The Dynamics of Networks

Definitions and mathematical statement

DARPA's Mathematical Challenge Two (2007), "The Dynamics of Networks," asks for the development of high-dimensional mathematical methods to model and predict the behavior of large-scale distributed networks. The supplied examples include the internet, social networks, and biological networks, and the background emphasizes network dynamics.

This is a research programme, not a specified mathematical proposition. The source does not fix a class of networks, a state space or evolution law, the meaning of "large-scale" or "high-dimensional," available observations, a prediction target, or a quantitative accuracy or performance requirement. Consequently it supplies no domain and quantifiers for a theorem and no formal completion criterion. A precise instance would require those choices; none is selected here.

The local registry labels the problem open. Its literature triage dated 2026-08-17 labels the agenda partially solved in the sense of partial progress, cites DARPA's later Models, Dynamics and Learning programme as pursuing rigorous data-driven models for non-equilibrium dynamics, including communication and social systems, and explicitly states that a full solution or refutation is not applicable without a success condition. These are dated source assessments, not a claim that the agenda has a mathematically defined solved status.

Short English statement

Develop mathematics for modelling and predicting the dynamics of massive distributed networks. The supplied challenge is an open-ended research agenda: a network model, prediction task, data regime, and quantitative guarantee must first be specified to obtain a precise mathematical problem.

Sources

• DARPA DSO, Broad Agency Announcement 07-68, Mathematical Challenge Two (2007), as supplied in the record: https://web.math.utk.edu/~vasili/refs/darpa07.MathChallenges.html.
• DARPA, Models, Dynamics and Learning programme page, accessed 2026-08-17 according to the supplied literature triage: https://www.darpa.mil/research/programs/models-dynamics-and-learning.
• Local registry data/unsolvedmath/problems.json, permanent catalogue entry, source code catalogue entry, "The Dynamics of Networks," attributed to DARPA (2007); statement, background, and literature triage dated 2026-08-17. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.