Multiplex Networks

• Multiplex networks are multi-layered systems where common nodes connect via distinct link types, preserving each layer's unique interaction.
• They enable nuanced modeling of cascading dynamics, percolation, and congestion by capturing inter-layer correlations and cooperative effects.
• Applications in social, biological, and transportation systems demonstrate their utility in robust control and vulnerability analysis.

A multiplex network is a network formed by a fixed set of nodes that are connected by multiple types of links, with each type constituting a separate network “layer.” In contrast to simplex (single‐layer) networks, multiplex networks retain the identity of each layer, thereby capturing the diversity of interactions or relationships (for example: social ties, modes of transportation, or communication channels) explicitly. Modeling networks in this way reveals structural, dynamical, and functional phenomena that are absent or obscured in aggregated, single‐channel representations.

1. Fundamental Principles of Multiplex Networks

A multiplex network is defined by a collection of layers $\mathcal{A} = [A^{(1)}, A^{(2)}, \ldots, A^{(M)}]$, where each $A^{(\alpha)}$ is an adjacency matrix representing a distinct kind of interaction among the common set of $N$ nodes. For binary, unweighted edges, $a_{ij}^{(\alpha)} = 1$ indicates a link of type $\alpha$ between nodes $i$ and $j$; otherwise, $a_{ij}^{(\alpha)} = 0$. Each node $i$ then has a degree vector $k_i = (k_i^{(1)},\ldots, k_i^{(M)})$ with $k_i^{(\alpha)} = \sum_j a_{ij}^{(\alpha)}$.

Distinctiveness arises from the preservation of layer structure: instead of merging every interaction into a single, undifferentiated network, multiplex modeling maintains explicit information on which type(s) of edge connect any given pair of nodes. Overlaps, interdependence, and correlations across layers serve as the basis for rich emergent properties.

Key measures defined for multiplex networks include:

• Edge overlap: For each pair $(i,j)$, $o_{ij} = \sum_\alpha a_{ij}^{(\alpha)}$ counts how many layers connect the pair.
• Aggregated adjacency: $a_{ij}$ indicates whether $i$ and $j$ are connected in any layer ($1$ if there exists at least one $\alpha$ with $a_{ij}^{(\alpha)}=1$, $0$ otherwise).
• Conditional degree: The conditional average degree for layer $\alpha'$ given degree $k^{(\alpha)}$ in layer $\alpha$, defined as $$\bar{k}^{(\alpha')}(k^{(\alpha)}) = \sum_{k^{(\alpha')}} k^{(\alpha')} P(k^{(\alpha')}|k^{(\alpha)})$$, quantifies inter-layer degree correlations.

The multiplex structure is essential in accurately modeling real-world systems such as social networks (multiple types of relationships), transportation systems (different modalities), neuronal networks (synaptic and gap junction layers), and infrastructure networks (power, communications, water).

2. Structural and Dynamical Implications

Preserving multiple layers introduces interplay and cooperativity that fundamentally alter both static properties and dynamical processes on the network. For instance, in the context of cascade phenomena (“multiplexity-facilitated cascades”) (Brummitt et al., 2011):

• Threshold cascades: In the classic Watts’ model on a simplex network, a node of degree $k$ and threshold $r$ activates if $m/k > r$ for $m$ active neighbors. In a multiplex, activation occurs if in any layer the fraction $m^{(\alpha)}/k^{(\alpha)} > r$. The response function for a duplex (two-layer) network is

$$F_{m_1, m_2}^{k_1, k_2} = \begin{cases} 0, & \max\left(\frac{m_1}{k_1}, \frac{m_2}{k_2}\right) \leq r \ 1, & \max\left(\frac{m_1}{k_1}, \frac{m_2}{k_2}\right) > r \end{cases}$$

• Recursion and cascade conditions: The propagation of activation is described recursively via $q^{(1)}$, $q^{(2)}$, the probabilities a vertex reached along layer-1 or layer-2 edge, respectively, becomes active:

$$\begin{pmatrix} q_{n+1}^{(1)} \ q_{n+1}^{(2)} \end{pmatrix} = \begin{pmatrix} g^{(1)}(q_n^{(1)}, q_n^{(2)}) \ g^{(2)}(q_n^{(1)}, q_n^{(2)}) \end{pmatrix}$$

Cascade vulnerability is determined by the maximum eigenvalue of the Jacobian matrix $J$ at the fixed point $q^{(i)} = 0$: $\lambda_{\max}(J) > 1$.

