发表机构
Instituto Nacional de Medicina Genómica; Departamento de Física, Facultad de Ciencias, Universidad Nacional Autónoma de México（墨西哥国家基因组医学研究所; 墨西哥国立自治大学理学院物理系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究探讨在幂律成本下导电网络的最优架构，发现成本凸性决定电导分配的唯一性与边存在性，凹性导致稀疏化，并揭示星形与完全图交叉及塌缩指数随规模变化。
AI 中文摘要
导电网络只有通过花费材料才能改善其内部的通达性，因此架构仅在预算相等时进行比较。我们在幂律资源法则 $\C_{\alpha}=\sum_e a_e w_e^{\alpha}$ 下重新分配电导，并通过平均成对有效电阻的倒数来衡量通达性。当成本严格凸时，分配是唯一的，且每条允许的边保持正电导；在 $\alpha=1$ 时，唯一性仍然存在，但一条边可能被定价出局。边的边际价值是其平方双调和距离，这是一种Murray型平衡，其平衡量是全局的而非结点局部的。同一最优值在其自身预算下对任何加权网络进行评分，将分配因子与拓扑因子分离。在凹成本下，新生通道的边际价格发散，可行集非凸，电导凝聚到稀疏骨干上。均匀星形和均匀完全图在 $\alpha_c(n)=\log(n/2)/\log(n-1)$ 处交叉，接近该点时中间状态优于两者。在 $n$ 高达64时，找到的最佳状态大致随 $n$ 线性增长，因此每个结点的通道数变化远小于活跃比例的变化。环路由一个塌缩指数 $\alpha^{*}(n)$ 控制，在每个尺寸下解析到0.005：它从 $n=8$ 时的0.555上升到 $n=64$ 时的0.675，且跨越该指数的塌缩是突然的，$n=64$ 时的支撑从133条活跃边下降到生成树的63条。因此，固定指数通过跨越此边界而随尺寸改变特性。一个指定的源和汇反而给出无环最优。
英文摘要
A conducting network improves access to its interior only by spending material, so architectures compare only at equal budget. We redistribute conductance under a power-law resource law $\C_α=\sum_e a_e w_e^α$ and measure access by the reciprocal of the mean pairwise effective resistance. When the cost is strictly convex the allocation is unique and every admissible edge keeps positive conductance; at $α=1$ uniqueness survives but an edge can be priced out. The marginal worth of an edge is then its squared biharmonic distance, a Murray-type balance whose balanced quantity is global rather than junction-local. The same optimum scores any weighted network at its own budget, separating an allocation factor from a topological one. Under concave cost the marginal price of an incipient channel diverges, the feasible set is nonconvex, and conductance condenses onto sparse backbones. The uniform star and the uniform complete graph cross at $α_c(n)=\log(n/2)/\log(n-1)$, near which intermediate states beat both. Up to $n=64$ the best states found grow approximately linearly in $n$, so channels per node change much less than the active fraction does. Loops are governed instead by a collapse exponent $α^{*}(n)$, resolved to $0.005$ at every size: it rises from $0.555$ at $n=8$ to $0.675$ at $n=64$, and the collapse across it is abrupt, the support at $n=64$ falling from $133$ active edges to the $63$ of a spanning tree. A fixed exponent therefore changes character with size by crossing this boundary. One prescribed source and sink instead gives a loopless optimum.
Comments13 pages, 7 figures, 6 tables
