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### igraph Reference Manual

Asked 8th Aug, Seema Aswani. Can anyone provide a short example of how the modularity is being calculated in networks? I have read the documents available on google and also gone through the Wikipedia's definition and formula about calculating the modularity for a particular network.

I am using r studio for implementation. For example if my network is like following. Then the value of modularity being calculated in r studio as an output for this network is 0. How the answer 0. Can any one explain the mathematical formulation of modularity for networks.

How the formula that is given in Wikipedia is being applied in this case that i am not getting. I hope my question make sense. Thanks in advance. Linear Algebra. Network Architecture. Bioinformatics Analysis. Most recent answer.

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Mohammad Radi. University of Tehran. Thanks a lot Matthew.

Popular Answers 1. Matthew Joseph Michalska-Smith. University of Minnesota Twin Cities. While the above address your first question explaining the mathematical formulation of modularityI don't think any sufficiently address the second how the formula is being applied in this caseso I will work through your example below:. This produces an igraph object which can be visualized through a matrix by running:. What is important to note here is that igraph does not sort the matrix by row-name even though you have named your nodes with integers.

Thus, when you assign the membership:. You are stating that nodes 1, 4, and 3 are in module 1 and nodes 2 and 5 are in module 2. Calculates Q as defined by Newman and referenced by both Christopher and Daniel above :. A ij is the element i, j of the adjacency matrix depicted above. In this way, Q only cares about links that fall within modules and ignores those which link separate modules to each other except as a normalizing factor.The modularity of a graph with respect to some division or vertex types measures how good the division is, or how separated are the different vertex types from each other.

Modularity on weighted graphs is also meaningful. See also Clauset, A. Finding community structure in very large networks, Physical Review E,70, Numeric vector which gives the type of each vertex, ie. It does not have to be consecutive, i. Note that self-loops are multiplied by 2 in this implementation. If weights are specified, the weighted counterparts are used. Edge weights, pointer to a vector. If this is a null pointer then every edge is assumed to have a weight of 1. This function calculates the optimal community structure for a graph, in terms of maximal modularity score.

The calculation is done by transforming the modularity maximization into an integer programming problem, and then calling the GLPK library to solve that. Please see Ulrik Brandes et al.

Note that modularity optimization is an NP-complete problem, and all known algorithms for it have exponential time complexity. This means that you probably don't want to run this function on larger graphs. Graphs with up to fifty vertices should be fine, graphs with a couple of hundred vertices might be possible.

Pointer to a real number, or a null pointer. If it is not a null pointer, then a optimal modularity value is returned here. Pointer to a vector, or a null pointer.

If not a null pointer, then the membership vector of the optimal community structure is stored here. Vector giving the weights of the edges. If it is NULL then each edge is supposed to have the same weight. This function creates a membership vector from a community structure dendrogram.

The matrix contains the merge operations performed while mapping the hierarchical structure of a network. If the matrix has n-1 rows, where n is the number of vertices in the graph, then it contains the hierarchical structure of the whole network and it is called a dendrogram.

This function performs steps merge operations as prescribed by the merges matrix and returns the current state of the network. If merges is not a complete dendrogram, it is possible to take steps steps if steps is not bigger than the number lines in merges.

The two-column matrix containing the merge operations. Pointer to an initialized vector, the membership results will be stored here, if not NULL. The vector will be resized as needed. Pointer to an initialized vector, or NULL.GitHub is home to over 40 million developers working together to host and review code, manage projects, and build software together.

Skip to content. Permalink Dismiss Join GitHub today GitHub is home to over 40 million developers working together to host and review code, manage projects, and build software together. Sign up. Branch: master. Find file Copy path. Raw Blame History. The matrix will be resized as needed. The vector will be resized as needed. The algorithm was invented by M. Newman, see: M. Girvan and M. It will be resized as needed. Now we do an inverse search, starting with the farthest nodes.

This is a trick to find multiple leading eigenvalues, because ARPACK is sometimes unstable when the first two eigenvalues are requested, but it does much better for the single principal eigenvalue. This is because for some hard cases it tends to fail. We need to suppress error handling for the first call. It must be an undirected graph. The weights are expected to be non-negative. We still might want the modularity score for that.

ISBN: J Stat Mech P, You signed in with another tab or window. Reload to refresh your session. You signed out in another tab or window.

IGraph library. This program is distributed in the hope that it will be useful. See the.Modularity is one measure of the structure of networks or 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. 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 real world problems that can be mathematically represented and topologically studied to reveal some unexpected structural features.

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.

Modularity is one such measure, which when maximized, leads to the appearance of communities in a given network. Modularity is the fraction of the edges that fall within the given groups minus the expected fraction if edges were distributed at random. For a given division of the network's vertices into some modules, modularity reflects the concentration of edges within modules compared with random distribution of links between all nodes regardless of modules.

**Graph Theory - An Introduction!**

There are different methods for calculating modularity. Also for simplicity we consider an undirected network. It is important to note that multiple edges may exist between two nodes, but here we assess the simplest case. Modularity Q is then defined as the fraction of edges that fall within group 1 or 2, minus the expected number of edges within groups 1 and 2 for a random graph with the same node degree distribution as the given network.

The expected number of edges shall be computed using the concept of a configuration model. Thus, even though the node degree distribution of the graph remains intact, the configuration model results in a completely random network.

We calculate the expected number of full edges between these nodes. If it does not, then its value is 0. Many texts then make the following approximations, for random networks with a large number of edges. Additionally, in a large random network, the number of self-loops and multi-edges is vanishingly small.

