# Connectivity vs. density for random, small world and scale-free networks

I am trying to understand a paper by reconstructing the the analyses discussed in the Validation study section, however, I am confused when it comes to how a network metric is used. Specifically, the authors discuss that they manipulated the connectivity metric in order to generate sparse networks.

I know that in graph theory connectivity refers to:

[...] the minimum number of elements (nodes or edges) that need to be removed to disconnect the remaining nodes from each other.

Question 1: What exactly does connectivity refer to in the paragraph below? Is it network density or degree centrality?

Question 2: Also, I don't understand how increasing the preferential attachment parameter (i.e., $P_{attach}$) affects the network sparsity in the case of scale-free networks. The authors mention that only one edge is added per node at each iteration, so, regardless of the preferential attachment, the resulting number of edges in the graph should remain the same, right?

This is the representative paragraph from the paper in question:

The level of connectivity of the networks was chosen to generate sparse networks. For this reason, in case of random networks, the probability of a connection ($P_{conn}$) between two nodes was set to 0.1, 0.2, and 0.3. For small world networks, the neighbourhood was set to 2, and for scale-free networks only one edge is added per node at each iteration in the graph generating process. To obtain awide variety of well known graph structures, the rewiring probability ($P_{rewire}$) in small world networks was set to 0.1, 0.5 and 1, and the power of preferential attachment ($P_{attach}$) in scale-free networks was set to 1, 2 and 3. For the condition with 100 nodes, we used different levels of connectivity for random and scale-free networks (random networks: $P_{conn}=$ .05, .1, and .15; scale-free networks: $P_{attach}=$ 1, 1.25, and 1.5). Otherwise, nodes will have too many connections.

• $P_{attach} = 1, 1.25, 1.5$. What, the scale-free attachers are so good that they have up to 150% chance of successfully attaching? Anyhow, what is the meaning of $P_{attach}$ If this is not a mistake, can you tell us what $P_{attach}$ means? – Mark L. Stone May 13 '18 at 19:22
• Thanks @MarkL.Stone for your input! I am quite sure that $P_{attach}$, as dsecribed in that paragraph, refers to the power of the preferential attachment, presumably, as discussed here (i.e., page 511, paragraph 2). I am confused because I don't understand how can $P_{attach}$ affect the network sparsity. – Mihai May 13 '18 at 19:39
• Whoops, i missed that. Anyhow, maybe (or not) www2.unb.ca/~ddu/6634/Lecture_notes/Lecture7_network_models.pdf will provide you some background to help understand this paper. – Mark L. Stone May 13 '18 at 19:55