Attributed Graphs are most commonly represented as an adjacency matrix or a list where nodes are considered first class citizens. There are many graph queries such as neighborhood, shortest path, page rank, connected component that operate on these matrix and list structures on nodes. The attributes of the node/edge can also be stored apart from the connections.
Another representation of the graph is an incidence matrix where the incident edges of a node are recorded in a matrix. I understand they represent exactly the same information as previous node-based methods.
My question is, are there any graph queries/workloads/algorithms that can benefit from the incidence matrix structure rather than using the node-based structures i.e. favoring an edge-based structure? When exactly are the incidence matrix used?
I can think of only one case where incidence matrix may prove faster:
Finding the degree of a node or finding adjacent nodes is an operation with complexity O(V) when using an adjacency matrix and O(E) when using an incidence matrix.
Usually E>V, but this may not be the case if the graph has many 0-degree nodes. Since finding adjacent nodes is a basic operation, many algorithms may prove to be faster on such graphs.
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I have a directed graph with non-negative weighted edges where there are multiple edges between two vertices.
I need to compute all pairs shortest path.
This graph is very big (20 milion vertices and 100 milion of edges).
Is Floyd–Warshall the best algorithm ? There is a good library or tool to complete this task ?
There exists several all-to-all shortest paths algorithms for directed graphs with non-negative cycles, Floyd-Warshall being probably the most famous, but with the figures you gave, I think you will have in any case memory issues (time could be an issue, but you can find all-to-all algorithm that can be easily and massively parallelized).
Independently of the algorithm you use, you will have to store the result somewhere. And storing 20,000,000² = 400,000,000,000,000 paths length (if not the full paths themselves) would use hundreds of terabytes, at the very least.
Accessing any of these results would probably be longer than calculating one shortest path (memory wall), which can be done in less than a milisecond (depending on the graph structure, you can find techniques that are much, much faster than Dijkstra or any priority queue based algorithm).
I think you should look for an alternative where computing all-to-all shortest paths is not required, to be honnest. Or, to study the structure of your graph (DAG, well structured graph easy to partition/cluster, geometric/geographic information ...) in order to apply different algorithms, because in the general case, I do not see any way around.
For example, with the figures you gave, an average degree of about 5 makes for a decently sparse graph, considering its dimensions. Graph partitioning approaches could then be very useful.
I have a list of about 100 igraph objects with a typical object having about 700 vertices and 3500 edges.
I would like to identify groups of vertices within which ties are more likely. My plan is to then use a mixed model to predict how many within-group ties vertices have using vertex and group attributes.
Some people may want to respond to other aspects of my project, which would be great, but the thing I'm most interested in is information about functions in igraph for grouping vertices. I've come across these community detection algorithms but I'm not sure of their advantages and disadvantages, or whether some other function would be better for my case. I saw the links here as well, but they aren't specific to igraph. Thanks for your advice.
Here is a short summary about the community detection algorithms currently implemented in igraph:
edge.betweenness.community is a hierarchical decomposition process where edges are removed in the decreasing order of their edge betweenness scores (i.e. the number of shortest paths that pass through a given edge). This is motivated by the fact that edges connecting different groups are more likely to be contained in multiple shortest paths simply because in many cases they are the only option to go from one group to another. This method yields good results but is very slow because of the computational complexity of edge betweenness calculations and because the betweenness scores have to be re-calculated after every edge removal. Your graphs with ~700 vertices and ~3500 edges are around the upper size limit of graphs that are feasible to be analyzed with this approach. Another disadvantage is that edge.betweenness.community builds a full dendrogram and does not give you any guidance about where to cut the dendrogram to obtain the final groups, so you'll have to use some other measure to decide that (e.g., the modularity score of the partitions at each level of the dendrogram).
fastgreedy.community is another hierarchical approach, but it is bottom-up instead of top-down. It tries to optimize a quality function called modularity in a greedy manner. Initially, every vertex belongs to a separate community, and communities are merged iteratively such that each merge is locally optimal (i.e. yields the largest increase in the current value of modularity). The algorithm stops when it is not possible to increase the modularity any more, so it gives you a grouping as well as a dendrogram. The method is fast and it is the method that is usually tried as a first approximation because it has no parameters to tune. However, it is known to suffer from a resolution limit, i.e. communities below a given size threshold (depending on the number of nodes and edges if I remember correctly) will always be merged with neighboring communities.
walktrap.community is an approach based on random walks. The general idea is that if you perform random walks on the graph, then the walks are more likely to stay within the same community because there are only a few edges that lead outside a given community. Walktrap runs short random walks of 3-4-5 steps (depending on one of its parameters) and uses the results of these random walks to merge separate communities in a bottom-up manner like fastgreedy.community. Again, you can use the modularity score to select where to cut the dendrogram. It is a bit slower than the fast greedy approach but also a bit more accurate (according to the original publication).
spinglass.community is an approach from statistical physics, based on the so-called Potts model. In this model, each particle (i.e. vertex) can be in one of c spin states, and the interactions between the particles (i.e. the edges of the graph) specify which pairs of vertices would prefer to stay in the same spin state and which ones prefer to have different spin states. The model is then simulated for a given number of steps, and the spin states of the particles in the end define the communities. The consequences are as follows: 1) There will never be more than c communities in the end, although you can set c to as high as 200, which is likely to be enough for your purposes. 2) There may be less than c communities in the end as some of the spin states may become empty. 3) It is not guaranteed that nodes in completely remote (or disconencted) parts of the networks have different spin states. This is more likely to be a problem for disconnected graphs only, so I would not worry about that. The method is not particularly fast and not deterministic (because of the simulation itself), but has a tunable resolution parameter that determines the cluster sizes. A variant of the spinglass method can also take into account negative links (i.e. links whose endpoints prefer to be in different communities).
