Yann Bramoullé (Aix-Marseille Université)

Altruism in Networks (joint work with Renaud Bourlès (Centrale Marseille (Aix-Marseille School of Economics)):We provide the first theoretical analysis of altruism in networks. Agents are embedded in a fixed, weighted network and care about their direct friends. They provide financial support to their friends in need. We analyze the resulting transfer game. We show that there is a unique profile of equilibrium incomes, for any network and any utility functions. There are typically multiple equilibria in transfers, even though uniqueness holds on trees. We highlight the importance of indirect gifts, where an agent gives to a friend because his friend himself has a friend in need. We show that a positive shock in one part of the network can never deteriorate agents’ situations in other parts. In contrast, a Pigou-Dalton redistribution from richer to poorer can end up increasing overall inequality. Finally, we study how ex-post inequality depends on the network structure. On a network with more altruistic links, the maximal income spread is lower but income variance can be higher. And homophily with respect to income tends to generate more inequality.

Vasco Carvalho (CREi, Universitat Pompeu Fabra)

Input Adoption and the Origins of Input-Output Networks (joint work with Nico Voigtlaender): A growing literature emphasizes the importance of input-output linkages for aggregate fluctuations. We study the evolution of these network linkages theoretically and empirically. We provide a model based on theories of dynamic network formation. A new product variety is introduced each period, and it initially draws a set of ’essential suppliers’. The latter in turn source their own inputs, and this forms the ’network neighborhood’ of the new variety, from which it subsequently adopts further inputs. The more prominent an input is in the new variety’s network neighborhood, the more likely it will be adopted. Thus, closer network proximity implies higher likelihood of input adoption. The mechanism also delivers a power law distribution of forward linkages. We show that both these predictions continue to hold when varieties are aggregated into sectors. We use detailed US input-output tables to test these predictions. We show that initial network proximity in 1967 predicts the formation of new linkages in the following four decades.

Thomas Chaney (Toulouse School of Economics)

The Gravity Equation in International Trade: An Explanation: The gravity equation in international trade is one of the most robust empirical finding in economics: bilateral trade between two countries is proportional to size, measured by GDP, and inversely proportional to the geographic distance between them. While the role of size is well understood, the role of distance remains a mystery. I propose the first explanation for the gravity equation in international trade, based on the emergence of a stable network of input-output linkages between firms. Over time, a firm acquires more suppliers and customers, which tend to be further away. I show that if, as observed empirically, (i) the distribution of firm sizes is well approximated by Zipf’s law and (ii) larger firms export over longer distances on average, then aggregate trade is inversely proportional to distance. Data on firm level, sectoral, and aggregate trade support further predictions of the model.

Habiba Djebbari (Aix-Marseille Université)

Accounting for Peer Effects in Treatment Response (joint work with Felipe Barrera-Osario and Rokhaya Dieye): We propose a new strategy for estimating peer effects on individual decisions of participants to a scholarship program. We study education decisions in the district of Bogota (Colombia) by the means of an experiment that was conducted in the 2005-2006 period. We use the structural social network model developed by BDF (2009) that helps correct for correlated effects due to self-selection in peer groups and common shocks. New identification conditions that mostly require balance in the characteristics of friends between treatment and control group are provided. We find that the simple difference in mean outcomes between experimental groups overestimate the reduction in student absenteeism resulting from the program.

Nial Friel (University College Dublin)

Estimating Bayes Factors for the Exponential Random Graph Model: Exponential random graph models are arguably the most popular statistical model for the analysis of network data. However, despite its popularity, this class of models poses significant problem from a statistical inferential viewpoint since the likelihood function is almost always impossible to evaluate. This talk will present some approaches to overcome this intractability and which also allows one to tackle the important issue of assessing model uncertainty through the estimation of Bayes factors.

Jiashun Jin (Carnegie Mellon University)

Fast Network Community Detection by SCORE: Consider a network where the nodes split into K different communities. The community labels for the nodes are unknown and it is of major interest to estimate them (i.e., community detection). Degree Corrected Block Model (DCBM) is a popular network model. How to detect communities with the DCBM is an interesting problem, where the main challenge lies in the degree heterogeneity. We propose a new approach to community detection which we call the Spectral Clustering On Ratios-of-Eigenvectors (SCORE). Compared to classical spectral methods, the main innovation is to use the entry-wise ratios between the first leading eigenvector and each of the other leading eigenvectors for clustering. The central surprise is, the effect of degree heterogeneity is largely ancillary, and can be effectively removed by taking entry-wise ratios between the leading eigenvectors. The method is successfully applied to the web blogs data and the karate club data, with error rates of 58/1222 and 1/34, respectively. These results are much more satisfactory than those by the classical spectral methods. Also, compared to modularity methods, SCORE is computationally much faster and has smaller error rates. We develop a theoretic framework where we show that under mild conditions, the SCORE stably yields successful community detection. In the core of the analysis is the recent development on Random Matrix Theory (RMT), where the matrix-form Bernstein inequality is especially helpful.

