Miniconference on the Mathematics of Computation 1st Symposium on Spatial Networks Oxford University The Geometry of Social Networks Anthony Bonato Ryerson University

Geometry of Social Networks Friendship networks network of on- and off-line friends form a large web of interconnected links Geometry of Social Networks

Geometry of Social Networks 6 degrees of separation (Stanley Milgram,67): famous chain letter experiment Geometry of Social Networks

Geometry of Social Networks 4 Degrees in Facebook 1.71 billion users (Backstrom,Boldi,Rosa, Ugander,Vigna,2012) 4 degrees of separation in Facebook when considering another person in the world, a friend of your friend knows a friend of their friend, on average similar results for Twitter and other OSNs Geometry of Social Networks

Are we really that similar? Geometry of Social Networks

Geometry of Social Networks Social distance 4 or 6 degrees of separation does not reflect our true social distance D. Liben-Nowell, J. Kleinberg, Tracing information flow on a global scale using Internet chain-letter data PNAS 105 (2008) 4633-4638. Geometry of Social Networks

Geometry of Social Networks Hidden geometry vs Geometry of Social Networks

Complex networks in the era of Big Data web graph, social networks, biological networks, internet networks, … Geometry of Social Networks

What is a complex network? no precise definition however, there is general consensus on the following observed properties large scale evolving over time power law degree distributions small world properties Geometry of Social Networks

Examples of complex networks technological/informational: web graph, router graph, AS graph, call graph, e-mail graph social: on-line social networks (Facebook, Twitter, LinkedIn,…), collaboration graphs, co-actor graph biological networks: protein interaction networks, gene regulatory networks, food networks, connectomes Geometry of Social Networks

Geometry of Social Networks Other properties densification power law (Leskovec, Kleinberg, Faloutsos,05): |(E(Gt)| ≈ |V(Gt)|a where 1 < a ≤ 2: densification exponent community structure spectral expansion Geometry of Social Networks

Geometry of Social Networks Blau space OSNs live in social space or Blau space: each user identified with a point in a multi-dimensional space coordinates correspond to socio-demographic variables/attributes homophily principle: the flow of information between users is a declining function of distance in Blau space Geometry of Social Networks

Geometry of Social Networks Dimensionality Question: What is the dimension of the Blau space of OSNs? what is a credible mathematical formula for the dimension of an OSN? Geometry of Social Networks

Geometry of Social Networks

Random geometric graphs n nodes are randomly placed in the unit square each node has a constant sphere of influence, radius r nodes are joined if their Euclidean distance is at most r G(n,r), r = r(n) Geometry of Social Networks

Some properties of G(n,r) Theorem (Penrose,97) Let μ = nexp(-πr2n). If μ = o(1), then asymptotically almost surely (a.a.s.) G is connected. If μ = Θ(1), then a.a.s. G has a component of order Θ(n). If μ →∞, then a.a.s. G is disconnected. many other properties studied of G(n,r): chromatic number, clique number, Hamiltonicity, random walks, … Geometry of Social Networks

Geometry of Social Networks Spatially Preferred Attachment (SPA) model (Aiello, Bonato, Cooper, Janssen, Prałat,08), (Cooper, Frieze, Prałat,12) volume of sphere of influence proportional to in-degree nodes are added and spheres of influence shrink over time a.a.s. leads to power laws graphs, low directed diameter, and small separators Geometry of Social Networks

Geometry of Social Networks Ranking models (Fortunato,Flammini,Menczer,06), (Łuczak,Prałat,06), (Janssen,Prałat,09) parameter: α in (0,1) each node is ranked 1,2, …, n by some function r 1 is best, n is worst at each time-step, one new node is born, one randomly node chosen dies (and ranking is updated) link probability r-α many ranking schemes a.a.s. lead to power law graphs: random initial ranking, degree, age, etc. Geometry of Social Networks

Geometric model for OSNs we consider a geometric model of OSNs, where nodes are in m-dimensional Euclidean space volume of spheres of influence variable: a function of ranking of nodes Geometry of Social Networks

