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This article was the subject of a Wiki Education Foundation-supported course assignment, between 19 January 2021 and 16 April 2021. Further details are available
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Neuroam.
Above undated message substituted from Template:Dashboard.wikiedu.org assignment by PrimeBOT ( talk) 09:31, 17 January 2022 (UTC)
Currently the definition given at the start of this article is entirely about how the diameter scales with N. There is no mention of clustering in this definition. I have always seen high clustering + short path lengths given as the definition. Clustering has always been a key part of the definition in the papers I've read, and in the discussion of properties, it already contains the quote "This follows from the defining property of a high clustering coefficient" (further numerous statements in this talk section mention things like "small average distance PLUS a high clustering coefficient", "Most prefer the term... 'small world network' for networks that has some or a lot of clustering") The reference given for the currently presented definition which ignores clustering is the 1998 Nature paper by Watts and Strogatz. This paper devoted significant attention to clustering, and focused on the existence of an interesting class of network for which the diameter scaled roughly like log(N), but the clustering coefficient is quite large. I have updated the definition to mention high clustering. If there is any disagreement, a different reference is needed for the current definition and a section should be included discussing the fact that often clustering is a key component of the definition. Joelmiller ( talk) 01:02, 7 October 2016 (UTC)
I can find no reference that states that degree-diameter graphs are small-worlds networks. I do not believe these have the clustering that is typically assumed for a small worlds network. I propose to delete this from the section on examples if no-one can provide a competing reference. Joelmiller ( talk) 01:23, 7 October 2016 (UTC)
Another point that is lingering around here is the mix-up between scale-free networks and small-worlds. As has already been discussed on this page, the small-world definition contains both, a small average distance PLUS a high clustering coefficient. It is agreed that scale-free networks (or other networks with a fat-tail degree distribution) have a very small average distance but they do not need to have a high clustering coefficient. Thus, I would suggest to remove this part or to state that often these two things are confused but that the agreement today is that small-worlds need to have both structures to classify. There is a thesis in which the author tries to dissect the small-world effect (low average distance) from the small-world phenomenon (low average distance DESPITE high clustering coefficient). If we pick this up, the messy situation can be neatly resolved by saying that scale-free networks show the small-world effect but not necessarily the small-world phenomenon. Netzwerkerin ( talk) 16:24, 6 April 2011 (UTC)
Road maps are very possibly one of the few real-world networks that are NOT small-worlds (which is actually not so easy! :-)). So, I'd like to delete this if no evidence/reference for this is provided and might even add it as one of those types of networks which are not small-worlds. Anyone up for discussion?
Netzwerkerin ( talk) 16:17, 6 April 2011 (UTC)
Actually this paper discusses how road and transportation networks can become small-world networks depending on how you define neighbors. I'm removing it from either section because its behavior is very dependent on edge defintion.
[1] — Preceding unsigned comment added by Atashsiah ( talk • contribs) 17:38, 12 August 2016 (UTC)
References
Does anyone else think this article might benefit from a picture. Just an example, or perhaps a demonstration of the algorithm for growing a small-world graph? Grj23 ( talk) 07:39, 2 December 2009 (UTC)
"If the clustering coefficient is significantly higher than would be expected for a random network, and the mean shortest-path length is lower than would be expected for a regular network, then the network is a small world."
I disagree. Also random networks are small world networks. They are a special case (or if you will an ideal case) of a small world network. What is important in the expression "small world network" is the Small world phenomenon. That is that the number of jumps between any two nodes in the network is short. The name does not imply that it is a network of several different but connected small worlds. It is about that the WHOLE world seems small due to short jump distances. This means a random network is very much a small world network, actually the ideal small world network since it does have shorter average jump distances then networks that has some or a lot of clustering.
Oh by the way, since random networks are part of the research I am doing here's some nice facts about random networks: If each node has about 4 randomly chosen connections the network almost never splits into separate islands. ("Never" as in very statistically unlikely.) If you lower the number of connections to 3 you see netsplits pretty often. Of course more connections means more robust. If the network is heavily clustered (not a random network) you might need more or even much more then 4 connections to avoid netsplits.
