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Correlated multiplexity and connectivity of multiplex random networks
Kyu-Min Lee, Jung Yeol Kim, Won-kuk Cho, K. -I. Goh, I. -M. Kim
TL;DR
The paper asks how correlations between a node’s degrees across multiplex layers affect network connectivity. It studies duplex Erdős–Rényi networks using joint-degree models, mean-field-like analysis, and numerical simulations. Positive correlations can produce a giant component at arbitrarily small nonzero density, whereas negative correlations delay emergence but can yield full connectivity at finite density.
Problem
The paper addresses how correlated multiplexity, observed when degrees across interaction layers are nonrandomly related, changes giant-component properties.
Method
The paper studies duplex Erdős–Rényi layers specified by joint degree distributions, using mean-field-like analytical calculations and extensive numerical simulations.
Results
Positive multiplexity correlations make the giant component emerge at a much lower critical density, approaching zero for MP, while negative correlations delay emergence and can produce full connectivity at z* = 1.14619322....
Takeaways & Limitations
Correlated multiplexity can dramatically alter connectivity: positive correlations yield more gradual giant-component growth, whereas negative correlations yield more abrupt growth and finite-density full connectivity.
Takeaways & Limitations
The total-degree formulation neglects overlapped links in the N →∞ limit for random sparse networks with largest degree at most O(N).
Abstract
from arXiv · showhide
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.
1. Introduction
Multiplex networks capture systems where nodes participate in multiple interaction types, and correlated multiplexity describes nonrandom relationships between layer degrees. The paper studies how these correlations affect multiplex connectivity.
- Real-world systems often contain multiple interaction types, motivating multiplex rather than isolated single-network models.
- Earlier interacting and interdependent network models coupled layers through random connections or pairings.
- Correlated multiplexity occurs when a node’s degree in one layer is nonrandomly related to its degree in another.Examples include social, trade, and transportation networks.
- The paper’s framework distinguishes maximally-positive and maximally-negative correlated multiplexity alongside uncorrelated multiplexity.MP and MN denote maximally-positive and maximally-negative correlated multiplexity.
2. Model and formalism
The paper models a duplex network through layer-specific degrees and their joint distribution, then applies generating-function connectivity analysis. Overlapping links are neglected in the sparse-network limit.
- A duplex multiplex network has N nodes, two link types, and layer degrees governed by intralayer distributions π^(1)(k_1) and π^(2)(k_2).Its multiplex structure is specified by the joint distribution Π(k_1,k_2) or conditional distribution Π(k_2|k_1).
- The total degree is k = k_1 + k_2 − k_o, where k_o counts links overlapped between the two layers.For random sparse networks with largest degree at most O(N), overlap can be neglected as N →∞.
- The total-degree distribution P(k) is obtained from the joint or conditional degree distribution and the layer-degree distribution.The construction uses a Kronecker delta to enforce the total-degree relation.
- The generating function g_0(x) of P(k) supports standard analysis of network structure, including giant-component size and susceptibility.The giant-component size S and mean component size ⟨s⟩ are derived through generating-function equations.
- A giant component exists when the generating-function equations admit a nontrivial solution u < 1, equivalent to the Molloy–Reed criterion.
3. Degree distributions
For uncorrelated duplex ER layers, total degrees follow an ER distribution with mean z_1 + z_2. The MP and MN constructions instead pair layer degrees in the same or opposite order, changing the resulting degree distributions.
- Uncorrelated multiplexity: Uncorrelated layer degrees factorize, so the multiplex total-degree distribution is the convolution of the two layer distributions.
- Uncorrelated multiplexity: The uncorrelated duplex ER generating function is e^((z_1+z_2)(x−1)), equivalent to an ER network with mean degree z_1 + z_2.
- Degree-distribution comparison: Figure 2 compares simulated and mean-field-predicted degree distributions for uncorrelated, MP, and MN multiplexity at z_1 = z_2 = 0.7 and 1.4.
- Maximally-positive multiplexity: In the MP case, nodes are matched by the same degree order across layers, pairing hubs with hubs and low-degree nodes with low-degree nodes.