Johan Koskinen (University of Manchester)

Hierarchical Stochastic Actor-Oriented Models for Longitudinal Analysis of Multiply Observed Networks Across Different Settings: We consider a hierarchical extension of class of continuous-time Markov models for longitudinal analysis of Social Networks called stochastic actor-oriented models (SAOM). SAOMs model repeated observations on social interactions between individuals along with changes in attributes of these individuals. In particular the relational ties of an actor is conceived of as a collection of binary indicator variables. A fixed set of actors constitutes a cross-classification of tie-variables as actors in the same set are both senders and receivers of ties, something which induces complex marginal dependencies. Modelling these dependencies is the main inferential target of SAOMs and this is accomplished through assuming incremental changes to the network as being conditioned on the current state of the network. The actors in a typical network are often selected based on some affiliation or membership and the fitted model will partly reflect effects due to the shared context and initial conditions. To account for heterogeneity across different settings multiply observed networks across different contexts have previous been investigated by pooling separate analyses using meta analysis. Here we develop a hierarchical model for independently observed networks, where group-level parameters are modeled as being drawn from a population distribution. The population means of the group-level parameters are thus the main targets of inference and heterogeneity may be investigated through network-level effects and predictive group-level distributions. Inference is performed through a straightforward hierarchical extension of a previously proposed Bayesian data-augmentation scheme for single network analysis.

Marc Lelarge (INRIA, ENS Paris)

Reconstruction in the Generalized Stochastic Block Model (joint work with Laurent Massoulie and Jiaming Xu): The labeled stochastic block model is a random graph model representing networks with community structure and interactions of multiple types. In its simplest form, it consists of two communities of approximately equal size, and the edges are drawn and labeled at random with probability depending on whether their two endpoints belong to the same community or not. It has been conjectured that this model exhibits a phase transition: reconstruction (i.e. identification of a partition positively correlated with the ‘true partition’ into the underlying communities) would be feasible if and only if a model parameter exceeds a threshold. We prove this conjecture for ‘not too sparse’ graphs. In the very sparse case (finite mean degree), we prove one half of this conjecture, i.e., reconstruction is impossible when below the threshold. In the converse direction, we introduce a suitably weighted graph. We show that when above the threshold by a specific constant, reconstruction is achieved by (1) minimum bisection, and (2) a spectral method combined with removal of nodes of high degree.

Catherine Matias (CNRS, Université d’Évry, Génopole)

Stochastic Block Models for Random Graphs: We will discuss in details the stochastic block model that allows to capture heterogeneity in random graphs, its properties and applications.

Aureo de Paula (CeMMAP, UCL)

Econometric Analysis of a Network Formation Game (joint work with Seth Richards-Shubik (Carnegie Mellon University) and Elie Tamer (Northwestern University)): This project deals with identification and inference of an empirical model of network formation. The model we suggest is based on a static game of complete information in which individuals propose links to each other. We assume that observed networks are equilibrium networks and use pairwise stability, proposed in Jackson and Wolinsky (1996), as the solution concept for this game. Two main issues arise in this analysis: the existence of multiple solutions (or even non-existence of a solution) and computational complexity. We suggest an estimation procedure that bypasses the selection of a particular equilibrium (when many are possible) using set identification and estimation techniques. To achieve a computationally feasible model, we then impose payoff restrictions that reduce the complexity of the problem. The resulting model resembles a familiar discrete choice model (with additional constraints) where individuals choose among “network types”. We illustrate the ideas above using examples and simulations.

Arvind Rangaswamy (Penn State, Smeal College of Business)

The Effects of Close and Distant Ties on Organizational and Consumer New Product Diffusion: Through two empirical studies, we explore the roles and effects of close and distant ties on new product diffusion. In the first study we focus on the diffusion of a technology among organizations. We draw upon the literature in institutional theory and population ecology of firms to propose that competitive and normative (i.e., legitimation) pressures play central roles in the diffusion of products and technologies across firms that can influence the business models of the adopting firms. Typically, competitive effects arise from close ties and legitimation effects arise from distant ties. For estimating the diffusion trajectory, we propose a binary choice model that accommodates both adoption and adoption timing decisions of German car dealers who had to decide whether to adopt a new online channel for listing of used cars. A key insight that emerges from the study is that competitive pressure acts as a “trigger” in inducing adoptions (i.e., contributes strongly to likelihood recovery), but normative pressure creates the pre-disposition for adoption (i.e., contributes strongly to the utility of adoption). In the second study, we use the formalisms of small world network and agent-based simulations to aggregate unobserved individual-level adoptions into aggregate diffusion patterns. We model the existence of potential connections between members of a target population, the mechanisms for the activation of connections, and the effect of an activation (i.e., whether or not it leads to an adoption). Using large multi-country data sets of past diffusions of several consumer durables, we decompose word-of-mouth effects and assess whether an innovation spreads due to the relatively strong collective influence of a small close circle of family members and friends or through the total influence of distant acquaintances, each with relatively weak influence. The results from our two studies provide new insights and more-nuanced interpretations of the effects of close and distant ties on new product diffusion, further extending the insights about an issue that was first articulated in the seminal work of Granovetter (1973).

Stéphane Robin (INRA, AgroParisTech)

Deciphiring and Modelling Heterogeneity in Interaction Networks: Network analysis has become a very active field of statistics within the last decade. Several models have been proposed to describe and understand the heterogeneity observed in real networks. We will present here several modeling involving latent variable. We will discuss the issues raised by their inference, focusing on the stochastic block model. We will describe a variational approach that allows to deal with the complex dependency structure of this model.

Nicolas Verzelen (INRA, Supagro Montpellier)

Community Detection in Random Networks: We formalize the problem of detecting a community in a network into testing whether in a given (random) graph there is a subgraph that is unusually dense. We observe an undirected and unweighted graph on N nodes. Under the null hypothesis, the graph is a realization of an Erdös-Rényi graph with probability p0. Under the (composite) alternative, there is a subgraph of n nodes where the probability of connection is p1 > p0. We derive a detection lower bound for detecting such a subgraph in terms of N, n, p0, p1 and exhibit a test that achieves that lower bound. We do this both when p0 is known and unknown. We also consider the problem of testing in polynomial-time. As an aside, we consider the problem of detecting a clique, which is intimately related to the planted clique problem.