Dynamic networks
Monday 12th December 2016 to Friday 16th December 2016
09:00 to 09:15  Registration  
09:15 to 09:20  Welcome from Christie Marr (INI Deputy Director)  INI 1  
09:20 to 09:30  Opening Remarks from SNAW04 Organisers  INI 1  
09:30 to 10:30 
Tom Snijders Continuoustime statistical models for network panel data
For
the statistical analysis of network panel data even with as little as 2 waves,
it is very fruitful to use models that assume a continuoustime Markov network
process, observed only at the moments of observation for the panel. This is
analogous to the use of continuoustime models for classical (nonnetwork)
panel data proposed by Bergstrom, Singer, and others. For network data such an
approach was proposed already by Coleman in 1964. The advantage of this
approach is that it provides a simple way to represent the feedback that is
inherent in network dynamics, and the model can be defined by just specifying
the conditional probability of a tie change, given the current state of the
network. This approach is used in the Stochastic ActorOriented Model of Snijders (2001) and in the Longitudinal Exponential Random Graph Model of Snijders & Koskinen (2013). The first of these is actororiented, i.e., tie changes are modelled as choices by actors, which among their outgoing tie variables to toggle; the second is tieoriented, i.e., tie changes are modelled as toggles of single tie variables. Both are generalized linear models for the (unobserved) continuoustime process, with all the practical modelling flexibility of such models. Estimation for panel data is more involved, requiring a simulation approach. Estimators have been developed along several lines, including Method of Moments, Generalized Method of Moments, Maximum Likelihood, and Bayesian, and are available in the R package RSiena. This package is widely applied in empirical social network studies in the social sciences. This presentation treats the basic definition of the model and some of its extensions, e.g., coevolution of multivariate networks. Some open problems, from a mathematical and from an applied perspective, will be mentioned. References

INI 1  
10:30 to 11:15  Morning Coffee  
11:15 to 12:00 
Eric Kolaczyk Dynamic causal networks with multiscale temporal structure
Coauthors: Xinyu Kang (Boston University), Apratim
Ganguly (Boston University) I will discuss a novel method to model multivariate time series using dynamic causal networks. This method combines traditional multiscale modeling and network based neighborhood selection, aiming at capturing the temporally local structure of the data while maintaining the sparsity of the potential interactions. Our multiscale framework is based on recursive dyadic partitioning, which recursively partitions the temporal axis into finer intervals and allows us to detect local network structural changes at varying temporal resolutions. The dynamic neighborhood selection is achieved through penalized likelihood estimation, where the penalty seeks to limit the number of neighbors used to model the data. Theoretical and numerical results describing the performance of our method will be presented, as well as an application in computational neuroscience. 
INI 1  
12:30 to 13:30  Lunch @ Wolfson Court  
13:30 to 14:30 
Jennifer Chayes Graphons and Machine Learning: Modeling and Estimation of Sparse Massive Networks  Part I
There are
numerous examples of sparse massive networks, in particular the Internet, WWW
and online social networks. How do we model and learn these
networks? In contrast to conventional learning problems, where we have
many independent samples, it is often the case for these networks that we can
get only one independent sample. How do we use a single snapshot today to
learn a model for the network, and therefore be able to predict a similar, but
larger network in the future? In the case of relatively small or
moderately sized networks, it’s appropriate to model the network
parametrically, and attempt to learn these parameters. For massive
networks, a nonparametric representation is more appropriate. In this
talk, we first review the theory of graphons, developed over the last decade to
describe limits of dense graphs, and the more the recent theory describing
sparse graphs of unbounded average degree, including powerlaw graphs. We
then show how to use these graphons as nonparametric models for sparse
networks. Finally, we show how to get consistent estimators of these
nonparametric models, and moreover how to do this in a way that protects the
privacy of individuals on the network.
Part I of this talk reviews the theory of graph convergence for dense and sparse graphs. Part II uses the results of Part I to model and estimate sparse massive networks. 
