Mutually Connected Giant Component (MCGC)

• MCGC is defined as a maximal set of nodes that are mutually connected across all network layers, with each node’s replicas also connected.
• The framework uses self-consistency equations and percolation theory to reveal both continuous and hybrid phase transitions in complex systems.
• Understanding MCGC is crucial for predicting the robustness and failure properties of interdependent infrastructures like power grids, communications, and neural networks.

A Mutually Connected Giant Component (MCGC) is a structural hallmark in multilayer, interdependent, or temporal networks, representing a maximal subset of nodes such that for every pair of nodes, paths exist connecting them in every relevant network layer or channel. In the MCGC, node membership is recursive: a node is present if it is connected to the component within each layer it participates in, and all its interdependent replicas (or counterparts) in other layers are also present. The existence, size, and stability of the MCGC are central to the robustness and failure properties of complex systems ranging from power grids and communications to neural and temporal networks, and are associated with rich percolation phenomena including @@@@1@@@@ phase transitions, multicriticality, and anomalous scaling.

1. Formal Definitions and Mathematical Frameworks

The MCGC encompasses several mathematical formulations depending on network architecture:

• Multiplex and Networks of Networks: Given $M$ layers (networks) $\alpha=1,\ldots,M$ and $N$ nodes $i=1,\ldots,N$ with a "replica" structure (i.e., node $i$ exists as $(i,\alpha)$ in each layer), mutually connected means two nodes belong to the MCGC if there exists a path between them in every layer they share. This requirement is global: all interdependent replicas must be present and connected within their own layer (Bianconi et al., 2014).
• Temporal Networks: A component is "mutually connected" if every ordered pair $u,v$ can reach each other via temporal paths, with conditions differing between open (paths may use external nodes) and closed (paths must remain inside the component) variants (Becker et al., 2022).
• Directed and Spatially Embedded Networks: For instance, in continuum disk-spin percolation, the "giant strongly connected component" (GSCC) is the MCGC analog, requiring mutual reachability via directed geometric paths among all its nodes (Caravelli et al., 2015).

Table: Key Mathematical Objects Defining the MCGC

Setting	Membership Criterion	Reference
Multiplex (replica nodes)	Node must be present in all connected layers and in each layer's GCC	(Bianconi et al., 2014)
Temporal open/closed component	For all $u,v$ in $Z$, temporal path $u\rightarrow v$ exists (path constraints)	(Becker et al., 2022)
Directed geometric network (GSCC)	Strong connectivity: paths $i\to j$ and $j\to i$ inside component	(Caravelli et al., 2015)

2. Percolation Theory and Self-Consistency Equations

Emergence of the MCGC is characterized by percolation transitions, controlled by self-consistency equations formulated in the configuration model or generating function frameworks. These equations express the fraction of nodes in the MCGC (order parameter) as a function of control parameters such as occupation probability $p$, mean degrees, and the multilayer structure.

In locally tree-like multiplexes with joint degree distribution $P(q_1,\ldots,q_M)$, the self-consistency for the MCGC size $S$ is

$$u_\alpha = 1 - p + p \sum_{q_1,\dots,q_M} \frac{q_\alpha P(q_1,\dots,q_M)}{\langle q_\alpha \rangle} (1-u_\alpha)^{q_\alpha-1} \prod_{\beta \neq \alpha} (1-u_\beta)^{q_\beta}$$

$$S = p \sum_{q_1,\dots,q_M} P(q_1,\ldots,q_M) \prod_{\alpha=1}^{M} \left[ 1 - (1-u_\alpha)^{q_\alpha} \right]$$

These equations generalize to networks of networks, edge-overlapping multiplexes, and temporal graphs with appropriate adjustments (Baxter et al., 2020).

In temporal Erdős–Rényi graphs, the emergence of a giant temporally connected component is a sharp threshold phenomenon, with the largest mutual component jumping from $o(n)$ to $n-o(n)$ at $p_c = (\log n)/n$ (Becker et al., 2022).