Multiplex structure increases vulnerability to cascades even when individual layers would be too sparse or too dense for activation to propagate—coupling can facilitate collective phenomena unattainable in any one layer alone. The maximum eigenvalue for the multiplex is always greater than or equal to that of the single layers.

Similarly, percolation transitions and k-core emergence are strongly modified (Cellai et al., 2013, Azimi-Tafreshi et al., 2014, Baxter et al., 2016). For k-core percolation, the largest subgraph in which each node has at least $k_\alpha$ edges of type $\alpha$ is generically realized by coupled recursive equations for order parameters $x_i$, leading to hybrid (discontinuous with critical scaling) transitions for $(k_a, k_b, ...)\geq2$. Overlap of links (multilinks) is critical for robustness and for the nature (continuous/discontinuous) of phase transitions (Cellai et al., 2013, Baxter et al., 2016).

3. Analytical and Modeling Frameworks

Multiplex networks necessitate generalized analytical tools.

Node- and link-level measures (Battiston et al., 2013):

• Node degree vectors, total and layer-specific.
• Overlapping and aggregated degree.
• Edge overlap and conditional probabilities for reinforcement across layers.
• Interdependence: For node $i$, $$\lambda_i = \sum_{j\neq i} (\psi_{ij} / \sigma_{ij})$$ where $\psi_{ij}$ is the count of shortest paths spanning multiple layers, $\sigma_{ij}$ the total count.

Growth models (Nicosia et al., 2013):

• Nodes arrive with prescribed numbers of “edge stubs” per layer.
• Preferential or mixed-attachment kernels allow new edges in one layer to depend on degrees in others:

$$\begin{bmatrix} F^{(1)}(k, q) \ F^{(2)}(k, q) \end{bmatrix} = \begin{bmatrix} c^{(1,1)} & c^{(1,2)} \ c^{(2,1)} & c^{(2,2)} \end{bmatrix} \begin{bmatrix} k \ q \end{bmatrix}$$

• Temporal arrival (simultaneous or delayed) of node replicas in different layers significantly shapes degree distributions, interlayer correlations, prevalence of “super-hubs,” and shortest path statistics.

Clustering and mesoscale analysis:

• Spectral and modularity-based measures are adapted via multi-layer Laplacians and partitioning heuristics (DeFord et al., 2017, Kao et al., 2017, Mondragon et al., 2017).
• Multilink-based communities (communities of edge-patterns rather than nodes) reveal mesoscale multiplex structure and differences in compressibility (whether communities may be represented after layer aggregation or require a full multiplex description).

4. Dynamical Processes and Cooperative Phenomena

Multiplex networks provide a substrate for a range of dynamical processes, many exhibiting compound phenomena not present in simplex networks.

• Cascading processes: Generalized threshold cascades show multiplexity-facilitated activation, with mathematical description in terms of recursive probability equations and local/global stability governed by the eigen-structure of the interlayer-coupled Jacobian (Brummitt et al., 2011).
• Percolation: Mutually connected giant component (MCGC) requires simultaneous connectivity in all layers; overlaps (multilinks) smooth transitions and increase robustness (Cellai et al., 2013, Baxter et al., 2016).
• Pattern formation: In reaction–diffusion systems where distinct species occupy different layers and react across layers, topology-driven Turing instability occurs even for equal mobility rates, controlled by the node degrees in the two layers and expressed by

$$(u) = \frac{f_u g_v - f_v g_u - f_u (v)(v)}{g_v (u) - (u)(v)(v)}$$

where $f_u, f_v, g_u, g_v$ are derivative entries in the Jacobian of the reaction terms (Kouvaris et al., 2014).

