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| {{Other uses|Modularity}}
| | Surely the second option would be more beneficial for any website. The next step is to visit your Word - Press blog dashboard. Wordpress Content management systems, being customer friendly, can be used extensively to write and manage websites and blogs. Word - Press also provides protection against spamming, as security is a measure issue. It's as simple as hiring a Wordpress plugin developer or learning how to create what is needed. <br><br>purcase and download - WPZOOM Tribune wordpress Theme, find and use the WPZOOM Discount Code. You do not catch a user's attention through big and large pictures that usually takes a millennium to load up. If you cherished this short article and you would like to obtain far more information pertaining to [http://roaaad.com/link//wordpress_backup_5150130 wordpress backup plugin] kindly take a look at our own web-page. Well Managed Administration The Word - Press can easily absorb the high numbers of traffic by controlling the server load to make sure that the site works properly. E-commerce websites are meant to be buzzed with fresh contents, graphical enhancements, and functionalities. But in case you want some theme or plugin in sync with your business needs, it is advisable that you must seek some professional help. <br><br>Digital photography is a innovative effort, if you removethe stress to catch every position and viewpoint of a place, you free yourself up to be more innovative and your outcomes will be much better. The following piece of content is meant to make your choice easier and reassure you that the decision to go ahead with this conversion is requited with rich benefits:. Possibly the most downloaded Word - Press plugin, the Google XML Sitemaps plugin but not only automatically creates a site map linking to everyone your pages and posts, it also notifies Google, Bing, Yahoo, and Ask. Nonetheless, with stylish Facebook themes obtainable on the Globe Broad Internet, half of your enterprise is done previously. Websites using this content based strategy are always given top scores by Google. <br><br>Additionally Word - Press add a default theme named Twenty Fourteen. In case you need to hire PHP developers or hire Offshore Code - Igniter development services or you are looking for Word - Press development experts then Mindfire Solutions would be the right choice for a Software Development partner. Specialty about our themes are that they are easy to load, compatible with latest wordpress version and are also SEO friendly. Word - Press is the most popular open source content management system (CMS) in the world today. This includes enriching the content with proper key words, tactfully defining the tags and URL. <br><br>As a open source platform Wordpress offers distinctive ready to use themes for free along with custom theme support and easy customization. Sanjeev Chuadhary is an expert writer who shares his knowledge about web development through their published articles and other resource. Just download it from the website and start using the same. This is because of the customization that works as a keystone for a SEO friendly blogging portal website. The 2010 voting took place from July 7 through August 31, 2010. |
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| '''Modularity''' is one measure of the structure of [[Complex network|networks]] or [[Graph (mathematics)|graphs]]. It was designed to measure the strength of division of a network into modules (also called groups, clusters or communities). Networks with high modularity have dense connections between the nodes within modules but sparse connections between nodes in different modules. Modularity is often used in optimization methods for detecting [[community structure]] in networks. However, it has been shown that modularity suffers a resolution limit and, therefore, it is unable to detect small communities. Biological networks, including animal brains, exhibit a high degree of modularity.
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| ==Motivation==
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| Many scientifically important problems can be represented and empirically studied using networks. For example, biological and social patterns, the World Wide Web, metabolic networks, food webs, neural networks and pathological networks are a few examples of real world problems that can be mathematically represented and topologically studied to reveal some unexpected structural features.<ref name="npnas">{{cite journal
| |
| | author = Newman, M. E. J.
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| | year = 2006
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| | title = Modularity and community structure in networks
| |
| | journal = Proceedings of the National Academy of Sciences of the United States of America
| |
| | volume = 103
| |
| | issue = 23
| |
| | pages = 8577–8696
| |
| | doi = 10.1073/pnas.0601602103
| |
| | pmid=16723398
| |
| | pmc=1482622
| |
| |arxiv = physics/0602124 |bibcode = 2006PNAS..103.8577N }}</ref> Most of these networks possess a certain community structure that has substantial importance in building an understanding regarding the dynamics of the network. For instance, a closely connected social community will imply a faster rate of transmission of information or rumor among them than a loosely connected community. Thus, if a network is represented by a number of individual nodes connected by links which signify a certain degree of interaction between the nodes, communities are defined as groups of densely interconnected nodes that are only sparsely connected with the rest of the network. Hence, it may be imperative to identify the communities in networks since the communities may have quite different properties such as node degree, clustering coefficient, betweenness, centrality.<ref name="mathnet">{{cite journal
| |
| | author = Newman, M. E. J.