3. Critical Phenomena and Transition Types

The MCGC can undergo continuous (second-order) or hybrid (discontinuous) transitions:

• $M=2$ Layers: The transition is generally continuous; for uncorrelated Poisson degree distributions, the threshold is $c_c=1$ (Baxter et al., 2020).
• $M \geq 3$ Layers: The transition is hybrid, exhibiting a finite jump $S_c>0$ in MCGC size at threshold $c_c$, with post-critical growth $S-S_c\propto (c-c_c)^{1/2}$. As $M\to\infty$, $c_c \sim \log M$ and $S_c\to 1$ (Baxter et al., 2020).
• Power-law Degree Distributions: The order of the transition depends sensitively on degree exponent $\gamma$. For $\gamma>1+ 1/(M-1)$, the hybrid jump persists; for $\gamma<1+1/(M-1)$, the transition becomes continuous from $p_c=0$ (Baxter et al., 2020).

Edge (overlap) correlations in multiplexes can lead to multiple, reentrant hybrid transitions and split loci in the phase diagram, reflecting complex underlying interlayer dependencies (Baxter et al., 2016).

4. Network Architecture and Correlations

The detailed topology of the supernetwork (network of networks), interlayer dependency architecture, and edge correlations fundamentally shape MCGC phenomenology:

• Replica Networks and Independence from Supernetwork Structure: For networks of networks with replica nodes and any connected supernetwork, the MCGC size and percolation properties are fully described by the corresponding $M$-layer multiplex; the supernetwork topology per se does not modify the MCGC (Bianconi et al., 2014).
• Correspondently Coupled Networks (CCN): When degrees are perfectly correlated, the MCGC is more robust; broader degree distributions lower the percolation threshold $p_c$ and enhance robustness (Buldyrev et al., 2010).
• Overlapping Multiplexes: In two-layer networks with partial edge overlap and tunable correlations, the nature of MCGC transitions changes dramatically. Assortative edge correlations induce repeated hybrid transitions, while strong disassortativity can split transitions and induce sequential jumps in mutual connectivity (Baxter et al., 2016).

5. Temporal and Directed Percolation Generalizations

Temporal graphs demand a temporally ordered notion of connectivity, distinguishing between open and closed mutually connected components. In the temporal Erdős–Rényi model, the threshold for the emergence of a giant temporally connected component is $p_c = (\log n)/n$, which is later than the static (undirected) percolation threshold of $p_c=1/n$ (Becker et al., 2022).

In directed spatial networks (e.g., disk-spin percolation models motivated by neuronal networks), the MCGC is realized as a GSCC, with percolation thresholds and critical exponents sensitive to local directionality (“temperature”) and spatial constraints (Caravelli et al., 2015). The GSCC fraction $\Delta(p)$ serves as the order parameter, and universality can be broken by anisotropy.

6. Hybrid Transitions, Robustness, and Multicriticality

Hybrid transitions, where the size of the MCGC jumps discontinuously at the percolation threshold and then displays square-root singular growth, are characteristic of many multiplex settings for $M\geq3$ (Baxter et al., 2020, Bianconi et al., 2014). The conditions for the emergence of hybrid versus continuous transitions are governed by the interplay of number of layers, degree distribution tails, and interlayer correlations.

Physical phenomena associated with the MCGC include cascading failures, robustness against random node or edge removal, and the coexistence of multiple transition points (multicriticality) under suitable interlayer correlations or partial edge overlap (Baxter et al., 2016). The presence, size, and stability of the MCGC are direct predictors of systemic resilience in interdependent infrastructures and communication networks.

7. Summary Table: MCGC Transition Properties by Network Class

Network Class	Transition Type	Threshold Expression	Notable Effects / Features
Duplex ($M=2$ Poisson)	Continuous	$c_c = 1$
Multiplex $M \ge 3$ Poisson	Hybrid (discontinuous)	$S=\left[1-e^{-cS}\right]^M$	$S_c>0$, square-root singularity
Temporal ER ($n$ nodes)	Sharp (w.h.p.)	$p_c = (\log n)/n$	Both open/closed coincide
Edge-overlapping/multiplex	Hybrid, multiple/reentrant possible	Dependent on overlap/correlations	Repeated, split transitions
Power-law degree ($\gamma$)	Continuous or hybrid	Depends on $\gamma$ and $M$	$p_c=0$ for $\gamma <1+1/(M-1)$

• Networks of networks with replica nodes, independence of supernetwork topology: (Bianconi et al., 2014)
• Mutual giant component in CCN: (Buldyrev et al., 2010)
• Temporal dynamics and thresholds in ER graphs: (Becker et al., 2022)
• Correlated overlap in multiplex, multicriticality: (Baxter et al., 2016)
• Exotic critical behavior, general theory: (Baxter et al., 2020)
• GSCC and spatial/directed percolation: (Caravelli et al., 2015)