Simulating the Dynamics of Scale-Free Networks via Optimization

We deal here with the issue of complex network evolution. The analysis of topological evolution of complex networks plays a crucial role in predicting their future. While an impressive amount of work has been done on the issue, very little attention has been so far devoted to the investigation of how information theory quantifiers can be applied to characterize networks evolution. With the objective of dynamically capture the topological changes of a network's evolution, we propose a model able to quantify and reproduce several characteristics of a given network, by using the square root of the Jensen-Shannon divergence in combination with the mean degree and the clustering coefficient. To support our hypothesis, we test the model by copying the evolution of well-known models and real systems. The results show that the methodology was able to mimic the test-networks. By using this copycat model, the user is able to analyze the networks behavior over time, and also to conjecture about the main drivers of its evolution, also providing a framework to predict its evolution.

where, n k is the number of nodes with degree k and Proof: By definition: Defining n k as the number of nodes with degree k, then: The first term on the right hand side of equation (3) is given by where, in the last equality, we use the fact that ∑ N −1 k=0 n k = N . The second term on the right hand side of equation (3) is given by where, in the last equality, we use the fact that ∑ N −1 k=0 n k = N . Combining equations (2), (4) and (5) we obtain the result. Corollary 1. Given a network G with size N and degree distribution P then: (2).
Proof: observe that f (n k + 1) − f (n k ) ≥ 0 for all n k = 0, 1, 2 · · · . Then, using the fact that in a network of size N , Thus, by equation (6) and Proposition 1 we obtain the result.
Proposition 1 offers an important tool in analysis of complex networks: if a new connection in a network is made, at most four terms on the right hand side of equation (1) should be computed. In other words, adding a new connection from a node with degree k 1 to a node with degree k 2 : then, to compute changes in the Jensen-Shannon divergence value of the network when a new connection is made, we do not need to look up all degree distribution of the network. Another consequence of this proposition stays in the fact that two different networks could exhibit the same Jensen-Shannon divergence value.

Measurements on complex networks
To validate our approach, we measure another properties of the network: related with distance (diameter and average path length), with degree distribution (link density, square root of the Jensen-Shannon divergence values and average neighbor degree), with clustering and cycles (clustering coefficient, transitive clustering coefficient and number of loops), with centrality (betweenness centrality, eigenvector centrality and closeness centrality) and spectral measurements (graph energy, graph spectrum and S-metric). Here we present a brief definition of each measure for an undirected and unweighted network G with size N and adjacency matrix A. Readers can refer to [1,2] for a deeper discussion on the topic.

Measurements related with distance
The distance, d(i, j) between the nodes i and j is defined as the minimum number of edges connecting them. The average path length (l) is the average of all possible distances in the network. Thus, for an undirected and unweighted network: .
The diameter of a network is the maximum distance of any two nodes on the network:

Measurements related with degree distribution
The square root of Jensen-Shannon divergence values is well defined in the main body of the paper. The link density of a network is the fraction between the total number of edges and N · (N − 1)/2 (the total number of possible edges). The average neighbor degree computes the average degree of neighboring nodes for every vertex.

Measurements related with clustering and cycles
The clustering coefficient, C, and the transitive clustering coefficient, C T , are well defined in the main body of the paper. The number of loops in a network G is the number of edges that need to be removed in order that the graph cannot have cycles.

Centrality measures
The betweenness centrality measure of a node u quantifies its importance in terms of interactions of nodes on the network. It is defined by: where, σ(i, u, j) is the number of shortest paths between i and j passing through u and σ(i, j) is the number of shortest paths between i and j.
The closeness centrality measure of a node i is given by: It can be viewed as the efficiency of each vertex (individual) in spreading information to all other vertices. The larger the closeness centrality of a vertex, the shorter is the average distance from the vertex to any other vertex, and thus the better positioned the vertex is in spreading information to other vertices. Let X be a normalized eigenvector with respect to the greater eigenvalue λ of the adjacency matrix A, we define the eigenvector centrality of a node i, simply by X(i).

Spectral measurements
The graph energy is, by definition, the sum of absolute values of the eigenvalues of A. The graph spectrum is defined as the eigenvalues of the Laplacian of the graph.