Evolution of social networks: the example of obesity

Abstract

The present paper deals with the effect of the social transmission of nutrition habits in a social and biological age-dependent context on obesity, and accordingly on type II diabetes and among its complications, the neurodegenerative diseases. The evolution of social networks and inside a network the healthy weight of a person are depending on the context in which this person has contacts and exchanges concerning his alimentation, physical activity and sedentary habits, inside the dominant social network in which the person lives (e.g., scholar for young, professional for adult, home or institution for elderly people). Three successive steps of evolution will be considered for social networks (like for neural one’s): initial random connectivity, destruction and consolidation of links following a new transition rule called homophilic until an asymptotic architectural organization and configuration of states. The application of such a network dynamics concerns the sequence overweight/obesity/type II diabetes and neurodegenerative diseases.

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Acknowledgments

The research was supported by the PHC Maghreb SCIM and by the French Program “Investissements d’Avenir” VHP (VisioHome Presence inter@ctive).

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Correspondence to Jacques Demongeot.

Appendix

Appendix

Description of the microscopic dynamics

Let us denote for the states x and y in the graph G at time t by L x,y (t) (resp. L x,x (t), L x (t) and L(t)) the number of heterophilic links (resp. homophilic links of type x, links coming from type x nodes and total links) and by τ a relaxation time. The mean connectivity C is equal to L/N, where N is the total number of nodes of G. The marginal connectivity C x is related to the nodes of type x. We suppose in each time lapse of duration τ, a certain proportion of nodes (agents) create (resp. cancel) directed links towards nodes being in same (resp. different) state, with a certain tolerance threshold h, supposed to be the same for each state. The simulation follows the successive steps:

  1. (1)

    At t = t 0, generate the random value τ from an exponential distribution of parameter 1/ß

  2. (2)

    At t = t 0 + τ,

    • choose a fraction ϕ of nodes in G. Let M = ϕN.

    • for each node i among these M nodes (i = 1,…, M), define its state x(i) (known initial conditions), its out-degree k i a non-negative integer (equal to the number of links exiting from i), generate its tolerance to the difference, a real 0 ≤ h i  ≤ 1, from a probability distribution g(h) and do the following operations:

      *for k i  = 0, connection from i to j:

      • choose node j by chance among N−1 others.

      • create a link from i to j with probability h d(i,j) i , where d(i,j) is direct distance between i and j, with 3 levels: 0, 1, 2 as follows:

        $$\begin{aligned}d\left( {i,j} \right) &= 0, \, {\text{if}}\,x\left( i \right) = x\left( j \right) \\ &= 1, \, {\text{if}}\,x\left( i \right) = S,x\left( j \right) = W\, {\text{and vice versa}} \\ &= 1, \, {\text{if}}\,x\left( i \right) = W,x\left( j \right) = O\, {\text{and vice versa}} \\ &= 2, \, {\text{if}}\,x\left( i \right) = S,x\left( j \right) = O\, {\text{and vice versa}} \end{aligned}$$

    *for k i  ≥ 1, connection/disconnection from i to j:

    • if V i denotes the set of neighbours of i, let choose a node j among the ∣V i ∣ neighbours of i with the probability 1/k i . We will denote by V i j the set of the neighbours of j, minus i.

    • let r(i,j) be the total similarity distance between nodes i and j. The link between i and j will be cut with the probability 1−h i r (i,j), where the total distance r is defined by:

      $$\begin{aligned}r\left( {i,j} \right) &= d\left( {i,j} \right), \, if \, c\left( {i,j} \right) = 0 \\ &= \alpha d\left( {i,j} \right) + (1 - \alpha )c\left( {i,j} \right), \, if \, c\left( {i,j} \right) \ne 0, \\ \end{aligned}$$

      where the indirect distance c is given by:

      $$\begin{aligned}c\left( {i,j} \right) &= \varSigma_{k \in V_{j}^{i} } \,{ d}\left( {i,k} \right)/\left( {k_{j} - 1} \right), if\,k_{j} > 1 \\ &= \, 0, \, if\,k_{j} = 1 \\ \end{aligned}.$$
    • if the link between i and j has been cut, we choose by chance a new node k in G \V i \ V j i and we create a link from i to k with the probability:

      $$P(i \to k) = f\left( {d\left( {i,k} \right)} \right)n_{x(k)} h_{i}^{d(i,k)} \big/\big[\varSigma_{l \in G\backslash Vi\backslash V_{j}^{i}} \,\,n_{x\left( l \right)} h_{i}^{d(i,l)} \big],$$

      where n x(k) is the number of nodes in G \V i \ V j i having the same state as k, i.e., n x(k) = n S (resp. n W and n O ) if k is susceptible (resp. overweight and obese). We will consider in the simulations 3 versions for function f:

      • Version 1: f(d(i,w)) = 1, if d(i,w) = 0; = 0 elsewhere

      • Version 2: f(d(i,w)) = 1, if d(i,w) = 0 or 1; = 0 elsewhere

      • Version 3: : f(d(i,w)) = 1, if d(i,w) = 0, 1 or 2,

      these versions being used in the individual centred network for representing three types of progressively increasing influence: exogenous heterogeneous (individual-cultural, Version 1), exogenous homogeneous (individual-social, Version 2), endogenous (individual–individual, Version 3).

  3. (3)

    Change the states x(j), for all j at the end of the links created, by increasing their obesity weight of one level (S to W, W to O, O to O).

  4. (4)

    Generate a new τ and go to 2.

  5. (5)

    Stop when the graph G is no more changing.

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Demongeot, J., Taramasco, C. Evolution of social networks: the example of obesity. Biogerontology 15, 611–626 (2014). https://doi.org/10.1007/s10522-014-9542-z

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Keywords

  • Obesity
  • Type II diabetes
  • Social network
  • Homophilic graph