Feedback arcs and node hierarchy in directed networks

Abstract

Directed networks such as gene regulation networks and neural networks are connected by arcs (directed links). The nodes in a directed network are often strongly interwound by a huge number of directed cycles, which lead to complex information-processing dynamics in the network and make it highly challenging to infer the intrinsic direction of information flow. In this theoretical paper, based on the principle of minimum-feedback, we explore the node hierarchy of directed networks and distinguish feedforward and feedback arcs. Nearly optimal node hierarchy solutions, which minimize the number of feedback arcs from lower-level nodes to higher-level nodes, are constructed by belief-propagation and simulated-annealing methods. For real-world networks, we quantify the extent of feedback scarcity by comparison with the ensemble of direction-randomized networks and identify the most important feedback arcs. Our methods are also useful for visualizing directed networks.

Optimal Segmentation of Directed Graph and the Minimum Number of Feedback Arcs

“…The total arc energy of a valid configuration h is defined as the total number of feedback arcs in this configuration. We should emphasize that, different from the model introduced in [4], a horizontal arc between two vertices of the same layer costs no energy in the D-segmentation problem. An optimal solution for the D-segmentation problem is a valid configuration h whose arc energy achieves the global minimum value.…”

“…For example, in Fig. 1 the directed path formed by arcs (1,5), (5,3), (3,4), (4,2), and (2, 1) is a directed cycle of length five connecting all the vertices of the digraph.…”

“…An issue of fundamental importance in digraph research is the minimum feedback arc set (FAS) problem, which is also encountered in many practical applications, such as inferring the hierarchical structure of the digraph [1,2,3,4] and detecting the dominant direction of information flow [5,6]. The minimum FAS problem aims at breaking all the directed cycles of an input digraph by deleting as few arcs as possible.…”

“…Various heuristic procedures have been investigated to solve it approximately but efficiently [1,2,3,5]. In a recent work [4], Zhao and one of the present authors proposed a spin glass model for this cycle-constrained optimization problem and derived a mean field theory for estimating the minimum number of feedback arcs and for constructing near-minimum feedback arc sets. The physics-inspired message-passing algorithm performs slightly worse in comparison with a simulated annealing (SA) algorithm adapted from [8].…”

Coevolution spreading in complex networks

“…The element of B is where δ jk = 1 if j = k, and δ jk = 0 otherwise. Recent studies showed that the dynamic message-passing approach predicts the epidemic spreading dynamics well on uncorrelated locally tree-like networks [56], and it has found wide applications in epidemic containment [52], locating spreading sources [57], and network dismantling [58]. However, the dynamic message-passing approach needs 2E + N differential equations, which is time-consuming for large networks.…”