Does exponential degree distribution entail Log-normal distance distribution in large complex graphs? We've been exploring the graph structure of a large genealogical data base (WikiTree) of which main connected component contains about 23 million nodes. The graph edges are defined by any direct family relationship (parent, child, sibling, spouse). See https://www.wikitree.com/wiki/Space:100_Circles for more details.
Based on preliminary results, the degree distribution seems to follow an exponential law, whereas the distance distribution seems to fit a Log-normal law, at least for central values, with noisy head and long-tail.
We are trying to figure explanations for both of those features, but I was wondering if one of them entails the other, and more generally if a specific degree distribution entails a specific distance distribution. I've not found yet literature on this question.
A side question is that one could expect "real life" large family networks to be scale-free, but WikiTree, like any genealogical data base, is far from equilibrium in the sense that the degree of nodes in the data base is often much smaller than the real-life value. People are missing everywhere and the growth is collaborative, distributed all over the graph and somehow random. The exponential degree distribution would agree with this result : exponential distribution is found in non-equilibrium networks with random growth, found at https://www.researchgate.net/publication/257218160_The_exponential_degree_distribution_in_complex_networks_Non-equilibrium_network_theory_numerical_simulation_and_empirical_data
 A: There is no direct relationship between the degree distribution and the distance distribution of a graph.
There are a lot of ways to think about this question, one of them is to consider the following model, very much inspired by Watts and Strogatz Small-World model:

*

*start from a regular graph (e.g. a grid-like graph where each node is connected to a fixed number of neighbours $k$),

*then draw some pairs of edges randomly and exchange their ends if it doesn't create multi-edges or loops (transform edges $(a,b)$ and $(c,d)$ in $(a,d)$ and $(c,b)$).

The degree distribution will remain strictly the same (every node has degree $k$), but shortcuts have been created in the graph, and pairs of nodes will be closer to each other on average, so the degree distribution can be drastically modified.
Of course, you might argue that your graph is very different from such a graph, but a similar reasoning stands: with a same degree distribution you can have a lot of cycles that will bring nodes close to each other or you can have a "tree-like" graph where distances tend to be longer on average.
An interesting property of genealogical graphs is that they probably have few cycles or at least few short cycles (although maybe not if siblings are connected to each other and to their parents), which makes me think that you might be able to devise some kind of model to explain the form of the distance distribution.
