On the origins of hierarchy in complex networks

Abstract

Hierarchy seems to pervade complexity in both living and artificial systems. Despite its relevance, no general theory that captures all features of hierarchy and its origins has been proposed yet. Here we present a formal approach resulting from the convergence of theoretical morphology and network theory that allows constructing a 3D morphospace of hierarchies and hence comparing the hierarchical organization of ecological, cellular, technological, and social networks. Embedded within large voids in the morphospace of all possible hierarchies, four major groups are identified. Two of them match the expected from random networks with similar connectivity, thus suggesting that nonadaptive factors are at work. Ecological and gene networks define the other two, indicating that their topological order is the result of functional constraints. These results are consistent with an exploration of the morphospace, using in silico evolved networks. modularity | evolution F ifty years ago (1), Herbert Simon defined complex systems as nested hierarchical networks of components organized as interconnected modules. Hierarchy seems a pervasive feature of the organization of natural and artificial systems (2, 3). The examples span from social interactions (4, 5), urban growth (6, 7), and allometric scaling (8) to cell function (9-13), development (14), ecosystem flows (15, 16), river networks (17), brain organization (18), and macroevolution (19,20). It also seems to pervade a coherent form of organization that allows reducing the costs associated to reliable information transmission (21) and to support efficient genetic and metabolic control in cellular networks (22). However, hierarchy is a polysemous word, involving order, levels, inclusion, or control as possible descriptors (23), none of which captures either its complexity or the problem of its measure and origins. Although previous work using complex networks theory has quantitatively tackled the problem (4, 24-34), some questions remain: Is hierarchy a widespread feature of complex systems organization? What types of hierarchies do exist? Are hierarchies the result of selection pressures or, conversely, do they arise as a by-product of structural constraints?A well-established concept where such questions are addressed involves the use of a morphospace (35-40), namely a phenotype space where a small set of quantitative traits can be defined as the axes. Here we take a step in this direction by combining morphospace and network theories, taking the intuitive idea of hierarchy as the starting point: a pattern of relations where there is no ambiguity in who controls whom with a pyramidal structure in which the few control the many. Formally, the picture of hierarchy matches a tree of relations (41), ideally represented by a directed graph. As shown in Fig. 1 A-D, the elements of the system are represented by nodes connected by arrows establishing the map of relations of who affects whom. Accordingly, a measure of hierarchy should account for the deviations from this ideal ...

Analysis of the hierarchical structure of the B. subtilis transcriptional regulatory network

“…We have first determined the extent of the hierarchical organization of the TRNs using a range of recently proposed measures, including Treeness, Feedforwardness and Orderability 25 . We have then combined decomposition approaches 19,20 to classify the transcriptional regulators in the TRNs of B.…”

“…Recently Corominas-Murtra et al 25 proposed three measures, Treeness (T), Feedforwardness (F) and Orderability (O), to quantify the extent of hierarchical organization in complex directed networks. In a given network, the treeness quantifies the extent of pyramidal structure and unambiguity in the chain of command, the feedforwardness measures the impact of feedback processes in the casual flow of information, and the orderability gives the fraction of nodes that does not belong to any cycle.…”

Feedback arcs and node hierarchy in directed networks

“…They are ubiquitous in biological and technological systems; for instance, neurons in the brain rely on directed synaptic connections to form an information-processing network [1], and cell regulatory networks contain directed interactions between genes, proteins and other small molecules [2][3][4]. Structural properties of directed networks at different scales have been studied in the literature for many years, especially on network small motifs [4,5], mesoscopic communities [6,7], strongly connected components [8,9], and network hierarchical structure [10][11][12][13][14]. A directed network can easily be decomposed into a set of strongly connected components (SCCs) and at this coarse-grained level is a directed acyclic feedforward graph of SCCs, with clear-cut hierarchical structure as directed cycle is absent [9,13].…”

“…Structural properties of directed networks at different scales have been studied in the literature for many years, especially on network small motifs [4,5], mesoscopic communities [6,7], strongly connected components [8,9], and network hierarchical structure [10][11][12][13][14]. A directed network can easily be decomposed into a set of strongly connected components (SCCs) and at this coarse-grained level is a directed acyclic feedforward graph of SCCs, with clear-cut hierarchical structure as directed cycle is absent [9,13]. Each SCC is itself a maximal subnetwork formed by some nodes and the arcs between them, and any node can reach and be reached by any another node of the same SCC through at least one directed path.…”

Network Hierarchy Evolution and System Vulnerability in Power Grids

“…Here, and in order to quantify this pyramidal property in the case of a power network, we adopt and simplify "treeness" [13] as our metric to analyze hierarchy evolution property of power networks, and to achieve balance between accuracy and computation burden. It's noticed that the hierarchy conception is based on a directed network which is also a natural feature for power network due to its power flow.…”