# An ensemble of random graphs with identical degree distribution

## TL;DR

This paper demonstrates that networks with identical degree distributions can have vastly different topological structures and explores the diversity and phase transitions within ensembles of such graphs, revealing their complex properties.

## Contribution

It introduces an ensemble of random graphs with a fixed power-law degree distribution and analyzes their structural diversity, phase transitions, and bounds on spanning trees.

## Key findings

- Networks with same degree distribution can have different topologies.
- The ensemble size appears large in the thermodynamic limit.
- Identifies phase transitions in graph properties over time.

## Abstract

Degree distribution, or equivalently called degree sequence, has been commonly used to be one of most significant measures for studying a large number of complex networks with which some well-known results have been obtained. By contrast, in this paper, we report a fact that two arbitrarily chosen networks with identical degree distribution can have completely different other topological structure, such as diameter, spanning trees number, pearson correlation coefficient, and so forth. Besides that, for a given degree distribution (as power-law distribution with exponent $\gamma=3$ discussed here), it is reasonable to ask how many network models with such a constraint we can have. To this end, we generate an ensemble of this kind of random graphs with $P(k)\sim k^{-\gamma}$ ($\gamma=3$), denoted as graph space $\mathcal{N}(p,q,t)$ where probability parameters $p$ and $q$ hold on $p+q=1$, and indirectly show the cardinality of $\mathcal{N}(p,q,t)$ seems to be large enough in the thermodynamics limit, i.e., $N\rightarrow\infty$, by varying values of $p$ and $q$. From the theoretical point of view, given an ultrasmall constant $p_{c}$, perhaps only graph model $N(1,0,t)$ is small-world and other are not in terms of diameter. And then, we study spanning trees number on two deterministic graph models and obtain both upper bound and lower bound for other members. Meanwhile, for arbitrary $p(\neq1)$, we prove that graph model $N(p,q,t)$ does go through two phase transitions over time, i.e., starting by non-assortative pattern and then suddenly going into disassortative region, and gradually converging to initial place (non-assortative point). Among of them, one "null" graph model is built.