Network Geometry

Finding the optimal flow through a given network is relevant in a wide range of systems, including urban transportation, power grids, river basins, plant and animal venation. Gaining insight into the mechanisms behind the structure of optimal networks is of crucial importance, especially in the context of designing transport systems. Many such systems involve multiple modes or layers of transport, motivating the study of optimal flow in multiplex networks.In this article we introduce an analytically solvable model for symmetry breaking in optimal networks. We provide a comprehensive solution for elementary model networks and demonstrate two different mechanisms of symmetry breaking. First, the optimal state can undergo a bifurcation, such that the symmetric state vanishes entirely. Second, the symmetric and the symmetry-broken state co-exist and exchange roles as global and local minima of the optimisation problem.We extend our fundamental results to (i) study the impact of uncertain supply and demand and (ii) the optimal structure of multilayer networks. Combining analytic and numeric tools, we show how noise impacts the emergence to symmetry breaking and construct the phase diagram. For multi-layer networks, we demonstrate symmetry breaking between layers, where one layer carries the entire flow despite a symmetry in supply and demand between the layers.Our results can hold importance for future energy systems, where electricity, gas and heat networks are integrated and generation becomes increasingly uncertain.

Multiplex networks provide a powerful multilayer framework for modeling complex systems with diverse interaction types. The different layers are expected to be correlated, a fact that has been confirmed in previous works. However, a comprehensive analysis of the overlap caused by such correlations has been lacking, especially for higher order structures. We show that the mutual clustering coefficient -- a measure for the amount of triangles present in all layers -- is high in many real networks. This result can be explained through an enhanced coupling of the overlapping edges to the underlying metric space, which conditions their connectivity.

Conventional network-based algorithms may not work efficiently in certain circumstances, including when facing enormous network sizes and incomplete or inaccurate networks, such as missing links. To address this challenge, one possible method is identifying the shortest path based on network geometry. In this work, we aim to investigate, in a direct way, the extent the shortest paths are aligned along geodesic curves in Euclidean geometric networks and if it can be used to infer shortest paths. We show that the deviation, i.e. the distance from a node to the geodesic, of a node belonging to the shortest path is smaller than the deviation of randomly selected nodes. Additionally, we demonstrate that the deviation of the nodes comprising the shortest path decreases as network size increases. However, an increment in the expected degree leads to a non-monotonic behaviour characterized by the presence of a local minimum deviation. We investigate the monotonic behaviour in three distinct phases: the initial growth arises from the connectivity transition, the tail is dominated by large degree and the middle phase reflects the interplay between these two factors. To investigate the significance of shortest path alignment, we ask if it can be used to infer shortest paths. To this end, we consider the shortest path problem in a geometric network where only nodes, node coordinates and connection probabilities are known to the observer. We try to find the shortest paths based on the vicinity to geodesic curves and also by first reconstructing networks. Our method shows promising potential: identifying shortest path nodes using deviation is not only computationally efficient, but also robust to blurred coordinates in SRGGs across various network topologies, which can benefit practical scenarios where the coordinates of nodes are not fixed or precisely known. Specifically, the benefits of our method become higher as noise increases.

When nodes can be divided into two disjoint sets and no links connect two nodes in the same set a network is called bipartite. There are many examples of such networks as collaboration networks (authors and articles), actor networks (actors and movies), metabolic networks (metabolites and chemical reactions), and many more. Despite their abundance, these networks are less studied compared to unipartite networks. Recent works show that network geometry framework can be extended to bipartite networks in which nodes of both types lie in a metric space, and the connection depends on their distance. In this work, we extend the current one-dimensional model into $D$ dimensions by introducing a bipartite-$\mathbb{S}^D/\mathbb{H}^{D+1}$ model when nodes of two kinds lie in the same $D$-dimensional similarity space. Moreover, we develop an embedding framework that allows the creation of multidimensional hyperbolic maps of real bipartite networks. Using this framework, we analyzed two datasets: Unicodelang and Flavour datasets. In the first one, countries are connected to the languages spoken in that region. In the second one, food ingredients are coupled with the food compounds. Moreover, we apply our approach in the machine learning context, where a node feature matrix is represented as a bipartite graph. Our tests on the node classification and link prediction tasks show a significant improvement when a correlation between nodes' labels and nodes' features is high. The findings pinpoint the cases in which a unipartite network is redundant or not when a feature matrix is known.

Key features of real-world networks include sparsity, the small-world property, inhomogeneous degree distribution, high clustering coefficients, and community structure. So-called Random Hyperbolich Graphs (RHG) can naturally explain many of these observed network features by assuming that nodes are embedded in hidden metric space having negative curvature.RHG models provide direct control on the degree distribution and transitivity of the generated networks, and can produce networks with strong community structures, even without explicitly incorporating community-generating mechanisms. However, they lack parameters to control the existence of blocks – or communities – with specific mixing patterns, another common feature in real-world networks.To address these limitations, we introduce the Random Hyperbolic Block Model (RHBM), a max-entropy network model obtained constraining the expected total energy of the system, degree sequence and mixing matrix. The RHBM extends both the degree-corrected stochastic block model and the S1 model, enjoying a combination of their properties.To highlight the utility of the RHBM, we use it to both generate synthetic networks and to randomize real networks. By using the d-mercator embedding tool, we find the maximum-likelihood underlying geometry of these networks, to show that some of them are “non-geometric”, in the sense that purely geometric models cannot reproduce the input community structure, while the RHBM can. Our findings suggest that the popularity-similarity principle underlying hyperbolic models cannot explain mesoscale structures like communities, and that an additional term in the expressionof edge probabilities is required.