https://digitalcivics.io

A computing research lab in

• Human-computer interaction
• Social and ubiquitous computing

Themes:

• Health and social care
• Education
• Politics

Online learning platform

Different from Massive Open Online Course (MOOC)

• passive learning

Students learn from interaction with each other and mentors/experts

• active learning
• the line between student and teacher becomes less clear

• Just like an ordinary class, but students take notes via twitter or write blogs
• #CClasses and the class specific hashtags for extracting and collecting data
• Can interact in the social network with each other, as well as people outside the classroom, e.g. experts in the subject
• Expand the learning space beyond the physical classroom

• An app for finding breastfeeding friendly places

• User review system

• How about other ideas that can be turned into a geolocation-based app?

• What if we let the community decide what app to develop?

Support/sharing stage

• Start a movement, and share to friends on social network
• Potential users support and share

Design stage

• Use the templates provided by the developers
• Vote on the design aspects of the app

Launch stage

• Similar to Feed Finder, users search for places and give reviews

Not all movements reach the target support and get launched

Some successful apps

• Drone Zones
• Gender Neutral Toilet Finder
• Feed Finder, Lebanese version

The sharing of a movement is like spreading an "epidemic" on the social network

This motivates us to develop a network epidemic model for the sharing data

Compartment models

• Each individual is at exactly one stage at any time
• Potentially goes from one stage to another

Classical assumptions governing the dynamics

• Markovian
• Homogeneous mixing

See e.g. Andersson and Britton (2000) (link)

Modelling heterogeneity

Multiple levels of mixing

• Ball, Mollison, and Scalia-Tomba (1997): two levels
• Britton, Kypraios, and O’Neill (2011): three levels

Not quite applicable to our data

• No information on such levels in the social network

Network modelling

Previous work

• Britton and O’Neill (2002) (link)
• Neal and Roberts (2005) (link)
• Focus on inference algorithms

Based on Bernoulli random graph (BRG)

• \(\Pr\)(any two individuals are connected) = \(p\)
• Such probability independent of all other connections
• Not quite realistic for social networks

Barabási and Albert (1999): Preferential attachment (PA) model

• Nodes join network sequentially, each with \(\mu\) new edges
• Each existing node gets an edge with probability proportional to current degree

image source

Power-law degree distribution

Small path lengths

• Six degrees of separation

High clustering

• Three typical nodes \(A,B\) and \(C\)
• Assume \(A\leftrightarrow B\) and \(B\leftrightarrow C\)
• Clustering coefficient: how likely that \(A\leftrightarrow C\)
• BRG: whether each pair is connected is independent of other pairs a priori

Epidemic modelling

Compartment models represented by ordinary differential equations

• \(\displaystyle\frac{dS}{dt}=-\beta IS, \quad\qquad\frac{dI}{dt}=\beta IS-\gamma I, \quad\qquad\frac{dR}{dt}=\gamma I\)

Incorporate network aspects as covariate

• E.g. same parameter values for nodes with same degree
• Comprehensive review by Pastor-Satorras et al. (2015)

Quite remote from previously introduced statistical models

Population: \(m\) nodes/individuals

The epidemic times \(\boldsymbol{I}=(I_1,I_2,\ldots,I_m)\)

• Susceptible-Infected (SI) model
• Markovian, constant infection rate \(\beta\)

The underlying graph/network \(\boldsymbol{G}=\{G_{ij}\}_{m\times m}\)

• PA model, parameter \(\mu\)
• \(G_{ij}=1\) if \(i\leftrightarrow j\) for \(i\neq j\), \(0\) otherwise

The transmission tree \(\boldsymbol{P}=\{P_{ij}\}_{m\times m}\)

• \(P_{ij}=1\) if node \(j\) is infected by \(i\), \(0\) otherwise

Step 1: Obtain new edges for the nodes

Index the nodes by their order of entering the network

One edge to start with: nodes \(1\leftrightarrow2\)

Node \(i~(>2)\) brings in \(x_i\) new edges

• \(x_i=\mu\) (constant) in original PA model too restrictive
• \(x_i=y_i \wedge (i-1)\) where \(y_i\sim\) Poission \((\mu)\)

Step 1: Obtain new edges for the nodes

Toy example: \(m=8,\mu=3\)

• \(x_3=2,~x_4=3,~x_5=4,~x_6=2,~x_7=3,~x_8=4\)

## [1] "G"
##                     
## [1,] . 1 1 1 1 . 1 1
## [2,] 1 . 1 1 1 . . 1
## [3,] 1 1 . 1 1 . 1 .
## [4,] 1 1 1 . 1 1 . 1
## [5,] 1 1 1 1 . 1 1 1
## [6,] . . . 1 1 . . .
## [7,] 1 . 1 . 1 . . .
## [8,] 1 1 . 1 1 . . .

