---
layout: default
lang: en
subtitle: FJMT.2024 / 23 April 2024
math: true
---
# Lancichinetti-Fortunato-Radicchi (N_mfl = 301)

{% assign sigma_array = "0.001, 0.004, 0.012" | split: ", " %}
{% assign mu_array = "0.001, 0.005, 0.01, 0.05, 0.5" | split: ", " %}
{% assign prefix = "https://gaspard.janko.fr/s/numerics/FJMT.2024/plots_23_04_2024/results/LFR_single_graph/N_micro=1000,N_mfl=301" %}

<style>
.manual_center_1500 { margin-left: -367px; margin-top: 1em; width: 1500px; height: 750px; }
.manual_center_510 { margin-left: 128px; margin-top: 1em; width: 510px; height: 510px; }
h3 { margin-top: 1em; margin-bottom: 0.5em; font-size: 110%; }
p { margin-top: 1em; }
.form { display: flex; align-items: center; margin-bottom: 0.5em; }
.form input { margin-right: 1em; margin-left: 1em; }
</style>

This page shows the differences in the dynamics between the microscopic and the kinetic (meanfield) model for Lancichinetti-Fortunato-Radicchi (LFR) type graphs,
depending on two parameters:

- $\sigma^2$, which is the variance of the $\beta$ distributions making up the initial distribution $f$,
- $\mu$, the so-called mixing parameter for the construction of the LFR graphs.

![f vs sigma](https://gaspard.janko.fr/s/numerics/FJMT.2024/plots_23_04_2024/f_init_vs_sigma.png){:width="750"}

<h3 id="LFR_graphs">Graphs</h3>

|----------|
| {% for mu in mu_array -%}
| [μ={{ mu | round: 4 }}]({{ prefix }}/reference/μ={{ mu }}/graph_LFR.png){% if forloop.last -%} | {% endif -%}
{% endfor %}

{% for sigma in sigma_array %}
<h2>$β$ distribution with $σ² = {{sigma}}$</h2>
<h3 id="LFR301_var=0.001_movies">Movies (ensemble averages)</h3>

|----------|
{% for mu in mu_array -%}
| μ={{ mu | round: 4 }}{% if forloop.last -%} | {% endif -%}
{% endfor %}
|----|
{% for mu in mu_array -%}
| [plain $w_i$]({{ prefix }}/σ²={{ sigma }}/μ={{ mu }}/movie_without_g.mp4){% if forloop.last -%} | {% endif -%}
{% endfor %}
{% for mu in mu_array -%}
| [centered $w_i$]({{ prefix }}/σ²={{ sigma }}/μ={{ mu }}/movie_without_g.mp4){% if forloop.last -%} | {% endif -%}
{% endfor %}
{:.headerline.center}

<h3>Dynamics</h3>

Convergence rates are computed over the time span marked in blue in the first plot.

**Parameters:**

- $\mu$: LFR mixing parameter
- $T^*$: time to consensus = $-1/\log(\vert\lambda_2\vert) \cdot \delta t$, where $\lambda_2$ is the second largest eigenvalue of the transition matrix for the associated time discrete model. See [here](https://doi.org/10.1016/j.ins.2019.02.028).
- [assortativity](https://en.wikipedia.org/wiki/Assortativity)
- [clustering coeff.](https://en.wikipedia.org/wiki/Clustering_coefficient)


![convergence]({{ prefix }}/σ²={{ sigma }}/comparison.svg){:.manual_center_1500}

<h3>Graph properties</h3>
![graph metrics]({{ prefix }}/σ²={{ sigma }}/graph_analysis.svg){:.manual_center_1500}

<h3>g(t=0), 1 row per run, p increasing →</h3>
<div class="form">
  <label>Brightness: </label><input id="slider_LFR301_var={{ sigma }}" type="range" min="1"
    max="10" value="1" step="1" /><span id="brightness_LFR301_var={{ sigma }}">1</span>
  <input id="check_LFR301_var={{ sigma }}" type="checkbox">
<label>Invert </label>
</div>
<img src="{{ prefix }}/σ²={{ sigma }}/g_init_montage.png" id="g_init_LFR301_var={{ sigma }}"/>{:.manual_center_510}

{% endfor %}

<script>
  const tags = [
    {% for sigma in sigma_array %}
      "LFR301_var={{ sigma }}",
    {% endfor %}
  ];
  tags.forEach((tag) => {
    const slider = document.getElementById("slider_" + tag);
    const check = document.getElementById("check_" + tag);
    const img = document.getElementById("g_init_" + tag);
    const disp = document.getElementById("brightness_" + tag);
    update = () => {
      disp.textContent = slider.value;
      const i_val = check.checked ? "100%" : "0%";
      const new_value = slider.value;
      const new_filter_value =  `brightness(${new_value}) invert(${i_val})`
      img.style.filter = new_filter_value;
    }
    slider.addEventListener("input", update);
    check.addEventListener("change", update);
  })
</script>