INI 1  
14:30 to 15:30 
Christian Borgs Graphons and Machine Learning: Modeling and Estimation of Sparse Massive Networks  Part II
There are
numerous examples of sparse massive networks, in particular the Internet, WWW
and online social networks. How do we model and learn these
networks? In contrast to conventional learning problems, where we have
many independent samples, it is often the case for these networks that we can
get only one independent sample. How do we use a single snapshot today to
learn a model for the network, and therefore be able to predict a similar, but
larger network in the future? In the case of relatively small or
moderately sized networks, it’s appropriate to model the network
parametrically, and attempt to learn these parameters. For massive
networks, a nonparametric representation is more appropriate. In this
talk, we first review the theory of graphons, developed over the last decade to
describe limits of dense graphs, and the more the recent theory describing
sparse graphs of unbounded average degree, including powerlaw graphs. We
then show how to use these graphons as nonparametric models for sparse
networks. Finally, we show how to get consistent estimators of these
nonparametric models, and moreover how to do this in a way that protects the
privacy of individuals on the network.
Part I of this talk reviews the theory of graph convergence for dense and sparse graphs. Part II uses the results of Part I to model and estimate sparse massive networks. 
INI 1  
15:30 to 16:00  Afternoon Tea  
16:00 to 16:45 
Jeanette Janssen Recognizing graphs formed by spatial random processes
In many real life applications, network formation can be modelled using a spatial random graph model: vertices are embedded in a metric space S, and pairs of vertices are more likely to be connected if they are closer together in the space. A general geometric graph model that captures this concept is G(n,w), where w is a symmetric "link probability" function from SxS to [0,1]. To guarantee the spatial nature of the random graph, we requite that this function has the property that, for fixed x in S, w(x,y) decreases as y is moved further away from x. The function w can be seen as the graph limit of the sequence G(n,w) as n goes to infinity. We consider the question: given a large graph or sequence of graphs, how can we determine if they are likely the results of such a general geometric random graph process? Focusing on the onedimensional (linear) case where S=[0,1], we define a graph parameter \Gamma and use the theory of graph limits to show that this parameter indeed measures the compatibility of the graph with a linear model. This is joint work with Huda Chuangpishit, Mahya Ghandehari, Nauzer Kalyaniwalla, and Israel Rocha 
INI 1  
17:00 to 18:00  Welcome Wine Reception at INI 
09:30 to 10:30 
Tom Britton A network epidemic model with preventive rewiring: comparative analysis of the initial phase
Coauthors: Joan Saldana
(Universitat de Girona), David Juher
(Universitat de Girona)
This talk is concerned with stochastic SIR and SEIR epidemic models on random networks in which individuals may rewire away from infected neighbors at some rate ω (and reconnect to noninfectious individuals with probability α or else simply drop the edge if α=0), socalled preventive rewiring. The models are denoted SIRω and SEIRω, and we focus attention on the early stages of an outbreak, where we derive expression for the basic reproduction number R0 and the expected degree of the infectious nodes E(DI) using two different approximation approaches. The first approach approximates the early spread of an epidemic by a branching process, whereas the second one uses pair approximation. The expressions are compared with the corresponding empirical means obtained from stochastic simulations of SIRω and SEIRω epidemics on Poisson and scalefree networks. To appear in Bull Math Biol. 
INI 1  
10:30 to 11:15  Morning Coffee  
11:15 to 12:00 
Maria Deijfen Birds of a feather or opposites attract  effects in network modelling
We study properties of some standard network models when the population is split into two types and the connection pattern between the types is varied. The studied models are generalizations of the ErdösRenyi graph, the configuration model and a preferential attachment graph. For the ErdösRenyi graph and the configuration model, the focus is on the component structure. We derive expressions for the critical parameter, indicating when there is a giant component in the graph, and study the size of the largest component by aid of simulations. For the preferential attachment model, we analyze the degree distributions of the two types and derive explicit expressions for the degree exponents. Joint work with Robert Fitzner (Eindhoven University of Technology). 
