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Timetable (SNAW04)

Dynamic networks

Monday 12th December 2016 to Friday 16th December 2016

Monday 12th 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 (University of Groningen); (University of Oxford)
Continuous-time 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 continuous-time Markov network process, observed only at the moments of observation for the panel. This is analogous to the use of continuous-time models for classical (non-network) 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 Actor-Oriented Model of Snijders (2001) and in the Longitudinal Exponential Random Graph Model of Snijders & Koskinen (2013). The first of these is actor-oriented, i.e., tie changes are modelled as choices by actors, which among their outgoing tie variables to toggle; the second is tie-oriented, i.e., tie changes are modelled as toggles of single tie variables. Both are generalized linear models for the (unobserved) continuous-time 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., co-evolution of multivariate networks. Some open problems, from a mathematical and from an applied perspective, will be mentioned.
           

References  
  • Ruth M. Ripley, Tom A.B. Snijders, Zsófia Boda, András Vörös, and Paulina Preciado, 2016. Manual for SIENA version 4.0. Oxford: University of Oxford, Department of Statistics; Nuffield College. http://www.stats.ox.ac.uk/siena/   
  • Tom A.B. Snijders, 2001. The statistical evaluation of social network dynamics. Sociological Methodology, 31, 361-395.
  • Tom A.B. Snijders and Johan Koskinen, 2013. "Longitudinal Models". Chapter 11 (pp. 130-140) in D. Lusher, J. Koskinen, and G. Robins, Exponential Random Graph Models for Social Networks, Cambridge: Cambridge University Press.  
  • Tom A.B. Snijders, Gerhard G. van de Bunt, G. G., and Christian E.G. Steglich, 2010. Introduction to actor-based models for network dynamics. Social Networks, 32, 44–60.  




INI 1
10:30 to 11:15 Morning Coffee
11:15 to 12:00 Eric Kolaczyk (Boston University)
Dynamic causal networks with multi-scale temporal structure
Co-authors: 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 multi-scale 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 multi-scale 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 (Microsoft (UK))
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 non-parametric 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 power-law 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 non-parametric 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 (Microsoft (UK))
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 non-parametric 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 power-law 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 non-parametric 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 (Dalhousie University)
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 one-dimensional (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
Tuesday 13th December 2016
09:30 to 10:30 Tom Britton (Stockholm University)
A network epidemic model with preventive rewiring: comparative analysis of the initial phase
Co-authors: 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 non-infectious individuals with probability α or else simply drop the edge if α=0), so-called preventive rewiring. The models are denoted SIR-ω and SEIR-ω, 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 SIR-ω and SEIR-ω epidemics on Poisson and scale-free 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 Erdös-Renyi graph, the configuration model and a preferential attachment graph. For the Erdös-Renyi 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 (Stanford University)
Study of the dynamics of bacterial communities in the Human Microbiome
Co-authors: 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, hypo-salivation 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 multi-table 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 (University of Pennsylvania)
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 spatio-temporal 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 ever-reconfiguring 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 (INRA - Institut National de la Recherche Agronomique)
Detecting change-points in the structure of a network: Exact Bayesian inference
Joint work with Loïc Schwaller

We consider the problem of change-point detection in multivariate time-series, 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 change-points 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 (Rutgers, The State University of New Jersey)
Markov process models for time-varying 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 time-varying networks and discuss relevant consequences.
INI 1
Wednesday 14th December 2016
09:30 to 10:30 Frank Granville Ball (University of Nottingham)
Epidemics on networks
In this talk we consider two extensions of the standard stochastic epidemic SIR (Susceptible-Infected-Recovered) 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 (Brunel University)
Sparse Gaussian graphical models for dynamic gene regulatory networks
Co-authors: 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 high-throughput data. The model allows including dynamic-related 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 high-resolution time-course microarray data from the Neisseria meningitidis bacterium, a causative agent of life-threatening 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 (ENS - Lyon); (CNRS (Centre national de la recherche scientifique))
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 face-to-face contacts. The extraction of sub-networks highlights significant structures decomposed on time intervals. 
INI 1
14:45 to 15:30 Marianna Pensky (University of Central Florida)
Non-parametric methods for the dynamic stochastic block model and the time-dependent 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 time-independent to the time-dependent 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 non-parametric 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 non-asymptotic and allow a variety of  extensions.  


INI 1
15:30 to 16:00 Afternoon Tea
16:00 to 16:45 Sidney Resnick (Cornell University)
Multivariate power laws in a preferential attachment network model; model calibration
We begin with a review of the multivariate regular variation of in- and out-degree 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 out-degree. 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 trade-off 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 (Columbia University)
Random walk models of network formation
INI 1
19:30 to 22:00 Formal Dinner at Christ's College
Thursday 15th December 2016
09:30 to 10:30 Andrew Nobel (University of North Carolina)
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 second-order 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 (Microsoft (UK))
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 between-community 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 latentcountry--community memberships, including the number of latentcommunities, as well as directed community--community 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 (Stanford University)
The Effect of Recommendations on Network Structure
Co-authors: 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 mid-2010 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 "friend-of-friend"-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 well-known individuals.

Related Links
INI 1
14:45 to 15:30 Peter Mörters (University of Bath)
The spread of infections on evolving scale-free networks
We study the contact process on a class of evolving scale-free 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 power-law 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 mean-field approximation, which has a phase transition at power-law 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 (CNRS (Centre national de la recherche scientifique)); (Université Pierre et Marie Curie Paris)
Statistical clustering of temporal networks through a dynamic stochastic block model
Co-author: 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
Friday 16th December 2016
09:30 to 10:30 George Michailidis (University of Florida)
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 multi-layer networks and illustrate the results with application​s
to financial stability monitoring and biological regulation.
INI 1
10:30 to 11:00 Morning Coffee
11:00 to 11:45 Miklos Racz (Microsoft (UK))
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 root-finding 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 (University College London)
Edge- and vertex-reinforced random walks with super-linear reinforcement on infinite graphs
Co-author: Debleena Thacker (Lund University)

In this talk we introduce a new simple but powerful general technique for the study of edge- and vertex-reinforced processes with super-linear 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 vertex-reinforced random walks with super-linear reinforcement on arbitrary infinite connected graphs of bounded degree. We stress that, unlike all previous results for processes with super-linear reinforcement, we make no other assumption on the graphs.

For edge-reinforced random walks, we complete the results of Limic and Tarres [Ann. Probab. (2007)] and we settle a conjecture of Sellke [Technical Report 94-26, 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 vertex-reinforced 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 (Universiteit Leiden)
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 non-decreasing, 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 power-law 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 (University of Wollongong)
Modeling and Simulation of Dynamic Networks using Egocentrically-Sampled Data
 In spread of infections over a sexual or other contact networks, the timing of the contacts can be as important as their cross-sectional 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 non-identifying 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' long-run properties, and proposes a network-size 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 (University College London)
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INI 1
16:15 to 16:30 Close of Workshop and Programme by Organisers INI 1
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons