Universal patterns in egocentric communication networks

TL;DR

The paper investigates universal patterns of tie-strength heterogeneity in egocentric communication networks across diverse channels. By analyzing numerous datasets and formulating a concise alter-activity model, the authors show that cumulative advantage versus random choice, governed by the preferentiality parameter $\beta=t_r/\alpha_r$, drives the universal distribution of alter activity. They derive an analytically tractable master equation and show two asymptotic regimes: a gamma-distributed heterogeneous regime for $\beta>1$ and a Poisson/Gaussian homogeneous regime for $\beta<1$, with a maximum-likelihood framework and goodness-of-fit tests confirming the model captures a substantial fraction of ego networks. Importantly, they reveal persistent individual 'social signatures' in activity allocation that endure despite turnover in alters, highlighting general mechanisms shaping social-network structure and individual behavior with broad implications for temporal networks modeling.

Abstract

Tie strengths in social networks are heterogeneous, with strong and weak ties playing different roles at both the network and the individual level. Egocentric networks, networks of relationships around a focal individual, exhibit a small number of strong ties and a larger number of weaker ties, a pattern that is evident in electronic communication records, such as mobile phone calls. Mobile phone data has also revealed persistent individual differences within this pattern. However, the generality and the driving mechanisms of this tie strength heterogeneity remain unclear. Here, we study tie strengths in egocentric networks across multiple datasets containing records of interactions between millions of people over time periods ranging from months to years. Our findings reveal a remarkable universality in the distribution of tie strengths and their individual-level variation across different modes of communication, even in channels that may not reflect offline social relationships. With the help of an analytically tractable model of egocentric network evolution, we show that the observed universality can be attributed to the competition between cumulative advantage and random choice, two general mechanisms of tie reinforcement whose balance determines the amount of heterogeneity in tie strengths. Our results provide new insights into the driving mechanisms of tie strength heterogeneity in social networks and have implications for the understanding of social network structure and individual behavior.

Figures (10)

• Figure 1: Basic properties of communication datasets. Complementary cumulative distribution functions (CCDFs) $P[\bullet' \geq \bullet]$ for several properties $\bullet$ of ego networks in each dataset: degree $k$ (number of alters of an ego), strength $\tau$ (number of events involving an ego), mean alter activity $t$ (average number of events per alter of an ego), minimum activity $a_0$ (minimum number of events with the same alter), and maximum activity $a_m$ (maximum number of events with the same alter). All properties are heterogeneously distributed across egos and alters, with some differences between datasets.
• Figure 2: Dispersion in communication activity. Complementary cumulative distribution function (CCDF) $P[d' \geq d]$ of the number of egos having at least dispersion index $d$ in all considered datasets, calculated only for ego networks with more than 10 events, i.e. $\tau>10$. The average dispersion $\langle d \rangle$ is displayed as a dashed line. Communication channels show broad variation in how egos allocate activity among alters.
• Figure 3: Connection kernel in communication. Probability $\langle \pi_a \rangle$ that an alter with current activity $a$ communicates once more with the ego, averaged over time and subsets of at least 50 egos with degree $k \geq 2$ for each $a$ value, shown here for all considered datasets. The dashed line corresponds to the average baseline $\langle \pi_a \rangle = \langle 1/k \rangle$ when communication events are distributed randomly. The growth of the connection kernel $\langle \pi_a \rangle$ with activity indicates cumulative advantage, where alters with high prior activity receive more communication.
• Figure 4: Simple model of alter activity. Probability $p_a(t)$ of a randomly selected alter having activity $a$ at time $t$, as a function of $a = a_r + a_0$ for varying $t = t_r + a_0$ and varying $\alpha_r = \alpha + a_0$, for fixed $a_0 = 1$ and $k = 10, 100$ (bottom/top rows), in both numerical simulations of the model [Eq. (\ref{['eq:CumAdvDef']}); dots] and its analytical solution [Eq. (\ref{['eq:ActDistExp']}); lines]. For given $t$ and $a_0$, the time evolution of $p_a$ [as defined by Eq. (\ref{['eq:MasterEqCont']})] reaches an $\alpha_r$-dependent asymptotic shape. (right) When $\alpha_r \to \infty$ ($\alpha \to \infty$), $p_a$ converges to a Poisson distribution with mean and variance $t_r$ [Eq. (\ref{['eq:ActDistLargeAlpha']})]. (left) When $\alpha_r \to 0$ ($\alpha \to -a_0$), $p_a$ approaches a gamma distribution with shape $\alpha_r$ and scale $\beta = t_r / \alpha_r$ [Eq. (\ref{['eq:ActDistSmallAlpha']})]. (middle) When $\alpha_r = 1$, the exponent of the power-law decay in the gamma distribution is $\alpha_r - 1 = 0$, so the activity distribution has a plateau of $a$ values with relatively constant $p_a$ that grows with $t$. Eq. (\ref{['eq:ActDistExp']}) approximates numerical simulations very well, but fails at the tail for sufficiently low $k$. Simulations are averaged over $10^4$ realizations.
• Figure 5: Crossover in scaling of alter activity. Probability $p_a(t)$ of a randomly selected alter having activity $a$ at time $t$, as a function of $a_r = a - a_0$ for varying $t = t_r + a_0$ and varying $\alpha_r = \alpha + a_0$, for fixed $a_0 = 1$ and $k = 100$, in both numerical simulations of the model [Eq. (\ref{['eq:CumAdvDef']}); dots] and its analytical solution [Eq. (\ref{['eq:ActDistExp']}); lines]. (top) By plotting $\beta p_a$ vs. $a_r / \beta$ for varying $t_r$ and fixed $\alpha_r$, curves collapse to the standard gamma distribution [Eq. (\ref{['eq:gammaScaling']}); dashed lines]. This $\alpha_r$-dependent, gamma scaling is valid in the heterogeneous regime $\beta > 1$, with $\beta = t_r / \alpha_r$ the scale parameter of the gamma distribution in Eq. (\ref{['eq:ActDistSmallAlpha']}). Though only asymptotically correct, gamma scaling is a good approximation even at the crossover $\beta = 1$. (bottom) By plotting $\sqrt{ t_r } p_a$ vs. $( a_r - t_r ) / \sqrt{ t_r }$ for varying $t_r$, curves collapse to the standard Gaussian distribution [Eq. (\ref{['eq:GaussianScaling']}); dashed lines]. This Gaussian scaling is valid in the homogeneous regime $\beta < 1$ and becomes asymptotically more accurate with $t_r$. Simulations are averaged over $10^4$ realizations.