INI 1  
12:30 to 13:30  Lunch @ Wolfson Court  
13:45 to 14:45 
Susan Holmes Study of the dynamics of bacterial communities in the Human Microbiome
Coauthors: Kris Sankaran
(Stanford), Julia Fukuyama
(Stanford), Lan Nguyen
(Stanford), Diana Proctor
(Stanford), David Relman
(Stanford), Sergio Bacallado
(Cambridge), Boyu Ren
(Stanford), Pratheepa Jeganathan
(Stanford)
The human microbiome is a complex assembly of bacteria that are sensitive to many perturbations. In several longitudinal analyses we study perturbations of bacterial community networks over time. For this, we have developed specific tools for modeling the vaginal, intestinal and oral microbiomes under these different perturbations (pregnancy, hyposalivation inducing medications and antibiotics are some examples). A suite of statistical tools written in R based on a Bioconductor package (phyloseq) allows for easy normalization, visualization and statistical testing of the longitudinal multitable data composed of 16sRNA reads combined with clinical data, transcriptomic and metabolomic profiles. Challenges we have had to address include information leaks, the heterogeneity of the data, multiplicity of choices during the analyses and validation of results. Each different body site requires a different modeling strategy as some sites form tight communities easily modeled with Stochastic Block Models whereas others show more diverse assemblies that require complex latent variable models. Related Links

INI 1  
14:45 to 15:30 
Danielle Bassett Dynamic networks in the human brain
Each area of the human brain plays a unique role in processing
information gleaned from the external world and in driving our responses
to that external world via behavior. However, the brain is far from a
set of disconnected building blocks. Instead, parts of the brain
communicate with one another in complex spatiotemporal patterns that
enable human behavior. Understanding this spatiotemporal complexity
requires a paradigmatic shift in our conceptual approaches, empirical
thrusts, and quantitative methods. In this talk, I will describe the
recent use of tools from network science to understand the structure and
function of the human brain. With these novel approaches, we can begin
to characterize the ``connectome’’, a model representation of
neurobiological data that encapsulates both constituent elements of the
brain (network nodes) and their interactions with one another (network
edges). In a critical innovation, we imbue network edges with temporal
dependence
to capture the dynamics of the everreconfiguring brain communication
patterns that support cognition. I will recount the utility of dynamic
network approaches in not only understanding, but also predicting
individual differences in adaptive functions such as learning, and in
delineating healthy versus diseased brain communication dynamics. An
emerging frontier, dynamic network neuroscience provides a powerful new
conceptual and mathematical framework with which to understand adaptive
human capabilities because it embraces the inherently evolving,
interconnected nature of neurophysiological phenomena underlying human
thought.

INI 1  
15:30 to 16:00  Afternoon Tea  
16:00 to 16:45 
Stéphane Robin Detecting changepoints in the structure of a network: Exact Bayesian inference
Joint work with Loïc Schwaller We consider the problem of changepoint detection in multivariate timeseries, typically the expression of a set of genes, or the activity of a set of brain regions over time. We adopt the framework of graphical models to described the dependency between the series. We are interested in the situation where the graphical model is affected by abrupt changes throughout time. In the above examples, such changes correspond to gene or brain region rewiring. We demonstrate that it is possible to perform exact Bayesian inference whenever one considers a simple class of undirected graphs called spanning trees as possible structures. We are then able to integrate on both the graph and segmentation spaces at the same time by combining classical dynamic programming with algebraic results pertaining to spanning trees. In particular, we show that quantities such as posterior distributions for changepoints or posterior edge probabilities over time can efficiently be obtained. We illustrate our results on both synthetic and experimental data arising from molecular biology and neuroscience. 
INI 1  
16:45 to 17:30 
Harry Crane Markov process models for timevarying networks
Many models for dynamic networks, such as the preferential attachment
model, describe evolution by sequential addition of vertices and/or
edges. Such models are not suited to networks whose connectivity varies
over time, as in social relationships and other kinds of temporally
varying interactions. For modeling in this latter setting, I develop
the general theory of exchangeable Markov processes for timevarying
networks and discuss relevant consequences.

