Take a look at hierarchical/multilevel/mixed-effects modelling. Essentially it is the inclusion of random-effects at hierarchical levels of data structures. It accounts for clustering and sample bias. For example in a longitudinal study there will be clustering along the time dimension (i.e. measurements at time 1 are similar, time 2 are similar) and clustering at the participant level (all measurements for ppt 1 are similar, ppt 2 are similar, etc). You can add groups to this - ppts in schools, in districts. All ppts in a school will be similar, all schools in a district will be similar. It allows for the creation of a complex structure of variance, or partitioning of common error.

I think.

It can be related also to Hierarchical games for the study of complex networks with interaction between players. The hierarchy comes from the type of information learnd by each player.

See these two paper by Leskovec and his group.

http://arxiv.org/abs/0810.1355

http://arxiv.org/abs/0812.4905

The first one studies hierarchical clustering on several naturally evolved large scale network. And the second (less relevant) describes a generative model which is essentially hierarchical and self-similar.

Hierarchy of complex network reflect in a certain degree the similar strategies in enlarging of complex networks. To cite an example, researchers generally cooperate in their works and form acedemic groups. The group is formed accroding to a certain rule. Differen groups can integrated into a super group. Though the size is lenlarged, but the structure is formed according to a similar strategy with that in each group. By this way, the acedemic society is formed.

If the rules are identical, a fractal structure will occur. But generally, at different scales the behaviors may be different. A private person's strategy in a group maybe different with that of a group in a group's group. This difference leads to disortion of perfect fractal structure, and the new structure is called hierarchical structure.

Dear Carlos, there are different ways to define hierarchy in complex networks. One definition that I frequently use refers to the ranking of centrality among vertices, i.e., which vertices are relatively more important for maintaining the whole network structure. You may also define hierarchy in relation to the structure of cohesive subgroups: depending on the similarity in their pattern of connections, you may classify different vertices successively in subgroups with different degrees of similarity.

Take a look at these two papers by Barabasi :

http://arxiv.org/abs/cond-mat/0206130,

http://www.sciencemag.org/content/297/5586/1551.abstract

One of the definitions of hierachy in complex networks corresponds to nested hierachy in which small groups of nodes (moduels) organize in a hierarchical manner to form larger groups and so on.

Dear Carlos, Hierarchy in trophic networks is an old problem in ecosystems theory. Hierarchical network clustering has been described in (See also Section 5 in .) The aggregation scheme is, of course, complicated by cycling. See . Good luck! Bob

Thank you for your answers. In my opinion, I miss in the concept of hierarchy (applied to complex networks) the idea of order. I see that neither barabasi's works as @Atieh Mirshahvalad points out nor the centrality suggested by @Marco Mello captures this idea. In my experience having worked in metabolic networks with the idea of hierarchical clustering as suggested @Rami Puzis is that most of you see (at least in metabolism) is a consequence of that metabolism is bipartite graph that when collapses in a metabolite netwrk introduces a bias that produce and scaling in clustering degree distribution (See Montañez et al 2011 Bioessays in my profile).

Concerning @Ulanowicz comment, I agree that the problem of cycles, especially in food webs is crucial in the definition of hierarchy. But intrinsically captures the idea that in a cycle there is no order and therefore hierarchy.

Thank you for the comments.

A hierarchical network has a distinct pattern of node groupings; the network is not homogeneous because there are subsets of nodes more densely connected. These groups of nodes are in turn linked among them, forming groups of groups up through all levels of organization in the network. Food webs in ecosystems are neat examples: primary producers, consumers, predators etc. are hierarchical levels of organization linking groups of species in each of these levels; in turn, these levels are connected among them.

@Jordano. But then... what is the difference with modularity? ;) According to you are saying, Does a network need to be modular to exhibit hierarchy?

Thank to all for this exciting and productive discussion!!!

Dear Carlos, I agree with Pedro. But there are really many concepts of hierarchy in network theory. Take a look at this website: http://cyvision.ifsc.usp.br/networks/papers.htm. Luciano Costa and his co-workers published a lot of papers on hierarchical structure. They developed some really interesting concepts. Look at this one too: http://cyvision.ifsc.usp.br/~bant/hierarchical/.

Carlos, yes, you need modules to have a hierarchical structure. The key point with hierarchy is that you get 'modules of modules', i.e., layers of additional connection up to all levels of the network. This 'modularity of modules' may generate fractal-like network structures.

Fort instance, we have examined the modularity of some super-complex plant-pollinator interaction networks, like those in tropical rainforest which involve hundreds of species. You get modules, but then groups of modules show up, where some of the modules appear more densely linked among them than with other modules. For example, some of these groups of modules link together distinct modules of pollinators of the canopy of the rainforest, while others group together pollinators inhabiting the understory of the forest. Less complex networks can still be modular without showing such a hierarchical structure.

Another nice paper on hierarchical structure; this one focuses on links instead of vertices: http://www.nature.com/nature/journal/v466/n7307/full/nature09182.html

@Jordano, I thought that the idea of "modules of modules" was more related to "nestedness" than hierarchy. I see that nesting groups into lager groups is a kind of order, as it happens with matrioska dolls but I wonder if this really fits with other ideas oh hierarchy such as a pure tree. According the view you explain a tree is not actually jierarchical, because you have not got mudules. I see that the problem when we talk about hierarchy is that there are many ideas of hierarchy. Don't you think?