Geometric Protean (GEO-P) Model (Bonato,Janssen,Prałat,12) parameters: α, β in (0,1), α+β < 1; positive integer m nodes live in an m-dimensional hypercube each node is ranked 1,2, …, n by some function r 1 is best, n is worst we use random initial ranking at each time-step, one new node v is born, one randomly node chosen dies (and ranking is updated) each existing node u has a region of influence with volume add edge uv if v is in the region of influence of u Geometry of Social Networks

Geometry of Social Networks Notes on GEO-P model models uses both geometry and ranking number of nodes is static: fixed at n order of OSNs at most number of people (roughly…) top ranked nodes have larger regions of influence Geometry of Social Networks

Geometry of Social Networks Simulation with 5000 nodes Geometry of Social Networks

Geometry of Social Networks Simulation with 5000 nodes random geometric GEO-P Geometry of Social Networks

Properties of the GEO-P model (BJP,2012) a.a.s. the GEO-P model generates graphs with the following properties: power law degree distribution with exponent b = 1+1/α average degree d = (1+o(1))n(1-α-β)/21-α densification diameter D = nΘ(1/m) small world: constant order if m = Clog n bad spectral expansion and high clustering coefficient Geometry of Social Networks

Geometry of Social Networks Dimension of OSNs given the order of the network n and diameter D, we can calculate m gives formula for dimension of OSN: Geometry of Social Networks

Logarithmic Dimension Hypothesis In an OSN of order n and diameter D, the dimension of its Blau space is posed independently by (Leskovec,Kim,11), (Frieze, Tsourakakis,11) Geometry of Social Networks

Few dimensions implies greater difference low dimensional separation high dimensional separation Geometry of Social Networks

Uncovering the hidden reality reverse engineering approach given network data (n, D), dimension of an OSN gives smallest number of attributes needed to identify users that is, given the graph structure, we can (theoretically) recover the Blau space Geometry of Social Networks

6 Dimensions of Separation OSN Dimension Facebook 7 YouTube 6 Twitter 4 Flickr Cyworld Geometry of Social Networks

Geometry of Social Networks

MGEO-P (Bonato,Gleich,Mitsche,Prałat,Tian,Young,14) time-steps in GEO-P form a computational bottleneck consider a GEO-P where we forget the history of ranks memoryless GEO-P (MGEO-P) place n points u.a.r. in the hypercube assign ranks from via a random permutation σ for each pair i > j, ij is an edge if j is in the ball of volume σ(i)–αn-β Geometry of Social Networks

Contrasting the models by considering the evolution of ranks in GEO-P, the probability that an edge is present in GEO-P and not in MGEO-P is: intuitively, the models generate similar graphs many a.a.s properties hold in MGEO-P with similar parameters Geometry of Social Networks

Properties of the MGEO-P model (BGMPTY,14) a.a.s. the MGEO-P model generates graphs with the following properties: power law degree distribution with exponent b = 1+1/α average degree d = (1+o(1))n(1-α-β)/21-α densification diameter D = nΘ(1/m) Geometry of Social Networks

Proof sketch: diameter eminent node: highly ranked: ranking greater than some fixed R partition hypercube into small hypercubes choose size of hypercubes and R so that each hypercube contains at least log2n eminent nodes sphere of influence of each eminent node covers each hypercube and all neighbouring hypercubes choose eminent node in each hypercube: backbone show all nodes in hypercube distance at most 2 from backbone Geometry of Social Networks

Geometry of Social Networks Back to question… How would we measure the dimensionality of Blau space? Geometry of Social Networks

Aside: machine learning machine learning is a branch of AI that infers structure from data examples: spam filters Netflix recommender systems text and image categorization especially useful when the data or number of decisions are too large for humans to process Geometry of Social Networks

