We introduce a cost-benefit model for network formation, where exchanges among nodes are based on reciprocity. Peers receive from the network an amount of resources commensurate with their contribution. When creating links is costly, peers tend to limit the number of their connections, and sparsity of the exchange network may be desirable. In a fully connected graph, finding the sparsest exchanges that achieve a desired level of reciprocity is in general NP-hard. To capture near optimal allocations, we consider variants of the Eisenberg–Gale program, and an equivalent formulation given by Shmyrev, with sparsity penalties. We propose two decentralized algorithms for reciprocation, whereby peers approximately compute the sparsest allocations, by submitting bids for each other's resources. The algorithms extend the proportional–response dynamics, and promote sparsity through nonlinear pricing either at the bidding or the allocation stage. Numerical results illustrate the network formation process by peers who achieve almost-perfect reciprocity, with a small number of active connections. The proposed self-organization model leverages the reciprocity vs sparsity tradeoff to generate cyclic motifs in exchange networks.

中文翻译：

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互惠与稀疏权衡的网络形成

我们为网络形成引入了成本效益模型，其中节点之间的交换基于互惠。对等点从网络中获得与其贡献相称的资源量。当创建链接的成本很高时，对等点往往会限制它们的连接数，并且交换网络的稀疏性可能是可取的。在全连接图中，找到达到所需互惠水平的最稀疏交换通常是 NP-hard。为了获得接近最优的分配，我们考虑了 Eisenberg-Gale 程序的变体，以及 Shmyrev 给出的等效公式，带有稀疏惩罚。我们提出了两种分散的互惠算法，通过对彼此的资源提交出价，对等方近似计算最稀疏的分配。这些算法扩展了比例响应动态，并通过在投标或分配阶段的非线性定价来促进稀疏性。数值结果说明了通过少量活动连接实现几乎完美互惠的对等体的网络形成过程。所提出的自组织模型利用互惠与稀疏权衡在交换网络中生成循环主题。