Diffusion geometry of multiplex and interdependent systems

Complex networks are characterized by latent geometries induced by their topology or by the dynamics on them. In the latter case, different network-driven processes induce distinct geometric features that can be captured by adequate metrics. Random walks, a proxy for a broad spectrum of processes, from simple contagion to metastable synchronization and consensus, have been recently used, Domenico [Phys. Rev. Lett. 118, 168301 (2017)] to define the class of diffusion geometries and pinpoint the functional mesoscale organization of complex networks from a genuine geometric perspective. Here we first extend this class to families of distinct random walk dynamics—including local and nonlocal information—on multilayer networks—a paradigm for biological, neural, social, transportation, and financial systems—overcoming limitations such as the presence of isolated nodes and disconnected components, typical of real-world networks. We then characterize the multilayer diffusion geometry of synthetic and empirical systems, highlighting the role played by different random search dynamics in shaping the geometric features of the corresponding diffusion manifolds.

Figure 1

Different multilayer networks and their supra-adjacency matrix representation [37, 40]: (a) an edge-colored multigraph, with layers corresponding to colors and no interlayer connections; (b) a multiplex network where the replicas in the different layers are interconnected sequentially, a type of intertwining or diagonal coupling; and (c) the most general interconnected case, where interlayer connections are not restricted to replicas (exogenous interactions). Latin letters denote nodes, and Greek letters are used for layers. $D(i;\alpha \beta )$ is the intensity of the intertwining between state nodes $i,\alpha$ and $i,\beta$. $M_{j\beta }^{i\alpha }$ describes the general interlayer connectivity.

Figure 2

A synthetic two-layer network with unitary diagonal coupling $D(i;\alpha ,\beta )=1$, for all $i$ and $\alpha \ne \beta$, and its functional characterization. (a) Isolated nodes, the largest connected component (LCC), and other different components have been highlighted in the two layers. (b) The supra-adjacency matrix of the multiplex. (c) The $NL\times NL$ diffusion supradistance matrices for three different RW dynamics and $t=1,2,5$. The PrRW is evaluated for $r=0.5$.

Figure 3

Average diffusion distance matrices of the two-layer synthetic network shown in Fig. 2, w.r.t. three RW dynamics: (top) classical, (middle) diffusive, and (bottom) physical with relaxation $r=0.5$. The state nodes $(i,\alpha )$ are colored in the corresponding dendrogram according to the layer they belong to, except for isolated nodes, highlighted in purple. Small squares with a black border indicate those state nodes grouped together, e.g., $(29,1),(29,2)$ in the top panel.

Figure 4

Comparing different types of ($N\times N$) average diffusion distance matrices between physical nodes, regardless of layers. (a) The weighted adjacency matrix of the aggregated network. The diffusion distance matrix (w.r.t. the CRW and averaged over time) (b) of the aggregated network corresponding to the synthetic multiplex of Figs. 2 and 2 of the edge-colored network. (d) The equivalent diffusion distance matrix obtained with the reduction of Eq. (8). The distances are rescaled to $[0,1]$, normalizing by the corresponding maximum over all node pairs [aggregated: $max\left({\overline{\mathbf{D}}}_t\right)\approx 0.13$, edge-colored: $max\left({\overline{\mathbf{D}}}_t\right)\approx 0.16$, equivalent distances: $max\left({\overline{\mathbf{D}}}_t^{\text{eq}}\right)\approx 0.11$].

Figure 5

Average diffusion distance ${\overline{D}}_t$ on two-layer multiplexes with different topologies, for fixed values of global average edge overlap (Barabasi-Albert and Watts-Strogatz) and partition overlap (Girvan-Newman) between layers. Dendrograms on the right-hand side of each distance matrix represent the corresponding hierarchical clustering to highlight the mesoscale organization of the system, with color encoding the planted node assignment in each layer. Distances have been rescaled $\frac{{\overline{D}}_t}{max{\overline{D}}_t}$. The Barabasi-Albert model is characterized by the presence of hubs, which are clearly recognizable in the supradistance matrix w.r.t. the DRW, despite the small overlap between layers. The Girvan-Newman two-layer multiplex has a mesoscale organized in strong communities, partly overlapping across layers. Note that at variance with edge overlapping, here partition overlapping is defined in terms of nodes belonging to the same group without requiring those nodes to be connected by an overlapping edge.

Figure 6

Frobenius norm of the supradistance matrices of a synthetic two-layer network, w.r.t. a DRW. Each layer is generated from a Barabasi-Albert model as in Fig. 5. As the fraction of interlayer links grows we move from two disconnected multiplexes to a fully interconnected multilayer whit all $N^2$ connections across layers. The heatmaps of $D_{t=5}$ are four representatives of the different regimes: (a) uncoupled layers; (b) a single-interlink between the replicas of a random node coupling the two layers, i.e., $\left[\frac{1}{N^2},\frac{1}{N}\right]$ partially interconnected multiplex, and [1/N] fully interconnected multiplex (all state nodes corresponding to the same physical node are interconnected); (c–d) $\left[\frac{1}{N},1\right]$ multilayer regime consisting of an interconnected multiplex with the addition of cross-links between state nodes of distinct physical nodes.

Figure 7

As in Fig. 6 for networks with a mesoscale organized in four strong communities in each layer and partition overlap of 1%. The heatmaps on the top correspond to $D_{t=1}$, while the others are $D_{t=5}$.

Figure 8

Average diffusion distance supramatrices ${\overline{D}}_t$ organized according to their hierarchical clustering as in previous analyses, and projection in ${\mathbb{R}}^3$ of the corresponding diffusion manifolds, in the case of the two real multiplex networks: the London public transportation network (top panels) and the social relationships of Indonesian terrorists (bottom panels). In both cases, nodes have been embedded in space, more specifically in ${[-1,1]}^3$, through multidimensional scaling, a metric-preserving embedding method that depends on two parameters: a distance (or dissimilarity) matrix and an embedding dimension [50] (see Appendix pp3 for details). To facilitate the visualization of the three-dimensional embeddings, we draw the surface which better approximates the cloud of nodes in ${\mathbb{R}}^3$ and project it on the plane, encoding the third dimension with colors. Nodes are shown as dots on the top of the surfaces.

Figure 9

Estimating the similarity between diffusion manifolds corresponding to different RW dynamics, evaluated on the London multimodal transportation network. We calculate the Pearson's correlation (encoded by size and color) between the entries of pairs of supradistance matrices (Mantel's statistic), which is then tested for significance by permutation (permutation test), with $\alpha =0.001$. The test is general, because it applies directly on distance matrices, whereas any test performed on the low-dimensional embedding of diffusion manifolds would be less precise because of the information loss during projections.

Figure 10

Same as in Fig. 9 for the Noordin Top Terrorists Network.

Figure 11

Average diffusion distance ${\overline{D}}_t$ for a two-layer multiplex with mesoscale organization in strong communities, partly overlapping across layers. Note that at variance with edge overlapping, here partition overlapping is defined in terms of nodes belonging to the same group without requiring those nodes to be connected by an overlapping edge.