Hyperbolic mapping of human proximity networks

Abstract

Human proximity networks are temporal networks representing the close-range proximity among humans in a physical space. They have been extensively studied in the past 15 years as they are critical for understanding the spreading of diseases and information among humans. Here we address the problem of mapping human proximity networks into hyperbolic spaces. Each snapshot of these networks is often very sparse, consisting of a small number of interacting (i.e., non-zero degree) nodes. Yet, we show that the time-aggregated representation of such systems over sufficiently large periods can be meaningfully embedded into the hyperbolic space, using methods developed for traditional (non-mobile) complex networks. We justify this compatibility theoretically and validate it experimentally. We produce hyperbolic maps of six different real systems, and show that the maps can be used to identify communities, facilitate efficient greedy routing on the temporal network, and predict future links with significant precision. Further, we show that epidemic arrival times are positively correlated with the hyperbolic distance from the infection sources in the maps. Thus, hyperbolic embedding could also provide a new perspective for understanding and predicting the behavior of epidemic spreading in human proximity systems.

Dynamics of cold random hyperbolic graphs with link persistence

“…Figure 3 validates the above analysis. We note that the average contact duration, tc = τ −2 t=1 tP c (t), depends on both the temperature T and the link-persistence probability ω = 1 − ξ, as dictated by (23). In particular, tc increases with decreasing T or with increasing ω, see Fig.…”

“…( 15) on p. 77). This expansion allows us to express the hypergeometric function in (23) for t → ∞, as…”

“…For instance, in human proximity networks, studies have reported power-law contact distributions with exponents larger than or close to two [17,18], and power-law intercontact distributions with exponents between one and two [19][20][21][22]. Based on the dynamic-S 1 , human proximity networks have been recently mapped to hyperbolic spaces and related applications have been explored [23]. We note that the dynamic-S 1 exhibits realistic dynamical properties only in the cold regime (T ∈ (0, 1)) but not in the hot (T > 1) [24].…”

Finding shortest and nearly shortest path nodes in large substantially incomplete networks

“…Recent works indicate that common topological properties of real networks, such as the hierarchical organization, the heterogeneity in the number of connections per node, strong clustering coefficient, and self-similarity (3), are best mapped into latent spaces, which are hyperbolic rather than Euclidean. Notable examples of real networks with effective hyperbolic geometries are the PPI networks (4), the Internet (5), and human proximity networks (6). At the same time, there is no consensus on the extent to which shortest paths in these networks align along geodesic curves.…”

Finding shortest and nearly shortest path nodes in large substantially incomplete networks by hyperbolic mapping

“…Recent works indicate that common topological properties of real networks, such as the hierarchical organization, the heterogeneity in the number of connections per node, strong clustering coefficient, and self-similarity 3 , are best mapped into latent spaces, which are hyperbolic rather than Euclidean . Notable examples of real networks with effective hyperbolic geometries are the PPI networks 4 , the Internet 5 , and social networks 6 , 7 . At the same time, there is no consensus on the extent to which shortest paths in these networks align along geodesic curves.…”