Model selection in complex networks (Middendorf,Ziv,Wiggins,05) used ADTs and motifs for model selection in protein networks predicted duplication/mutation model (Memišević,Milenković,Pržulj,10) model selection predicting random geometric graphs as best fit for protein networks (Janssen,Hurshman,Kalyaniwalla,12) ADT with motif classifiers predict PA and SPA models best fit Facebook 100 graphs Geometry of Social Networks

Support Vector Machine (SVM) Sec. 15.1 Support Vector Machine (SVM) support vectors maximizes margin SVM maximizes the margin around the separating hyperplane solving SVMs is a quadratic programming problem successful text and image classification method Geometry of Social Networks

Geometry of Social Networks Facebook100 Geometry of Social Networks

Geometry of Social Networks Validating the LDH we tested the dimensionality of large-scale samples from real OSN data FB100 and LinkedIn (sampled over time) Idea: use machine learning (SVM) to predict dimensions features: small subgraph counts (3- and 4-vertex subgraphs) compared sampled data vs simulations of MGEO-P with dimensions 1 through 12 Geometry of Social Networks

Miniconference on the Mathematics of Computation Motifs/Graphlets Geometry of Social Networks

Miniconference on the Mathematics of Computation Experimental design Geometry of Social Networks

Miniconference on the Mathematics of Computation Sample: Michigan Geometry of Social Networks

Miniconference on the Mathematics of Computation Stanford3: n: 11621 edges: 568330 avgdeg: 97.81086 plexp: 3.730000 GeoP parameters alphabeta: 0.510389 alpha: 0.366300 beta: 0.144089 python geop_dim_experiment.py --logcount -s 50 -t 0 --mmax 12 --prob 0.001 Stanford3 11621 568330 0.366300 0.144089 M-GeoP dimensions: LADTree: 2 J48: 3 Logistic: 5 SVM: 5 Geometry of Social Networks

Miniconference on the Mathematics of Computation FB and LinkedIn - SVM Geometry of Social Networks

FB and LinkedIn - Eigenvalues Miniconference on the Mathematics of Computation FB and LinkedIn - Eigenvalues Geometry of Social Networks

Miniconference on the Mathematics of Computation Figure 6. For three of the Facebook networks, we show the eigenvalue histogram in red, the eigenvalue histogram from the best fit MGEO-P network in blue, and the eigenvalue histograms for samples from the other dimensions in grey. Bonato A, Gleich DF, Kim M, Mitsche D, et al. (2014) Dimensionality of Social Networks Using Motifs and Eigenvalues. PLoS ONE 9(9): e106052. doi:10.1371/journal.pone.0106052 http://www.plosone.org/article/info:doi/10.1371/journal.pone.0106052 Geometry of Social Networks

Miniconference on the Mathematics of Computation Underlying geometry Feature space thesis (B,16+) every complex network has an underlying metric (or feature) space, where nodes are identified with points in the feature space, and edges are influenced by node similarity and proximity in the space For e.g.: web graph: topic space OSNs: Blau space PPIs: biochemical space Geometry of Social Networks

Miniconference on the Mathematics of Computation Implications of FST new way of viewing complex networks not just graph structure, but underlying, hidden geometry that matters graph structure can help uncover this hidden geometry Geometry of Social Networks

Miniconference on the Mathematics of Computation Future directions other data sets fractal or other dimension underlying metric? what are the attributes? what implications does LDH have for OSNs or social networks in general? Geometry of Social Networks

Miniconference on the Mathematics of Computation Character networks cultural work: fictional works such novels or short stories, movies, biographies, historical works, religious texts character networks: nodes: characters or persona in a cultural work edges: co-occurrence edges may be weighted Geometry of Social Networks

Miniconference on the Mathematics of Computation E.g.: Marvel universe 10K nodes diameter 9 10 communities average degree 41 Character networks

Miniconference on the Mathematics of Computation moviegalaxies.com moviegalaxies.com,catalogues the social networks in 800+ movies Geometry of Social Networks

Dimensionality of character networks? Miniconference on the Mathematics of Computation Dimensionality of character networks? Geometry of Social Networks