Coexistence of phases and the observability of random graphs

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Coexistence of phases and the observability of random graphs

Antoine Allard, Laurent Hébert-Dufresne, Jean-Gabriel Young, and Louis J. Dubé

Phys. Rev. E 89, 022801 – Published 6 February 2014

Abstract

In a recent Letter, Yang et al. [Phys. Rev. Lett. 109, 258701 (2012)] introduced the concept of observability transitions: the percolationlike emergence of a macroscopic observable component in graphs in which the state of a fraction of the nodes, and of their first neighbors, is monitored. We show how their concept of depth- $L$ percolation—where the state of nodes up to a distance $L$ of monitored nodes is known—can be mapped onto multitype random graphs, and use this mapping to exactly solve the observability problem for arbitrary $L$ . We then demonstrate a nontrivial coexistence of an observable and of a nonobservable extensive component. This coexistence suggests that monitoring a macroscopic portion of a graph does not prevent a macroscopic event to occur unbeknown to the observer. We also show that real complex systems behave quite differently with regard to observability depending on whether they are geographically constrained or not.

Received 30 September 2013

DOI:https://doi.org/10.1103/PhysRevE.89.022801

Authors & Affiliations

Antoine Allard, Laurent Hébert-Dufresne, Jean-Gabriel Young, and Louis J. Dubé

Département de physique, de génie physique et d'optique, Université Laval, Québec (Québec), Canada G1V 0A6

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Vol. 89, Iss. 2 — February 2014

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Figure 1
Illustration of depth-1 percolation on a graph generated through the configuration model. Directly occupied, indirectly occupied, and nonoccupied nodes are in red (type 0), blue (type 1), and black (type 2), respectively. Occupied components are identified with orange edges, and nonoccupied components with black edges. Edges linking occupied and nonoccupied components—the ones removed by setting $x_{02} = x_{12} = x_{20} = x_{21} = 1$ in Eqs. (4), (8), and (9)—are shown in cyan.
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Figure 2
Validation of the theoretical formalism for different depth of percolation ( $L$ ). Size of the occupied and nonoccupied components in function of $φ$ for graph ensembles with a different average degree. The degrees of both ensembles are exponentially distributed according to $P (k) = (1 - e^{- λ}) e^{- λ (k - 1)}$ with $k \geq 1$ and $λ = - ln (1 - 1 / 〈 k 〉)$ . The size of the giant component in the original graph ensemble, $S_{cm}$ , is shown for comparison. It is equal to $S_{cm} = 1 - G_{0} (a)$ where $a$ is the solution of $a = G_{1} (a)$ [16]. Curves are the solutions of Eqs. (12), (13), (15), and (16) ( $L = 1$ ), and Eqs. (22, 23, 24, 25) ( $L = 3$ ). Symbols are the relative size of the largest occupied and nonoccupied components averaged over at least 100 graphs of at least $5 \times 10^{5}$ nodes each. Threshold values were obtained from Eqs. (14), (17), and (26), and by analyzing the stability of Eqs. (23) around $a = 1$ . Note the change of scale of the abscissa of (c).
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Figure 3
Effect of varying the depth $L$ on the coexistence regime. (a)–(e) The size of the nonoccupied giant component ( ${\bar{S}}^{(L)}$ , dash lines) and of the occupied giant component ( $S^{(L)}$ , solid lines) are shown as a function of $φ$ for different values of the depth $L$ and different degree distributions. The power-law degree distribution is defined as $P (k) = k^{- α} e^{- k / κ} / {Li}_{α} (e^{1 / κ})$ with $k \geq 1$ and ${Li}_{α} (x)$ denoting the polylogarithm. The Poisson degree distribution is defined as $P (k) = λ^{k} e^{- λ} / k!$ with $k \geq 0$ and $λ = 〈 k 〉$ . See the caption of Fig. 2 for the definition of the exponential degree distribution. All curves were obtained by solving Eqs. (22, 23, 24, 25). Figures (a)–(e) are a representative subset of the behaviors obtained with many realistic and commonly used degree distributions. (f) Behavior of ${\bar{φ}}_{c}^{(L)}$ (circles) and $φ_{c}^{(L)}$ (squares) as a function of $L$ using the degree distributions of (a)–(e). Values were obtained from (26), and by analyzing the stability of Eqs. (23) around $a = 1$ . Lines have been added to guide the eye.
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Figure 4
(a)–(e) Depth-1 percolation on graphs extracted from real nongeographically constrained complex systems (see Table 1). Symbols represent the average (100 simulations minimum) relative size of the largest occupied ( $S^{(1)}$ ) and nonoccupied ( ${\bar{S}}^{(1)}$ ) components found in these graphs where directly occupied nodes were selected randomly with probability $φ$ . Lines were obtained by solving Eqs. (12), (13), (15), and (16) with the degree distribution extracted from each graph [shown in (f)].
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Figure 5
(a) and (b) Depth-1 percolation on graphs extracted from real geographically constrained complex systems (see Table 1). Symbols represent the average (100 simulations minimum) relative size of the largest occupied ( $S^{(1)}$ ) and nonoccupied ( ${\bar{S}}^{(1)}$ ) components found in these graphs where directly occupied nodes have been selected randomly with probability $φ$ . Lines were obtained by solving Eqs. (12), (13), (15), and (16) with the degree distribution extracted from each graph [shown in (c)]. (d) Depth-1 percolation on square $L \times L$ lattices (circles: $L = 70$ , squares: $L = 1000$ ) where edges are randomly removed with probability $p = 0.30$ ( $〈 k 〉 = 2.8$ ). Symbols represent the average (100 simulations minimum) relative size of the largest occupied ( $S^{(1)}$ ) and nonoccupied ( ${\bar{S}}^{(1)}$ ) components. Lines (with no symbols) are the predictions of Eqs. (12), (13), (15), and (16) with the binomial degree distribution $P (k) = (\begin{matrix} 4 \\ k \end{matrix}) {(1 - p)}^{k} p^{4 - k}$ with $0 \leq k \leq 4$ . Lines have been added between symbols in (c) and (d) to guide the eye.
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