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| | year = 2007
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| | title = Mathematics of networks
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| | journal = The New Palgrave Encyclopedia of Economics
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| | edition = 2
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| | editors = Palgrave Macmillan, Basingstoke
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| }}</ref> etc., from that of the average network. Modularity is one such measure, which when maximized, leads to the appearance of communities in a given network.
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| | |
| ==Definition==
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| Modularity is the fraction of the edges that fall within the given groups minus the expected such fraction if edges were distributed at random. The value of the modularity lies in the range [−1/2,1). It is positive if the number of edges within groups exceeds the number expected on the basis of chance. For a given division of the network's vertices into some modules, modularity reflects the concentration of nodes within modules compared with random distribution of links between all nodes regardless of modules.
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| | |
| There are different methods for calculating modularity.<ref name="npnas"/> In the most common version of the concept, the randomization of the edges is done so as to preserve the [[Degree (graph theory)|degree]] of each vertex. Let us consider a graph with <math>n</math> [[Vertex (graph theory)|nodes]] and <math>m</math> links ([[Edge (graph theory)#Graph|edges]]) such that the graph can be partitioned into 2 communities using a membership variable <math>s</math>. If a node <math>i</math> belongs to community 1, <math>s_i = 1</math>, or if <math>i</math> belongs to community 2, <math>s_i = -1</math>. Let the [[adjacency matrix]] for the network be represented by <math>A</math>, where <math>A_{ij} = 0</math> means there's no edge (no interaction) between nodes <math>i</math> and <math>j</math> and <math>A_{ij} = 1</math> means there is an edge between the two. Also for simplicity we consider an undirected network. Thus <math>A_{ij} = A_{ji}</math>. (It is important to note that multiple edges may exist between 2 nodes and here we assess the simplest case).
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| Modularity Q is then defined as the fraction of edges that fall within group 1 or 2, minus the expected number of edges within group 1 and 2 for a random graph with same node degree distribution as the given network.
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| The expected number of edges shall be computed using the concept of Configuration Models.<ref name="config">{{cite journal
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| | author = Remco van der Hofstad | |
| | year = 2013
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| | title = Random Graphs and Complex Networks
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| | url = http://www.win.tue.nl/~rhofstad/NotesRGCN.pdf#page=149
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| | chapter = 7
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| }}</ref> The configuration model is a randomized realization of a particular network. Given a network with ''n'' nodes, where each node ''i'' has a node degree ''k''<sub>''i''</sub>, the configuration model cuts each edge into 2 halves, and then each half edge, called a stub, is rewired randomly with any other stub in the network even allowing self loops. Thus, even though the node degree distribution of the graph remains intact, the configuration model results in a completely random network.
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| Let the total number of stubs be ''l''<sub>''n''</sub>
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| :<math>
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| l_{n}= \sum_{i}^{n} k_{i} =2m
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| </math>
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| Now, if we randomly select two nodes ''i'' and ''j'' with node degrees ''k''<sub>''i''</sub> and ''k''<sub>''j''</sub> respectively and rewire the stubs for these 2 nodes, then,
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| Expected [Full edges between ''i'' and ''j''] = (Full edges between ''i'' and ''j'')/(Total number of rewirings possible) ''(2)''
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| Total number of rewirings possible = number of stubs remaining after choosing a particular stub
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| = ''l''<sub>''n-1''</sub>= ''l''<sub> ''n'' </sub> for large ''n''
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| Thus, Expected [Number of full edges between ''i'' and ''j'']=(''k''<sub>''i''</sub>* ''k''<sub>''j''</sub>)/''l''<sub>''n''</sub> =(''k''<sub>''i''</sub> ''k''<sub>''j''</sub>)/2''m''.
