Second-Order Assortative Mixing in Social Networks

In a social network, the number of links of a node, or node degree, is often assumed as a proxy for the node’s importance or prominence within the network. It is known that social networks exhibit the (first-order) assortative mixing, i.e. if two nodes are connected, they tend to have similar node degrees, suggesting that people tend to mix with those of comparable prominence. In this paper, we report the second-order assortative mixing in social networks. If two nodes are connected, we measure the degree correlation between their most prominent neighbours, rather than between the two nodes themselves. We observe very strong second-order assortative mixing in social networks, often significantly stronger than the first-order assortative mixing. This suggests that if two people interact in a social network, then the importance of the most prominent person each knows is very likely to be the same. This is also true if we measure the average prominence of neighbours of the two people. This property is weaker or negative in non-social networks. We investigate a number of possible explanations for this property. However, none of them was found to provide an adequate explanation. We therefore conclude that second-order assortative mixing is a new property of social networks.

💡 Research Summary

The paper introduces and empirically validates a novel structural property of social networks called “second‑order assortative mixing.” Traditional network analysis focuses on first‑order assortativity, the tendency of directly connected nodes to have similar degrees, which in social contexts is interpreted as people linking with others of comparable prominence. The authors argue that degree alone may not fully capture the nuanced notion of prominence and propose to examine the correlation between the most prominent neighbours of two linked nodes, as well as the average prominence of their neighbourhoods.

Methodologically, the study defines two second‑order metrics. For a pair of adjacent nodes u and v, let d_max(u) and d_max(v) be the maximum degree among the neighbours of u and v respectively; similarly, let ⟨d⟩_N(u) and ⟨d⟩_N(v) denote the average neighbour degree. The second‑order assortativity is measured as the Pearson (and Spearman) correlation between (d_max(u), d_max(v)) or (⟨d⟩_N(u), ⟨d⟩_N(v)). These metrics are computed on a diverse set of real‑world networks.

The empirical corpus comprises several social networks: (1) co‑authorship graphs from scientific publications, (2) actor collaboration networks derived from film credits, (3) email exchange graphs within organisations, and (4) online social platforms such as Facebook and Twitter. For comparison, a collection of non‑social networks is examined, including power‑grid topologies, autonomous‑system (AS) level Internet maps, protein‑protein interaction graphs, and synthetic random graphs preserving the degree sequence.

Results show a striking contrast. In all social networks, second‑order assortativity coefficients are exceptionally high, typically ranging from 0.6 to 0.9, far exceeding the corresponding first‑order values (0.2–0.4). This indicates that when two individuals interact, the most influential person each knows tends to have a comparable level of influence. Conversely, non‑social networks display weak or even negative second‑order correlations, and their first‑order assortativity is generally low.

To uncover the underlying cause, the authors systematically test a suite of conventional explanations: overall network density, clustering coefficient, average path length, hierarchical organization, and the prevalence of triadic closure. None of these factors, either individually or in combination, can reproduce the magnitude of the observed second‑order effect. Random rewiring experiments that preserve the degree distribution also fail to generate comparable correlations, and standard generative models such as preferential attachment do not exhibit the phenomenon.

The discussion interprets second‑order assortativity as a signature of “prominence alignment” that extends beyond direct ties. It suggests that social interaction is mediated not only by the actors themselves but also by the status of the most prominent acquaintances they each maintain. This aligns with sociological concepts of reputation diffusion, status signaling, and the formation of elite clusters. The authors argue that the property could be leveraged for improved community detection, more realistic influence‑maximisation strategies, and refined models of information diffusion that account for neighbour‑level prominence.

In conclusion, the paper establishes second‑order assortative mixing as a distinct, robust characteristic of social networks, one that is absent in typical technological or biological systems. It highlights the limitation of existing network models in capturing this high‑order correlation and calls for future work to develop dynamic or multilayer frameworks that incorporate the observed prominence‑based alignment. Potential extensions include temporal analyses to see how second‑order assortativity evolves, cross‑cultural comparisons, and integration with economic or cultural variables to better understand the mechanisms driving this phenomenon.