Multiplexity is temporal: effects of social times on network structure

TL;DR

This work tackles the problem of inferring context-specific multiplexity in large-scale communication data by reconstructing a multilayer network from latent social times derived from population-level weekly activity using orthogonal NNMF.The method yields four latent social times and shows that individual ties allocate their contacts across these times in a nonrandom, tie-specific way, enabling analysis of social foci at a society-wide scale.Key findings include that monoplex (single-layer) ties function as efficient bridges and that temporal multiplexity influences both local clustering and global connectivity, with transitive usage of social times among ego-alter triads, especially around weekends.The framework provides a flexible, data-driven approach to study temporal structure in multiplex networks, with potential applications in link prediction, diffusion modeling, and understanding social organization across diverse communication channels.

Abstract

Large-scale social networks constructed using contact metadata have been invaluable tools for understanding and testing social theories of society-wide social structures. However, multiplex relationships explaining different social contexts have been out of reach of this methodology, limiting our ability to understand this crucial aspect of social systems. We propose a method that infers latent social times from the weekly activity of large-scale contact metadata, and reconstruct multilayer networks where layers correspond to social times. We then analyze the temporal multiplexity of ties in a society-wide communication network of millions of individuals. This allows us to test the propositions of Feld's social focus theory across a society-wide network: We show that ties favour their own social times regardless of contact intensity, suggesting they reflect underlying social foci. We present a result on strength of monoplex ties, which indicates that monoplex ties are bridging and even more important for global network connectivity than the weak, low-contact ties. Finally, we show that social times are transitive, so that when egos use a social time for a small subset of alters, the alters use the social time among themselves as well. Our framework opens up a way to analyse large-scale communication as multiplex networks and uncovers society-level patterns of multiplex connectivity.

Figures (4)

• Figure 1: Social times as a basis for multiplexity. Different ties are active at different times. Using NNMF we obtain population-level latent signals of social activity and reconstruct a multilayer network based on how they explain the calling patterns of each tie. In our approach, all nodes (people) are in all layers, but ties (relationships) can be present in one, some or all layers. (a) We obtain social times by decomposing the weekly activity matrix of the population using NNMF. The plot depicts the signal strength for each hour of the week. We name signals based on the times when they are strongest. (b) Each tie can be modeled as a linear combination of different signals. For four example ties (rows), we show a histogram of the number of contacts placed during the week (each bin is 6 hours, with $w$ total contacts). Each tie is described by a unique linear combination of social times, or layer weights. The stylized densities (lines, scaled to match histogram counts) depict tie-specific weights for each layer. (c) Visual depiction of how an aggregate network of four ties can be reconstructed as a multilayer network, with ties having different weights, if any, for each social time. As an example, tie a-b is present in all layers as all coefficients are positive, while tie a-c is only active in work-week layers.
• Figure 2: Ties favour their own social times. We test our assumption that social ties have a temporal expression by controlling for other factors that may affect coefficients for social times. (a) The number of layers as a function of the number of contacts shows that low-contact ties tend to be present in few layers, while high-contact ties in many or all layers. A total of 19.1% of ties are present in one layer, 29.49% in two layers, 25.05% in three, and 26.36% in all layers. (b) We measure the multiplexity induced by contact volume by shuffling calls according to the population activity (fraction of calls during each hour), and classifying ties of $w$ shuffled contacts into layers. Higher contacts trivially induce multiplexity, but while low-contacts are heterogeneous, higher-contacts trivially induce a stable level of multiplexity. Mean entropy in solid line, shaded regions capture 1.5 standard deviations, and dotted lines capture the 1st and 9th deciles. (c) We define the focal multiplexity as the difference between the observed and the (average) induced multiplexity. The observed multiplexity is consistently lower than the induced (-0.2 units on average) and stable across contact counts. Focal multiplexity displays higher variance than the shuffled profiles, including for high-contact ties. (d) We use JSD to compare the layer coefficients between the full activity profiles and profiles of active hours; and the full activity with shuffled profiles. Shaded regions contain 80% of the distribution. Using the active hours suffices to broadly capture weight allocation patterns. In contrast, reference weight-allocation patterns under the shuffling model differ substantially from the observed data.
• Figure 3: Monoplex ties are locally and globally bridging. We measure multiplexity via entropy, assessing its effect on local tie bridginess and global network connectivity. (a) Within the framework of social foci, monoplex ties (blue link) are more bridging than multiplex ties (multi-colored link). We measure local bridginess with topological overlap $O$, the ratio of common neighbors (dark nodes) to all neighbors (dark and light). (Bottom) Ties with lower entropy have consistently low overlap, i.e., they serve as topological bridges, while high-entropy ties have high overlap. These results hold regardless of the number of contacts $w$. Ties are binned based on their contact count, colors depict the lower bin border. (b) Effect of link addition on the appearance of a giant connected component. Starting with all the nodes and no edges, we add links based on total contacts $w$ and entropy, sorted by low-first (dashed lines) and high-first (solid lines). The parameter $f$ denotes the fraction of added links, while $P_{GC}$ denotes the fraction of nodes that belong in the giant connected component. Adding low-entropy ties first leads to the appearance of a GCC faster than adding low-contact ties. After an initial faster appearance, the sizes of GCCs largely align. Both adding high-entropy and high-contact leads to a slower appearance of the GCC.
• Figure 4: Layers capture social behaviour around egos. We assess whether layers contain relevant topological characterizations around egos. (a) Layers capture some ego-specific behaviour dependent on degrees. We compare the observed and expected number of alters per layer under randomized calls. Higher-degree egos tend to over-represent more alters on weekend evenings, and under-represent alters on daytime layers. (b) Egos allocate alters to layers in unique ways. Pearson correlation for layer sizes $r\left(\frac{n_L}{k}, \frac{n_{L'}}{k}\right)$ show little correlation between the layer sizes. Under the shuffling model, the fractions of alters in layers are highly correlated. (c) We test whether the fraction of alters in a layer is revealing of triangles around an ego. Given the aggregate network $A$ and $n_L$ the number of alters in layer $L$ (in the example $n_L=5$), we sample $n_L$ alters independently (probability dependent on contacts $w_a$). (Left) Our first null hypothesis captures the number of triangles $t^L_a$ in a layer $L$ assuming transitivity --if two sampled alters $a$ are connected in the aggregate, they are also connected in the sampled layer; empirically such links might exist in another layer only (dashed line). (Right) The triangles around alters $t_a^A$ in the aggregate network $A$ captures whether the alters are embedded in the ego network regardless of the contact time. (d) Layers display a negative association between the fraction of alters and the ratio of observed and expected triangles. Egos that contact many of their alters within a layer have less dense personal networks in that layer than expected, but there is also layer-specific behaviour. Weekend layers tend to preserve more triangles, while workweek layers consistently preserve less triangles than expected. Dots denote deciles of $n_L/k$ calculated over the layer. (e) All layers have more embedded alters than expected. Weekend layers over-represent such alters at higher rates. Ratios tend to 1 as selecting all alters implies observing all aggregate connections empirically and in expectation.