Superlinear Connectivity Lower Bounds

• Superlinear connectivity lower bounds are proven guarantees that connectivity measures grow faster than linearly, often realized as Ω(n²/log n) or higher in key models.
• They leverage diverse techniques—combinatorial constructions, circuit decomposition, communication complexity reductions, and topological mappings—to expose inherent barriers in processing and structure.
• These bounds impact applications across polytope abstractions, streaming algorithms, distributed sketching, topological methods, and quantum circuit complexity, offering deep insights into computational limitations.

A superlinear connectivity lower bound refers to any proven guarantee that the connectivity measure of a mathematical or computational object—often a graph, complex, or circuit—grows faster than linearly (i.e., exceeds O(n)) in a natural parameter such as size, dimension, or degree. These bounds are fundamental in combinatorics, topology, computational complexity, optimization, and distributed computing, as they pinpoint inherent barriers in the structure or processing of connected systems. This survey synthesizes key developments and methods used to establish such bounds across a variety of domains.

1. Structural Graph and Polytope-Based Lower Bounds

Superlinear connectivity and diameter lower bounds arise in settings where combinatorial or topological surrogates for geometric objects are studied. In particular, subset partition graphs and their relation to the Linear Hirsch Conjecture illustrate this connection.

• Subset Partition Graphs and Polytope Abstractions: Constructed family graphs satisfying dimension reduction, (strong) adjacency, and endpoint count properties can have diameter as large as Ω(n²/log n), far exceeding the linear conjecture of Hirsch. These results are achieved by layering, doubling, and specialized set interpolations which strictly preserve the “polytope-like” properties at the combinatorial level, yet force superlinear traversability (Bogart et al., 2014).

Setting	Lower Bound	Notes
Subset partition graph	Ω(n²/log n) diameter	Retains “natural” polytope properties
Hypercube Vietoris–Rips	⎣2ⁿ/∑₍ᵢ₌r₊₁₎ⁿ (n choose i)⎦–connectivity	Disproves prior linearity conjecture (Bendersky et al., 2023)

2. Circuit Complexity: Cancellation-Free Linear Circuits

Achieving superlinear lower bounds in circuit complexity remains a central pursuit, particularly in restricted models such as cancellation-free linear circuits.

• Cancellation-Free Linear Circuits: In this class, once input variables appear in a computation, they are never canceled. For explicit matrices, such as the Sierpinski gasket matrix, any cancellation-free circuit requires at least (½)·n·log₂n gates, a bound achieved via careful circuit decomposition and induction. Furthermore, for certain incidence matrices derived from extremal graphs (e.g. those avoiding large complete bipartite subgraphs), nearly quadratic lower bounds of Ω(n^{2–ε}) are proven (Boyar et al., 2012). Though every matrix admits a cancellation-free realization and the overhead compared to the optimal general circuit is only constant-factor, restricted structure enables these strong lower bounds.

Circuit Type	Lower Bound	Achieved For
Canc.-free (Sierpinski)	½·n·log₂n	Sierpinski gasket matrix
Canc.-free (bipartite/Andreev)	Ω(n^{2–ε})	Explicit family of matrices

3. Algorithmic and Communication Models: Graph and Streaming Lower Bounds

Fundamental limitations appear in space, round, and message complexity for connectivity or matching in various models.

• Multipass Graph Processing: Space lower bounds of n^{1+Ω(1/p)}/poly(p) apply to p-pass streaming algorithms solving problems such as perfect matching, bounded distance queries, or directed s-t connectivity. These results transfer lower bounds from pointer-chasing communication problems, where information must propagate over long graph distances and across player boundaries, even when allowing multiple passes (Guruswami et al., 2012).
• Distributed Sketching: In the broadcast congested clique and distributed sketching models, deciding (single-component) connectivity requires Ω(log³n) bits per message, even for decision problems (as opposed to full spanning forest construction). The proof routes through lower bounds on the “universal relation” communication problem and shows no shortcut is possible by compression or by focusing on decision outputs (Yu, 2020). For more general k-edge connectivity (k > 1), the first super-polylogarithmic (in fact, Ω(k)) deterministic lower bound for sketch length has been proven (Robinson et al., 15 Jul 2025).

Model	Lower Bound	Problem
Multipass streaming	n^{1+Ω(1/p)}/p^{O(1)} space	Matching, distance, s-t-reach
Distributed sketching	Ω(log³n) bits	Connectivity
Distributed sketching	Ω(k) bits	k-edge connectivity (deterministic)

4. Topological and Geometric Superlinear Lower Bounds

• Percolation and Connectivity Exponents: For critical site percolation on ℤᵈ, the point-to-point connection probability decays no faster than c·n^{–d²}, surpassing previously conjectured exponents. This is accomplished by constructing multivariate “good point” paths and converting the probabilistic problem into a topological mapping one, using Brouwer's fixed point theorem (Berg et al., 2019).
• Convex Geometry—Level Set Connectivity Radius: In convex sets, the connectivity radius CR(O) at a boundary point O—i.e., the supremum r > 0 such that level sets S_r(O) remain connected—is lower-bounded by the least distance to a Grove-Shiohama-Gromov-Cheeger critical point (LCD(O)), providing a geometric “superlinear” range for connectivity before topological transitions must occur (Katz, 2019).

