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Interacting simple contagions are complex contagions (and why that matters) Laurent H´ ebert-Dufresne [email protected] :: @LHDnets Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne

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Simple contagions An edge is enough information. • All edges between susceptible and infectious nodes are identical. • Known rate of transmission βdt. • Includes more complicated models, such as non-Markovian dynamics. Example: Most models of disease transmission. Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne

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Interacting simple contagions An edge is enough information. • All edges between susceptible and infectious nodes are not all identical. • The rate of transmission βdt depends on the state of another process. • The interaction alone depends on a lot of parameters. Example: Most real diseases interact, like influenza and pneumonia. Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne

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Complex contagions An edge is not enough information. • All edges between susceptible and infectious nodes are not identical. • The rate of transmission βdt depends on the neighbourhood of the susceptible node. • The transmission rate per edge is now a function of mesoscopic information. Example: Social contagions with social reinforcement. Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne

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Important properties of contagion dynamics Let’s focus on ease of spread in clustered networks. Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne

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LHD et al., Phys. Rev. E (2010) Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne

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Important properties of contagion dynamics Let’s focus on ease of spread in clustered networks. 1 Simple contagions emerge continuously. 2 Clustering slows down simple contagions. Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne

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Dodds & Watts, PRL (2004) Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne

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Centola, Science (2010) Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne

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O’Sullivan et al., Frontiers in Physics (2015) Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne

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Important properties of contagion dynamics Let’s focus on ease of spread in clustered networks. 1 Simple contagions emerge continuously. 2 Clustering slows down simple contagions. 3 Complex contagions can emerge discontinuously. 4 Clustering can speed up complex contagions. Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne

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LHD & Althouse, Proc. Natl. Acad. Sci. U.S.A. (2015) Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne

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Important properties of contagion dynamics Let’s focus on ease of spread in clustered networks. 1 Simple contagions emerge continuously. 2 Clustering slows down simple contagions. 3 Complex contagions can emerge discontinuously. 4 Clustering can speed up complex contagions. 5 Interacting simple contagions can emerge discontinuously. 6 Clustering can speed up interacting simple contagions. Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne

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Important properties of contagion dynamics Let’s focus on ease of spread in clustered networks. 1 Simple contagions emerge continuously. 2 Clustering slows down simple contagions. 3 Complex contagions can emerge discontinuously. 4 Clustering can speed up complex contagions. 5 Interacting simple contagions can emerge discontinuously. 6 Clustering can speed up interacting simple contagions. Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne

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Can we tell complex contagions apart from interacting simple contagions? Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne

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Two types of approaches in the literature 1 Phenomenological: Based on structural properties 2 Statistical: Based on model selection Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne

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Phenomenological and statistical experiments Study properties of all SIS contagions on a cliquish graph Fit complex/simple contagions to interacting SI contagions on a clique graph Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne

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Results 1: Phenomenology of SIS contagions Study properties of all SIS contagions on a cliquish graph Simple SIS: transmission rate β and recovery rate α. Complex SIS: transmission rate β(k) = βmin + βmax − βmin 1 + exp (−w(k − k0)) and recovery rate α. Interacting SIS: transmission rates β1 and β2, recovery rates α1 and α2, and 10 interaction parameters! Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne

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Results 1: Phenomenology of SIS contagions Study properties of all SIS contagions on a cliquish graph 10−3 10−2 10−1 Fraction I of infected 0 2 4 6 8 10 I relative to rand. Sim. Com. Inter. 1.0 1.5 2.0 2.5 # inf. neigh. on inf. Sim. Com. Inter. 1.0 1.5 2.0 2.5 # inf. neigh. on rec. Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne

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Results 2: Fitting to interacting simple contagions Fit complex/simple dynamics to interacting SI contagions on a clique graph d dt Si,j = − β(i + j)Si,j − (n − 1 − i)β(i + Θi)Si,j − (n − 1 − j)β(j + Θj)Si,j + (n − i)β(i − 1 + Θi−1)Si−1,j + (n − j)β(j − 1 + Θj−1)Si,j−1 Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne

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Results 2: Fitting to interacting simple contagions Fit complex/simple dynamics to interacting SI contagions on a clique graph 0 5 10 15 20 25 30 Interaction factor −0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 0.20 normalized ∆ BIC Simple contagion Complex contagion Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne

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Results 2: Fitting to interacting simple contagions Fit complex/simple dynamics to interacting SI contagions on a clique graph 0 5 10 15 20 25 30 Interaction factor 0.00 0.05 0.10 0.15 0.20 0.25 0.30 βmax Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne

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Results 2: Fitting to interacting simple contagions Fit complex/simple dynamics to interacting SI contagions on a clique graph 0 5 10 15 20 25 30 Interaction factor 0 2 4 6 8 10 k0 Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne

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Interacting contagions are complex contagions An edge is enough information. • All edges between susceptible and infectious nodes are not all identical. • The rate of transmission βdt depends on the state of another process. • The interaction alone depends on a lot of parameters. Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne

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Interacting contagions are complex contagions An edge is enough information, if you are aware of the interaction. • Edges that look identical are not truly identical. • The rate of transmission βdt depends on some hidden process. • The interactions are potentially impossible to parametrize. Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne

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Interacting contagions are complex contagions An edge is enough information, if you are aware of the interaction. • Edges that look identical are not truly identical. • The rate of transmission βdt depends on some hidden process. • The interactions are potentially impossible to parametrize. Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne

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Interacting contagions are complex contagions An edge is enough information, if you are aware of the interaction. • Edges that look identical are not truly identical. • The rate of transmission βdt depends on some hidden process. • The interactions are potentially impossible to parametrize. Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne

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Conclusions (“and why that matters”) 1 Interacting contagions look like complex contagions, and they are complex contagions when using hidden Markov models. → Similar macro properties: discontinuous emergence, speed up with clustering. → Similar micro properties: heterogeneous spreading rates, more clustered cases. Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne

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Conclusions (“and why that matters”) 2 Bad news: Phenomenology and model fitting do not distinguish spreading mechanisms. → Measurements of social reinforcement “in the wild” are confounded by interactions with billions of spreading memes. → Very controlled experiments (` a la Centola) can help, but results are hard to generalize. Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne

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Conclusions (“and why that matters”) 3 Good news: Complex contagion models can detect novel pathogen interactions. → Many diseases interact significantly, and model complexity scales superlinearly with the number of pathogens. → Complex contagion offer an effective model where “social reinforcement” correlates with interaction strength. Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne

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If you have any questions Preprint will be available eventually. Until then: [email protected] or @LHDnets Interacting simple contagions are complex contagions Laurent H´ ebert-Dufresne