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

Improved Piggery Effluent Management Systems Incorporating Highly Loaded Primary Ponds

This project has demonstrated the technical feasibility and benefits to the Australian pig industry of utilising highly loaded (and significantly smaller) primary effluent ponds for the treatment of effluent from piggery sheds. In comparison to conventional effluent ponds, the project results indicate that highly loaded ponds offer comparable levels of treatment (solids reduction) along with a range of practical and financial benefits including easier desludging, lower overall odour emissions, reduced construction costs, reduced lining and covering costs, and improved potential to establish or expand piggeries at sites limited by separation distance to sensitive receptors

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

Pork water balance model development

Piggery effluent storage and use systems need to be designed and managed to minimise the risk of overtopping (or spilling) and thereby releasing effluent containing nutrients and pathogens into adjoining properties or downstream aquatic environments. This is particularly critical following major rainfall events and during periods of extended wet weather, when the soil in the effluent reuse area is too wet to allow effective effluent irrigation. State and local government regulatory agencies generally specify a minimum average spill recurrence interval (typically ten years) to minimise the risk of spilled effluent contaminating surface water and/or groundwater resources. This design standard may be varied depending on the sensitivity of the receiving environment

Regulating risk and the boundaries of state conduct: a relational perspective on home birth in Australia

The concept of motivated reasoning and conflicting moral domains behind the state’s conduct towards pregnant women, as described by Minkoff and Marshall (2015), can also be observed in the apparent attitudes towards homebirth in Australia. In this commentary, we briefly outline the status of homebirth in Australia and provide some examples of motivated reasoning in the Australian context. Despite this, some commentators have refrained from risk-based judgments to instead emphasize the importance of communication with, and making ‘reasonable accommodation’ for, pregnant women; even in high-risk situations. We consider that a relational approach might work better than Minkoff and Marshall’s conclusion that pregnant women are best situated to decide on risk. Indeed, their paper hints to a relational approach at several points, but this is not explicitly taken up. We also claim that a relational approach provides a way to give rise to a principled compromise of conflicts in this contested space