Node-layer duality in networked systems

Abstract

Real-world networks typically exhibit several aspects, or layers, of interactions among their nodes. By permuting the role of the nodes and the layers, we establish a new criterion to construct the dual of a network. This approach allows to examine connectivity from either a node-centric or layer-centric viewpoint. Through rigorous analytical methods and extensive simulations, we demonstrate that nodewise and layerwise connectivity measure different but related aspects of the same system. Leveraging node-layer duality provides complementary insights, enabling a deeper comprehension of diverse networks across social science, technology and biology. Taken together, these findings reveal previously unappreciated features of complex systems and provide a fresh tool for delving into their structure and dynamics.

Fig. 1. Conceptual model of duality in complex networks.

Real networks typically consist of entities (nodes) interacting across different modes or aspects (layers). The unitary element of such multilayer networks is the node-layer duplet. In the left side illustration, a duplet identifies a node i = 1, 2, 3 and a layer α = I, II. By opportunely permuting the indices ($P^{\mathsf{T}}XP=Y$) nodes become layers (blue) and layers become nodes (red) without altering the local connectivity (right side). By construction, replica links become intralayer, intralayer links become replica, and interlayer links stay interlayer. This transformation is symmetric and it is always possible to go back to the primal version $X=PY{P}^{\mathsf{T}}$. The same system can be, therefore, equivalently represented as entities connected through aspects or aspects connected through entities. Two complementary descriptions can be then obtained depending on the representation side. In the primal nodewise (left), connectivity is integrated across aspects and one looks at how entities are interacting. In the dual layerwise (right), connectivity is integrated across entities and one rather looks instead at how their aspects are interconnected.

Fig. 2. Complementary properties of nodewise and layerwise connectivity.

a Schematic illustration of the different types of edge rewiring. The type of perturbation is determined by the probability to make a link displacement while i) keeping the layers unaltered (p_node), ii) keeping the nodes unaltered (p_layer), iii) altering both nodes and layers (p_tel). Dotted lines = old position, solid lines = new position. b Linear relation between layerwise and nodewise global connectivity changes. Global changes are computed as Euclidean distances (d) between multidegree centrality vectors. Lower slopes (higher $d_{\mathcal{X}}$) are obtained for p_node > p_layer. Higher slopes (higher $d_{\mathcal{Y}}$) are obtained for p_layer > p_node. Solid lines correspond to the theoretical formulas in the case of random networks with N = M = 200, connection density q = 0.0005, for the entire rewiring range r (color line). Scattered points correspond to synthetic random networks simulated with the same parameters. c Global connectivity changes as a function of rewiring parameters. Nodewise distances ($d_{\mathcal{X}}$) increase linearly with p_node (x-axis) and p_tel (white diagonals) but they cannot see edge displacements that keep nodes unvaried (p_layer → 1). Layerwise distances ($d_{\mathcal{Y}}$) increase linearly with p_layer (y-axis) and p_tel (white diagonals) but they are blind to edge displacements that keep layers unvaried (p_node → 1).

Fig. 3. Different size effects for multilayer and multiplex configurations.

In multilayer random networks, the largest global connectivity changes obtained after uniform rewiring occur in both nodewise and layerwise representations when the number of nodes N equal the number of layers M (black lines). However, in multiplex random networks the maximum change, as measured by the Euclidean distance (d) between multidegree centrality vectors, is reached when there are more nodes than layers (i.e., N = M + (N + M)/3, Text S1.10). In this plot N and M vary in a way to ensure the condition N + M = 200 so that N = M + 200/3. In addition, layerwise distances (orange) are by construction higher than nodewise distances (blue). These findings suggest that layerwise representations might be a-priori better candidates to spout out topological differences in multiplex networks, as compared to standard nodewise (Text S1.7). Solid lines correspond to the theoretical formulas for random networks with connection density q = 0.0005, rewiring ratio r = 0.5, and uniform rewiring probability (Text S1.9).

Fig. 4. Dual characterization of real-world multiplex networks.

a Log-log scatter plot of the nodewise (x-axis) and layerwise (y-axis) connectivity distances from uniformly rewired counterparts (Text S1.8). To avoid network-size biases, all values are further divided by the distances obtained rewiring equivalent random networks. The layer representation enables a better classification of networks that would be otherwise indistinguishable (e.g., Arxiv versus German transport highlighted by dashed circles). Networks optimally group into two clusters almost perfectly matching the spatial (violet) and non-spatial (green) nature of the systems (k-means=2, Silhouette score = 0.70, Supplementary Fig. S4). b Projection plots of two representative multiplex networks (i.e., German transport and Arxiv). In the nodewise, markers correspond to nodes and gray lines to layers. In the layerwise, markers correspond to layers and gray lines to nodes (Methods). Differently from Arxiv (top), the markers in the German network (bottom) tend to accumulate on few main lines meaning that both nodes and layers tend to contribute preferentially to few components. Also, values in the layerwise tend to span larger intervals in comparison with nodewise, indicating the presence of more heterogenous multidegree centrality distributions in the layerwise representation (Text S2). The standard deviation of the Arxiv’s layer multidegree centrality (σ_Y = 8931) is significantly higher than the German transport (σ_Y = 80.39), and this is eventually reflected by the relative higher layerwise distance d_Y in panel a).

Fig. 5. Alzheimer’s disease multilayer brain network disruption in region- and frequency-wise representations.

Multilayer brain networks are inferred from source-reconstructed MEG signals using cross-frequency coupling. In the primal representation, nodes correspond to brain regions (N = 70) and layers to different frequency bins (M = 77). Both intralayer and interlayer links are provided, estimating the amount of activity interaction (Methods). a Statistical difference (Wilcoxon test, Z-score) between brain region multidegree centralities of AD patients and healthy controls (HC). b Statistiscal difference (Wilcoxon test, Z-score) between brain frequency multidegree centralities of AD patients and healthy controls (HC). c Group-averaged nodewise and layerwise distances between AD and HC for different frequency resolutions (from M = 77 to M = 3). The asterisk marks the number of layers (M≤31) for which distances become significantly different (p < 0.05, FDR corrected). Vertical bars denote standard deviations. d Spearman correlation is computed between the frequency multidegree centralities of AD patients and their cognitive decline scores (MMSE). Colored areas show the significant ranges obtained from a cluster-based permutation procedure for multiple correlations. Darker color p = 0.0374; lighter color p = 0.049). The inset spots out the AD patients' MMSE as a function of the multidegree centrality at 9 Hz, giving the highest significant correlation R = 0.601 as indicated by the asterisk. Regressing curves resulting from a square fit are shown for illustrative purposes.