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| Hence, the actual number of edges between ''i'' and ''j'' minus expected number of edges between them is ''A''<sub>''ij''</sub>-(''k''<sub>''i'' </sub> ''k''<sub>''j'' </sub>)/2''m''. Thus using <ref name="npnas" />
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| :<math>
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| Q = \frac{1}{2m} \sum_{ij} \left[ A_{ij} - \frac{k_i*k_j}{2m} \right] \frac{s_{i} s_{j}+1}{2} (3)
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| </math>
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| It is important to note that (3) holds good for partitioning into 2 communities only. Hierarchical partitioning (i.e. partitioning into 2 communities, then the 2 sub-communities further partitioned into 2 smaller sub communities only to maximize ''Q'') is a possible approach to identify multiple communities in a network. Additionally, (3) can be generalized for partitioning a network into ''c'' communities.<ref name="community">{{cite journal
| |
| | author = Clauset, Aaron and Newman, M. E. J. and [[Cris Moore|Moore, Cristopher]]
| |
| | year = 2004
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| | title = Finding community structure in very large networks
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| | journal = Phys. Rev. E
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| | volume = 70
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| | issue = 6
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| | pages = 066111
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| | doi = 10.1103/PhysRevE.70.066111
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| |arxiv = cond-mat/0408187 |bibcode = 2004PhRvE..70f6111C }}</ref>
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| :<math>
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| Q = \sum_{ij} \left[ \frac {A_{ij}}{2m} - \frac{k_i*k_j}{(2m)(2m)} \right] \delta(c_{i}, c_{j})
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| =\sum_{i=1}^{c} (e_{ii}-a_{i}^2) (4)
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| </math>
| |
| | |
| where ''e''<sub>''ii''</sub> is the fraction of edges with both end vertices in the same community ''i'':
| |
| :<math>
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| e_{ii}= \sum_{j} \frac{A_{ij}}{2m} \delta(c_i,c_j)
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| </math>
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|
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| and ''a''<sub>''i''</sub> is the fraction of edges with at least one end vertex in community ''i'':
| |
| :<math>
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| a_i=\frac{k_i}{2m}
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| = \sum_{j} e_{ij}
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| </math>
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| | |
| ==Example of multiple community detection==
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| We consider an undirected network with 10 nodes and 12 edges and the following adjacency matrix.
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| [[File:Sample Network.jpg|thumb|Fig 1. Sample Network corresponding to the Adjacency matrix with 10 nodes, 12 edges.]]
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| [[File:Partitioned network.jpg|thumb|Fig 2. Network partitions that maximize Q. Maximum Q=0.4895]]
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| {| class="wikitable"
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| |-
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| ! Node ID !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 10
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| |-
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| | 1 || 0 || 1 || 1 || 0 || 0 || 0|| 0 || 0 || 0 || 1
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| |-
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| | 2 || 1 || 0|| 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0
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| |-
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| | 3 || 1 || 1 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 0
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| |-
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| | 4 || 0 || 0 ||0 || 0 || 1 || 1|| 0 || 0 || 0 || 1
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| |-
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| | 5|| 0 || 0 || 0 || 1 || 0 || 1 || 0 || 0 || 0 || 0
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| |-
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| | 6 || 0 || 0 || 0 || 1 || 1 || 0 || 0 || 0 || 0 || 0
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| |-
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| | 7 || 0 || 0 || 0 || 0 || 0 || 0 || 0 || 1 || 1 || 1
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| |-
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| | 8 || 0 || 0 || 0 || 0 || 0 || 0 || 1 || 0 || 1 || 0
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| |-
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| | 9 || 0 || 0 || 0 || 0 || 0 || 0 || 1 || 1 || 0 || 0
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| |-
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| | 10 || 1 || 0 || 0 || 1 || 0 || 0 || 1 || 0 || 0 || 0
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| |}
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| The communities in the graph are represented by the red, green and blue node clusters in Fig 1. The optimal community partitions are depicted in Fig 2.
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| ==Matrix formulation==
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| An alternative formulation of the modularity, useful particularly in spectral optimization algorithms, is as follows.<ref name="npnas" /> Define ''S''<sub>''ir''</sub> to be 1 if vertex ''i'' belongs to group ''r'' and zero otherwise. Then
| |
| | |
| :<math>
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| \delta(c_i,c_j) = \sum_r S_{ir} S_{jr}
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| </math>
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| and hence | |
| | |
| :<math>
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| Q = \frac{1}{2m} \sum_{ij} \sum_r \left[ A_{ij} - \frac{k_i k_j}{2m} \right] S_{ir} S_{jr}
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| = \frac{1}{2m} \mathrm{Tr}(\mathbf{S}^\mathrm{T}\mathbf{BS}),
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| </math>
| |
| | |
| where '''S''' is the (non-square) matrix having elements ''S''<sub>''ir''</sub> and '''B''' is the so-called modularity matrix, which has elements
| |
| | |
| :<math>
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| B_{ij} = A_{ij} - \frac{k_i k_j}{2m}.
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| </math>
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| All rows and columns of the modularity matrix sum to zero, which means that the modularity of an undivided network is also always zero.
