Modularity of minor-free graphs

We prove that a class of graphs with an excluded minor and with the maximum degree sublinear in the number of edges is maximally modular, that is, modularity tends to 1 as the number of edges tends to infinity.Comment: 7 pages, 1 figur

Modularity of regular and treelike graphs

Clustering algorithms for large networks typically use modularity values to test which partitions of the vertex set better represent structure in the data. The modularity of a graph is the maximum modularity of a partition. We consider the modularity of two kinds of graphs. For $r$-regular graphs with a given number of vertices, we investigate the minimum possible modularity, the typical modularity, and the maximum possible modularity. In particular, we see that for random cubic graphs the modularity is usually in the interval $(0.666, 0.804)$, and for random $r$-regular graphs with large $r$ it usually is of order $1/\sqrt{r}$. These results help to establish baselines for statistical tests on regular graphs. The modularity of cycles and low degree trees is known to be close to 1: we extend these results to `treelike' graphs, where the product of treewidth and maximum degree is much less than the number of edges. This yields for example the (deterministic) lower bound $0.666$ mentioned above on the modularity of random cubic graphs.Comment: 25 page

Post-processing partitions to identify domains of modularity optimization

We introduce the Convex Hull of Admissible Modularity Partitions (CHAMP) algorithm to prune and prioritize different network community structures identified across multiple runs of possibly various computational heuristics. Given a set of partitions, CHAMP identifies the domain of modularity optimization for each partition ---i.e., the parameter-space domain where it has the largest modularity relative to the input set---discarding partitions with empty domains to obtain the subset of partitions that are "admissible" candidate community structures that remain potentially optimal over indicated parameter domains. Importantly, CHAMP can be used for multi-dimensional parameter spaces, such as those for multilayer networks where one includes a resolution parameter and interlayer coupling. Using the results from CHAMP, a user can more appropriately select robust community structures by observing the sizes of domains of optimization and the pairwise comparisons between partitions in the admissible subset. We demonstrate the utility of CHAMP with several example networks. In these examples, CHAMP focuses attention onto pruned subsets of admissible partitions that are 20-to-1785 times smaller than the sets of unique partitions obtained by community detection heuristics that were input into CHAMP.Comment: http://www.mdpi.com/1999-4893/10/3/9

Modularité asymptotique de quelques classes de graphes

International audienceDe nombreuses disciplines scientifiques font appel au clustering pour l'analyse de leurs réseaux d'interaction. Pami les très nombreux algorithmes existants, toute une famille utilise la modularité de Newman-Girvan comme objectif à maximiser, et cette valeur est devenue un paramètre de graphe standard. Dans cette étude nous prenons à rebours l'approche empirique usuelle et nous posons la question théorique de la modularité de classes de graphes. Nous montrons que des classes très régulières et sans ''clusters'' naturels (grilles, hypercubes,...) ont une modularité asymptotiquement 1 (le maximum possible), soit bien plus que les valeurs usuelles des données ''bien clusterisées''. Nous montrons que sous réserve d'une condition sur le degré maximum, les arbres ont aussi modularité asymptotique 1. Résultat qui nous permet de fournir une borne inférieure pour la modularité des graphes connexes peu denses et de certains power-law graphs, qui ont modularité asymptotique au moins 2/degré moyen, ainsi qu'un algorithme garantissant cette performance

Modularity of nearly complete graphs and bipartite graphs

It is known that complete graphs and complete multipartite graphs have modularity zero. We show that the least number of edges we may delete from the complete graph $K_n$ to obtain a graph with non-zero modularity is $\lfloor n/2\rfloor +1$. Similarly we determine the least number of edges we may delete from or add to a complete bipartite graph to reach non-zero modularity. We give some corresponding results for complete multipartite graphs, and a short proof that complete multipartite graphs have modularity zero. We also analyse the modularity of very dense random graphs, and in particular we find that there is a transition to modularity zero when the average degree of the complementary graph drops below 1

Modularity and Graph Expansion

We relate two important notions in graph theory: expanders which are highly connected graphs, and modularity a parameter of a graph that is primarily used in community detection. More precisely, we show that a graph having modularity bounded below 1 is equivalent to it having a large subgraph which is an expander. We further show that a connected component $H$ will be split in an optimal partition of the host graph $G$ if and only if the relative size of $H$ in $G$ is greater than an expansion constant of $H$. This is a further exploration of the resolution limit known for modularity, and indeed recovers the bound that a connected component $H$ in the host graph~$G$ will not be split if~$e(H)<\sqrt{2e(G)}$.Comment: Accepted to Innovations in Theoretical Computer Science (ITCS) 202

The parameterised complexity of computing the maximum modularity of a graph

The maximum modularity of a graph is a parameter widely used to describe the level of clustering or community structure in a network. Determining the maximum modularity of a graph is known to be NP-complete in general, and in practice a range of heuristics are used to construct partitions of the vertex-set which give lower bounds on the maximum modularity but without any guarantee on how close these bounds are to the true maximum. In this paper we investigate the parameterised complexity of determining the maximum modularity with respect to various standard structural parameterisations of the input graph G. We show that the problem belongs to FPT when parameterised by the size of a minimum vertex cover for G, and is solvable in polynomial time whenever the treewidth or max leaf number of G is bounded by some fixed constant; we also obtain an FPT algorithm, parameterised by treewidth, to compute any constant-factor approximation to the maximum modularity. On the other hand we show that the problem is W[1]-hard (and hence unlikely to admit an FPT algorithm) when parameterised simultaneously by pathwidth and the size of a minimum feedback vertex set

On the modularity of 3-regular random graphs and random graphs with given degree sequences

The modularity of a graph is a parameter introduced by Newman and Girvan measuring its community structure; the higher its value (between $0$ and $1$), the more clustered a graph is. In this paper we show that the modularity of a random $3-$regular graph is at least $0.667026$ asymptotically almost surely (a.a.s.), thereby proving a conjecture of McDiarmid and Skerman stating that a random $3-$regular graph has modularity strictly larger than $\frac{2}{3}$ a.a.s. We also improve the upper bound given therein by showing that the modularity of such a graph is a.a.s. at most $0.789998$. For a uniformly chosen graph $G_n$ over a given bounded degree sequence with average degree $d(G_n)$ and with $|CC(G_n)|$ many connected components, we distinguish two regimes with respect to the existence of a giant component. In more detail, we precisely compute the second term of the modularity in the subcritical regime. In the supercritical regime, we further prove that there is $\varepsilon > 0$ depending on the degree sequence, for which the modularity is a.a.s. at least \begin{equation*} \dfrac{2\left(1 - \mu\right)}{d(G_n)}+\varepsilon, \end{equation*} where $\mu$ is the asymptotically almost sure limit of $\dfrac{|CC(G_n)|}{n}$.Comment: 41 page