N : P J.-G. Young1 L. Hébert-Dufresne1,2, E. Laurence1, C. Murphy1 G. St-Onge1 and P. Desrosiers1,3 June th — NetSci — Theory I . Département de physique, Université Laval, Québec, QC, Canada . Vermont Complex Systems Center, University of Vermont, Burlington, VT, USA . Centre de recherche de CERVO, QC, Québec, Canada

/ Erdős-Rényi Complex networks Grid

/ Growth models One of the great tools of Network Science

/ Great fit of macro quantities via A.L. Barabási & R. Albert, Science, L. Hébert-Dufresne et al., Phys. Rev. E, JGY et al., Phys. Rev. E, L. Hébert-Dufresne et al., Phys. Rev. E,

/ Natural question : What else can we learn from growth models?

/ Natural question : What else can we learn from growth models? Some answers : Micro-evolution [J. Leskovec et al., ACM SIGKDD ( ) Prominence of growth rules [T. Pham et al., PLoS ONE ( )]

/ Natural question : What else can we learn from growth models? Some answers : Micro-evolution [J. Leskovec et al., ACM SIGKDD ( ) Prominence of growth rules [T. Pham et al., PLoS ONE ( )] First node(s) [S. Bubeck et al., Random Struct. Algor., ( )] History [J. Pinney et al., PNAS ( ); A. Magner et al. WWW ( )]

T /

/ The network archaeology problem : concept W ? Older edges : thick, dark strokes

/ The network archaeology problem : definitions G : Unannotated network X : Modifications to G, ordered in time E Possible history : X (e 1 , e 2 , e 3 , e 4 , e 5 ) in T 5 steps.

/ The network archaeology problem : Bayesian formulation S I We assume that the parameters θ are known, such that P(X|G, θ) P(G|X, θ)P(X|θ) P(G|θ) ∝ P(G|X, θ)P(X|θ) . Probabilities defined by a model : Likelihood P(G|X, θ) : Prob. of G given history X (logical) Prior P(X|θ) : Prob. of producing X Evidence P(G|θ) X P(G|X, θ)P(X|θ)

A /

/ Parametrized random attachment model : concept Preferential attachment with general attachment kernel g(k) kγ (γ ∈ R); events between existing nodes (prob. 1 − b). Each discrete time t : new edge, choose site with prob. ∝ g(k)

/ Parametrized random attachment model : network zoo γ 0 γ 0 γ 0

A . /

/ Algorithms for network archaeology Our goal : Order the edges of G, assuming the G generated by PA We compare three methods : . Degree ordering. Higher degree = older. . Onion decomposition (generalizes k-core). Central = older. . Principled inference by sampling. Evaluate expected arrival time of each edge according to P(X|G, θ). Onion decomposition : [L. Hébert-Dufresne, J. Grochow, and A. Allard, Sci Rep., ( )]

E /

/ Experiments and results : real system Social network built with emails ( day) Nodes ( ) : Researchers Edges ( ) : Reciprocated emails (40+) [Paranjape et al., ACM Web Search and Data Mining ( )]

/ Experiments and results : real system 0 200 400 True arrival time X (e) 0 200 400 Estimated arrival time (e) (a) 0 200 400 True arrival time X (e) 0 200 400 (b) 0 200 400 True arrival time X (e) 0 200 400 (c) Degree (ρ 0.39) Onion (ρ 0.41) Sampled (ρ 0.62)

/ But what are the limits of inference?

/ Experiments and results : artificial networks ( of ) E Generate artificial networks with fixed loopiness b and vary the strength of the rich-get-richer mechanism via γ.

/ Experiments and results : artificial networks ( of ) Tree networks (b 1) Loopy networks (b < 1) 10 5 0 5 10 0.00 0.25 0.50 0.75 1.00 Correlation (a) Bayesian Degree Onion 10 5 0 5 10 0.00 0.25 0.50 0.75 1.00 Correlation (b) Bayesian Degree Onion

/ Experiments and results : artificial networks ( of ) 10 5 0 5 10 0.00 0.25 0.50 0.75 1.00 Correlation Bayesian Degree Onion Condensation begins: Possible but imperfect Chains: Easy Nearly star-graphs: Impossible Phenomenology of the model [Krapivsky et al., Phys. Rev. Lett., ]

C /

/ Take-home message

/ Take-home message Network archaeology : Recover history encoded in structure. Reference : arxiv.org/ . Software : github.com/jg-you/network-archaeo ogy

/ Take-home message Network archaeology : Recover history encoded in structure. Best inference results rely on a full knowledge of the model and a Bayesian formulation, but ∃ efficient approximation. Reference : arxiv.org/ . Software : github.com/jg-you/network-archaeo ogy

/ Take-home message Network archaeology : Recover history encoded in structure. Best inference results rely on a full knowledge of the model and a Bayesian formulation, but ∃ efficient approximation. There are fundamental limits to inference. Reference : arxiv.org/ . Software : github.com/jg-you/network-archaeo ogy

/ Take-home message Network archaeology : Recover history encoded in structure. Best inference results rely on a full knowledge of the model and a Bayesian formulation, but ∃ efficient approximation. There are fundamental limits to inference. Imperfect but non-trivial inference on real systems. Reference : arxiv.org/ . Software : github.com/jg-you/network-archaeo ogy

/ Reference : arxiv.org/1803.09191 Software : github.com/jg-you/network-archaeo ogy info@jgyoung.ca jgyoung.ca @_jgyou

/ Selected references O ( ) J.-G. Young, L. Hébert-Dufresne, E. Laurence, C. Murphy, G. St-Onge and P. Desrosiers arxiv : . Archaeology in PPI networks ( ) J. W. Pinney et al., PNAS , ( ) ( ) S. Navlakha and C. Kingsford, PLoS Comput. Biol. , ( ) Archaeology in SF networks ( ) S. Bubeck et al., Random Struct. Algor., , ( ) ( ) A. Magner et al., WWW ( )