{
"cells": [
{
"cell_type": "markdown",
"id": "c952a6fb",
"metadata": {},
"source": [
"# Building CTMCs\n",
"A Continuous Time Markov Chain (CTMC) is similar to a DTMC. The differences are that time is *continuous*, not discrete, and that transitions have *rates* instead of probabilities. So to summarize, a CTMC has:\n",
"* states (includig an initial state)\n",
"* transitions with rates. The rate is an indication of the speed in which a transition occurs. To be precise, the probability that state $s_i$ goes to state $s_j$ in $t$ time steps is $1- e^{R(s_i,s_j)\\cdot t}$, where $R(s_i,s_j)$ is the *rate* between $s_i$ and $s_j$.\n",
"* labels\n",
"\n",
"As an example for a CTMC, we have a model of a star. It will first fuse hydrogen into helium until there is no hydrogen left, then it will fuse helium into carbon, etc. until there is only iron left and the star goes supernova."
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "3787d8d7",
"metadata": {
"execution": {
"iopub.execute_input": "2026-03-26T10:47:15.983878Z",
"iopub.status.busy": "2026-03-26T10:47:15.983634Z",
"iopub.status.idle": "2026-03-26T10:47:16.379861Z",
"shell.execute_reply": "2026-03-26T10:47:16.379317Z"
}
},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n",
"\n",
" \n",
" Network\n",
" \n",
" \n",
" \n",
" \n",
" \n",
" \n",
" \n",
" \n",
" \n",
"\n"
],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"from stormvogel import *\n",
"\n",
"# Create a new model\n",
"ctmc = stormvogel.model.new_ctmc()\n",
"\n",
"# hydrogen fuses into helium\n",
"ctmc.get_state_by_id(0).set_choice([(3, ctmc.new_state(\"helium\"))])\n",
"# helium fuses into carbon\n",
"ctmc.get_state_by_id(1).set_choice([(2, ctmc.new_state(\"carbon\"))])\n",
"# carbon fuses into iron\n",
"ctmc.get_state_by_id(2).set_choice([(7, ctmc.new_state(\"iron\"))])\n",
"# supernova\n",
"ctmc.get_state_by_id(3).set_choice([(12, ctmc.new_state(\"Supernova\"))])\n",
"\n",
"# we add the rates which are equal to whats in the transitions since the probabilities are all 1\n",
"rates = [3, 2, 7, 12, 0]\n",
"for i in range(5):\n",
" ctmc.set_rate(ctmc.get_state_by_id(i), rates[i])\n",
"\n",
"# we add self loops to all states with no outgoing transitions\n",
"ctmc.add_self_loops()\n",
"vis = show(ctmc, layout=Layout(\"layouts/star.json\"))"
]
},
{
"cell_type": "markdown",
"id": "c85326c9",
"metadata": {},
"source": [
"Or using the bird API"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "128a5982",
"metadata": {
"execution": {
"iopub.execute_input": "2026-03-26T10:47:16.387099Z",
"iopub.status.busy": "2026-03-26T10:47:16.386780Z",
"iopub.status.idle": "2026-03-26T10:47:16.417881Z",
"shell.execute_reply": "2026-03-26T10:47:16.417309Z"
}
},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n",
"\n",
" \n",
" Network\n",
" \n",
" \n",
" \n",
" \n",
" \n",
" \n",
" \n",
" \n",
" \n",
"\n"
],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"from stormvogel import *\n",
"\n",
"init = bird.State(x=\"\")\n",
"\n",
"\n",
"def delta(s: bird.State):\n",
" if s == init:\n",
" return [(3, bird.State(x=[\"helium\"]))]\n",
" elif \"helium\" in s.x:\n",
" return [(2, bird.State(x=[\"carbon\"]))]\n",
" elif \"carbon\" in s.x:\n",
" return [(7, bird.State(x=[\"iron\"]))]\n",
" elif \"iron\" in s.x:\n",
" return [(12, bird.State(x=[\"Supernova\"]))]\n",
" else:\n",
" return [(0, s)]\n",
"\n",
"\n",
"labels = lambda s: s.x\n",
"\n",
"bird_star = bird.build_bird(\n",
" delta=delta, init=init, labels=labels, modeltype=ModelType.CTMC\n",
