## Wednesday, 27 February 2019

### Guessing the spreading time of rumours

Social scientists are fascinated by social influence. That is, how people's beliefs, opinions and actions are influenced by others. This is relevant for understanding voting, health behaviour or opinions on issues like vaccination and climate change (topics our group is interested in). Mathematically inclined social scientists often interpret social influence using network theory.  Networks or graphs are used to represent systems consisting of many individual units, known as nodes, and the interactions between them, which are referred to as edges or links. In social networks the nodes represent people and the links represent social ties such as friendships.

Given a particular graph there are tools for modelling how opinions and beliefs can spread through a graph. However, in practice we often don’t know the structure of the social network itself. This could be because: i) the data we would like is unavailable ii) privacy concerns about social network data mean we can't share it even if we have it iii) the data exists but is full of errors or omissions. Fortunately, we know a lot about the structure of social networks from decades of past research by social scientists and statisticians. For example, many social networks are known to be homophilous - this means that people who are similar to each other are more likely to share a social connection (e.g many of your friends are probably a similar age to you).

Inspired by this, we consider a simple mathematical model for homophilous networks known as a Random Geometric Graph (RGG). In an RGG the nodes are assigned random positions in a (unit) box. Nodes are connected to all the nodes which are within a set distance (see figure), which we refer to as the connection radius. Positions of nodes may represent the positions of individuals in geographic space or in some “social space” where the coordinate axis might represent attributes such as age, income and education level. Since social networks are homophilous we will expect those who are closer together in “social space” to share a social tie.
 Example of a Random Geometric Graph with 100 nodes and a connection radius of 0.2.
One basic question we can ask about a network is: “how long does it take something to spread across it?” We refer to this as the diffusion timescale. The diffusion timescale in a graph is indicative of how well connected the graph is and governs how quickly we might expect a disease, rumour or the adoption of a new behaviour to spread through it (or even how long it will take a zombie apocalypse to take hold). In our recent research we focus on the question:

“If we do not know the network (but perhaps know some of its properties) how precisely can we know the diffusion timescale?”

We show that different RGGs drawn at random with the same number of nodes and connection radius can have very different diffusion timescales. This implies that if we don’t have a good grasp of the graph structure then it could be difficult to predict the outcomes of processes such as the spread of an opinion through a social network. Or alternatively we can gain lots of extra information about diffusion timescales if we happen to know the social co-ordinates of individuals. On the other hand, we do find some classes of RGGs where the diffusion time scale is very predictable given only knowledge of the number of nodes and the connection radius.

Our work helps put limitations on how accurately we can forecast the outcome of processes on networks given the available data (which is always imperfect). Future work may involve asking the same questions for real world datasets. In addition, most of our new results were obtained through computer simulations, meaning that there is also scope for more theory.

You can read about our research in the paper Large algebraic connectivity fluctuations in spatial network ensembles imply a predictive advantage from node location information” for free here or for not-free here in Physical Review E. Matt and Nick.

### How mitochondria can vary, and consequences for human health

Mitochondria are components of the cell which are involved in generating “energy currency” molecules called ATP across much of complex life. Since many mitochondria exist within single cells (often hundreds or thousands), it is possible for the characteristics of individual mitochondria to vary within cells, and within tissues. This variation of mitochondrial characteristics can affect biological function and human health.

Since mitochondria possess their own, small, circular, DNA molecules (mtDNA), we can split mitochondrial characteristics into two categories: genetic and non-genetic. In our review, we discuss a number of aspects in which mitochondria vary, from both genetic and non-genetic perspectives.

In terms of mitochondrial genetics, the amount of mtDNA per cell is variable. When a cell divides, its daughters receive a share of its parents mtDNA, but the split isn’t precisely 50/50, so cell division can cause variability in the number of mtDNAs per cell. As mtDNAs are replicated and degraded over time, errors in the copying process may give rise to mtDNA mutations, which may spread throughout a cell. Factors such as: the total amount, the rate of degradation/replication, the mean fraction of mutants, and the extent of fragmentation in the mitochondrial network, can all influence how variable the fraction of mutated mtDNAs becomes through time (see here for a preview of some upcoming work on this topic). The total amount, and mutated fraction of mtDNAs, are implicated in diseases such as neurodegeneration, as well as the ageing process.

Apart from genetic variations, there are many non-genetic features of mitochondria which also vary within and between cells. Changes in mtDNA sequence can change the amino-acid sequence of the proteins encoded by mtDNA, causing structural changes in the molecular machines which generate ATP. The shape of the membranes of mitochondria are also highly variable, and respond to mitochondrial activity through quantities such as pH, where mitochondrial activity itself may depend on mtDNA sequence. The previous two examples (mitochondrial protein and membrane structure) demonstrate how the genetic state of mitochondria may influence their non-genetic characteristics. Mitochondrial non-genetic characteristics may also influence the genetic state: for instance, mitochondrial membrane potential can influence the probability of a mitochondria being degraded, along with its mtDNA.

The inter-dependence of genetic and non-genetic characteristics demonstrate the complex feedback loops linking these two aspects of mitochondrial physiology. We suggest here that, since changes in mitochondrial genetics occur more slowly than most physical aspects of mitochondrial physiology, understanding mitochondrial genetics may be especially important in explaining phenomena such as ageing, which appears to be closely related to mitochondrial heterogeneity. You can freely access our work, which has recently been published in Frontiers in Genetics, as “Mitochondrial Heterogeneity” https://www.frontiersin.org/articles/10.3389/fgene.2018.00718/full Juvid, Iain and Nick.

### catch22 features of signals

By taking repeated measurements over time we can study the dynamics of our environment – be it the mean temperature of the UK by month, th...