INI 1 
09:30 to 10:30 
Frank Granville Ball Epidemics on networks
In this talk we consider two extensions of the standard stochastic epidemic SIR (SusceptibleInfectedRecovered) on a configuration model network. The first extension, which is joint work with Peter Neal (Lancaster University), incorporates casual contacts, i.e. with people chosen uniformly at random from the population. The second extension, which is joint work with Tom Britton (Stockholm University) and David Sirl (University of Nottingham), involves the spread of an epidemic on a random network model which allows for tunable clustering, degree correlation and degree distribution. For each model, approximating branching processes are used to obtain a threshold parameter, which determines whether or not an epidemic with few initial infectives can become established and lead to a major outbreak, and the (approximate) probability and relative final size of a major outbreak. For the model with casual contacts, an embedding argument is used to derive a central limit theorem for the size of a major epidemic; similar methods lead to the asymptotic variance of the giant component in a configuration model random graph. The theory is illustrated by numerical studies, which explore the impact of network properties on the outcome of an epidemic.

INI 1  
10:30 to 11:15  Morning Coffee  
11:15 to 12:00 
Veronica Vinciotti Sparse Gaussian graphical models for dynamic gene regulatory networks
Coauthors: Luigi Augugliaro
(University of Palermo), Antonino Abbruzzo
(University of Palermo), Ernst Wit
(University of Groningen)
In this talk, I will present a factorial Gaussian graphical model for inferring dynamic gene regulatory networks from genomic highthroughput data. The model allows including dynamicrelated equality constraints on the precision matrix as well as imposing sparsity constraints in the estimation procedure. I will discuss model selection and present an application on a highresolution timecourse microarray data from the Neisseria meningitidis bacterium, a causative agent of lifethreatening infections such as meningitis. The methodology described in this paper is implemented in the R package sglasso, freely available from CRAN. 
INI 1  
12:30 to 13:30  Lunch @ Wolfson Court  
14:00 to 14:45 
Pierre Borgnat Networks as signals: Extraction of dynamical network structures
Joint work with Ronan Hamon (LIF, Marseille, France), P. Flandrin (CNRS, LP, ENS de Lyon, France) and C. Robardet (LIRIS, INSA de Lyon, France) We have proposed a new framework to track the structure of temporal networks, using a signal processing approach: the method is based on the duality between static networks and signals using a multidimensional scaling technique. For temporal networks, it enables a tracking of the network structure over time. To extract the most significant patterns of the networks and their activation over time, we use nonnegative matrix factorization of the temporal spectra. This framework, inspired by audio decomposition, allows transforming back these frequency patterns into networks, so as to highlight the evolution of the underlying structure of the network over time. The effectiveness of the method is first evidenced on a toy example, prior being used to study a temporal network of facetoface contacts. The extraction of subnetworks highlights significant structures decomposed on time intervals. 
INI 1  
14:45 to 15:30 
Marianna Pensky Nonparametric methods for the dynamic stochastic block model and the timedependent graphon
The Dynamic Stochastic Block Model (DSBM) and the dynamic graphon are natural extensions of the, respectively, Stochastic Block Model and the graphon, from the timeindependent to the timedependent setting. The objective of the present talk is estimation of the tensor of the connection probabilities when it is generated by the DSBM and the dynamic graphon. In particular, in the context of the DSBM, under very few simple nonparametric assumptions, we derive a penalized least squares estimator and show that it satisfies an oracle inequality and also attains the minimax lower bounds for the risk. We extend those results to estimation in the context of the dynamic graphon. The estimators are adaptive to the unknown number of blocks in the context of DSBM or of the smoothness of the graphon function. The technique relies on the vectorization of the model and leads to to much simpler mathematical arguments than the ones used previously in the stationary set up. In addition, all our results are nonasymptotic and allow a variety of extensions. 