It is important to note that Eq. Hierarchical partitioning i. Additionally, 3 can be generalized for partitioning a network into c communities. 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. An alternative formulation of the modularity, useful particularly in spectral optimization algorithms, is as follows.

All rows and columns of the modularity matrix sum to zero, which means that the modularity of an undivided network is also always zero. This function has the same form as the Hamiltonian of an Ising spin glassa connection that has been exploited to create simple computer algorithms, for instance using simulated annealingto 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. Modularity compares the number of edges inside a cluster with the expected number of edges that one would find in the cluster if the network were a random network with the same number of nodes and where each node keeps its degree, but edges are otherwise randomly attached.

This random null model implicitly assumes that each node can get attached to any other node of the network.A problem we see in psychological network papers is that authors sometimes over-interpret the visualization of their data.

This pertains especially to the layout and node placement of the graph, for instance: do nodes in the networks cluster in certain communities. Below I will discuss this problem in some detail, and provide a basic R-tutorial on how to identify communities of items in networks.

You can find the data and syntax I use for this tutorial here — feedback is very welcome in the comments section below. The bottom line makes the graph pretty. When there are only nodes in a network, the algorithm will always place them in the same way where the length of the edges between the nodes represent how strongly they are relatedand the only degree of freedom the algorithm has is the rotation of the graph.

But especially in graphs with many nodes, the placement tells us only a very rough story, and should not be over-interpreted.

While the edges between items are obviously the same, the placement of the nodes differs considerably. They investigated the networks structure of 14 depression symptoms in about psychiatric outpatients diagnosed with Major Depression at two timepoints 12 weeks apart.

A very cool contribution of the paper is that they examined the change of the network structure over time, in a somewhat different way than we did previously in the same dataset. Similar to the network example above, they used regularized partial correlation networks to estimate the cross-sectional network models at both timepoints, and plotted the networks using the Fruchterman-Reingold algorithm. They concluded from visually inspecting the graphs that there are 4 symptom clusters present, and that these did not change over time:.

But these findings and conclusions are merely based on visual inspection of the resulting graphs — and has we have learned above, these should be interpreted with great care. So what might be a better way to do this, and how can we do it in R? There are numerous possibilities, and I introduce three: one very well-established method from the domain of latent variable modeling eigenvalue decomposition ; one well-established algorithm from network science spinglass algorithm ; and a very new tool that is under development exploratory graph analysis which uses the walktrap algorithm.

Traditionally, we would want to describe the 20 items above in a latent variable framework, and the question arises: how many latent variables do we need to explain the covariance among the 20 items?

A very easy way to do so is to look at the eigenvalues of the components in the data. This shows us the value of each eigenvalue of each components on the y-axis; the x-axis shows the different components.

A high eigenvalue means that it explains a lot of the covariance among the items. There are better ways to do so, such as parallel analysis, which you can also do in R.

In any case, depending on the rule we use now, we would probably decide to extract components. We do not yet know what item belongs to what component — for that, we need to run, for instance, an exploratory factor analysis EFA and look at the factor loadings.

And yes, you guessed correctly: you can obviously also do that in R … life is beautiful! Why is this related to networks at all? Numerous papers have now shown that latent variable models and network models are mathematical equivalentwhich means that the numbers of factors that underlie data will in most cases translate to the number of communities you can find in a network.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

The dark mode beta is finally here. Change your preferences any time. Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. It works perfectly for undirected graph objects, but I am getting the Modularity is implemented for undirected graphs only error when attempting to use this function on directed graph objects. I have found a similar question herebut it does not completely resolve my issue.

They suggest that I However, I cannot figure out how to do this. May somebody please point me in the correct direction? NOTE: The result is the exact same as an undirected graph run with the same directed graph object i. The only difference is that I get Modularity is implemented for undirected graphs only printed multiple times before the result.

Learn more. Asked 7 months ago. Active 12 days ago. Viewed times. EDIT: The following is a reproducible example of my issue. I produce a graph object for the function. Let's call that G for now. Modularity is implemented for undirected graphs only. Could you add a reproducible example? JamesMartherus I have added a code example of events leading up to my error.

### Can anyone provide a short example of how the modularity is being calculated in networks?

Please let me know if I can help clarify anything. Thank you very much! Are you using the most recent version of the igraph package? I get no such warning. G5W R igraph version 1. Could this affect the outcome? Active Oldest Votes.

Sign up or log in Sign up using Google. Sign up using Facebook.This function implements the multi-level modularity optimization algorithm for finding community structure, see references below. It is based on the modularity measure and a hierarchial approach.

Optional positive weight vector. If the graph has a weight edge attribute, then this is used by default. Supply NA here if the graph has a weight edge attribute, but you want to ignore it. Larger edge weights correspond to stronger connections. Initially, each vertex is assigned to a community on its own. In every step, vertices are re-assigned to communities in a local, greedy way: each vertex is moved to the community with which it achieves the highest contribution to modularity.

When no vertices can be reassigned, each community is considered a vertex on its own, and the process starts again with the merged communities. The process stops when there is only a single vertex left or when the modularity cannot be increased any more in a step. Vincent D. See communities for extracting the membership, modularity scores, etc. R igraph manual pages Use this if you are using igraph from R. Finding community structure by multi-level optimization of modularity Description This function implements the multi-level modularity optimization algorithm for finding community structure, see references below.

This function was contributed by Tom Gregorovic. Author s Tom Gregorovic, Tamas Nepusz ntamas gmail.

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