")\n",
"\n",
"vis2 = show(bird_star, layout=Layout(\"layouts/star.json\"))"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3 (ipykernel)",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.12.13"
},
"widgets": {
"application/vnd.jupyter.widget-state+json": {
"state": {
"15e7e72dddfe45eb801918f85f048df3": {
"model_module": "@jupyter-widgets/output",
"model_module_version": "1.0.0",
"model_name": "OutputModel",
"state": {
"_dom_classes": [],
"_model_module": "@jupyter-widgets/output",
"_model_module_version": "1.0.0",
"_model_name": "OutputModel",
"_view_count": null,
"_view_module": "@jupyter-widgets/output",
"_view_module_version": "1.0.0",
"_view_name": "OutputView",
"layout": "IPY_MODEL_561eaaf760c143748bc3cbf623f09ee4",
"msg_id": "",
"outputs": [],
"tabbable": null,
"tooltip": null
}
},
"1ccd5063077e46aca600cde8c2a9f4ce": {
"model_module": "@jupyter-widgets/output",
"model_module_version": "1.0.0",
"model_name": "OutputModel",
"state": {
"_dom_classes": [],
"_model_module": "@jupyter-widgets/output",
"_model_module_version": "1.0.0",
"_model_name": "OutputModel",
"_view_count": null,
"_view_module": "@jupyter-widgets/output",
"_view_module_version": "1.0.0",
"_view_name": "OutputView",
"layout": "IPY_MODEL_74129b69248544f18672dddb876af111",
"msg_id": "",
"outputs": [],
"tabbable": null,
"tooltip": null
}
},
"561eaaf760c143748bc3cbf623f09ee4": {
"model_module": "@jupyter-widgets/base",
"model_module_version": "2.0.0",
"model_name": "LayoutModel",
"state": {
"_model_module": "@jupyter-widgets/base",
"_model_module_version": "2.0.0",
"_model_name": "LayoutModel",
"_view_count": null,
"_view_module": "@jupyter-widgets/base",
"_view_module_version": "2.0.0",
"_view_name": "LayoutView",
"align_content": null,
"align_items": null,
"align_self": null,
"border_bottom": null,
"border_left": null,
"border_right": null,
"border_top": null,
"bottom": null,
"display": null,
"flex": null,
"flex_flow": null,
"grid_area": null,
"grid_auto_columns": null,
"grid_auto_flow": null,
"grid_auto_rows": null,
"grid_column": null,
"grid_gap": null,
"grid_row": null,
"grid_template_areas": null,
"grid_template_columns": null,
"grid_template_rows": null,
"height": null,
"justify_content": null,
"justify_items": null,
"left": null,
"margin": null,
"max_height": null,
"max_width": null,
"min_height": null,
"min_width": null,
"object_fit": null,
"object_position": null,
"order": null,
"overflow": null,
"padding": null,
"right": null,
"top": null,
"visibility": null,
"width": null
}
},
"6fc6cfff8f9146019319b68e4ba04ec5": {
"model_module": "@jupyter-widgets/base",
"model_module_version": "2.0.0",
"model_name": "LayoutModel",
"state": {
"_model_module": "@jupyter-widgets/base",
"_model_module_version": "2.0.0",
"_model_name": "LayoutModel",
"_view_count": null,
"_view_module": "@jupyter-widgets/base",
"_view_module_version": "2.0.0",
"_view_name": "LayoutView",
"align_content": null,
"align_items": null,
"align_self": null,
"border_bottom": null,
"border_left": null,
"border_right": null,
"border_top": null,
"bottom": null,
"display": null,
"flex": null,
"flex_flow": null,
"grid_area": null,
"grid_auto_columns": null,
"grid_auto_flow": null,
"grid_auto_rows": null,
"grid_column": null,
"grid_gap": null,
"grid_row": null,
"grid_template_areas": null,
"grid_template_columns": null,
"grid_template_rows": null,
"height": null,
"justify_content": null,
"justify_items": null,
"left": null,
"margin": null,
"max_height": null,
"max_width": null,
"min_height": null,
"min_width": null,