The tree is hierarchical because you can detect subtrees (groups of branch tips) that are properly joined within larger subtrees.

The nested structure is simpler because it implies a linear hierarchy (one set within a larger set within a larger set...). There are not GROUPS of sets more connected and then nested again within 'higher' sets. This would be like having groups of matrioska dolls included in larger dolls and then included within more groups of dolls...

Together with Jens Olesen we've found that things are a bit more complicated in real-world plant-animal interaction networks: modules within a nested network show nested structures internally, especially when they are species-rich. This leads to a fractal-like structure in these pollination networks, with self-similar arrangement of the interaction patterns.

My understanding is based on social networks where I believe hierarchy is essentially as described above in that you have clear groupings (the specific terminology for these various groupings varies across disciplines and studies) within a larger network. These groupings can be based on various metrics (e.g., strength, reach, and betweenness) but essentially a particular grouping consists of nodes (or individuals) with the highest number of connections to one another based on some threshold. Nodes with the highest connections are grouped together at one level; nodes with fewer connections are grouped at the next level and so on. In the end you effectively have clusters of nodes, with some nodes connected between clusters. Modularity, on the other hand, refers to how connected these various groups or clusters are to one another. For instance, high modularity occurs when groups are connected by very few nodes (i.e., only a few individuals move among groups) whereas low modularity occurs when there are many nodes connected among groups (i.e., many individuals move among groups). At least that's my understanding of things. Therefore, you can still have a hierarchy with low modularity.

I realize my post is pretty late, but hopefully this helps.

Thank your for the very helpful discussion. I want to share with you our contribution from the perspective of complex networks to the topic of hierarchy: http://www.pnas.org/content/110/33/13316.abstract

I hope you find it interesting.

Best

Carlos

Hi Carlos,

The theoretical foundations of your question was a topic of discussion at the IUBS symposium in the late 60's organized by Waddington and published as a series of publications under the title, "Towards a Theoretical Biology".

I think the confusing matter is that the world is still more complex than computers, so it remains "beyond description".  Therefore, we attempt salient reconstructions of the part with which we are interested, for example, by characterizing modules.  If one is good, we can model modules of modules but the connectivity is extremely complex.

This is borne out by Waddington's statements regarding a "complete" description needing to address the different timescales (physiology, development, evolution). 

Here is a little blurb that addresses a part of your question:

https://www.researchgate.net/post/Can_you_communicate_the_concept_of_nonholonomic_constraints_to_a_biologist

hth,

Jerry

You should also consider the concept of heterarchy, which is commonly used in social networks, but might be applied to any other types of networks (Cumming 2016: http://dx.doi.org/10.1016/j.tree.2016.04.009).

I am working on several fields of natural and artificial complex systems. For me hierarchy is a basic characteristic of the complex system. Complex system contains constancies. In a multi-level hierarchical system the constancies in all level represent an embedding sequence starting from the top level, and closing at the bottom level structures. I have a paper on the RG with title: Structural hierarchy. There are drawings which help clearing the written sructure. Structural hierarchy concept helps in unifying several decomposed joint systems in a complex natural phenomenon: i.e. the soil. The detailed description of the soil contains the structural hierarchies of the mineralogical, the chemical, the physical and biological systems.

Dr. Berczi, By way of comparison, you might be interested in (https://people.clas.ufl.edu/ulan/files/Lindeman.pdf). Best of success, Bob Ulanowicz

Dr. Ulanowicz, Thank you very much.

Hierarchical thermodynamics studies evolutionary changes in real biological systems. Physical-chemical approaches allowed us to explain many facts and make predictions that were confirmed.

Gladyshev G.P. Life – a complex spontaneous process takes place against the background of non-spontaneous process processes initiated by the environment. J Thermodyn Catal 2017, 8: 2 DOI: 10,4172 / 2157-7544.100018

And

1. Gladyshev G.P., Thermodynamics of the origin of life, evolution, and aging. International Journal of Natural Science and Reviews, 2018; 2:7. USA. http://escipub.com/ijnsr-2018-01-1001/

2. Gladyshev G. P. On General Physical Principles of Biological Evolution, International Journal of Research Studies in Biosciences. Volume 5, Issue 3, 2017, Page No: 5-10. https://www.arcjournals.org/pdfs/ijrsb/v5-i3/2.pdf

https://www.researchgatenet/publication/314187646_On_General_Physical_Principles_of_Biological_Evolution

3. Gladyshev G. P. Hierarchical Thermodynamics: Foundation of Extended Darwinism. Imperial Journal of Interdisciplinary Research (IJIR) Vol-3, Issue-2, 2017, ISSN: 2454-1362, https://www.researchgate.net/publication/314082150_Hierarchical_Thermodynamics_Foundation_of_Extended_Darwinism

my understanding is that a complex networks represents one type of complex thermal system, and many systems structured hierarchically.

There exists of a network hierachies which require that some black lives retain more value than others? DO YOU AGREE DISCUS