- Maximally-negative multiplexity: In the MN case, nodes are matched by opposite degree orders, pairing hubs in one layer with smallest-degree nodes in the other.
4. Duplex ER networks with equal link densities
In duplex ER networks with equal layer densities, multiplexity correlations strongly alter giant-component onset and connectivity: positive correlation permits emergence at any nonzero density, while negative correlation delays emergence but can yield complete connectivity at finite density.
- Uncorrelated case: The uncorrelated duplex ER network is equivalent to an ER network with mean degree z = 2z1 and has critical intralayer degree zc = 1/2.Its giant component follows standard mean-field scaling with β = 1 and γ = 1.
- MP case: Maximally positive multiplexity satisfies the Molloy-Reed criterion for every nonzero z1, so a giant component exists at any nonzero link density.The criterion evaluates to 4z1^2 > 0 for z1 ≠ 0.
- MP case: For maximally positive multiplexity, the giant component grows linearly near the origin and all linked nodes form a single giant component, leaving only isolated nodes outside it.The scaling is S ∼ z1, corresponding to β = 1, and the susceptibility is ⟨s⟩ = 1 for z1 > 0.
- MN case: Maximally negative multiplexity has no giant component for 0 ≤ z1 ≤ ln 2 because linked nodes are paired with degree-0 nodes across layers.The conditional degree distribution is correspondingly complicated in this regime.
- MN case: At z∗ = 1.14619322..., the MN network becomes fully connected at finite link density, despite its delayed giant-component emergence.The intermediate MN regime has a significantly higher onset density than the uncorrelated case and a more abrupt growth after emergence.
- MN case: Despite differing onset and growth profiles, the MN critical behavior remains conventional mean-field with β = 1 and γ = 1.Figure 4 reports data collapse using β = 1 and ν = 3.
5. Imperfect correlated multiplexity
The paper models imperfect correlated multiplexity by mixing maximally correlated and uncorrelated nodes, finding that correlation affects the critical degree across the full range of correlation fractions.
- Critical degree: The critical degree zc changes with q, with separate formulae across q = 2 −1/ ln 2.Figure 5 plots equation (15) and marks this threshold with a vertical dotted line.
- Model: A fraction q of nodes is assigned maximally correlated multiplexity, while the remaining fraction 1 −q is randomly multiplexed.The resulting degree distribution is Ppartial(k) = qPmaximal(k) + (1 −q)Puncorr(k).
- Critical degree: The effect of correlated multiplexity persists for general q, not only for maximally correlated networks.This conclusion applies to the partially correlated cases between q = 0 and q = 1.
6. Duplex ER networks with general link densities
For unequal layer link densities, correlated multiplexity changes both giant-component emergence and growth, while inducing degree correlations that explain deviations from mean-field predictions.
- For MP, the giant component emerges at lower link densities but grows more slowly than in the uncorrelated case.
- For MN, the giant component emerges at higher link densities, grows more abruptly, and connects all nodes at finite link density.
- For unequal z1 and z2, the mean-field-like approximation remains qualitatively correct but fails quantitatively against numerical simulations.
- With z2 = 0.4, uncorrelated multiplexity gives r = 0 and exact agreement between numerical simulation and mean-field calculation.
- For MP and MN, positive assortativity produces simulation–mean-field deviations; these vanish at z1 = z2 when assortativity also vanishes.
- Correlated multiplexity changes P(k) and introduces higher-order correlations despite no degree correlations within individual network layers.
7. Conclusion
The paper shows that correlated multiplexity can dramatically alter giant-component structure in multiplex networks. Positive correlation lowers the emergence threshold but slows growth, whereas negative correlation delays emergence but enables abrupt growth and finite-density full connectivity.
- Positive correlated multiplexity makes the giant component emerge at a much lower critical link density, approaching zero for MP.
- After emergence under positive correlation, the giant component grows much more gradually than in uncorrelated multiplex networks.
- Negative correlated multiplexity raises the giant-component emergence threshold but makes subsequent growth more abrupt.
- At finite link density, negative correlated multiplexity can establish full connectivity by connecting the entire network into a single component.
- Multiplex networks can exhibit structural properties that cannot be represented by individual network-layer properties alone.