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Organization and evolution of synthetic idiotypic networks

Elena Agliari, Lorenzo Asti, Adriano Barra, and Luca Ferrucci
Phys. Rev. E 85, 051909 – Published 16 May 2012
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  • INTRODUCTION
  • A GLANCE AT ADAPTIVE IMMUNITY
  • THE MODEL
  • GLOBAL TOPOLOGY
  • COUPLING AND WEIGHTED DEGREE…
  • BIAS AND SPECIFICITY
  • CONCLUSIONS
    • References

    Abstract

    We introduce a class of weighted graphs whose properties are meant to mimic the topological features of idiotypic networks, namely, the interaction networks involving the B core of the immune system. Each node is endowed with a bit string representing the idiotypic specificity of the corresponding B cell, and the proper distance between any couple of bit strings provides the coupling strength between the two nodes. We show that a biased distribution of the entries in bit strings can yield fringes in the (weighted) degree distribution, small-world features, and scaling laws, in agreement with experimental findings. We also investigate the role of aging, thought of as a progressive increase in the degree of bias in bit strings, and we show that it can possibly induce mild percolation phenomena, which are investigated too.

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    • Received 12 December 2011

    DOI:https://doi.org/10.1103/PhysRevE.85.051909

    ©2012 American Physical Society

    Authors & Affiliations

    Elena Agliari1,2, Lorenzo Asti3,4,*, Adriano Barra4, and Luca Ferrucci4

    • 1Dipartimento di Fisica, Università degli Studi di Parma, Parma, Italia
    • 2INFN, Gruppo Collegato di Parma, Parma, Italia
    • 3Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sezione di Matematica, Sapienza Università di Roma, Via Antonio Scarpa 16, 00161 Roma, Italia
    • 4Dipartimento di Fisica, Sapienza Università di Roma, Roma, Italia

    • *lorenzo.asti@sbai.uniroma1.it

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    Issue

    Vol. 85, Iss. 5 — May 2012

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    Images

    • Figure 1
      Figure 1
      Link probability Plink(ρi,ρj;α,L) as a function of ρi and ρj with L=10 and α=0.5 (left) or α=1.1 (right).Reuse & Permissions
    • Figure 2
      Figure 2
      Link probability Plink(ρi;a,α,L) as a function of ρi and a, while α and L are fixed. In the left panel α=0.5, while in the right panel α=1.1; in both cases L=10. It is straightforward to see that on the line a=0 the probability link is independent of ρi. When a≠0, the link probability changes with varying ρi. Due to symmetry, only the range a∈[0,1] is shown.Reuse & Permissions
    • Figure 3
      Figure 3
      Degree distribution for different values of a and fixed α=1.2, N=10000, γ=10. From left to right: a=0.1 (unimodal behavior; the mean degree is about 0.8N), a=0.3 (multimodal behavior; the mean degree is 0.5N), and a=0.6 (there is no extensive network; the average degree is 0.03N).Reuse & Permissions
    • Figure 4
      Figure 4
      Number of triangles and of squares averaged over 100 realizations in our idiotypic network as a function of a. Parameters characterizing the network are N=1000, γ=7, and α=0.7. Curves represent the number of triangles (dashed line) and of squares (solid line) expected for an analogous ER graph. Inset: Number of isolated nodes present in the system; again, the idiotypic network (circles) and ER graph (line) are compared. The latter displays a larger number of triangles as long as a giant component can be detected.Reuse & Permissions
    • Figure 5
      Figure 5
      Average value of the coupling strength E[J], and of its variance Var[J], versus L for several choices of parameters, depicted in different colors. We considered α=0.7 (circles), α=1.0 (squares), and α=1.3 (triangles) for different values of a pertaining to connected (upper panels) and disconnected (lower panels) regimes (see the legend). Symbols refer to data obtained from exact numerical calculations, while curves are drawn according to the approximation, Eq. (27).Reuse & Permissions
    • Figure 6
      Figure 6
      Average value of the coupling strength E[J], and of its variance Var[J], versus a for several choices of parameters, depicted in different colors. We considered α=0.7 (circles), α=1.0 (squares), and α=1.3 (triangles) and L=120 (see the legend). Symbols refer to data obtained from exact numerical calculations, while curves are drawn according to the approximation, Eq. (27).Reuse & Permissions
    • Figure 7
      Figure 7
      Semilogarithmic plot for the distribution P(w;a,α,L) obtained by averaging a 100 system made of N=2×105 nodes with γ=7; results shown were averaged over 103 different realizations. We considered several values of a and α, corresponding to either over- or underpercolated regimes.Reuse & Permissions
    • Figure 8
      Figure 8
      Each point shown corresponds to a node of the idiotypic network, and its coordinates correspond to its weighted degree w and its bare degree z, respectively. The 100 system simulated is made up of N=2×105 nodes and we fixed γ=7. We considered the same values of a and α used in Fig. 7. Note that nodes displaying a different degree (and therefore a different ρ) may display the same weighted degree. The reshuffling among weighted and nonweighted peaks is evident.Reuse & Permissions
    • Figure 9
      Figure 9
      Average weighted degree E[w] and its variance Var[w] versus a for a system of N=104 sites, γ=7, and α=0.7 (circles), α=1.0 (squares), and α=1.3 (triangles). Symbols represent data from numerical simulations, while curves are the best fit obtained from Eq. (32).Reuse & Permissions
    • Figure 10
      Figure 10
      Realization of idiotypic networks made up of N=10000 nodes with γ=3 and α=0.7. Different values of a have been considered: from left to right a=0.50, 0.55, 0.59, and 0.62. Although these plots refer to one single realization, we have checked that the system displays robustness in this sense.Reuse & Permissions
    • Figure 11
      Figure 11
      These plots describe the evolution of the network as a is tuned while we fix the system size N=10000, γ=7, and α=0.7 (solid line) and α=1 (dashed line). Left: Average size of the giant component S/N. Right: Relative number of isolated nodes N(1,a)/N (upper panel) and average size of the nongiant components excluding isolated nodes (lower panel).Reuse & Permissions
    • Figure 12
      Figure 12
      Left: Number of k stars (a dimer is a 1-star and a trimer is a 2-star) averaged over 100 numerical realizations of N=10000 and α=0.7 idiotypic networks, while a (circles) is varied. Solid lines represent the ER prediction for the same quantity at the relative link probability. Right: Mean coupling value within a star.Reuse & Permissions
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