In order to unveil the value of network connectivity, discounted both in space and time, we formalize the construction of networks as an optimal control dynamic graph-theoretic problem. The network is based on a set of leaders and followers linked through edges. The node dynamics, built upon the consensus protocol, form a time evolutive Mahalanobis distance weighted by the opportunity costs. The results show that the network equilibrium depends on the influence of leader nodes, while the network connectivity depends on the cohesiveness among followers. Through numerical simulations, we find that — past a threshold level of opportunity costs — the values of shadow prices become stationary. Likewise, the model outputs show that, at a fixed level of foregone gains, agents value the safeguard of connections less in time than in space.

中文翻译：

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网络连接的时空折扣价值

为了揭示网络连接在空间和时间上的价值，我们将网络的构建形式化为最优控制动态图论问题。该网络基于一组通过边缘链接的领导者和追随者。建立在共识协议之上的节点动态形成了一个由机会成本加权的时间演化马氏距离。结果表明，网络平衡取决于领导者节点的影响，而网络连通性取决于追随者之间的凝聚力。通过数值模拟，我们发现——超过机会成本的阈值水平——影子价格的值变得稳定。同样，模型输出表明，在已放弃收益的固定水平上，代理在时间上比在空间上更重视连接的保护。