The document discusses the origin of sparsity in interaction networks of living systems. It proposes that sparsity emerges from mechanisms that maximize the volume of stable attractors and promote robustness to perturbations. Sparse, tree-like networks provide greater explorability of stable states and asymptotic stability of dynamics compared to more dense networks. The optimal network structure is found to be self-similar and able to deal with real-world complexity through aggregation of communities.

1. The origin of sparsity in the interaction
networks of living systems
Daniel M. Busiello, S. Suweis, J. Hidalgo
and A. Maritan

2. Sparsity as an emergent pattern
Proposed underlying mechanisms:
- maximization of the volume of stable attractors
- robustness respect to perturbations of the dynamics

3. The interaction network
Tree-like sparse network More dense network
Increasing connectivity
𝝐 𝟏
𝝐 𝟐
𝐴𝑖𝑗(𝒙∗
, 𝜶) 𝐴𝑖𝑗 𝒙∗, 𝜶, 𝝐 𝟏, 𝝐 𝟐, …

4. What is Explorability?
𝑥𝑖 = 𝑥𝑖 𝛼𝑖 +
𝑗=1
𝑆
𝐴𝑖𝑗 𝑥𝑗 ℰ 𝐴|𝐶 = { 𝑥∗
∶ 𝐽𝑖𝑗 = 𝑥𝑖
∗
𝐴𝑖𝑗 𝑠𝑡𝑎𝑏𝑙𝑒}

5. What is Asymptotic stability?
𝐽𝑖𝑗 = 𝜉𝑖 𝐴𝑖𝑗 𝜉 ≡ 𝐷𝑦𝑛𝑎𝑚𝑖𝑐𝑠 𝜎 𝐴, 𝜉 𝐶 = Max Re 𝜆𝐽

6. Further results and conclusions
Tree-like structures 𝐶 ≈
1
𝑠
provide quasi-optimal robustness respect to fixed points
(Explorability) and dynamics (Asymptotic stability)
The optimal solution is self-similar when we deal with real-world complexity
(aggregation of communities)
Convergent optimality
Generalized dynamics 𝑥𝑖 = 𝐺(𝑥𝑖)𝐹
𝑗=1
𝑆
𝐴𝑖𝑗 𝑥𝑗

7. Thanks for your attention