Correlated multiplexity and connectivity of multiplex random networks

Nodes in a complex networked system often engage in more than one type of interactions among them; they form a multiplex network with multiple types of links. In real-world complex systems, a node’s degree for one type of links and that for the other are not randomly distributed but correlated, which we term correlated multiplexity. In this paper we study a simple model of multiplex random networks and demonstrate that the correlated multiplexity can drastically affect the properties of giant component in the network. Specifically, when the degrees of a node for different interactions in a duplex Erdos-Renyi network are maximally correlated, the network contains the giant component for any nonzero link densities. In contrast, when the degrees of a node are maximally anti-correlated, the emergence of giant component is significantly delayed, yet the entire network becomes connected into a single component at a finite link density. We also discuss the mixing patterns and the cases with imperfect correlated multiplexity.

💡 Research Summary

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The paper investigates how correlations between node degrees across different layers of a multiplex network influence the emergence of a giant connected component. The authors focus on the simplest setting: a duplex (two‑layer) network where each layer is an Erdős–Rényi (ER) random graph with mean degrees (z_1) and (z_2). They introduce the concept of “correlated multiplexity,” i.e., statistical dependence between a node’s degree in one layer and its degree in the other. Three limiting cases are studied: (i) uncorrelated multiplexity, where the joint degree distribution factorizes; (ii) maximally‑positive (MP) correlation, where the degree ordering is identical in both layers; and (iii) maximally‑negative (MN) correlation, where the ordering is reversed.

Using the generating‑function formalism for percolation, the authors derive the total degree distribution (P(k)) from the joint distribution (\Pi(k_1,k_2)) and then apply the Molloy‑Reed criterion (\sum_k k(k-2)P(k)>0) to locate the percolation threshold. In the uncorrelated case the duplex behaves like a single ER graph with mean degree (z = z_1+z_2); the giant component appears at (z_c = 1) (i.e., (z_1 = 0.5) per layer) with mean‑field critical exponents (\beta = \gamma = 1).

In the MP case the conditional distribution collapses to (\Pi(k_2|k_1)=\delta_{k_2,k_1}). Consequently the total degree is always even, and the Molloy‑Reed condition is satisfied for any non‑zero (z_1). The percolation threshold drops to zero: as soon as a single link exists in either layer, a giant component spanning a finite fraction of the network forms. Analytically one finds (u=0) (the smallest solution of (x=g_1(x))), leading to (S=1-e^{-z_1}) and a susceptibility (\chi=1). Thus the network is essentially fully connected apart from isolated nodes, a result confirmed by extensive simulations.

The MN case is more intricate. For small (z_1) (up to (\ln 2)) more than half the nodes have degree zero in each layer, preventing any giant component. The percolation threshold is pushed up to (z_c \approx 0.8386). As (z_1) increases further, a giant component emerges abruptly at a higher density than in the uncorrelated case, and when (z_1) exceeds a second critical value ((z^* \approx 1)), the network becomes fully connected ((S=1)). Despite the delayed onset, the critical behavior remains mean‑field ((\beta=\gamma=1), correlation length exponent (\nu=3)).

Recognizing that real systems rarely exhibit perfect correlation, the authors introduce a mixing parameter (q) (0 ≤ q ≤ 1) representing the fraction of nodes that are maximally correlated while the remainder are uncorrelated. The total degree distribution becomes a convex combination (P_{\text{partial}} = q P_{\text{max}} + (1-q) P_{\text{uncorr}}). For the MP scenario the percolation threshold varies linearly as (z_c = (1-q)/2). For the MN scenario the dependence is piecewise: (z_c = 1/(2-q)) for (q < 2-1/\ln 2) and a more complex implicit relation for larger (q). This interpolation shows that even modest levels of correlation can substantially shift the percolation point.

Methodologically, the work relies on generating‑function calculations, the Molloy‑Reed condition, and large‑scale Monte‑Carlo simulations (network sizes up to (N=10^6)). The authors also discuss connections to earlier studies on interacting and interdependent networks, emphasizing that correlated multiplexity introduces a distinct mechanism for altering connectivity that is not captured by simple inter‑layer coupling models.

In summary, the paper demonstrates that degree correlations across layers dramatically affect the connectivity of multiplex networks. Maximally positive correlation eliminates the percolation threshold, while maximally negative correlation raises it and can even produce a finite density at which the entire network becomes a single component. Partial correlations interpolate smoothly between these extremes. These findings have direct implications for robustness, epidemic spreading, and information diffusion in real‑world systems where multiple types of relationships coexist and are often correlated.