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Homophily Rearrangement Algorithms and Similarity Based Diffusion on Networks Thesis defense Benedek András Rózemberczki Central European University Supervisor: Professor Rosario Nunzio Mantegna 2016.06.13.

Introduction Informal model descriptions Simulations Summary References Overview Introduction Informal model descriptions Homophily rearrangement algorithms Similarity based diffusion Simulations Homophily rearrangement simulations Similarity based diffusion simulations Summary Homophily Rearrangement Algorithms and Similarity Based Diffusion on Networks Thesis defense

Introduction Informal model descriptions Simulations Summary References Introduction Research questions 1. Univariate homophily rearrangement algorithms. 2. Multivariate homophily rearrangement algorithms. 3. Similarity based diffusion model. Diffusion process ended Ordered state Randomization Diffusion Diffusion process started Initial state Pseudo-ordered state Homophily rearrangement Diffusion initialized Figure 1: The schematics of the modeling framework used in my thesis Homophily Rearrangement Algorithms and Similarity Based Diffusion on Networks Thesis defense

Introduction Informal model descriptions Simulations Summary References The context of homophily ◮ Birds of a feather (McPherson et al., 2001). ◮ Not just a social phenomenon. ◮ Micro-level similarity results in a macro-level outcome (Jackson et al., 2016). It is present in numerous socio-economic and non socio-economic networks, such as: ◮ Corporate governance networks (Kogut et al., 2012). ◮ Friendships (Epstein, 1986). ◮ Labor market referrals (Fernandez & Fernandez-Mateo, 2006). ◮ Blogs and webpages (Bisgin et al., 2010). ◮ Interactomes (Navlakha & Kingsford, 2010). Homophily Rearrangement Algorithms and Similarity Based Diffusion on Networks Thesis defense

Introduction Informal model descriptions Simulations Summary References Homophily rearrangements and diffusion ◮ Homophilous network generation (van Eck & Jager, 2010; Quayle et al., 2006). ◮ Homophily rearrangement is used for randomized experiments (Centola, 2011). ◮ Peer-effects are measurable – it would be nice to have large-scale homophily rearrangement algorithms. ◮ The Schelling (1969) and Fagiolo et al. (2007) models are actually homophily rearrangement algorithms. ◮ Later diffusion can be initiated on the network (Yavas & Yusel, 2014). ◮ The diffusion in my model is probabilistic not relative threshold based (Yavas & Yusel, 2014; Halberstam & Knight, 2014). ◮ The seeders have multiple infection trials – unlike in Kempe et al. (2003). Homophily Rearrangement Algorithms and Similarity Based Diffusion on Networks Thesis defense

Introduction Informal model descriptions Simulations Summary References Informal model descriptions Homophily rearrangement algorithms I. (a) Perfect heterophily (b) Homophily (c) Strong homophily Figure 2: Different levels of universal homophily on a 4 × 4 square lattice without periodic boundary conditions The network is defined by the adjacency matrix ( W ) and the generic feature vector or matrix ( x and X respectively). Homophily Rearrangement Algorithms and Similarity Based Diffusion on Networks Thesis defense

Introduction Informal model descriptions Simulations Summary References Homophily rearrangement algorithms II. The algorithm design has to include: ◮ Homophily measurement function – H ( x , W ) or H ( X , W ) . ◮ The type of the generic vertex feature matters – for example continuous or categorical. ◮ The target homophily level(s) – φ or Φ . ◮ If there are multiple homophily targets they must have the same sign. The following homophily rearrangement algorithms were implemented for univariate and multivariate systems: ◮ Heuristic ◮ Heuristic with bag of indices ◮ Greedy Homophily Rearrangement Algorithms and Similarity Based Diffusion on Networks Thesis defense

Introduction Informal model descriptions Simulations Summary References Similarity based diffusion I. The model setup consists: 1. Agents (vertices) who are connected by a network (edges). 2. A binary information that agents transmit among each other. 3. Agents have generic vertex features denoted by X . 4. The transmission of the information is probabilistic. The probability that i transmits the information to agent j is epxressed by the pairwise transmission probability equations. See Equation (1). P i , j = P 0 · Ψ( − γ · d ( X i , X j )) (1) � �� � Base function Specifically Equation (2) describes the pairwise transmission proba- bility equation that I use later during the simulations. P i , j = P 0 · exp ( − γ · | x i − x j | ) (2) Homophily Rearrangement Algorithms and Similarity Based Diffusion on Networks Thesis defense