Step 1: Obtain new edges for the nodes

Toy example: \(m=8,\mu=3\)

• \(x_3=2,~x_4=3,~x_5=4,~x_6=2,~x_7=3,~x_8=4\)

## [1] "G"
##                     
## [1,] . 1 1 1 1 . 1 1
## [2,] 1 . 1 1 1 . . 1
## [3,] 1 1 . 1 1 . 1 .
## [4,] 1 1 1 . 1 1 . 1
## [5,] 1 1 1 1 . 1 1 1
## [6,] . . . 1 1 . . .
## [7,] 1 . 1 . 1 . . .
## [8,] 1 1 . 1 1 . . .

Step 1: Obtain new edges for the nodes

Likelihood: \[ L_1(\boldsymbol{G};\mu)=\prod_{i=3}^{m} \left[\frac{e^{-\mu}\mu^{x_i}}{x_i!}\right]^{\boldsymbol{1}\{0~\leq~x_i~<~i-1\}} \left[\sum_{z=i-1}^\infty\frac{e^{-\mu}\mu^{z}}{z!}\right]^{\boldsymbol{1}\{x_i~=~i-1\}}\\ =\frac{e^{-\mu(m-2)}}{\prod_{i=3}^{m}(x_i!)}\prod_{i=3}^{m} \mu^{\left[\sum_{j=1}^{i-1}G_{ij}\boldsymbol{1}\left\{0\leq\sum_{j=1}^{i-1}G_{ij}<i-1\right\}\right]}\\ \times\prod_{i=3}^{m}\left[(i-1)!\sum_{z=i-1}^{\infty}\frac{\mu^z}{z!}\right]^{\boldsymbol{1}\left\{\sum_{j=1}^{i-1}G_{ij}=i-1\right\}} \]

Step 2: Preferentially attach the edges to build the network

When node \(i\) enters

• \(x_i\) existing nodes chosen from \(\{1,2,\ldots,i-1\}\)
• Weighted sampling without replacement
• Weight of existing node \(j = \frac{\sum_{k=1}^{i-1}G_{kj}}{\sum_{j=1}^{i-1}\sum_{k=1}^{i-1}G_{kj}}\)

Exact likelihood

• Go through all \(x_i!\) permutations of the selected nodes
• When \(x=1,2,3,4,5,6,\ldots,~x!=1,2,6,24,120,720,\ldots\)
• Computationally not feasible

Step 2: Preferentially attach the edges to build the network

When node \(i\) enters

• \(x_i\) existing nodes chosen from \(\{1,2,\ldots,i-1\}\)
• Weighted sampling without replacement
• Weight of existing node \(j = \frac{\sum_{k=1}^{i-1}G_{kj}}{\sum_{l=1}^{i-1}\sum_{k=1}^{i-1}G_{kl}}\)

Approximate likelihood

• Weighted sampling with replacement

Step 2: Preferentially attach the edges to build the network

Contribution by node \(i\)'s new edges: \[ L_{2i}=x_i!\times\prod_{j=1}^{i-1}\left(\frac{\sum_{k=1}^{i-1}G_{kj}} {\sum_{l=1}^{i-1}\sum_{k=1}^{i-1}G_{kl}}\right)^{G_{ij}} \]

Likelihood by the process of adding new edges: \[ L_2(\boldsymbol{G}):=\prod_{i=3}^{m}L_{2i}= \prod_{i=3}^{m}(x_i!)\times \prod_{i=3}^{m}\prod_{j=1}^{i-1}\left(\frac{\sum_{k=1}^{i-1}G_{kj}} {\sum_{l=1}^{i-1}\sum_{k=1}^{i-1}G_{kl}}\right)^{G_{ij}} \]

Step 3: Spread the epidemic on the given network

Index the nodes by their epidemic (temporal) order

Infected node \(i\) makes infectious contacts

• with its network neighbours
• at points of Poisson process with rate \(\beta\sum_{j=1}^{m}G_{ij}\)

Likelihood independent of transmission tree \(\boldsymbol{P}\)

Step 3: Spread the epidemic on the given network

So where is \(\boldsymbol{P}\) gone?