INI 1  
15:30 to 16:00  Afternoon Tea  
16:00 to 16:45 
Sidney Resnick Multivariate power laws in a preferential attachment network model; model calibration
We begin with a review of the multivariate regular variation of in and outdegree in a preferential attachment model. The problem can be approached in a variety of ways: (i) Multivariate Tauberian theory; (ii) Direct approach via asymptotics to get a limit measure; (iii) proving multivariate regular variation of the limiting mass function of normalized in and outdegree. We then turn to model calibration comparing various information sources and methods. If a full history of network growth is available, full MLE implementation is possible and performs well on simulated data. If a single snapshot in time is all that is available, then approximate MLE can be used. Comparison with MLE and use of asymptotic methods relying on heavy tail estimators is also made and predictably there is a tradeoff between robustness and accuracy. Methods generally perform well on simulated data but real data creates problems with model error and we can illustrate this with wikipedia data obtained from Konect.

INI 1  
16:45 to 17:30 
Peter Orbanz Random walk models of network formation 
INI 1  
19:30 to 22:00  Formal Dinner at Christ's College 
09:30 to 10:30 
Andrew Nobel Mining differential correlation
Given data obtained under two sampling
conditions, it is often of interest to identify variables that behave
differently in one condition than in the other. The talk will describe a method
for differential analysis of secondorder behavior called Differential
Correlation Mining (DCM). DCM is a special case of differential analysis for
weighted networks, and is distinct from standard analyses of first order
differential behavior, for example studies of differential expression. The DCM method identifies differentially correlated sets of variables, with the property that the average pairwise correlation between variables in a set is higher under one sample condition than the other. DCM is based on an iterative testing procedure that adaptively updates the size and elements of a candidate variable set. Updates are performed via hypothesis testing of individual variables, based on the asymptotic distribution of their average differential correlation. The method does not assume that the sample or population correlation matrices are sparse, or have any particular structure. I will present both simulation results and applications of DCM to genomics and brain imaging. As time permits, I will also present a brief overview of some additional network related work being done with collaborators at UNC. 
INI 1  
10:30 to 11:15  Morning Coffee  
11:15 to 12:00 
Hanna Wallach Bayesian Poisson Tensor Decomposition for International Relations
Like their inhabitants, countries interact with one another: theyconsult, negotiate, trade, threaten, and fight. These interactions areseldom uncoordinated. Rather, they are connected by a fabric ofoverlapping communities, such as security coalitions, treaties, tradecartels, and military alliances. A single country can belong tomultiple communities, reflecting its many overlapping identities, andcan engage in both within and betweencommunity interactions,depending on the capacity in which it is acting. In this talk, I willintroduce two tensor decomposition models for modeling interactionevents of the form "country i took action a toward country j at timet." The first model (Bayesian Poisson CP decomposition) discoverscoherent threads of events, characterized by sender countries,receiver countries, action types, and time steps; the second model(Bayesian Poisson Tucker decomposition) discovers latentcountrycommunity memberships, including the number of latentcommunities, as well as directed communitycommunity interactionnetworks that are specific to "topics" of similar action types. I willdemonstrate that these models infer interpretable latent structuresthat conform to and inform our knowledge of international relations.Many existing models for discrete data (such as networks and text) arespecial cases of these models, including infinite relational models,stochastic block models, and latent Dirichlet allocation. As a result,Bayesian Poisson tensor decomposition is a general framework foranalyzing and understanding discrete data sets in the social sciences. 
