Functional Multiplex PageRank

• Functional Multiplex PageRank is a centrality measure that generalizes classic PageRank by incorporating multilink weights across multiple network layers.
• It employs a nonlinear, tunable weight model to capture redundancy and synergy among different connection types, enabling context-dependent node assessments.
• The method uses efficient power iteration for convergence and has been validated on real-world networks like airport systems, neuronal connectomes, and academic collaborations.

Functional Multiplex PageRank (FMP) generalizes the classic PageRank centrality to multiplex networks, where nodes are connected by multiple types of links across several layers. Unlike single-layer measures, FMP allows quantification of node importance as a continuous, multivariate function of how much weight is assigned to each possible type of layer-overlap (multilink) between node pairs. This approach accounts for nonlinear and context-dependent effects arising from link redundancy—where links may exist in different subsets of layers—and enables a nuanced, tunable ranking of nodes according to the specific relevance ascribed to each pattern of multilayer connectivity (Iacovacci et al., 2016).

1. Multiplex and Multilink Structures

A multiplex network consists of a fixed set of nodes $V=\{1,\dots,N\}$ and $M$ distinct layers, each representing a specific connection type (e.g., communication, trust, transport mode). The connections in layer $\alpha$ are described by an adjacency matrix $a_{ij}^{[\alpha]}$. For any node pair $(i, j)$, the structure distinguishes all ways in which they can be linked across layers via the concept of multilinks: vectors $\vec m = (m_1,\dots,m_M)$ where $m_\alpha=1$ if $a_{ij}^{[\alpha]}=1$, otherwise $0$. Thus, every node pair has one of $2^M$ multilink types, capturing all overlap patterns from “no link” ($\vec 0$) to “present in every layer.”

The multi-adjacency matrix $A_{ij}^{\vec m}$ encodes whether the $j\to i$ link belongs to multilink $\vec m$: $$A_{ij}^{\vec m} = \prod_{\alpha=1}^M \left[m_\alpha\, a_{ij}^{[\alpha]} + (1 - m_\alpha)(1 - a_{ij}^{[\alpha]})\right]$$ so that exactly one $A_{ij}^{\vec m}=1$ for each pair (Iacovacci et al., 2016).

2. Mathematical Formulation of Functional Multiplex PageRank

FMP assigns a nonnegative weight $w_{\mu}$ to each nontrivial multilink type $\mu$. The centrality $\Pi_i(\mathbf w)$ for node $i$ is the equilibrium probability for a random walk that, at each step, chooses among outgoing links from $j$ with probabilities proportional to the multilink weights $w_\mu$, thus: $$\boxed{ \Pi_i(\mathbf w) = \alpha \sum_{j=1}^N \sum_{\mu \neq \vec0} A_{ij}^{\mu}\, w_{\mu}\, \frac{\Pi_j(\mathbf w)}{\kappa_j} + \beta\, v_i }$$ where:

• $\alpha \in (0,1)$ is the damping factor (typically $\approx 0.85$),
• $$\kappa_j = \sum_{i,\mu \neq 0} A_{ij}^\mu\, w_\mu + \delta_{0, \sum_{i,\mu \neq 0} A_{ij}^\mu w_\mu}$$ is the out-strength normalization,
• $v_i = 1/N$ (or normalized over all active nodes) for the teleportation vector,
• $\beta = 1-\alpha$ ensures stochastic normalization,
• $\delta_{a,b}$ is the Kronecker delta.

The critical feature is that $\Pi(\mathbf w)$ is homogeneous: scaling all $w_{\mu}$ by a common nonzero factor leaves the result unchanged. Only the ratios of the weights matter, yielding $2^M-2$ effective parameters for $M$ layers (Iacovacci et al., 2016).

3. Algorithmic Computation and Convergence

The FMP vector $\Pi(\mathbf w)$ is computed by straightforward power iteration:

1. Initialize $\Pi_i^{(0)} = 1/N$ for all $i$.
2. For each iteration $t$:
   • Compute $\kappa_j^{(t)}$ for all $j$.
   • Update

   $$\Pi_i^{(t+1)} = \alpha \sum_{j,\;\mu\neq 0} A_{ij}^\mu\, w_\mu\, \frac{\Pi_j^{(t)}}{\kappa_j^{(t)}} + (1-\alpha)\, v_i.$$

3. Stop when $\|\Pi^{(t+1)} - \Pi^{(t)}\|_1 < \varepsilon$.

This method converges geometrically due to the irreducibility and aperiodicity of the induced transition matrix, mirroring the guarantees for standard PageRank (Iacovacci et al., 2016). Each update is equivalent to applying PageRank to a single weighted directed network with aggregated adjacency $\sum_\mu A^\mu w_\mu$. Computational complexity per parameter choice is thus $O(T(N+E))$, with $T$ iterations and $E$ nonzero links.