• "Uniform distribution on the set of all possible infection pathways" (Britton and O’Neill 2002)

Posterior of \(\boldsymbol{G}\) involves \(\boldsymbol{P}\)

• If \(P_{ij}=1\), \(G_{ij}(=G_{ji})=1\) with probability \(1\) a posteriori
• If \(P_{ij}=0\), posterior of \(G_{ij}\) derived in inference

Step 4: Connect the network and the epidemic

Epidemic order not necessarily the same as network order

• Fix the labelling of nodes by epidemic order
• Introduce variable \(\boldsymbol{\sigma}\) - permutation of \(\{1,2,\ldots,m\}\)

Convert from epidemic order to network order

• Replace \(\boldsymbol{G}\) by \(\boldsymbol{G}_{\boldsymbol{\sigma}}=f(\boldsymbol{G},\boldsymbol{\sigma})\) when computing \(L_1(\cdot;\mu)\) and \(L_2(\cdot)\)

Step 4: Connect the network and the epidemic

Toy example continued: \(m=8,\mu=3\)

• \(\boldsymbol{\sigma}=(5,3,8,2,6,4,7,1)\)

## [1] "G"
##                     
## [1,] . 1 1 1 1 . 1 1
## [2,] 1 . 1 1 1 . . 1
## [3,] 1 1 . 1 1 . 1 .
## [4,] 1 1 1 . 1 1 . 1
## [5,] 1 1 1 1 . 1 1 1
## [6,] . . . 1 1 . . .
## [7,] 1 . 1 . 1 . . .
## [8,] 1 1 . 1 1 . . .

## [1] "G_sigma = f(G, sigma)"
##                     
## [1,] . 1 1 1 1 1 1 1
## [2,] 1 . . 1 . 1 1 1
## [3,] 1 . . 1 . 1 . 1
## [4,] 1 1 1 . . 1 . 1
## [5,] 1 . . . . 1 . .
## [6,] 1 1 1 1 1 . . 1
## [7,] 1 1 . . . . . 1
## [8,] 1 1 1 1 . 1 1 .

\(\boldsymbol{G}\) is usually unknown

\(~~~~\pi(\boldsymbol{G},\boldsymbol{\sigma},\beta,\mu|\boldsymbol{P},\boldsymbol{I})\\ \propto\pi(\boldsymbol{P},\boldsymbol{I},\boldsymbol{G},\boldsymbol{\sigma},\beta,\mu)\\ =\pi(\boldsymbol{P},\boldsymbol{I}|\boldsymbol{G},\boldsymbol{\sigma},\beta,\mu)~\pi(\boldsymbol{G},\boldsymbol{\sigma},\beta,\mu)\\ =\pi(\boldsymbol{P}|\boldsymbol{G})~\pi(\boldsymbol{I}|\boldsymbol{G},\beta)~\pi(\boldsymbol{G}|\boldsymbol{\sigma},\beta,\mu)~\pi(\boldsymbol{\sigma},\beta,\mu)\qquad(\boldsymbol{P}~\bot~\boldsymbol{I}~\text{given}~\boldsymbol{G})\\ =\pi(\boldsymbol{P}|\boldsymbol{G})~\pi(\boldsymbol{I}|\boldsymbol{G},\beta)~ L_1(\boldsymbol{G}_{\boldsymbol{\sigma}};\mu)~L_2(\boldsymbol{G}_{\boldsymbol{\sigma}})~\pi(\boldsymbol{\sigma})~\pi(\beta)~\pi(\mu)\)

Markov Chain Monte Carlo (MCMC) algorithm straightforward

• Similar latent approach if \(\boldsymbol{P}\) is also unknown

Uninformative priors

\(\beta\sim\text{Gamma}(a_\beta, \text{rate}=b_\beta)\)

\(\mu\sim\text{Gamma}(a_\mu, \text{rate}=b_\mu)\)