INI 1  
12:30 to 13:30  Lunch @ Wolfson Court  
14:00 to 14:45 
Sharad Goel The Effect of Recommendations on Network Structure
Coauthors: Jessica Su
(Stanford University), Aneesh Sharma
(Twitter)
Online social networks regularly offer users personalized, algorithmic suggestions of whom to connect to. Here we examine the aggregate effects of such recommendations on network structure, focusing on whether these recommendations increase the popularity of niche users or, conversely, those who are already popular. We investigate this issue by empirically and theoretically analyzing abrupt changes in Twitter's network structure around the mid2010 introduction of its "Who to Follow" feature. We find that users across the popularity spectrum benefitted from the recommendations; however, the most popular users profited substantially more than average. We trace this "rich get richer" phenomenon to three intertwined factors. First, as is typical of network recommenders, the system relies on a "friendoffriend"style algorithm, which we show generally results in users being recommended proportional to their degree. Second, we find that the baseline growth rate of users is sublinear in degree. This mismatch between the recommender and the natural network dynamics thus alters the structural evolution of the network. Finally, we find that people are much more likely to respond positively to recommendations for popular users  perhaps because of their greater name recognition  further amplifying the cumulative advantage of wellknown individuals. Related Links 
INI 1  
14:45 to 15:30 
Peter Mörters The spread of infections on evolving scalefree networks
We study the contact process
on a class of evolving scalefree networks, where each node updates its connections
at independent random times. We give a rigorous mathematical proof that there is a
transition between a phase where for all infection rates the infection survives for a long time, at least
exponential in the network size, and a
phase where for sufficiently small infection rates extinction
occurs quickly, at most polynomially in the network size.
The phase transition occurs when the powerlaw exponent crosses the value four. This behaviour is in
contrast to that of the contact process on the corresponding static model,
where there is no phase transition, as well as that of a classical meanfield approximation,
which has a phase transition at powerlaw exponent three.
The new observation behind our result is that temporal variability of networks can simultaneously increase the rate at which
the infection spreads in the network, and decrease the time which the infection
spends in metastable states. This is joint work with Emmanuel Jacob (ENS Lyon). 
INI 1  
15:30 to 16:00  Afternoon Tea  
16:00 to 16:45 
Catherine Matias Statistical clustering of temporal networks through a dynamic stochastic block model
Coauthor: Vincent MIELE
(CNRS / LBBE / Univ. Lyon 1)
Statistical node clustering in discrete time dynamic networks is an emerging field that raises many challenges. Here, we explore statistical properties and frequentist inference in a model that combines a stochastic block model (SBM) for its static part with independent Markov chains for the evolution of the nodes groups through time. We model binary data as well as weighted dynamic random graphs (with discrete or continuous edges values). Our approach,motivated by the importance of controlling for label switching issues across the different time steps, focuses on detecting groups characterized by a stable within group connectivity behavior. We study identifiability of themodel parameters, propose an inference procedure based on a variational expectation maximization algorithm as well as a model selection criterion to select for the number of groups. We carefully discuss our initialization strategy which plays an important role in the method and compare our procedure with exi sting ones on synthetic datasets. We also illustrate our approach on dynamic contact networks, one of encounters among high school students and two others on animal interactions. An implementation of the method is available as a R package called dynsbm. Related Links

INI 1 
09:30 to 10:30 
George Michailidis Vector Autoregressive based Network Models
Vector autoregressions represent a popular class of time series models that aim to capture temporal interconnections between temporally evolving entities. They have been widely used in macroeconomic and financial modeling and more recently they have found novel applications in functional genomics and neuroscience. In this presentation, we discuss modeling and estimation issues in the high dimensional setting under different constrains on the transition matrices  sparsity, low rankness. We also provide extensions to multilayer networks and illustrate the results with applications to financial stability monitoring and biological regulation. 
INI 1  
10:30 to 11:00  Morning Coffee  
11:00 to 11:45 
Miklos Racz Finding and hiding the seed
I will present an overview of recent developments in source detection and estimation in randomly growing graphs and diffusions on graphs. Can one detect the influence of the initial seed graph? How good are rootfinding algorithms? Can one engineer messaging protocols that hide the source of a rumor? I will explore such questions in the talk. This is based on joint works with Sebastien Bubeck, Ronen Eldan, Elchanan Mossel, Jacob Richey, and Tselil Schramm.

