# 3rd UK Network Science Workshop

## 16th July, University of Leeds, 2019

## Aim of the workshop

### Following recent workshops in Bristol and London, we aim to bring together Network Science researchers in the UK, particularly those from the mathematical and statistical mechanical communities. This one-day workshop will feature talks on recent Network Science research and encourage discussion regarding the strategic direction of the UK Network Science community.

Please note that this website will continue to be updated as the programme is finalised.

**Registration (no fee)**: For catering purposes, please let Jonathan Ward know via email j.a.ward@leeds.ac.uk if you are planning to attend and if you have any dietary requirements.

**Venue**: Roger Stevens LT 24 (10.24), Roger Stevens building, University of Leeds

**Directions**: The Roger Stevens building is about a 20 minute walk from Leeds station. Directions to the University of Leeds can be found here:

Note that these directions take you to the Parkinson building and the main entrance to the University but this is not the most direct route to Roger Stevens from the station. Please use the map below.

If you are planning to arrive by car, please contact Jonathan Ward for parking information.

## Provisional Programme

**10:00-10:30** Arrival and refreshments

**10:30-11:00** Otti D'Huys (Aston University Birmingham)

*"Interplay of delay, topology and noise in coupled oscillator networks"*

**11:00-11:30** Naomi Arnold and Richard Clegg (Queen Mary University of London)

*"Mixed and time varying models for evolving complex networks"*

**11:30-12:30** Lightning talks

- David O'Sullivan (University of Oxford), "Social Structure, Sentiment and Voter Alignment: The Irish Marriage Referendum"
- Jens Christian Claussen (Aston University Birmingham), "Complexity of complex networks"
- Andrew Mellor (University of Oxford), "Graph Comparison via the Non-backtracking Spectrum"
- Alice Tapper (University of Leeds), "Inference on multilayer networks with latent layers"
- Annabel Davies (University of Manchester), "Network meta-analysis: Topology investigations"

**12:30-14:00** Lunch

**14:00-14:30** James Gleeson (University of Limerick, Ireland)

"*Branching processes as models of cascade dynamics on networks*"

**14:30-15:00** Francesco Di Lauro (University of Sussex)

*"Network Inference from Population-Level Observation of Epidemics"*

**15:00-15:30** Refreshments

**15:30-16:00** Danica Vukadinovic-Greetham (The Open University)

"Weighted dominating set problems on graphs"

**16:00-17:00 **Group discussion

**17:00 onwards **Socialising in Leeds

## Abstracts

## Otti D'Huys - Interplay of delay, topology and noise in coupled oscillator networks

Time-delay networks are omnipresent in nature, as information transmission takes a finite time, for example between neurons in the brain, optical elements, in gene regulatory or communication networks. Such a coupling delay can have profound effects on the network behaviour, but is not always taken into account due to mathematical complexity.

Here, we study small network motifs of oscillators coupled with delay. The coupling delay gives rise to coexisting stable oscillation patterns, under influence of noise, the network can switch between these coexistent orbits with different frequencies and different oscillation patterns. For continuously coupled phase oscillators, we reduce the delay system to a non-delayed Langevin equation, which allows to analytically compute the distribution of frequencies, and their corresponding residence times. In a unidirectional ring, we show that the dynamics can be embedded in that of a single oscillator with delayed feedback, and that all oscillatory patterns are equally stable (in terms of relative attendance and lifetime), while the interplay of coupling delay and noise can induce in-phase synchronisation in other network motifs. Finally, we investigate the role of the coupling function, and show that noise can induce a shift towards orbits with a higher frequency in pulse-coupled networks.

## Naomi Arnold and Richard Clegg (Queen Mary University of London)

*"Mixed and time varying models for evolving complex networks"*

A number of models have been proposed to explain the way that dynamic networks grow and evolve, the most famous of these being the Barabási Albert model [1] but many others exist [2][3]. In fact for real network data, the best explanatory model is unlikely to be a “pure” model but some mixture of these. Here we show how models can be mixed in a way analogous to regression analysis in traditional statistics and how we can develop a likelihood measure to show which mixture has the highest likelihood for a given set of observations. We also consider that the best model for a given data set may evolve over the lifetime of the network. We will present work showing how the framework can recover parameters on artificial data and fit statistics to real data. All the code is freely available to researchers and we encourage other complex networks researchers to use it [4].