5. Conditional and Complexity-Theoretic Superlinear Lower Bounds

• Log-CircuitSAT, ETH, and SETH Barriers: Conditional lower bounds based on ETH show that certain polynomial-time problems (log-CircuitSAT, k-Clique) do not admit essentially-linear time algorithms unless 3-SAT admits subexponential time solutions. Any truly superlinear lower bound for decision time in these “plain” problems would break known complexity hypotheses (Salamon et al., 2020). This route is extended in (Belova et al., 2023), which argues proving even n^{1+ε} SETH-based lower bounds for problems like k-SUM or triangle detection would yield major circuit lower bounds for explicit arithmetic/Boolean functions, a milestone long out of reach.
• Space Complexity for Connectivity: Under SETH, every deterministic algorithm for s-t connectivity (st-CON) requires Θ(log²n) working space, matching Savitch's theorem, and separating L from NL under this plausible assumption (Czerwinski, 2023).

6. Curvature, Graph Geometry, and Connectivity Product Lower Bounds

Connectivity lower bounds can also arise from “combinatorial curvature” concepts.

• Lin–Lu–Yau Curvature: For a connected graph with minimum degree δ(G) and Lin–Lu–Yau curvature κₗₗy(G), the connectivity k(G) satisfies k(G) ≥ δ(G)·κₗₗy(G). When both parameters are “large” (e.g., dense regular graphs with uniform positive curvature), this multiplcative estimate can be sharply superlinear (Chen et al., 19 Apr 2025). In contrast, alternative curvature-based bounds using Bakry–Émery curvature involve only additive combinations and may not capture this potential for product amplification.

Curvature Notion	Lower Bound	Comment
Lin–Lu–Yau (κₗₗy)	k(G) ≥ δ(G)·κₗₗy(G)	Multiplicative; can be sharply superlinear
Bakry–Émery (κ_BE)	k(G) ≥ [2·κ_BE(G)+δ(G)+5]/8	Additive; different sign/behavior possible

7. Applications: Synchronization and Quantum Circuit Complexity

• Kuramoto Synchronization: In circulant oscillator networks, the critical connectivity μ_c above which only in-phase synchronization is stable is sharply lower-bounded by analyzing the stability of twisted states via a linear-stability condition encoded as an integer programming problem. Through this, μ_c is established to exceed 0.6838..., showing that dense (superlinear) connectivity is necessary to preclude multistability (Yoneda et al., 2021).
• Quantum Circuits (QAC⁰): Any depth-d constant-depth quantum circuit (QAC⁰) requires at least n^{1+3^{-d}} ancillary qubits to compute parity, majority, or high-approximate-degree functions. This is the first superlinear lower bound for QAC⁰ beyond linear size, paralleling classical AC⁰ barriers and extending to quantum state/channel synthesis and agnostic learning (Anshu et al., 9 Oct 2024).

Domain	Functionality	Superlinear Lower Bound
Quantum circuits (QAC⁰)	Parity, Majority, MODₖ, synthesis	Ω(n^{1+3^{-d}}) ancilla qubits
Synchronization	In-phase uniqueness	μ_c > 0.6838 (superlinear density)

References Table

Domain	Main Parameter	Lower Bound Form	Citation
Subset partition/Polytope	symbols n	Ω(n²/log n) diameter	(Bogart et al., 2014)
Cancellation-free circuits	n	½n log n, Ω(n^{2–ε})	(Boyar et al., 2012)
Streaming/Distributed	n, passes p, k	n^{1+Ω(1/p)}, Ω(k)	(Guruswami et al., 2012, Robinson et al., 15 Jul 2025)
Percolation/Probability	d, n	Ω(n^{-d²})	(Berg et al., 2019)
Curvature-based (graph)	δ(G), κₗₗy(G)	δ(G)·κₗₗy(G)	(Chen et al., 19 Apr 2025)
Quantum circuits (QAC⁰)	n, depth d, ancillae a	Ω(n^{1+3^{-d}})	(Anshu et al., 9 Oct 2024)

Conclusion

Superlinear connectivity lower bounds are derived by a wide range of specialized methods: combinatorial constructions, circuit decomposition, communication complexity reductions, recursive geometric/graph transformations, integer programming, and topological/curvature analysis. They serve not only as technical milestones but as signposts for where further progress—unconditional or conditional—may be fundamentally limited by deep phenomena in computation, topology, and geometry. These results frequently reveal sharp phase transitions and demonstrate that even “local” or “natural” constraints can generate global complexity that grows much faster than size, profoundly impacting algorithm design, structural graph theory, and complexity theory at large.