In complex networks, the 'rich-get-richer' effect (nodes with high degree at one point in time gain more degree in their future) is commonly observed. In practice this is often studied on a static network snapshot, for example, a preferential attachment model assumed to explain the more highly connected nodes or a rich-club effect that analyses the most highly connected nodes. In this paper, we consider temporal measures of how success (measured here as node degree) propagates across time. By analogy with social mobility (a measure of people moving within a social hierarchy through their life) we define hierarchical mobility to measure how a node's propensity to gain degree changes over time. We introduce an associated taxonomy of temporal correlation statistics including mobility, philanthropy and community. Mobility measures the extent to which a node's degree gain in one time period predicts its degree gain in the next. Philanthropy and community measure similar properties related to node neighbourhood. We apply these statistics both to artificial models and to 26 real temporal networks. We find that most of our networks show a tendency for individual nodes and their neighbourhoods to remain in similar hierarchical positions over time, while most networks show low correlative effects between individuals and their neighbourhoods. Moreover, we show that the mobility taxonomy can discriminate between networks from different fields. We also generate artificial network models to gain intuition about the behaviour and expected range of the statistics. The artificial models show that the opposite of the 'rich-get-richer' effect requires the existence of inequality of degree in a network. Overall, we show that measuring the hierarchical mobility of a temporal network is an invaluable resource for discovering its underlying structural dynamics.
In many complex networks, well-connected nodes tend to get even more connected over time — a "rich get richer" effect usually studied on a single network snapshot. Here we borrow the idea of social mobility (how much someone's position in a social hierarchy changes over their life) to define hierarchical mobility for temporal networks: how much a node's tendency to gain new connections changes from one time period to the next. We introduce related measures — mobility, philanthropy and community — and apply them to 26 real temporal networks as well as artificial models. Most real networks show individuals and their neighbourhoods staying in similar hierarchical positions over time, but the mobility measures can still distinguish networks from different fields, and the artificial models show that some inequality of degree is needed to escape the "rich get richer" effect.
author = {Matthew Russell Barnes and Vincenzo Nicosia and Richard G. Clegg},
title = {Measuring Social Mobility in Temporal Networks},
journal = {Scientific Reports},
year = {2025},
volume = {15},
number = {1},
doi = {10.1038/s41598-025-89090-1}
}