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| | |
| For networks divided into just two communities, one can alternatively define ''s''<sub>''i''</sub> = ±1 to indicate the community to which node ''i'' belongs, which then leads to
| |
| | |
| :<math>
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| Q = {1\over 2m} \sum_{ij} B_{ij} s_i s_j = {1\over 2m} \mathbf{s}^\mathrm{T}\mathbf{Bs},
| |
| </math>
| |
| | |
| where '''s''' is the column vector with elements ''s''<sub>''i''</sub>.<ref name="npnas" />
| |
| | |
| This function has the same form as the [[Hamiltonian (quantum mechanics)|Hamiltonian]] of an Ising [[spin glass]], a connection that has been exploited to create simple computer algorithms, for instance using [[simulated annealing]], to maximize the modularity. The general form of the modularity for arbitrary numbers of communities is equivalent to a Potts spin glass and similar algorithms can be developed for this case also.<ref name="rb06">{{cite journal
| |
| | author = Joerg Reichardt and Stefan Bornholdt
| |
| | year = 2006
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| | title = Statistical mechanics of community detection
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| | journal = Physical Review E
| |
| | volume = 74
| |
| | issue = 1
| |
| | pages = 016110
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| | doi = 10.1103/PhysRevE.74.016110
| |
| |arxiv = cond-mat/0603718 |bibcode = 2006PhRvE..74a6110R }}</ref>
| |
| | |
| == Resolution limit ==
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| Modularity compares the number of edges inside a cluster with the expected number of edges that
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| one would find in the cluster if the network were a random network with the same number of nodes and where
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| each node keeps its degree, but edges are otherwise randomly attached. This random null model implicitly assumes that
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| each node can get attached to any other node of the network. Such assumption is however unreasonable if the network
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| is very large, as the horizon of a node includes a small part of the network, ignoring most of it.
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| Moreover, this implies that the expected number of edges between two groups of nodes decreases
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| if the size of the network increases. So, if a network is large enough, the expected number of edges between two groups
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| of nodes in modularity's null model may be smaller than one. If this happens, a single edge between the two clusters
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| would be interpreted by modularity as a sign of a strong correlation between the two clusters, and optimizing modularity
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| would lead to the merge of the two clusters, independently of the clusters' features. So, even weakly interconnected complete graphs, which
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| have the highest possible density of internal edges, and represent the best identifiable communities,
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| would be merged by modularity optimization if the network is sufficiently large.<ref>{{cite journal
| |
| | author = Santo Fortunato and Marc Barthelemy
| |
| | year = 2007
| |
| | title = Resolution limit in community detection
| |
| | journal = Proceedings of the National Academy of Sciences of the United States of America
| |
| | volume = 104
| |
| | pages = 36–41
| |
| | url = http://www.pnas.org/content/104/1/36.abstract
| |
| | pmid = 17190818
| |
| | doi = 10.1073/pnas.0605965104
| |
| | issue = 1
| |
| | pmc = 1765466
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| |arxiv = physics/0607100 |bibcode = 2007PNAS..104...36F }}</ref>
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| For this reason, optimizing modularity in large networks would fail to resolve small communities, even when they are well defined. This bias
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| is inevitable for methods like modularity optimization, which rely on a global null model.<ref>{{cite journal
| |
| | author = J.M. Kumpula, J. Saramäki, K. Kaski , and J. Kertész
| |
| | year = 2007
| |
| | title = Limited resolution in complex network community detection with Potts model approach
| |
| | journal = European Physical Journal B
| |
| | volume = 56
| |
| | issue = 1
| |
| | pages = 41–45
| |
| | doi = 10.1140/epjb/e2007-00088-4
| |
| |arxiv = cond-mat/0610370 |bibcode = 2007EPJB...56...41K }}</ref>
| |
| | |
| == Multiresolution methods ==
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| There are two main approaches which try to solve the resolution limit within the modularity context: the addition of a resistance ''r'' to every node, in the form of a [[self-loop]], which increases (''r>0'') or decreases (''r<0'') the aversion of nodes to form communities;<ref>{{cite journal
| |
| | author = Alex Arenas, Alberto Fernández and Sergio Gómez
| |
| | year = 2008
| |