\(\pi(\boldsymbol{\sigma})=(m!)^{-1}\)

Posteriors

\(\beta|\ldots\sim \text{Gamma}\left(a_\beta+m-1,\text{rate}=b_\beta+\sum_{i=1}^{m-1}\sum_{j=i+1}^{m}G_{ij}(I_j-I_i)\right)\)

\(\pi(\mu|\ldots)\propto L_1(\boldsymbol{G}_{\boldsymbol{\sigma}};\mu)~\pi(\mu)\)

\(\pi(\boldsymbol{\sigma}|\ldots)\propto L_1(\boldsymbol{G}_{\boldsymbol{\sigma}};\mu)~L_2(\boldsymbol{G}_{\boldsymbol{\sigma}})\)

Exploring permutation space

• Bezáková, Kalai, and Santhanam (2006): link
• Random insertion
• More efficient than random swap

Posteriors

\(\Pr(G_{ij}=1|P_{ij}=1,\ldots)=1\)

\(\boldsymbol{G}_0\) the same as \(\boldsymbol{G}\) except \(G_{ij}\) (and \(G_{ji}\)) is set to \(0\)

\(\boldsymbol{G}_1\) the same as \(\boldsymbol{G}\) except \(G_{ij}\) (and \(G_{ji}\)) is set to \(1\)

\(\Pr(G_{ij}=0|P_{ij}=0,\boldsymbol{G}_{-ij},\ldots)\propto \displaystyle\frac{\quad\pi(\boldsymbol{G}_0|\boldsymbol{\sigma},\beta,\mu)\quad} {\sum_{k=1,k\neq i}^{j-1}G_{kj}}\)

\(\Pr(G_{ij}=1|P_{ij}=0,\boldsymbol{G}_{-ij},\ldots)\propto \displaystyle\frac{\pi(\boldsymbol{G}_1|\boldsymbol{\sigma},\beta,\mu)~e^{-\beta(I_j-I_i)}} {\sum_{k=1,k\neq i}^{j-1}G_{kj}+1}\)

Scenarios

Simulate network only, estimate \((\mu,\boldsymbol{\sigma})\)

• Good

Simulate network & epidemic, estimate \((\mu,\beta,\boldsymbol{\sigma})\) given \(\boldsymbol{G}\) & \(\boldsymbol{I}\)

• Good

Simulate network & epidemic, estimate \((\mu,\beta,\boldsymbol{\sigma},\boldsymbol{G})\) given \(\boldsymbol{P}\) & \(\boldsymbol{I}\)

• Problematic

Identifiability

Posterior of \(\mu\) does not depend not on its true value

Identifiability

\(\beta\) not good either

Identifiability

What about \(\ldots \alpha=\beta\times\mu\)?

Observations

Inverse relationship between epidemic rate \(\beta\) and parameter characterising network connectedness

• Average number new edges \(\mu\) in PA model
• Edge inclusion probability \(p\) in BRG model
  • Echoing observation by Britton and O’Neill (2002)

Identifiability

• Identifying one parameter \((\alpha)\) as good as we can get
• Interpretation: network scaled epidemic rate

Review on network epidemics

• Statistical models assume BRG, unrealistic
• Physical models focus on dynamics, difficult to do inference

Model and inference

• PA model grows the network, SI model spreads the epidemic
• Gibbs steps for \(\beta\) and individual edges \(G_{ij}\), Metropolis steps for \(\mu\) and \(\boldsymbol{\sigma}\)

Simulation study

• Product of \(\beta\) and \(\mu\) identifiable but not individually

Still waiting for results

Typical times taken per iteration:

• \(m=100: 0.88\text{s}\)
• \(m=200: 14\text{s}\)

Computational time \(O(m^4)\)

• # potential edges \((G_{ij})\) to update \(O(m^2)\)
• # computations involved in updating one edge \(O(m^2)\)

Data sets worth applying the model to

• \(m = 350\) and up

Apply to App Movement data

• Sample sizes \(m\) much larger than in simulation study
• Current MCMC algorithm not scalable with \(m\)

Compare with models which use BRG

• One possible way: calculate the marginal likelihoods

Marginal MCMC methods by simulating the network

• Direct according to the preferential attachment rule
• Takes much less time than updating the edges one by one