• Congestion: Multiplex organization can induce congestion in flow processes at lower injection rates than would occur in separate layers. Critical rates are determined by multiplexed “interconnected betweenness” $\mathcal{B}_{i\alpha}$ and critical injection rate

$\rho_c = (\tau/L) \frac{N-1}{\mathcal{B}^\star}$

where $\tau$ is the node-layer processing capacity and $L$ the number of layers. Increases in multiplexity may induce phenomena similar to Braess’ paradox (Solé-Ribalta et al., 2016).

5. Control, Robustness, and Predictability

Multiplex structure offers both challenges and opportunities for network control.

Facilitating or Suppressing Cascades (Brummitt et al., 2011):

• Introducing sparse layers (additional, even weak, channels of interaction) can dramatically increase susceptibility to global cascades, acting as a “catalyst.”
• Conversely, removing vulnerable (often sparser) layers is an efficient means to suppress unwanted cascading processes.

Robustness to Damage (Cellai et al., 2013, Baxter et al., 2016, Azimi-Tafreshi et al., 2014):

• Overlap among layers (multilinks) generally increases the robustness of the MCGC.
• In k-core percolation, depth of the core (resilience) depends on both layer degree distributions and overlap.
• Hybrid transitions are common: abrupt macroscopic failures accompanied by critical fluctuations and diverging avalanche sizes near percolation points.

Model-Hidden Multiplexity:

• Predictions based on simplex (aggregated) models can dramatically mischaracterize vulnerability, robustness, and critical points. Knowledge of all relevant interaction layers is necessary; neglect of even sparse or seemingly subcritical layers leads to errors due to missing cooperative effects.

6. Applications, Measures, and Methodological Developments

Multiplex networks underpin modeling and analysis in a variety of domains:

• Social networks: Analysis of trust, communication, operational, or business relations may reveal reinforcement effects—strong ties in one layer increase the probability of ties in others (Battiston et al., 2013).
• Biological networks: Layers reflect different types of cellular interactions. Multilink community analysis identifies key nodes (e.g., interneurons in C. elegans) participating in multiple communities or across high-complexity mesoscale structures (Mondragon et al., 2017).
• Transportation systems: Multiplex percolation, congestion, and efficiency studies inform optimal resource allocation and vulnerability analysis (Solé-Ribalta et al., 2016).
• Centrality and inference: New frameworks generalize centrality to account for semantics of each layer, including cyclic dependency (“configuration”) models (Spatocco et al., 2018), Functional Multiplex PageRank (assigning a node a function-valued centrality across multilink patterns) (Iacovacci et al., 2016), and data-driven inference of multiplex topology from dynamical time series (Ma et al., 2018).

Advanced measures have been proposed for clustering (community and layer similarities), centrality, multilink-based modularity, and for quantifying the similarity of mesoscale organizations across layers (e.g., via entropy/specificity-based indicators) (Iacovacci et al., 2016, Kao et al., 2017).

7. Open Problems and Research Directions

Work to date has established the fundamental influence of multiplex structure on both network topology and dynamics, but numerous research challenges remain (Lee et al., 2015):

• Systematic development of universality classes and critical phenomena for multiplex-specific transitions (including hybrid and discontinuous percolation).
• Extending analytical frameworks to spatial or clustered multiplexes, where local tree-likeness assumptions may fail.
• Incorporating broader types of layer coupling (cooperative, antagonistic, directional, hierarchical) and semantics in models.
• Full characterization of mesoscale compressibility: deciding which communities or functions require the full multiplex description.
• Scalable and context-aware centrality measures and inference methodologies.

In summary, multiplex networks yield a mathematically and conceptually rich modeling framework critical for accurate understanding and engineering of complex systems with diverse, interdependent interactions. Their paper entails advances in network theory, dynamics, and data-driven methodologies, with broad implications across scientific and technological domains.