Many hierarchically modular systems are structured in a way that resembles an hourglass. This "hourglass effect" means that the system generates many outputs from many inputs through a relatively small number of intermediate modules that are critical for the operation of the entire system, referred to as the waist of the hourglass. We investigate the hourglass effect in general, not necessarily layered, hierarchical dependency networks. Our analysis focuses on the number of source-to-target dependency paths that traverse each vertex, and it identifies the core of a dependency network as the smallest set of vertices that collectively cover almost all dependency paths. We then examine if a given network exhibits the hourglass property or not, comparing its core size with a "flat" (i.e., non-hierarchical) network that preserves the source dependencies of each target in the original network. As a possible explanation for the hourglass effect, we propose the Reuse Preference (RP) model that captures the bias of new modules to reuse intermediate modules of similar complexity instead of connecting directly to sources or low complexity modules. We have applied the proposed framework in a diverse set of dependency networks from technological, natural and information systems, showing that all these networks exhibit the general hourglass property but to a varying degree and with different waist characteristics.
We approach the C. elegans connectome as an information processing network that receives input from about 90 sensory neurons, processes that information through a highly recurrent network of about 80 interneurons, and it produces a coordinated output from about 120 motor neurons that control the nematode's muscles. We focus on the feedforward flow of information from sensory neurons to motor neurons, and apply a recently developed network analysis framework referred to as the "hourglass effect". The analysis reveals that this feedforward flow traverses a small core ("hourglass waist") that consists of 10-15 interneurons. These are mostly the same interneurons that were previously shown (using a different analytical approach) to constitute the "rich-club" of the C. elegans connectome. This result is robust to the methodology that separates the feedforward from the feedback flow of information. The set of core interneurons remains mostly the same when we consider only chemical synapses or the combination of chemical synapses and gap junctions. The hourglass organization of the connectome suggests that C. elegans has some similarities with encoderdecoder artificial neural networks in which the input is first compressed and integrated in a low-dimensional latent space that encodes the given data in a more efficient manner, followed by a decoding network through which intermediate-level sub-functions are combined in different ways to compute the correlated outputs of the network. The core neurons at the hourglass waist represent the information bottleneck of the system, balancing the representation accuracy and compactness (complexity) of the given sensory information.
Graph computation approaches such as GraphChi and TurboGraph recently demonstrated that a single PC can perform efficient computation on billion-node graphs. To achieve high speed and scalability, they often need sophisticated data structures and memory management strategies. We propose a minimalist approach that forgoes such requirements, by leveraging the fundamental memory mapping (MMap) capability found on operating systems. We contribute: (1) a new insight that MMap is a viable technique for creating fast and scalable graph algorithms that surpasses some of the best techniques; (2) the design and implementation of popular graph algorithms for billion-scale graphs with little code, thanks to memory mapping; (3) extensive experiments on real graphs, including the 6.6 billion edge YahooWeb graph, and show that this new approach is significantly faster or comparable to the highly-optimized methods (e.g., 9.5× faster than GraphChi for computing PageRank on 1.47B edge Twitter graph). We believe our work provides a new direction in the design and development of scalable algorithms. Our packaged code is available at http://poloclub.gatech.edu/mmap/.
We approach the C. elegans connectome as an information processing network that receives input from about 90 sensory neurons, processes that information through a highly recurrent network of about 80 interneurons, and it produces a coordinated output from about 120 motor neurons that control the nematode's muscles. We focus on the feedforward flow of information from sensory neurons to motor neurons, and apply a recently developed network analysis framework referred to as the "hourglass effect". The analysis reveals that this feedforward flow traverses a small core ("hourglass waist") that consists of 10-15 interneurons. These are mostly the same interneurons that were previously shown (using a different analytical approach) to constitute the "rich-club" of the C. elegans connectome. This result is robust to the methodology that separates the feedforward from the feedback flow of information. The set of core interneurons remains mostly the same when we consider only chemical synapses or the combination of chemical synapses and gap junctions. The hourglass organization of the connectome suggests that C. elegans has some similarities with encoder-decoder artificial neural networks in which the input is first compressed and integrated in a low-dimensional latent space that encodes the given data in a more efficient manner, followed by a decoding network through which intermediate-level sub-functions are combined in different ways to compute the correlated outputs of the network. The core neurons at the hourglass waist represent the information bottleneck of the system, balancing the representation accuracy and compactness (complexity) of the given sensory information. Author SummaryThe C. elegans nematode is the only species for which the complete wiring diagram ("connectome") of its neural system has been mapped. The connectome provides architectural constraints that limit the scope of possible functions of a neural system. In this work, we identify one such architectural constraint: the C. elegans connectome includes a small set (10-15) of neurons that compress and integrate the information provided by the much larger set of sensory neurons. These intermediate-level neurons encode few sub-functions that are combined and re-used in different ways to activate the circuits of motor neurons, which drive all higher-level complex functions of the organism such as feeding or locomotion. We refer to this encoding-decoding structure as "hourglass architecture" and identify the core neurons at the "waist" of the hourglass. We also discuss the similarities between this property of the C. elegans connectome and artificial neural networks. The hourglass architecture opens a new way to think about, and experiment with, intermediate-level neurons between input and output neural circuits.
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