hierarchies in Sec. \[sec:flexible\] and for implementing the graphs in software, e.g. as used to generate the numerical results in Sec. \[sec:probabilistic\].

### Spectral Properties {#sec:spectral}

One of the tools frequently used in analyzing large networks is the spectral decomposition of the Laplacian. The behavior of the largest eigenvalue, the first eigenvalue gap, and the distribution of eigenvalues as a function of the network parameters are some of the diagnostics that can provide key information about dynamical processes on the network, and can also be used as points of comparison between competing network topologies [@Newman2000].

The smallest eigenvalue of a Laplacian is always $\lambda_1=0$, which corresponds to the uniform eigenvector ${\vec{e}}_1 = {\left(1, 1, \ldots, 1\right)}$. In ascending order, the eigenvalues of $L$ may be denoted by $0=\lambda_1 \le \lambda_2 \le \cdots \le \lambda_N$. We now state some graph properties that can be related to the spectrum of $L$ [@Newman2000; @Mohar1991].

The second eigenvalue $\lambda_2$ is known as the *algebraic connectivity* of the graph and is closely related to the expansion and connectivity properties of the graph. Broadly, the larger the value of $\lambda_2$, the better the connectivity of the network. To illustrate this point, consider the graph diameter, $\delta(H)$, 