Linked: How everything is connected to everything else and what it means for business, science, and everyday life

June-1-2017

Chapter 7-1:  Rich get richer-Barabási–Albert model

1. Barabási–Albert generates the first scale-free network by combining this two laws:

In this graph, the network start from 2 nodes, in each panel a new node (empty circle) is added to the network and add two links to existing nodes, when decides which to link, the new node prefer the more connected ones.

Law A. Growth: for each given period of time we add a new node to the network. This step underscores the fact that networks are assembled one node at a time.

Why growth is important: because early nodes have more time than the latecomers to acquire links.

Law B. Preferential attachment: each new node is assumed to connect to the existing nodes with two links. The probability that it will choose a given node is proportional to the number of links the chosen node has. That is, given the choice between two nodes, one with twice as many links as the other, it is twice as likely that the now ndoe will connect to the more connected node.

Why Preferential attachment is necessary: with only the first law (only growth, new node randomly choose two existing nodes to link with), we will end up with a Bell curve like distribution instead of power laws.

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2. Only growth or preferential attachment would not generate the power laws and hubs:

Model A

Model A retains growth but does not include the preferential attachment. The probability of a new node connecting to any pre-existing node is equal. The resulting degree distribution in this limit is geometric, indicating that growth alone is not sufficient to produce a scale-free structure.

Model B

Model B retains preferential attachment but eliminates growth. The model begins with a fixed number of disconnected nodes and adds links, preferentially choosing high degree nodes as link destinations. Though the degree distribution early in the simulation looks scale-free, the distribution is not stable, and it eventually becomes nearly Gaussian as the network nears saturation. So preferential attachment alone is not sufficient to produce a scale-free structure.

The failure of models A and B to lead to a scale-free distribution indicates that growth and preferential attachment are needed simultaneously to reproduce the stationary power-law distribution observed in real networks.

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3. The evolution of Barabási–Albert 's scale-free network model

Internal links, rewiring, removal of nodes and links, aging, nonlinear effects, and many other processes affecting network topology and alter the way networks grow and evolve, inevitably changing the number and size of the hubs.

No matter how large and complex a network becomes, as long as growth and preferential attachment are simultaneously present, it will maintain its hub-dominated scale-free topology (hubs and power laws emerge as well).

Chapter 7-2: Properties of Barabási–Albert model 

1. Degree distribution

The degree distribution of the BA Model, which follows a power law. In log-log scale the power law function is a straight line.

The degree distribution resulting from the BA model is scale free, in particular, it is a power law of the form:

2. Average path length

The average path length of the BA model increases approximately logarithmically with the size of the network. The actual form has a double logarithmic correction and goes as:

The BA model has a systematically shorter average path length than a random graph.

3. Clustering coefficient

While there is no analytical result for the clustering coefficient of the BA model, the empirically determined clustering coefficients are generally significantly higher for the BA model than for random networks. The clustering coefficient also scales with network size following approximately a power law.

This behavior is still distinct from the behavior of small-world networks where clustering is independent of system size.