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

Guessing Numbers of Odd Cycles

For a given number of colours, $s$, the guessing number of a graph is the base $s$ logarithm of the size of the largest family of colourings of the vertex set of the graph such that the colour of each vertex can be determined from the colours of the vertices in its neighbourhood. An upper bound for the guessing number of the $n$-vertex cycle graph $C_n$ is $n/2$. It is known that the guessing number equals $n/2$ whenever $n$ is even or $s$ is a perfect square \cite{Christofides2011guessing}. We show that, for any given integer $s\geq 2$, if $a$ is the largest factor of $s$ less than or equal to $\sqrt{s}$, for sufficiently large odd $n$, the guessing number of $C_n$ with $s$ colours is $(n-1)/2 + \log_s(a)$. This answers a question posed by Christofides and Markstr\"{o}m in 2011 \cite{Christofides2011guessing}. We also present an explicit protocol which achieves this bound for every $n$. Linking this to index coding with side information, we deduce that the information defect of $C_n$ with $s$ colours is $(n+1)/2 - \log_s(a)$ for sufficiently large odd $n$. Our results are a generalisation of the $s=2$ case which was proven in \cite{bar2011index}.Comment: 16 page

Random tree recursions: which fixed points correspond to tangible sets of trees?

Let $\mathcal{B}$ be the set of rooted trees containing an infinite binary subtree starting at the root. This set satisfies the metaproperty that a tree belongs to it if and only if its root has children $u$ and $v$ such that the subtrees rooted at $u$ and $v$ belong to it. Let $p$ be the probability that a Galton-Watson tree falls in $\mathcal{B}$. The metaproperty makes $p$ satisfy a fixed-point equation, which can have multiple solutions. One of these solutions is $p$, but what is the meaning of the others? In particular, are they probabilities of the Galton-Watson tree falling into other sets satisfying the same metaproperty? We create a framework for posing questions of this sort, and we classify solutions to fixed-point equations according to whether they admit probabilistic interpretations. Our proofs use spine decompositions of Galton-Watson trees and the analysis of Boolean functions.Comment: 41 pages; small changes in response to referees' comments; to appear in Random Structures & Algorithm

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

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

Universal lower bound for community structure of sparse graphs

We prove new lower bounds on the modularity of graphs. Specifically, the modularity of a graph $G$ with average degree $\bar d$ is $\Omega(\bar{d}^{-1/2})$, under some mild assumptions on the degree sequence of $G$. The lower bound $\Omega(\bar{d}^{-1/2})$ applies, for instance, to graphs with a power-law degree sequence or a near-regular degree sequence. It has been suggested that the relatively high modularity of the Erd\H{o}s-R\'enyi random graph $G_{n,p}$ stems from the random fluctuations in its edge distribution, however our results imply high modularity for any graph with a degree sequence matching that typically found in $G_{n,p}$. The proof of the new lower bound relies on certain weight-balanced bisections with few cross-edges, which build on ideas of Alon [Combinatorics, Probability and Computing (1997)] and may be of independent interest.Comment: 25 pages, 2 figure

A branching process with deletions and mergers that matches the threshold for hypercube percolation

We define a graph process G(p,q) based on a discrete branching process with deletions and mergers, which is inspired by the 4-cycle structure of both the hypercube Qd and the lattice Zd for large d. Individuals have Poisson offspring distribution with mean 1+p and certain deletions and mergers occur with probability q; these parameters correspond to the mean number of edges discovered from a given vertex in an exploration of a percolation cluster and to the probability that a non-backtracking path of length four closes a cycle, respectively.We prove survival and extinction under certain conditions on p and q that heuristically match the known expansions of the critical probabilities for bond percolation on the lattice Zd and the hypercube Qd. These expansions have been rigorously established by Hara and Slade in 1995, and van der Hofstad and Slade in 2006, respectively. We stress that our method does not constitute a branching process proof for the percolation threshold. However, it can provide a conjecture for other high-dimensional, odd-cycle free transitive graphs such as the body-centered cubic lattice.The analysis of the graph process survival is considerably more challenging than for branching processes in discrete time, due to the interdependence between the descendants of different individuals in the same generation. In fact, it is left open whether the survival probability of G(p,q) is monotone in p or q; we discuss this and some other open problems regarding the new graph process

Assigning times to minimise reachability in temporal graphs

Temporal graphs (in which edges are active at specified times) are of particular relevance for spreading processes on graphs, e.g. the spread of disease or dissemination of information. Motivated by real-world applications, modification of static graphs to control this spread has proven a rich topic for previous research. Here, we introduce a new type of modification for temporal graphs: the number of active times for each edge is fixed, but we can change the relative order in which (sets of) edges are active. We investigate the problem of determining an ordering of edges that minimises the maximum number of vertices reachable from any single starting vertex; epidemiologically, this corresponds to the worst-case number of vertices infected in a single disease outbreak. We study two versions of this problem, both of which we show to be -hard, and identify cases in which the problem can be solved or approximated efficiently