  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.
