Assortative mixing

Assortative mixing is the tendency of nodes in a network to connect to other nodes with similar properties — most commonly degree, but also any measurable attribute such as age, ethnicity, political affiliation, or functional role. In a degree-assortative network, high-degree nodes connect preferentially to other high-degree nodes, and low-degree nodes connect to low-degree nodes. The opposite pattern, disassortative mixing, occurs when high-degree nodes connect preferentially to low-degree nodes, producing a hub-and-spoke topology that is common in technological and biological networks.

The measure of assortativity was introduced by Newman in 2002 and is typically computed as a Pearson correlation coefficient between the degrees of nodes at either end of each edge. A positive coefficient indicates assortative mixing; a negative coefficient indicates disassortative mixing; a coefficient near zero indicates no degree-degree correlation. The measure is deceptively simple: a single number that summarizes a global pattern of preference. But like all summary statistics, it can obscure local structure — a network may be assortative globally but disassortative in particular subgraphs, or vice versa.

Assortative mixing has significant consequences for network dynamics. In social networks, assortativity by demographic attributes can accelerate the spread of behaviors and beliefs within groups while insulating groups from each other — a structural precondition for echo chambers and polarization. In epidemic networks, assortativity can either accelerate or slow disease spread depending on whether high-degree nodes (the natural superspreaders) are clustered together or distributed across the periphery. In technological networks, disassortativity often reflects design constraints: the internet's backbone routers have high degree and connect to many low-degree edge devices, not to each other.

The relationship between assortative mixing and core-periphery structure is complex and underexplored. A core-periphery network is structurally disassortative in the sense that core nodes connect to periphery nodes, but it may be assortative by degree if the core itself is densely interconnected and the periphery is not. The two concepts capture different dimensions of network organization: assortativity is a correlation, core-periphery is a partition. Networks can exhibit both, neither, or either alone, and conflating the two has led to mistaken claims about the vulnerability and efficiency of real-world systems.

Assortative mixing is not merely a statistical property of networks — it is a fossil record of the selection pressures that shaped them. Every assortative pattern encodes a history of who was allowed to connect with whom, which opportunities were visible to which nodes, and which similarities were treated as relevant. To measure assortativity without asking what produced it is to read the sedimentary layers without asking what river deposited them.