In general, that's sort of a defining characteristic of complex networks (see attached, which is undoubtedly to remedial but makes my point for me). Granted, most of the networks I'm concerned with are neural (either artificial or models), and I generally use climate sciences & systems biology for fooling around with possible approaches, so community structure in terms of actual people isn't my forte. That said, one of the things that makes graph theory and network analysis so powerful are the many ways in which similar features can be analyzed, or the same topological structures be represented in different ways. Also, while I haven't used social network analysis or network analysis in general when it comes to human communities, I have studied human communities, and I find the work on community structure to be sometimes over-dominated by the representation or search for structures without sufficiently motivated theoretical bases. That is, I can define communities in terms similar to the classic Karate class model or using ecological models, but there are good theoretical reasons for believing that online communities will differ structurally and dynamically than will communities in places in the world where kinship dominates social dynamics and pervades all sociocultural practices, institutions, mores, etc.

Dear Andrew Messing,

Many thanks for your insightful comments!

I tend to believe that simple network are like mechanical watches that are decomposable, while complex network like human brains that are hard to decompose. Communities in a complex network are not only nested, but also overlap each other. There are far more small communities than large ones. The scaling or fractal pattern of far more small communities than large ones can be compared to dropping a wine glass on stone ground, leading to far more small pieces than large ones.

Dear Bin Jiang,

I am sorry but this question is not within my research field.

Best Regards

Anna-Greta M

Hi, Following the remark by Christopher Alexander that "the details of a building cannot be made alive when they are made from modular parts." (p. 25-26 of the following source), I would argue that a real complex network cannot be made of communities that are not nested. In other words, a real complex network is full of far more small communities than large ones, nested and overlapped parts/units of various sizes.

https://www.dreamsongs.com/Files/AlexanderPresentation.pdf

To supplement, this set of slides provides a picture of Alexander's work:

https://www.researchgate.net/publication/299337109_A_Mathematical_Model_of_Beauty_for_Sustainable_Urban_Design

Presentation A Mathematical Model of Beauty for Sustainable Urban Design

You mean in all kinds of complex network or only in city networks? In the case of ecological networks, it looks like hierarchical levels of organization (individual, population, community, ecosystem) are not nested within each other, but rather form a much more complex topology. See:

http://www.pnas.org/content/104/50/19891.abstract

http://link.springer.com/article/10.1007%2Fs00442-011-1984-2

Hi,  For me, nested modularity typically refers to hierarchical networks (see link below) which are examples of networks with modular structure - often cited in the neuroscience literature - but other community structures are possible.

http://www.barabasilab.com/pubs/CCNR-ALB_Publications/200302-14_PhysRevE-HierarchicalOrg/200302-14_PhysRevE-HierarchicalOrg.pdf

@Marco Aurelio Ribeiro Mello

You mean in all kinds of complex network or only in city networks? In the case of ecological networks, it looks like hierarchical levels of organization (individual, population, community, ecosystem) are not nested within each other,....

I am not a ecologist, but I tend to think naively that the ecosystems at the city scale are included in the ecosystems at the country scale, which are further included in ecosystem at a larger scale. Does it make sense to you? Thanks.  

Dear Bin, yes, but only conceptually. When you analyze nestedness and modularity in ecological networks, you do not see a relationshio between those hierarchical levels of organization and the communities detected.

Dear Marco,

Yes, I meant conceptually. Taking a country ecosystem as a whole, it consists of many communities defined at regional level, even more communities at city level, and further more communities at neighborhood level.

I have just found that the following diagram is very much in line with what I said earlier simple networks like mechanical watches, while complex networks like human brains. It comes from this source: http://permaliv.blogspot.com/2015/01/design-for-en-levende-planet.html

Dear Bin Jiang,

Nested communities provide benefits akin to entanglement in quantum theory, greatly augmenting the utility of spaces and resources, because of the possibility of duality, multiplicity that a, decoupled ,non-nested structure lacks. Nevertheless, nested structures may also increase the fragility of the system exponentially. The crucial issue is whether the nested structure has the right kind of diversity and interconnection/intertwining structures.

I tend to think in metaphorical terms of bloatware versus "goodware" in software. Software by definition consist of nested algorithms, algorithms that run recursively. Bloatware multiplies these nested structures needlessly (approximates 1/f noise?) while "goodware" maintains just the right level of complexity and diversity in it. That is why thinking about cities and social networks from an information-theoretic perspective can make typical problems more tractable. Piece-meal, incremental improvements added to a complex system without understanding the implications of these actions (through simulations- to consider the different possible trajectories) can lead to self-organized criticality (SOC)- Bak et.al sandpile model comes to mind.

Cheers

http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.59.381