶𝑖 =𝑘|[𝑟,𝑠]|
𝑖(𝑘𝑖−1) (2-11)
if edge [vr, vs] exists, and both nodes vr and vs are also connected to node vi, then we have |[𝑟,𝑠]|
= 1.
Coreness Centrality
Recently, Kitsak et al. (Kitsak et al., 2010) argued that the position of a node is more effective than its degree. Specifically, the nodes located at the core part of a network, are more influential compared to the nodes in the periphery. Coreness of a node can be procured by using the k-core decomposition (Dorogovtsev et al., 2006) that repetitively decomposes the network according to the nodes residual degree. To calculate coreness, one needs global topological information of the network, which limits its usage to very large-scale networks (Lu et al., 2016). The k-core decomposition method applied to calculate the coreness centrality can be briefed as follows. Initially the nodes with degree zero are removed with Ci=0 before we begin the k-core
decomposition. In the first step, all nodes with degree equal to 1 are removed, until no more nodes with degree one remain in the network. The nodes removed in this stage have the coreness centrality Ci=1. Likewise, in the second step, repetitively the nodes with degree equal to 2 are
20 Complex Network Models removed, which are assigned to 2-shell and their coreness centrality is 2. This process keeps on going till all the nodes are removed.