Structural Polarization: Why Signed Social Networks Invert at the Hub Level

The finding

Two signed social trust networks -- Slashdot Zoo (28,574 users, 430k friend/foe edges) and Epinions (33,001 users, 702k trust/distrust edges) -- were analyzed using the IRDME loop. Both produced the same structural class: INVERSION.

Slashdot Zoo: r(friend_degree, foe_degree) at full network = +0.3704. At the top 1% hub level (286 users): r = -0.3572. The sign flips. Pre-registration hash: 242c15c6.

Epinions: r at full network = +0.3829. At the top 1% hub level (330 users): r = -0.0696. Same class. Independent replication.

Both results were pre-registered before the loop was run. The hypotheses were locked by SHA-256 hash before any loop iteration was executed.

What the numbers mean

At the population level, a positive r (+0.37) says: popular users tend to attract both friends and foes. If you have many followers in a community, you also attract detractors. This is the well-known social dynamics of visibility -- more exposure means more of everything, positive and negative.

At the hub level, the same correlation flips to -0.36 (Slashdot) and -0.07 (Epinions). The users with the most combined friend+foe activity are NOT both types simultaneously. The extreme hubs bifurcate: some are loved with few enemies; others are distrusted by many but trusted by few. Hub identity inverts between the two layers.

Social polarization is not visible at the population level. It is a hub-level structural phenomenon.

Why this is INVERSION, not BC_INVERSION

BC_INVERSION (documented earlier for cobra) requires Var(d2) above threshold and r < 0 -- a structural inversion at the full-network level. That is not what Slashdot and Epinions show. Their full-network r is positive (+0.37). The inversion only appears at the hub level, under iterative structural focusing.

This is the Hub Trajectory INVERSION class: r is positive at the full network and crosses zero when hub-focusing concentrates on the highest-degree nodes. It is the same trajectory type observed in mouse V1 cortex (r: +0.43 -> -0.92 at the top hub fraction), SIGNOR human signaling (r: +0.27 -> -0.42), and larval Drosophila (r: +0.36 -> -0.40). Signed social networks join this class across a completely different domain.

The structural mechanism

In Slashdot's Zoo system, users tag other users as friends (+1) or foes (-1). The average user's friend count and foe count are mildly correlated -- being active on the platform in any direction produces both types of tags. This gives r_full > 0.

The top hub users are selected by total degree (friend_count + foe_count). Among this set, two types emerge: users who are universally liked (beloved community figures: high friend degree, low foe degree) and users who are universally disliked or controversial (high foe degree, but not necessarily high friend degree). When the hub subgraph is dominated by these two types, the correlation between friend and foe degree inverts.

At the extreme hub clique, the most-trusted and most-distrusted users are different people. This is the structural signature of a polarized community: the very top of the social hierarchy is split between beloved and notorious, not distributed uniformly.

Replication and pre-registration

The Slashdot result was pre-registered as H2 with prediction "INVERSION or DISSIPATING" under the stable_trajectory classification rule. The Epinions result was pre-registered as H3 (independent replication of H2).

The stable_trajectory rule is new: it excludes BC_RADIAL terminal steps from classification. BC_RADIAL occurs when the top-k hub clique is fully connected (all nodes have identical degree, variance = 0). This is a measurement singularity, not a network property. The stable_trajectory rule identifies the trajectory from the stable compression window only.

For both social networks, stable_trajectory == trajectory: the INVERSION signal is robust and not affected by the BC_RADIAL correction. The same INVERSION classification is produced with or without the new rule.

All four hypotheses in the pre-registration (M_STRUCTURAL_MINING_REPLICATION_v2, hash 242c15c6) were confirmed: yeast PPI AMPLIFYING, Slashdot INVERSION, Epinions INVERSION, size-independence holding.

The contrast: biological vs social hubs

The structural fingerprint of signed social networks is r_full = +0.37-0.38, INVERSION trajectory. This is distinct from every biological network tested:

Yeast S. cerevisiae PPI: r_full = +0.94, AMPLIFYING. The 52-protein hub nucleus reaches r = +1.000 (perfect co-hub structure in both coexpression and experimental binding layers).

Human PPI STRING: r_full = +0.83, AMPLIFYING at 143-protein hub level (r = +0.995).

Florida cypress wetland food web: r_full = +0.996, SELF_SIMILAR. Stable ecological structure.

Social networks are structurally distinct not just in their lower r_full, but in the direction of hub compression: hubs become more specialized as you zoom in, not less. Biological systems produce generalist hub cores; social systems produce specialist hub cores.