Limits...
Spatially embedded growing small-world networks.

Zitin A, Gorowara A, Squires S, Herrera M, Antonsen TM, Girvan M, Ott E - Sci Rep (2014)

Bottom Line: Motivated by the growth and development of neuronal networks, we propose a class of spatially-based growing network models and investigate the resulting statistical network properties as a function of the dimension and topology of the space in which the networks are embedded.In particular, we consider two models in which nodes are placed one by one in random locations in space, with each such placement followed by configuration relaxation toward uniform node density, and connection of the new node with spatially nearby nodes.We find no qualitative differences in these properties for two different topologies, and we suggest that results for these properties may not depend strongly on the topology of the embedding space.

View Article: PubMed Central - PubMed

Affiliation: Institute for Research in Electronics and Applied Physics University of Maryland, College Park, Maryland 20742, USA.

ABSTRACT
Networks in nature are often formed within a spatial domain in a dynamical manner, gaining links and nodes as they develop over time. Motivated by the growth and development of neuronal networks, we propose a class of spatially-based growing network models and investigate the resulting statistical network properties as a function of the dimension and topology of the space in which the networks are embedded. In particular, we consider two models in which nodes are placed one by one in random locations in space, with each such placement followed by configuration relaxation toward uniform node density, and connection of the new node with spatially nearby nodes. We find that such growth processes naturally result in networks with small-world features, including a short characteristic path length and nonzero clustering. We find no qualitative differences in these properties for two different topologies, and we suggest that results for these properties may not depend strongly on the topology of the embedding space. The results do depend strongly on dimension, and higher-dimensional spaces result in shorter path lengths but less clustering.

Show MeSH

Related in: MedlinePlus

The characteristic graph path length (ℓ) versus network size N on semilogarithmic axes for the sphere model (blue markers) and the plum pudding model (red markers).The path length shows the desired scaling, ℓ ~ logN. Results are shown for d = 1 (circles), 2 (triangles), 3 (squares), and 4 (inverted triangles), all using m = 4. Errors are smaller than the point size. In general, the average shortest path is shorter for higher dimensions d in both models.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
getmorefigures.php?uid=PMC4231322&req=5

f3: The characteristic graph path length (ℓ) versus network size N on semilogarithmic axes for the sphere model (blue markers) and the plum pudding model (red markers).The path length shows the desired scaling, ℓ ~ logN. Results are shown for d = 1 (circles), 2 (triangles), 3 (squares), and 4 (inverted triangles), all using m = 4. Errors are smaller than the point size. In general, the average shortest path is shorter for higher dimensions d in both models.

Mentions: The path length ℓ scales as logN with a coefficient that decreases with dimension (Fig. 3) for fixed average degree.


Spatially embedded growing small-world networks.

Zitin A, Gorowara A, Squires S, Herrera M, Antonsen TM, Girvan M, Ott E - Sci Rep (2014)

The characteristic graph path length (ℓ) versus network size N on semilogarithmic axes for the sphere model (blue markers) and the plum pudding model (red markers).The path length shows the desired scaling, ℓ ~ logN. Results are shown for d = 1 (circles), 2 (triangles), 3 (squares), and 4 (inverted triangles), all using m = 4. Errors are smaller than the point size. In general, the average shortest path is shorter for higher dimensions d in both models.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4231322&req=5

f3: The characteristic graph path length (ℓ) versus network size N on semilogarithmic axes for the sphere model (blue markers) and the plum pudding model (red markers).The path length shows the desired scaling, ℓ ~ logN. Results are shown for d = 1 (circles), 2 (triangles), 3 (squares), and 4 (inverted triangles), all using m = 4. Errors are smaller than the point size. In general, the average shortest path is shorter for higher dimensions d in both models.
Mentions: The path length ℓ scales as logN with a coefficient that decreases with dimension (Fig. 3) for fixed average degree.

Bottom Line: Motivated by the growth and development of neuronal networks, we propose a class of spatially-based growing network models and investigate the resulting statistical network properties as a function of the dimension and topology of the space in which the networks are embedded.In particular, we consider two models in which nodes are placed one by one in random locations in space, with each such placement followed by configuration relaxation toward uniform node density, and connection of the new node with spatially nearby nodes.We find no qualitative differences in these properties for two different topologies, and we suggest that results for these properties may not depend strongly on the topology of the embedding space.

View Article: PubMed Central - PubMed

Affiliation: Institute for Research in Electronics and Applied Physics University of Maryland, College Park, Maryland 20742, USA.

ABSTRACT
Networks in nature are often formed within a spatial domain in a dynamical manner, gaining links and nodes as they develop over time. Motivated by the growth and development of neuronal networks, we propose a class of spatially-based growing network models and investigate the resulting statistical network properties as a function of the dimension and topology of the space in which the networks are embedded. In particular, we consider two models in which nodes are placed one by one in random locations in space, with each such placement followed by configuration relaxation toward uniform node density, and connection of the new node with spatially nearby nodes. We find that such growth processes naturally result in networks with small-world features, including a short characteristic path length and nonzero clustering. We find no qualitative differences in these properties for two different topologies, and we suggest that results for these properties may not depend strongly on the topology of the embedding space. The results do depend strongly on dimension, and higher-dimensional spaces result in shorter path lengths but less clustering.

Show MeSH
Related in: MedlinePlus