[1] Barabási, A.-L.; Albert, R..* Emergence of scaling in random networks*. Science. 286 (5439): 509–512, 1999

[2] Jackson, M. O., Rogers, B. W.. *Meeting strangers and friends of friends: How random are social networks?* American Economic Review, 97(3), 890-915, 2007

[3] Fortunato, S., Flammini, A., & Menczer, F.. *Scale-free network growth by ranking*. Physical Review Letters, *96*(21), 218701, 2006

[4] https://github.com/narnolddd/FETA3

## James Gleeson - Branching processes as models of cascade dynamics on networks

Network models may be applied to describe many complex systems, and in the era of online social networks the study of dynamics on networks is an important branch of computational social science. Cascade dynamics can occur when the state of a node is affected by the states of its neighbours in the network, for example when a Twitter user is inspired to retweet a message that she received from a user she follows, with one event (the retweet) potentially causing further events (retweets by followers of followers) in a chain reaction. In this talk I will review some mathematical models that can help us understand how social contagion (the spread of cultural fads and the viral diffusion of information) depends upon the structure of the social network and on the dynamics of human behaviour. Although the models are simple enough to allow for mathematical analysis, I will show examples where they can also provide good matches to empirical observations of cascades on social networks.

## Francesco Di Lauro - Network Inference from Population-Level Observation of Epidemics

The network paradigm is widely accepted as the gold standard in modelling complex systems such as epidemics or neuronal activity in the brain; however, in most cases, the exact nature of the network on which such dynamics unfold is unknown. This has motivated a significant amount of work on network inference. Whilst a large body of work is concerned with inferring network structure based on detailed node-level temporal data, in this work we tackle the more challenging scenario of inferring the family of the underlying network when only population-level temporal incidence data are available. A key obstacle is the forbiddingly high dimensionality of the resulting stochastic epidemic model. To tackle this, we approximate the susceptible-infected-susceptible (SIS) model on networks by a Birth-Death process, whose rates encode the structure of the underlying network and disease dynamics. Using systematic simulations, we propose a parsimonious (three-parameter) model of these rates and show that different well-known network families map onto distinct regions of the parameter space of this model. This result provides an a priori characterisation of different network families. Then, given population-level temporal epidemic data, we employ a Bayesian classifier to derive posterior distributions over different network families. We show that the proposed methodology yields excellent results when tested on synthetic and real-world networks. Our framework extends readily to many network families and spreading processes and it could provide a new benchmark in network inference from population-level data.

## Danica Vukadinovic-Greetham - Weighted dominating set problems on graphs

In complex networks, we often want to control or support a dynamic process unfolding on the network. For instance, for a social network intervention to work on an individual, it is beneficial to have a support community, so that change affirmative messages can come from multiple sources. In distributed systems, we might assign a cost of running a distributed application on each node. Then, our aim is to find the most cost effective subset of nodes such that each node would have a guaranteed percentage of its neighbourhood running the app. A set of nodes such that all other nodes are connected to that set is called a dominating set. The related optimisation problem of finding a dominating set of minimum size is NP-complete. From the 1950s onward, different variants of this problem have been investigated. I’ll discuss weighted variants and different algorithms to tackle these problems.

## Lightning talks - more information

### Jens Christian Claussen: "Complexity of complex networks"

Network complexity is an intriguing topic across all sciences. A vast variety of biological, social, and economical networks shows topologies drastically differing from random graphs; yet the quantitative characterization remains unsatisfactory from a conceptual point of view. How to characterize complexity of graphs, given just the adjacency matrix? To address this question, I have suggested Offdiagional Complexity [1], a link correlation entropy, which takes an extremum for a power law distribution. In the last years, several other complexity measures have been investigated. I will briefly comment how these do, or do not, characterize the complexity of a network.

[1] Jens Christian Claussen, Offdiagonal Complexity: A computationally quick complexity measure for graphs and networks, Physica A 375, 365 (2007)

https://doi.org/10.1016/j.physa.2006.08.067

### Andrew Mellor: "Graph Comparison via the Non-backtracking Spectrum"

The comparison of graphs is a vitally important, yet difficult task which arises across a number of diverse research areas including biological and social networks. There have been a number of approaches to define graph distance, however, often these are not metrics (rendering standard data-mining techniques infeasible) or are computationally infeasible for large graphs. In this work we define a new pseudometric based on the spectrum of the nonbacktracking graph operator and show that it cannot only be used to compare graphs generated through different mechanisms but can reliably compare graphs of varying size. We observe that the family of Watts-Strogatz graphs lie on a manifold in the nonbacktracking spectral embedding and show how this metric can be used in a standard classification problem of empirical graphs.

Links to article: