WAW 19 Modeling Adversaries in Networks Anthony Bonato

WAW’ 19 Modeling Adversaries in Networks Anthony Bonato Ryerson University

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Social networks • nodes: people • edges: social interaction – usually studied from perspective of positive social interaction Anthony Bonato 3

What about negative social interaction? Zachary Karate club Anthony Bonato 4

Dislikes Anthony Bonato 5

Structural balance theory • considers triads of nodes • triads seek balance (Heider, 58), (Easley, Kleinberg, 10) balanced unbalanced Anthony Bonato 6

Key Questions 1. models? – much greater emphasis on positive interaction 2. where to extract data on negative interactions? – challenging, especially for social networks 3. what does the data tell us about the network structure? Anthony Bonato 7

Graph parameters • average distance: • • clustering coefficient: Complex Networks 8

Properties of Complex Networks Small world property (Watts, Strogatz, 98) • low distances • diam(G) = O(log n) • L(G) = O(loglog n) • higher clustering coefficient than random graph with same expected degree Complex Networks 9

Properties of Complex Networks Densification power law (Leskovec, Kleinberg, Faloutsos, 05): • networks become more dense over time: average degree is increasing: |(E(Gt)| ~ |V(Gt)|a where 1 < a ≤ 2: densification exponent Complex Networks 10

Examples of negative interaction networks 1. Market graphs • nodes: stocks • edges: negative correlation • properties (Boginski, Butenko, Pardalos, 03) – power law – small world – cliques and co-cliques Anthony Bonato 11

Examples of negative interaction networks 2. Survivor • nodes: players • edges: votes • properties (B, Eikmeier, Gleich, Malik, 18) – alliances – leaders/top players Anthony Bonato 12

Examples of negative interaction networks 3. Food webs • nodes: species • edges: feeding-pathway • properties (Dunne, Neo, Martinez, 02): – small world – lack power laws Anthony Bonato 13

Transitivity B C A Anthony Bonato 14

Iterated Local Transitivity (ILT) model (Bonato, Hadi, Horn, Prałat, Wang, 11) • key paradigm is transitivity – “friends of friends are friends” • nodes have local influence • evolves over time • retains memory of initial graph Anthony Bonato 15

ILT model • Anthony Bonato 16

G 0 = C 4 Anthony Bonato 17

Densification • Anthony Bonato 18

Sketch of proof of densification Gt+1 y Gt x • now use induction Anthony Bonato 19

Average distance • Anthony Bonato 20

Distances, continued Gt+1 x’ y x Gt Anthony Bonato 21

Clustering Coefficient Theorem (BHHPW, 11): If t > 0, then c(Gt) = ntlog(7/8)+o(1). • higher clustering than in a random graph G(nt, p) with same order and average degree as Gt, which satisfies c(G(nt, p)) = ntlog(3/4)+o(1) Anthony Bonato 22

Anti-transitivity B A C Anthony Bonato 23

Iterated Local Anti-Transitivity (ILAT) model (Bonato, Infeld, Pokhrel, Prałat, 17) • key paradigm is anti-transitivity: “enemies of enemies are friends” – enemies/competitors: non-edges • model for adversarial network formation (Question 1) Anthony Bonato 24

ILAT model • start with a graph of order n • to form the graph Gt+1 for each node x from time t, add a node x*, the anti-clone of x, so that xx* is a non-edge, and x* is joined to each node non-joined to x Anthony Bonato 25

G 0 = C 4 Anthony Bonato 26

Properties (BIPP, 17) • Anthony Bonato 27

Sketch of proof of densification Gt+1 y x Gt Anthony Bonato 28

Iterated Local Model (ILM) (B, Chuangpishit, English, Kay, Meger, 19+) • incorporates transitivity and anti-transitivity • input includes a binary sequence • incorporates both friends and adversaries into network formation (Question 1) Anthony Bonato 29

ILM model (BCEKM, 19+) • start with a graph G and an infinite, binary sequence S = (sn) • if sn = 0, then at the nth step, anti-clone each vertex • if sn = 1, then at the nth step, clone each vertex Anthony Bonato 30

Densification • Anthony Bonato 31

Clustering coefficient • Anthony Bonato 32

Graph theoretic properties • Anthony Bonato 33

Graph theoretic properties Theorem (BCEKM, 19+) 1. An ILM graph is eventually Hamiltonian. 2. Every finite graph eventually appears as an induced subgraph in an ILM graph, regardless of the initial graph or sequence. Anthony Bonato 34

Sketch of proof of (2) • first show this holds for ILT (i. e. only 1’s/transitive steps) • find a copy of Kn by cloning cliques • can build Kn – e in one step: x Kn y t x’ y’ t+1 • iteratively delete edges from a sufficiently large clique to give any fixed graph Anthony Bonato 35

Sketch of proof of (2) • If there are infinitely many ILT steps, we are done • however, two ILAT steps “simulate” an ILT step: x x* x** Anthony Bonato t t+1 t+2 36

Key Questions 1. models? – much greater emphasis on positive interaction 2. where to extract data on negative interactions? – challenging, especially for social networks 3. what does the data tell us about the network structure? Anthony Bonato 37

Dynamic competition networks (B, Eikmeier, Gleich, Malik, 18) • nodes: agents • edges: u→v if v is a competitor of u • network evolves and edges added over time – NB: not necessarily tournaments Anthony Bonato 38

Survivor: Heroes vs Healers vs Hustlers Anthony Bonato 39

Key measures • Anthony Bonato 40

Near independent sets • Anthony Bonato 41

Alliances and Leaders • alliances: – groups of agents who pool capital towards mutual goals – in Survivor, alliances work together to vote off players outside the alliance and are less likely to vote for each other • leaders: – high social standing in the network – in Survivor, leaders may be the winner of a given season, but may also be non-winning players with a strong influence on the outcomes of the game Anthony Bonato 42

Dynamic Competition Hypothesis (DCH) (BEGM, 18) Dynamic competition networks have the following properties: 1. Alliances are near independent sets. 2. Strong alliances have low edge density. 3. Members of an alliance with high CON score are more likely leaders. 4. Leaders exhibit high closeness, high CON scores, and low in-degree. Anthony Bonato 43

DCH, visualized Anthony Bonato 44

Data and predictions in Survivor HHH Anthony Bonato 45

Outcomes • Sole Survivor: Ben – less likely Sole Survivor based on network science, but he played the Hidden Immunity Idol 3 x (!) • Chrissy and Ryan were runners up Anthony Bonato 46

Other Survivor seasons Survivor: Borneo Survivor: Game Changers Survivor: China • in all 3 cases, Sole Survivor and finalists have high CON scores, high closeness and low in -degree • finalists in low ED alliances 47

Statistics across seasons • 36 US seasons in Survivor and 20 for Big Brother • percentage of winners with various top scores: Anthony Bonato 48

Alliances across seasons • 60% of the Survivor seasons have an alliance with a lower ED than the ED for the total graph • 95% for BB Anthony Bonato 49

Other data: Food webs Anthony Bonato 50

Conflict graphs Anthony Bonato 51

Directions • spectral properties of ILAT, ILM – improved spectral gaps; eigenvalues • randomized ILM model • ILM and graph limits • DCH validation and analysis – geo-politics & food webs • DCH local properties (eg Epinions graph) Anthony Bonato 52

Contact • Web: http: //www. math. ryerson. ca/~abonato/ • Blog: https: //anthonybonato. com/ • @Anthony_Bonato • https: //www. facebook. com/anthony. bonato. 5