| | title = Analysis of the structure of complex networks at different resolution levels
| |
| | journal = New Journal of Physics
| |
| | volume = 10
| |
| | issue = 5
| |
| | pages = 053039
| |
| | doi = 10.1088/1367-2630/10/5/053039
| |
| |arxiv = physics/0703218 |bibcode = 2008NJPh...10e3039A }}</ref> or the addition of a parameter ''γ>0'' in front of the null-case term in the definition of modularity, which controls the relative importance between internal links of the communities and the null model.<ref name="rb06" /> Optimizing modularity for values of these parameters in their respective appropriate ranges, it is possible to recover the whole mesoscale of the network, from the macroscale in which all nodes belong to the same community, to the microscale in which every node forms its own community, thus the name ''multiresolution methods''. However, it has been recently demonstrated that these methods are intrinsically deficient and their use will not produce reliable solutions.<ref>{{cite journal
| |
| | author = Andrea Lancichinetti and Santo Fortunato
| |
| | year = 2011
| |
| | title = Limits of modularity maximization in community detection
| |
| | journal = Physical Review E
| |
| | volume = 84
| |
| | pages = 066122
| |
| | doi = 10.1103/PhysRevE.84.066122
| |
| |arxiv = 1107.1155 |bibcode = 2011PhRvE..84f6122L }}</ref>
| |
| | |
| ==See also==
| |
| * [[Complex network]]
| |
| * [[Community structure]]
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| * [[Null model]]
| |
| * [[Surprise (networks)|Surprise]]
| |
| | |
| ==References==
| |
| <references/>
| |
| | |
| ==External links==
| |
| * {{cite journal
| |
| | author = M. E. J. Newman
| |
| | year = 2006
| |
| | title = Modularity and community structure in networks
| |
| | journal = Proc. Natl. Acad. Sci. U.S.A.
| |
| | volume = 103
| |
| | pages = 8577–8582
| |
| | url = http://www.pnas.org/content/103/23/8577.full
| |
| | accessdate = 2008-07-11
| |
| | doi = 10.1073/pnas.0601602103
| |
| | pmid = 16723398
| |
| | issue = 23
| |
| | pmc = 1482622
| |
| | bibcode=2006PNAS..103.8577N
| |
| |arxiv = physics/0602124 }}
| |
| | |
| [[Category:Network theory]]
| |
| [[Category:Algebraic graph theory]]
| |
Surely the second option would be more beneficial for any website. The next step is to visit your Word - Press blog dashboard. Wordpress Content management systems, being customer friendly, can be used extensively to write and manage websites and blogs. Word - Press also provides protection against spamming, as security is a measure issue. It's as simple as hiring a Wordpress plugin developer or learning how to create what is needed.
purcase and download - WPZOOM Tribune wordpress Theme, find and use the WPZOOM Discount Code. You do not catch a user's attention through big and large pictures that usually takes a millennium to load up. If you cherished this short article and you would like to obtain far more information pertaining to wordpress backup plugin kindly take a look at our own web-page. Well Managed Administration The Word - Press can easily absorb the high numbers of traffic by controlling the server load to make sure that the site works properly. E-commerce websites are meant to be buzzed with fresh contents, graphical enhancements, and functionalities. But in case you want some theme or plugin in sync with your business needs, it is advisable that you must seek some professional help.
Digital photography is a innovative effort, if you removethe stress to catch every position and viewpoint of a place, you free yourself up to be more innovative and your outcomes will be much better. The following piece of content is meant to make your choice easier and reassure you that the decision to go ahead with this conversion is requited with rich benefits:. Possibly the most downloaded Word - Press plugin, the Google XML Sitemaps plugin but not only automatically creates a site map linking to everyone your pages and posts, it also notifies Google, Bing, Yahoo, and Ask. Nonetheless, with stylish Facebook themes obtainable on the Globe Broad Internet, half of your enterprise is done previously. Websites using this content based strategy are always given top scores by Google.
Additionally Word - Press add a default theme named Twenty Fourteen. In case you need to hire PHP developers or hire Offshore Code - Igniter development services or you are looking for Word - Press development experts then Mindfire Solutions would be the right choice for a Software Development partner. Specialty about our themes are that they are easy to load, compatible with latest wordpress version and are also SEO friendly. Word - Press is the most popular open source content management system (CMS) in the world today. This includes enriching the content with proper key words, tactfully defining the tags and URL.
As a open source platform Wordpress offers distinctive ready to use themes for free along with custom theme support and easy customization. Sanjeev Chuadhary is an expert writer who shares his knowledge about web development through their published articles and other resource. Just download it from the website and start using the same. This is because of the customization that works as a keystone for a SEO friendly blogging portal website. The 2010 voting took place from July 7 through August 31, 2010.