leading.eigenvector.community is a top-down hierarchical approach that optimizes the modularity function again. In each step, the graph is split into two parts in a way that the separation itself yields a significant increase in the modularity. The split is determined by evaluating the leading eigenvector of the so-called modularity matrix, and there is also a stopping condition which prevents tightly connected groups to be split further. Due to the eigenvector calculations involved, it might not work on degenerate graphs where the ARPACK eigenvector solver is unstable. On non-degenerate graphs, it is likely to yield a higher modularity score than the fast greedy method, although it is a bit slower.
label.propagation.community is a simple approach in which every node is assigned one of k labels. The method then proceeds iteratively and re-assigns labels to nodes in a way that each node takes the most frequent label of its neighbors in a synchronous manner. The method stops when the label of each node is one of the most frequent labels in its neighborhood. It is very fast but yields different results based on the initial configuration (which is decided randomly), therefore one should run the method a large number of times (say, 1000 times for a graph) and then build a consensus labeling, which could be tedious.
igraph 0.6 will also include the state-of-the-art Infomap community detection algorithm, which is based on information theoretic principles; it tries to build a grouping which provides the shortest description length for a random walk on the graph, where the description length is measured by the expected number of bits per vertex required to encode the path of a random walk.
Anyway, I would probably go with fastgreedy.community or walktrap.community as a first approximation and then evaluate other methods when it turns out that these two are not suitable for a particular problem for some reason.
A summary of the different community detection algorithms can be found here: http://www.r-bloggers.com/summary-of-community-detection-algorithms-in-igraph-0-6/
Notably, the InfoMAP algorithm is a recent newcomer that could be useful (it supports directed graphs too).
I'm preparing for a coding interview, and was refreshing my mind on graphs. I was wondering the following : in all places I've seen, it is assumed that adjacency lists are more memory efficient than adjacency matrices for large sparse graphs, and should thus be preferred in that case. In addition, computing the number of outgoing edges from a node requires O(N) in a matrix while it's O(1) in a list, as well as which are the adjacent nodes in O(num adjacent nodes) for the list instead of O(N) for the matrix.
Such places include Cormen et al.'s book, or StackOverFlow : Size of a graph using adjacency list versus adjacency matrix? or Wikipedia.
However, using a sparse matrix representation like with Compressed Row Storage representation, the memory requirement is just in O(number of non-zeros) = O(number of edges), which is the same as using lists. The number of outgoing edges from a node is O(1) (it is directly stored in CRS), and the adjacent nodes can be listed in O(num adjacent nodes).
Why isn't it discussed ? Should I assume that CSR is a kind of adjacency list representation of the graph represented by the matrix ? Or is the argument that matrices are memory intensive flawed because they don't consider sparse matrix representations ?
Thanks!
Not everyone uses sparse matrix representations every day (I just happen to do so :), so I guess nobody thought of them. They are a kind of intermediate between adjacency lists and adjacency matrices, with performance similar to the first if you pick the right representation, and are very convenient for some graph algorithms.
E.g., to get a proximity matrix over two hops, you just square the matrix. I've successfully done this with sparse matrix representations of the Wikipedia link structure in modest amounts of CPU time.
I have a JUNG graph containing about 10K vertices and 100K edges, and I'd like to get a measure of similarity between any pair of vertices.
The vertices represent concepts (e.g. dog, house, etc), and the links represent relations between concepts (e.g. related, is_a, is_part_of, etc).
The vertices are densely inter-linked, so a shortest-path approach doesn't give good results (the shortest paths are always very short).
What approaches would you recommend to rank the connectivity between vertices?
JUNG has some algorithms to score the importance of vertices, but I don't understand if there are measures of similarity between 2 vertices.
SimPack seems also promising.
Any hints?
The centrality scores don't measure similarity of pairs of vertices, but some kind of (depending on the method) centrality of single nodes of the network in general. Therefore this approach is possibly not what you want.
SimPack indeed has a nice goal set out, but for graphs it implements isomorphism-based comparations, which rather compare multiple graphs for similarity than pairs of nodes of one given graph. Therefore this is out of scope for now.
What you are seeking are so-called graph clustering methods (also called network module determination or network community determination methods), which divide the graph (network) into multiple partitions so that the nodes in each partition are more strongly interconnected with each other than with nodes of other partitions.
The most classic method is maybe the betweenness centrality clustering of Newman & Girvan where you can exploit the dendrogram for similarity calculation, and it is in JUNG. Of course there are throngs of methods nowadays. You may want to try (shameless plug) our ModuLand method, or read the fine table of module detection algorithms at the end of the Electronic Supplementary Material. That is an overlapping graph clustering method family, that is its result for each node is a vector containing the strengths of belonging to any respective cluster of the network. Pairwise node similarity is easy to derive from pairs of these node-to-cluster vectors.
Graph clustering is non-trivial, and possible you would need to adapt any method for very precise domain-specific results, but that's up to the reader ;) Good luck!
I have an undirected, unweighted graph, which doesn't have to be planar. I also have a subset of graph's nodes (true subset) and I need to find a node not belonging to the subset, with minimum sum of distances to all nodes in the subset.
So far, I have implemented breath-first search starting from each node in the subset, and the intersection that occurs first is the node I am looking for. Unfortunately, it is running too slow since the graph contains a large number of nodes.
An all-pair shortest path algorithm allows you to find the distance of all nodes to each other in O(V^3) time, see Floyd-warshall. Then summing afterwards will at least be quadratic and I believe worst case cubic as well. It's a very straightforward and not terribly fast way of doing it, but it sounds like it might be an order of magnitude faster than what you're doing right now.