4. Special Cases and Connections to Multiplex PageRank

Setting specific forms for the weights $w_\mu$ recovers various known centrality measures:

• Ordinary PageRank on layer $\alpha$: Take $w_\mu = w^\star$ for each multilink $\mu$ corresponding to presence only in layer $\alpha$, all others zero.

• Aggregated network PageRank: Set $w_\mu = w^\star$ for all nontrivial $\mu$.

• Complete overlap only: Weight only the all-one multilink $\vec{1} = (1,1,\dots,1)$.

• Multiplex PageRank variants: By constraining $w_{\mu}$ to depend linearly or nonlinearly on the number of overlapping layers, one obtains additive, multiplicative, and combined bias schemes (Iacovacci et al., 2016, Halu et al., 2013):

  • Additive: link selection is uniform, teleportation is biased by another layer’s centrality.
  • Multiplicative: links are weighted by the centrality in a reference layer, teleportation is uniform.
  • Combined: both steps include the external layer bias.

A key advance of the FMP formalism is that beyond these linear biases, it allows full non-linear dependence of node centrality on multilink overlap, e.g., $w_{(1,1)} \neq w_{(1,0)} + w_{(0,1)}$ in a duplex, thus capturing synergies or redundancies beyond additive aggregation (Iacovacci et al., 2016).

5. Empirical Examples and Interpretations

FMP provides rich insight when applied to real-world multiplex networks:

• Airport multiplex (Lufthansa vs. British Airways): By mapping FMP surface $\Pi_i(\phi, \theta)$ as a function of relative multilink weights, one observes for certain airports—e.g., Frankfurt—that centrality peaks at exclusive Lufthansa links, while others such as Düsseldorf derive more centrality from enhanced overlap links (Iacovacci et al., 2016). Absolute and average multiplex rankings are defined respectively by maximizing or averaging over all weightings, giving robust measures of node influence.
• C. elegans connectome: In the duplex of chemical synapses and gap junctions, central interneurons show distinct “patterns to success”—some achieve prominence mainly via one interaction type, others through overlap. Correlational clustering of the FMP surfaces recovers known neuron classes, demonstrating the framework’s discriminative capacity (Iacovacci et al., 2016).
• PRE author network (collaboration/citation): Distinct scientists (e.g., Newman, Stanley, Strogatz) show unique FMP “pattern to success” profiles: some are driven almost purely by citations, while others retain high rank even when overlap between collaboration and citation is accentuated, revealing multilayer paths to prominence (Iacovacci et al., 2016, Iacovacci et al., 2016).

A summary of applications and parameterizations is provided below:

Dataset	Layers	Main Finding
Lufthansa–BA airports	Flight operator duplex	Centrality depends on overlap and layer exclusive links; hub correlations revealed by FMP surface scanning
C. elegans connectome	Synapse, gap junction	Central neurons show distinct overlap dependencies; clustering recovers functional classes
PRE authors	Collaboration, citation	“Pattern to success” informs on the sources of scientific prominence

6. Parameter Dependence and Sensitivity

Since only the ratios of $w_\mu$ matter, practical analyses span the surface of nontrivial weight vectors, often via parametrization in terms of angles (duplex) or overlap-bias parameters (e.g., $w_\mu = q^{\nu(\mu)-1}$, where $\nu(\mu)$ is the overlap count for multilink $\mu$) (Iacovacci et al., 2016). This enables sensitivity analysis, detection of redundancy or antagonism between layers, and robust node ranking even under variation in connection pattern importance.

Absolute FMP rankings (maximum over weights) and average FMP rankings (mean over sampled weights), as well as “pattern to success” profiles, can be systematically computed and interpreted, providing a multidimensional centrality landscape for each node.

7. Theoretical Extensions and Related Frameworks

FMP forms part of a broader landscape of multiplex centrality measures. Notably, supracentrality frameworks construct a global $NL \times NL$ matrix incorporating both intra- and inter-layer coupling, and define PageRank-type scores via dominant eigenvectors (Taylor et al., 2019). In the limit of strong coupling, all layers collapse to an aggregated network; for weak coupling, the scores reflect individual layer PageRanks. This spectral approach connects with FMP by emphasizing the interplay of spectral properties and coupling topologies, and encompasses PageRank as a special case (Taylor et al., 2019).

Functional Multiplex PageRank facilitates principled, nonlinear, and parameter-dependent assessment of node importance in multiplex networks, enabling empirical analyses that reveal the mechanisms and synergies by which nodes achieve prominence under multilayer interaction regimes (Iacovacci et al., 2016, Iacovacci et al., 2016, Halu et al., 2013, Taylor et al., 2019).