Abstract
Most biological networks are modular but previous work with small model networks has indicated that modularity does not necessarily lead to increased functional efficiency. Most biological networks are large, however, and here we examine the relative functional efficiency of modular and non-modular neural networks at a range of sizes. We conduct a detailed analysis of efficiency in networks of two size classes: ‘small’ and ‘large’, and a less detailed analysis across a range of network sizes. The former analysis reveals that while the modular network is less efficient than one of the two non-modular networks considered when networks are small, it is usually equally or more efficient than both non-modular networks when networks are large. The latter analysis shows that in networks of small to intermediate size, modular networks are much more efficient that non-modular networks of the same (low) connective density. If connective density must be kept low to reduce energy needs for example, this could promote modularity. We have shown how relative functionality/performance scales with network size, but the precise nature of evolutionary relationship between network size and prevalence of modularity will depend on the costs of connectivity.
Highlights
Modularity in a network of interactions occurs when the network is subdivided into relatively autonomous, internally highly connected components [1]
The performance of the modular perfectly modular network (PMN) relative to the non-modular FCNMN is much greater in large networks compared with small networks
We have identified two effects here, that depend on scale of the networks, and that could influence the evolution of modularity in networked systems
Summary
Modularity in a network of interactions occurs when the network is subdivided into relatively autonomous, internally highly connected components [1]. We assume that each input stream is processed by distinct network modules across network layers and all information is integrated late in the information processing sequence Neural systems of this general form are represented in nature by, for example, the columnar organization of the somatic sensory cortex of mammals [2], the processing of different image attributes within distinct areas of the retina, superior colliculus, lateral geniculate nucleus and early visual cortical areas of primates [16], and the early visual processing apparatus of insects [17]. The same input set as system state 1 was employed but all 1 s within 140 rows of the 256 row input set were converted to a number from a random uniform distribution between 0 and 0.5, with each row receiving a different random number This procedure was repeated 20 times to produce inputs for the 20 replicates per network architecture type (SNMN, PMN or FCNMN). Simulations were run exactly as before but using only five replicates per network type/size combination owing to computational constraints
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