Introduction Informal model descriptions Simulations Summary References Similarity based diffusion II. 1. Initially only a single agent has the information. 2. The time is discrete. 3. This is a modification of the susceptible-infected model. 4. Convergence to a fully infected state only happens when the network has one single component. 5. There is no recovering. 6. The γ value can be breed specific – discrimination is built in. Homophily Rearrangement Algorithms and Similarity Based Diffusion on Networks Thesis defense

Introduction Informal model descriptions Simulations Summary References Homophily rearrangement simulations The notion of unstable results (a) Simulation run 1. (b) Simulation run 2. Inbreeding homophily Inbreeding homophily 0.05 F 0.05 M 0 0 -0.05 -0.05 -0.1 -0.1 -0.15 -0.15 0 2500 5000 0 1500 3000 Iterative steps Iterative steps Figure 3: The convergence of gender based homophily to a target in separate simulation runs – friendship network from Harris (2009) Homophily Rearrangement Algorithms and Similarity Based Diffusion on Networks Thesis defense

Introduction Informal model descriptions Simulations Summary References Relaxation of switching conditions (a) Simulation run 1. (b) Simulation run 2. Inbreeding homophily Inbreeding homophily 9 th 10 th -0.15 -0.15 11 th 12 th -0.20 -0.20 0 250 500 0 125 250 Iterative steps Iterative steps Figure 4: The convergence of grade based inbreeding homophily to a target vector in two separate simulation runs – based on the school friendship network Homophily Rearrangement Algorithms and Similarity Based Diffusion on Networks Thesis defense

Introduction Informal model descriptions Simulations Summary References Increased target homophily I. (a) Mean solution time (b) Median solution time E ( t ) Me ( t ) 400 400 300 300 200 200 100 100 φ φ -0.5 -0.25 0.25 0.5 -0.5 -0.25 0.25 0.5 Figure 5: Expected average and median convergence times of the heuristic algorithm on a square lattice with periodic boundary conditions as a function of target homophily Homophily Rearrangement Algorithms and Similarity Based Diffusion on Networks Thesis defense

Introduction Informal model descriptions Simulations Summary References Increased target homophily II. (a) Mean solution time (b) Median solution time E ( t ) Me ( t ) 10 10 5 5 φ φ -0.5 -0.25 0.25 0.5 -0.5 -0.25 0.25 0.5 Figure 6: Expected average and median convergence times of the greedy algorithm on a square lattice with periodic boundary conditions as a function of target homophily Homophily Rearrangement Algorithms and Similarity Based Diffusion on Networks Thesis defense

Introduction Informal model descriptions Simulations Summary References System size and feature distribution E ( t ) N = 121 1 , 600 N = 196 N = 256 1 , 200 800 400 P ( X = 1 ) 0 . 2 0 . 25 0 . 3 0 . 35 0 . 4 0 . 45 0 . 5 Figure 7: Expected convergence time of the heuristic algorithm as a function of system size and balancedness of the feature distribution Homophily Rearrangement Algorithms and Similarity Based Diffusion on Networks Thesis defense

Introduction Informal model descriptions Simulations Summary References Multiple features 3 , 000 N = 100 E ( t ) 2 , 500 N = 144 N = 196 2 , 000 N = 256 1 , 500 1 , 000 ρ − 0 . 5 − 0 . 4 − 0 . 3 − 0 . 2 − 0 . 1 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 Figure 8: Expected solution time of the multivariate heuristic homophily rearrangement algorithms as a function of feature correlation and lattice size Homophily Rearrangement Algorithms and Similarity Based Diffusion on Networks Thesis defense

Introduction Informal model descriptions Simulations Summary References Simulation of diffusion Pairwise transmission probability distributions 4 Density 2 0 0 0.1 0.2 0.3 0.4 Transmission probability φ = − 0 . 5 φ = 0 φ = 0 . 5 Figure 9: The distribution of pairwise transmission probabilities Homophily Rearrangement Algorithms and Similarity Based Diffusion on Networks Thesis defense

Introduction Informal model descriptions Simulations Summary References Ratio of infected nodes E ( Y t ) / N 1 φ = − 0 . 8 0 . 5 φ = 0 φ = 0 . 8 t 5 10 15 20 25 30 Figure 10: The ratio of infected nodes as a function of time Homophily Rearrangement Algorithms and Similarity Based Diffusion on Networks Thesis defense

Introduction Informal model descriptions Simulations Summary References Sensitivity to dissimilarity increase 60 E ( t ) φ = − 0 . 8 φ = 0 50 φ = 0 . 8 40 30 20 γ 0 . 2 0 . 4 0 . 6 0 . 8 Figure 11: Expected solution time as a function of sensitivity to dissimilarity Homophily Rearrangement Algorithms and Similarity Based Diffusion on Networks Thesis defense