INI 1  
11:45 to 12:30 
Codina Cotar Edge and vertexreinforced random walks with superlinear reinforcement on infinite graphs
Coauthor: Debleena Thacker (Lund University) In this talk we introduce a new simple but powerful general technique for the study of edge and vertexreinforced processes with superlinear reinforcement, based on the use of order statistics for the number of edge, respectively of vertex, traversals. The technique relies on upper bound estimates for the number of edge traversals, proved in a different context by Cotar and Limic [Ann. Appl. Probab. (2009)] for finite graphs with edge reinforcement. We apply our new method both to edge and to vertexreinforced random walks with superlinear reinforcement on arbitrary infinite connected graphs of bounded degree. We stress that, unlike all previous results for processes with superlinear reinforcement, we make no other assumption on the graphs. For edgereinforced random walks, we complete the results of Limic and Tarres [Ann. Probab. (2007)] and we settle a conjecture of Sellke [Technical Report 9426, Purdue University (1994)] by showing that for any reciprocally summable reinforcement weight function w, the walk traverses a random attracting edge at all large times. For vertexreinforced random walks, we extend results previously obtained on Z by Volkov [Ann. Probab. (2001)] and by Basdevant, Schapira and Singh [Ann. Probab. (2014)], and on complete graphs by Benaim, Raimond and Schapira [ALEA (2013)]. We show that on any infinite connected graph of bounded degree, with reinforcement weight function w taken from a general class of reciprocally summable reinforcement weight functions, the walk traverses two random neighbouring attracting vertices at all large times. Related Links

INI 1  
12:30 to 13:30  Lunch @ Wolfson Court  
13:30 to 14:15 
Fengnan Gao On the Statistical Estimation of the Preferential Attachment Network Model
The preferential attachment (PA) network is a popular way of modeling the social networks, the collaboration networks and etc. The PA network model is an evolving network model where new nodes keep coming in. When a new node comes in, it establishes only one connection with an existing node. The random choice on the existing node is via a multinomial distribution with probability weights based on a preferential function f on the degrees. f maps the natural numbers to the positive real line and is assumed apriori nondecreasing, which means the nodes with high degrees are more likely to get new connections, i.e. "the rich get richer". We proposed an estimator on f. We show, with techniques from branching process, our estimator is consistent. If f is affine, meaning f(k) = k + delta, it is well known that such a model leads to a powerlaw degree distribution. We proposed a maximum likelihood estimator for delta and establish a central limit result on the MLE of delta. If f belongs to a parametric family no faster than linear, we show the MLE will also yield optimal performance with the asymptotic normality results. We will also talk about the potential extensions of the model (with borrowed strength from nonparametric Bayesian statistics) and interesting applications. This is joint work with Aad van der Vaart. 
INI 1  
14:15 to 15:00 
Pavel Krivitsky Modeling and Simulation of Dynamic Networks using EgocentricallySampled Data
In spread of infections over a sexual or other contact networks, the
timing of the contacts can be as important as their crosssectional
structure. However, their modeling and simulation is complicated by the
difficulty of collecting data about these networks. Rather than the more
traditional panel (repeated observations) or event data, in which all
individuals are identified, these networks are often observed only in
the form of an egocentric survey: a sample of individuals in the network
reporting nonidentifying demographic information (e.g., age, sex,
race/ethnicity) about their contacts, as well as and contact history
(e.g., start and end of past contacts).
This work develops a generalized method of moments approach to simulation and inference for dynamic networks models from such data by using the models' longrun properties, and proposes a networksize invariant parametrization to facilitate using these models to simulate populations with changing sizes and compositions. These techniques are applied to egocentric data from the 1992 US National Health and Social Life Survey, and other applications are demonstrated as well, produced in collaboration Martina Morris and Steven Goodreau and others. 
INI 1  
15:00 to 15:30  Afternoon Tea  
15:30 to 16:15 
Patrick Wolfe tba 
INI 1  
16:15 to 16:30  Close of Workshop and Programme by Organisers  INI 1 