Numerical Assessment of the Percolation Threshold Using Complement Networks

Abstract

Models of percolation processes on networks currently assume locally tree-like structures at low densities, and are derived exactly only in the thermodynamic limit. Finite size effects and the presence of short loops in real systems however cause a deviation between the empirical percolation threshold $p_c$ and its model-predicted value $\pi _c$. Here we show the existence of an empirical linear relation between $p_c$ and $\pi _c$ across a large number of real and model networks. Such a putatively universal relation can then be used to correct the estimated value of $\pi _c$. We further show how to obtain a more precise relation using the concept of the complement graph, by investigating on the connection between the percolation threshold of a network, $p_c$, and that of its complement, $\bar{p}_c$.

Appendix: Definition and Properties of the Complement Network

Formally, let $a_{ij}$ be the generic element of the adjacency matrix $\mathsf {A}$ associated with a given binary undirected graph G of N vertices, such that $a_{ij}=1$ if an edge between vertices i and j exists, and $a_{ij}=0$ otherwise. The adjacency matrix of the complement graph $\bar{G}$ is defined through $\bar{a}_{ij}=1-\delta _{ij}-a_{ij}$, where $\delta _{ij}$ is the Kronecker delta which excludes self loops from $\bar{G}$, and $1-\delta _{ij}$ defines the adjacency matrix of the complete graph. It follows trivially that $M+\bar{M}=\left( {\begin{array}{c}N\\ 2\end{array}}\right)$, $\rho +\bar{\rho }=1$ and $k_i+\bar{k}_i=N-1$ $\forall i$, where M, $\rho$ and $k_i$ denote the number of edges, the edge density, and the degree of (number of edges incident with) generic vertex i, respectively. Thus, given the degree distribution P(k), the distribution of the complement degree is obtained as $\bar{P}(\bar{k})=P(N-1-\bar{k})$, i.e., as the reflection of P(k) on the $\frac{N-1}{2}$ vertical axis. Notably, the degree distribution of both a regular graph and an Erdös-Rényi graph (ER) are invariant under this transformation: the complement of a regular graph is a regular graph, as the complement of an ER is an ER. In particular, the complement of an ER with connection probability f is an ER with connection probability $1-f$.

Moving to higher-order properties, the number of triangles (closed loop of length 3) of a graph and of its complement is

$$\begin{aligned} \Sigma _{\triangle } = \frac{\text {Tr}\mathsf {A}^3+\text {Tr}\mathsf {\bar{A}}^3}{6}=\left( {\begin{array}{c}N\\ 3\end{array}}\right) -\frac{1}{2}\sum _i k_i\bar{k}_i. \end{aligned}$$

(4)

As such, both cases $k_i=0$ and $\bar{k}_i=0$ $\forall i$ (empty and complete graph) lead to $\Sigma _{\triangle }=\left( {\begin{array}{c}N\\ 3\end{array}}\right)$ as expected. As for transitivity, a complementarity relation can be written also for the local clustering coefficient $c_i= \frac{\sum _{j\ne i}\sum _{k\ne i,j}a_{ij}a_{jk}a_{ik}}{k_i(k_i-1)}$:

$$\begin{aligned} c_ik_i(k_i -1) + \bar{c}_i \bar{k}_i (\bar{k}_i -1) = k_i^{nn}k_i + \bar{k}_i^{nn}\bar{k}_i - k_i -\bar{k}_i - k_i\bar{k}_i, \end{aligned}$$

(5)

where $k_i^{nn}= \frac{\sum _{j\ne i } a_{ij}k_j}{k_i}$ is the average nearest-neighbors degree.