Fractality and the percolation transition in complex networks

On heights and plains: How rodents from different habitats cope with three-dimensional environments?

Dwelling in a specific habitat requires adaptation to the habitat physical and biological properties in order to maximize fitness. Adaptations that are manifested in the organization of behavior in time and space reflect how the environment is perceived and utilized. Testing species from different habitats in the same laboratory environment can uncover the differences in their behavior and their adaptations to specific habitats. The question posed in this study is that of how two rodent species, one occupying flatlands (Tristram’s jird; Meriones tristrami) and the other occupying structured rocky habitats (common spiny mouse; Acomys dimidiatus), differ in the way that they explore the same three-dimensional laboratory environment. Individuals of these two species were introduced into an arena with a five-level ziggurat in the center, and their behavior was followed for 60 min. We found that both species preserved the typical spatiotemporal rodents’ behavior of establishing a home-base—a location that is a terminal from which they set out to explore the environment. However, the jirds, which live in flatlands, mainly travelled on the arena floor and the lower levels of the ziggurat; while, in contrast, the spiny mice, which live in rocky habitats and are used to climbing, mostly remained and travelled on the ziggurat, with some of them hardly descending to the arena floor. We suggest that the distinction in spatial behavior between the two species reflects their different motor abilities, different depth perception, and different umvelt (perceived world), in accordance with their different natural habitats.

"It's all in their head": hierarchical exploration of a three-dimensional layered pyramid in rats

Wayfinding in a three-dimensional (3D) environment is intricate, and surface-bounded animals may overcome this complexity by breaking it down into horizontal layers along with the vertical location of each layer. Here, we examined how rats explored a layered pyramid placed in a large open field. We found that exploration presented a hierarchical (or fractal) shape of three types of roundtrips: (1) from the primary home-base to the open-field floor; (2) from the floor up and down the pyramid levels; and (3) from local home-base on each pyramid level. Ascent was slow and interrupted, whereas descent was fast. This difference was a result of level altitude, remaining after data were normalized proportionally to level area. In contrast, the time spent and the distance traveled on each level were dependent on level area, not on level altitude. This structure of spatial behavior accords with multilevel exploration, presenting a relatively independent exploration of each level. The vertical dimension in this experiment thus did not alter the typical spatiotemporal behavior, and the 3D environment was explored by application of the same spatiotemporal approach as that of a horizontal open field. We suggest that this lack of alteration is due to the horizontal posture of the animal's head and trunk during progression on the pyramid. This behavior also seems to fit the bicoding hypothesis, in which the vertical information is virtually contextual (non-metric), and so, when the rat progresses to a new level, it explores it as a newly accessed horizontal floor area.

Reliable Multi-Fractal Characterization of Weighted Complex Networks: Algorithms and Implications

Through an elegant geometrical interpretation, the multi-fractal analysis quantifies the spatial and temporal irregularities of the structural and dynamical formation of complex networks. Despite its effectiveness in unweighted networks, the multi-fractal geometry of weighted complex networks, the role of interaction intensity, the influence of the embedding metric spaces and the design of reliable estimation algorithms remain open challenges. To address these challenges, we present a set of reliable multi-fractal estimation algorithms for quantifying the structural complexity and heterogeneity of weighted complex networks. Our methodology uncovers that (i) the weights of complex networks and their underlying metric spaces play a key role in dictating the existence of multi-fractal scaling and (ii) the multi-fractal scaling can be localized in both space and scales. In addition, this multi-fractal characterization framework enables the construction of a scaling-based similarity metric and the identification of community structure of human brain connectome. The detected communities are accurately aligned with the biological brain connectivity patterns. This characterization framework has no constraint on the target network and can thus be leveraged as a basis for both structural and dynamic analysis of networks in a wide spectrum of applications.

Determination of multifractal dimensions of complex networks by means of the sandbox algorithm

Complex networks have attracted much attention in diverse areas of science and technology. Multifractal analysis (MFA) is a useful way to systematically describe the spatial heterogeneity of both theoretical and experimental fractal patterns. In this paper, we employ the sandbox (SB) algorithm proposed by Tél et al. (Physica A 159, 155-166 (1989)), for MFA of complex networks. First, we compare the SB algorithm with two existing algorithms of MFA for complex networks: the compact-box-burning algorithm proposed by Furuya and Yakubo (Phys. Rev. E 84, 036118 (2011)), and the improved box-counting algorithm proposed by Li et al. (J. Stat. Mech.: Theor. Exp. 2014, P02020 (2014)) by calculating the mass exponents τ(q) of some deterministic model networks. We make a detailed comparison between the numerical and theoretical results of these model networks. The comparison results show that the SB algorithm is the most effective and feasible algorithm to calculate the mass exponents τ(q) and to explore the multifractal behavior of complex networks. Then, we apply the SB algorithm to study the multifractal property of some classic model networks, such as scale-free networks, small-world networks, and random networks. Our results show that multifractality exists in scale-free networks, that of small-world networks is not obvious, and it almost does not exist in random networks.

Network analysis of rat spatial cognition: behaviorally-established symmetry in a physically asymmetrical environment

Background

We set out to solve two inherent problems in the study of animal spatial cognition (i) What is a “place”?; and (ii) whether behaviors that are not revealed as differing by one methodology could be revealed as different when analyzed using a different approach.

Methodology

We applied network analysis to scrutinize spatial behavior of rats tested in either a symmetrical or asymmetrical layout of 4, 8, or 12 objects placed along the perimeter of a round arena. We considered locations as the units of the network (nodes), and passes between locations as the links within the network.

Principal Findings

While there were only minor activity differences between rats tested in the symmetrical or asymmetrical object layouts, network analysis revealed substantial differences. Viewing ‘location’ as a cluster of stopping coordinates, the key locations (large clusters of stopping coordinates) were at the objects in both layouts with 4 objects. However, in the asymmetrical layout with 4 objects, additional key locations were spaced by the rats between the objects, forming symmetry among the key locations. It was as if the rats had behaviorally imposed symmetry on the physically asymmetrical environment. Based on a previous finding that wayfinding is easier in symmetrical environments, we suggest that when the physical attributes of the environment were not symmetrical, the rats established a symmetric layout of key locations, thereby acquiring a more legible environment despite its complex physical structure.

Conclusions and Significance

The present study adds a behavioral definition for “location”, a term that so far has been mostly discussed according to its physical attributes or neurobiological correlates (e.g. - place and grid neurons). Moreover, network analysis enabled the assessment of the importance of a location, even when that location did not display any distinctive physical properties.