-- Davidgothberg 09:13, 25 May 2005 (UTC)
Hello All. It is highly likely this article is going to become next week's Mathematics Collaboration of the Week, and so it is slated to undergo a rapid expansion. Just to seed some of the discussion, I would like to start listing possible sections for the expanded article:
(Debivort)
It seems implied in the latter part of this article that power law distribution networks with hubs and such are the same thing as small-world networks, but this isn't really true. For example, randomly rewiring a lattice is a popular way of creating a small-world network, but does not create any hubs (in fact all nodes have the same degree). Meekohi 13:28, 17 January 2006 (UTC)
Small-world networks and power law networks are two different things. A small world network has a short average path length relative to its average degree, while a power law/scale free network simply has a degree distribution that is the same independent of size. Preferential attachment will create scale free networks, but will not automatically lead to a small world network. A small world network can be, as discussed above, created by taking a low-dimensional latice and randomly bridging a few nodes. This is by analogy to the original small world experiment of Milgram: the assumption is that we know our neighbors (hence, the 2D lattice), but a few people also know people who are physically distant. Creating a network with both scale-free and small world properties can be done by selecting random nodes in proportion to their degree (scale free), and then duplicating a fixed number of their neighbors (small world). See Steyvers & Tanenbaum (2005) [Cognitive Science 29 41–78]. 203.86.44.40 04:37, 27 July 2007 (UTC)
This edit switched small world networks from types of graph to types of random graphs. I think this is incorrect, as one can construct small world networks non-randomly, such as the one in this image. Debivort 02:21, 14 May 2007 (UTC)
How do these networks relate to the kind of networks that are used in electrical engineering, e.g. resistor networks, from a mathematical perspective?
That a "small world network of neurons can exhibit short-term memory" is not surprising. Two neurons can do this. It's a property of neurons, not small world networks. I propose removing this section. de Bivort 20:41, 14 November 2010 (UTC)
In which publication was the small-worldness measure proposed? I didn't find that formula in neither the references 4 nor 5. I'm new to this field of network science as well as talks on the wiki, so if this is not appropriate way to publish my idea please delete it anyway. — Preceding unsigned comment added by 123.149.144.27 ( talk) 01:23, 21 January 2014 (UTC)
I've made an illustration - example of small-world network. Do you have any comments for illustration? Do you have any objections not to include illustration to article?
Schulllz ( talk) 13:51, 30 January 2014 (UTC)
How is being a nieghbour rigourously defined? Could someone add a link in the article? — Preceding unsigned comment added by Alrichardbo1 ( talk • contribs) 17:50, 6 September 2014 (UTC)
Regarding the small-worldness defined below, is it defined for undirected graph only? How does the definition look like for directed graph, e.g. internet?-- 128.92.121.91 ( talk) 20:55, 27 October 2020 (UTC)
This edit would provide evidence from the research literature to 'Applications' > 'Small world neural networks in the brain'. Information on short-term memory exists, but the section lacks general background of how the model is applied to neural networks. Key points are as follows: application of the small-world model reflects functional brain regions (nodes), highly connected regions (hubs), and characteristic short path length between hub regions indicates functional connectivity. The balance of global and local networks is supported by the application of the small-world network model. The model is associated with efficient brain function (high processing/integration with low energy cost). Signed neuroam ( talk) 15:19, 17 April 2021 (UTC)
The paragraph beginning with, "small-world network of neurons can exhibit short-term memory" is misleading as it is describing a computer modeling application that can be applied to short-term memory or seizures. This convolutes the application of small-world networks in the brain because there is no transition or introduction to the new topic of computer models. It would be the most effective to limit this section to the direct application of small-world network to functional and structural connections in the brain. See Talk post above for the general application as used in Neuroscience research. Signed neuroam ( talk) 9:49, 28 April 2021 (UTC)
While the visual system / brainstem exhibit small world characteristics, that is not more general than entire brain connectivity. It should be noted that these systems do exhibit small-world structure, but ultimately because those systems function within the whole-brain network. Any suggestions to clarifying? Or any oppositions to removing "on a more general level" or the entire last sentence until it can be properly introduced? *the cited paper refers to application of the network to the reticular formation and asserts small-world properties in terms of the network's rapid processing and adaptability. Signed neuroam ( talk) 10:19, 28 April 2021 (UTC)
This small section at the end has no context and seems out of place after the applications section. Might this section be a better fit after network robustness? Signed neuroam ( talk) 10:29, 28 April 2021 (UTC)
The definition of small-world networks does not look good to me. I am not sure what is the correct definition, but "[L] grows proportionally to the logarithm of the number of nodes N" only makes sense if we are talking about a series of networks. It does not make sense if we are talking about one specific graph and that seems to be the case now.
The formula means that N is an infinitely growing variable. Thus an infinite sequence of graphs is defined, but not a graph in any way. Сурбас ( talk) 11:34, 30 April 2022 (UTC)
Perhaps I am misreading something badly but the very first sentence just doesn't seem correct "A small-world network is a mathematical graph in which most nodes are not neighbors of one another, but the neighbors of any given node are likely to be neighbors of each other" . It describes a sparse network with high clustering, for example triangular lattice. The second sentence does not look accurate either. I will wait a couple of days for any objections and then I'll update the introduction. Guggger ( talk) 06:46, 16 November 2023 (UTC)