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Network morphospace - PubMed

  • ️Thu Jan 01 2015

Review

Network morphospace

Andrea Avena-Koenigsberger et al. J R Soc Interface. 2015.

Abstract

The structure of complex networks has attracted much attention in recent years. It has been noted that many real-world examples of networked systems share a set of common architectural features. This raises important questions about their origin, for example whether such network attributes reflect common design principles or constraints imposed by selectional forces that have shaped the evolution of network topology. Is it possible to place the many patterns and forms of complex networks into a common space that reveals their relations, and what are the main rules and driving forces that determine which positions in such a space are occupied by systems that have actually evolved? We suggest that these questions can be addressed by combining concepts from two currently relatively unconnected fields. One is theoretical morphology, which has conceptualized the relations between morphological traits defined by mathematical models of biological form. The second is network science, which provides numerous quantitative tools to measure and classify different patterns of local and global network architecture across disparate types of systems. Here, we explore a new theoretical concept that lies at the intersection between both fields, the 'network morphospace'. Defined by axes that represent specific network traits, each point within such a space represents a location occupied by networks that share a set of common 'morphological' characteristics related to aspects of their connectivity. Mapping a network morphospace reveals the extent to which the space is filled by existing networks, thus allowing a distinction between actual and impossible designs and highlighting the generative potential of rules and constraints that pervade the evolution of complex systems.

Keywords: Pareto optimality; brain connectivity; complexity; evolution; graph theory.

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Figures

Figure 1.
Figure 1.

Theoretical morphospaces allow organization of morphological complexity for a given group of organisms (usually focusing on some external, anatomical traits) within a limited phenotypic space. Here, we show a three-dimensional theoretical foraminiferal morphospace. The potential repertoire of Foraminifera shells is generated by a three-parameter model of form, whose parameters are: Δφ, deviation angle; translation factor, TF; growth factor, GF. Adapted with permission from reference [21].

Figure 2.
Figure 2.

Illustration of a theoretical morphospace: an N-dimensional hyperspace where each dimension (or axis) represents a network topological trait. Extrinsic constraints define the boundaries between the GIT region and the GPT region and the boundary between the functionally possible topology (FPT) region and the non-functionally possible topology (NPT) region. Note that the spatial distribution of the distinct regions may not be contiguous, as long as the non-overlapping set properties are preserved. Modified from McGhee [30].

Figure 3.
Figure 3.

Four steps to conduct a morphospace analysis. Step 1: defining the dimensions of a morphospace. Dimensions are given by the number of structural traits considered in the model; structural traits can be measured directly from the network's adjacency matrix (e.g. connection cost and characteristic path length) or given by the parameters of a growth model (e.g. the parameters of a spatial-growth model [55]). Step 2: generating network topologies that correspond to the distinct combinations of structural trait values. Step 3: placing empirical networks within the morphospace by measuring the pre-selected set of structural traits. Step 4: morphospace analysis. This step aims to answer the question of why real networks are located in particular regions of the morphospace, and not in other regions that are both possible and functional.

Figure 4.
Figure 4.

Hierarchy morphospace. The axes of this three-dimensional morphospace are given by treeness (T), feed-forwardness (F) and orderability (O). Coordinates of 125 real-world networks are indicated with filled circles, coloured according to network type (TECH, electronic circuits; GRN, GRNs; ECO, food webs; LANG, world corpora; MET, metabolisms; NEU, neuronal); four clusters can be identified, according to network's location within the morphospace. Non-coloured spheres represent ensemble of random networks of various sizes and degree distributions. Reproduced with permission from reference [70].

Figure 5.
Figure 5.

Communication-efficiency morphospace. Every point represents a network generated by the optimization algorithm. The regions explored by four independent simulations are shown. Each simulation corresponds to a different combination of objective functions which drive a population of networks towards four quadrants or the morphospace. Green points indicate the location of the initial population; blue and red points indicate the location of the lattice and random networks, respectively; orange points show the location of the final population of networks generated by each simulation. A sample of networks selected from the final population of networks shows that distinct topologies can be associated with different morphospace regions. Reproduced with permission from reference [72].

Figure 6.
Figure 6.

Efficiency-complexity morphospace of brain networks. Grey and orange points indicate the regions of the morphospace that have been explored by evolving a population of 500 brain networks. Networks are evolved employing a multi-objective optimization algorithm with eight distinct objective functions that drive networks towards eight quadrants of the morphospace. All objective functions impose distinct selective pressures over an evolving population of brain-like networks, resulting in eight final populations (orange points), called fronts, with distinctive structural properties. The greyscale assigned to each network indicates the epoch in which it was created, with light grey corresponding to early epochs and darker grey to later epochs. Blue and red points show the average trajectory of a randomized and latticized brain network, respectively, which are not subjected to any selective pressures. The spatial distribution of explored regions indicates that the accessibility of the morphospace is severely restricted: there are no networks found in the region Ediff < 1. Adapted with permission from reference [73].

Figure 7.
Figure 7.

Brain network morphospace. Axes of the morphospace are given by the parameters γ and η of the network growth model formula image where Pij is the probability that a functional connection exists between nodes i and j; kij is the number of nearest neighbours that nodes i and j have in common; dij is the spatial distance between nodes i and j. Distinct combinations of parameter values generate specific network topologies, which are indicated by the different shaded regions in the plot. The parameters values that best capture the topological features of healthy brain networks (HV) and of participants with childhood-onset schizophrenia (COS) are indicated with white dots within the region containing networks with heavy-tailed degree distributions. Adapted with permission from reference [78].

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