"object_fit": null,
"object_position": null,
"order": null,
"overflow": null,
"padding": null,
"right": null,
"top": null,
"visibility": null,
"width": null
}
},
"74129b69248544f18672dddb876af111": {
"model_module": "@jupyter-widgets/base",
"model_module_version": "2.0.0",
"model_name": "LayoutModel",
"state": {
"_model_module": "@jupyter-widgets/base",
"_model_module_version": "2.0.0",
"_model_name": "LayoutModel",
"_view_count": null,
"_view_module": "@jupyter-widgets/base",
"_view_module_version": "2.0.0",
"_view_name": "LayoutView",
"align_content": null,
"align_items": null,
"align_self": null,
"border_bottom": null,
"border_left": null,
"border_right": null,
"border_top": null,
"bottom": null,
"display": null,
"flex": null,
"flex_flow": null,
"grid_area": null,
"grid_auto_columns": null,
"grid_auto_flow": null,
"grid_auto_rows": null,
"grid_column": null,
"grid_gap": null,
"grid_row": null,
"grid_template_areas": null,
"grid_template_columns": null,
"grid_template_rows": null,
"height": null,
"justify_content": null,
"justify_items": null,
"left": null,
"margin": null,
"max_height": null,
"max_width": null,
"min_height": null,
"min_width": null,
"object_fit": null,
"object_position": null,
"order": null,
"overflow": null,
"padding": null,
"right": null,
"top": null,
"visibility": null,
"width": null
}
},
"b86a014c7b834d6786ad49122062069e": {
"model_module": "@jupyter-widgets/base",
"model_module_version": "2.0.0",
"model_name": "LayoutModel",
"state": {
"_model_module": "@jupyter-widgets/base",
"_model_module_version": "2.0.0",
"_model_name": "LayoutModel",
"_view_count": null,
"_view_module": "@jupyter-widgets/base",
"_view_module_version": "2.0.0",
"_view_name": "LayoutView",
"align_content": null,
"align_items": null,
"align_self": null,
"border_bottom": null,
"border_left": null,
"border_right": null,
"border_top": null,
"bottom": null,
"display": null,
"flex": null,
"flex_flow": null,
"grid_area": null,
"grid_auto_columns": null,
"grid_auto_flow": null,
"grid_auto_rows": null,
"grid_column": null,
"grid_gap": null,
"grid_row": null,
"grid_template_areas": null,
"grid_template_columns": null,
"grid_template_rows": null,
"height": null,
"justify_content": null,
"justify_items": null,
"left": null,
"margin": null,
"max_height": null,
"max_width": null,
"min_height": null,
"min_width": null,
"object_fit": null,
"object_position": null,
"order": null,
"overflow": null,
"padding": null,
"right": null,
"top": null,
"visibility": null,
"width": null
}
},
"bbae1b711b7343a5ab0af022c6a2e8a7": {
"model_module": "@jupyter-widgets/output",
"model_module_version": "1.0.0",
"model_name": "OutputModel",
"state": {
"_dom_classes": [],
"_model_module": "@jupyter-widgets/output",
"_model_module_version": "1.0.0",
"_model_name": "OutputModel",
"_view_count": null,
"_view_module": "@jupyter-widgets/output",
"_view_module_version": "1.0.0",
"_view_name": "OutputView",
"layout": "IPY_MODEL_b86a014c7b834d6786ad49122062069e",
"msg_id": "",
"outputs": [],
"tabbable": null,
"tooltip": null
}
},
"f76bb223f3f1459fa6b906476932972c": {
"model_module": "@jupyter-widgets/output",
"model_module_version": "1.0.0",
"model_name": "OutputModel",
"state": {
"_dom_classes": [],
"_model_module": "@jupyter-widgets/output",
"_model_module_version": "1.0.0",
"_model_name": "OutputModel",
"_view_count": null,
"_view_module": "@jupyter-widgets/output",
"_view_module_version": "1.0.0",
"_view_name": "OutputView",
"layout": "IPY_MODEL_6fc6cfff8f9146019319b68e4ba04ec5",
"msg_id": "",
"outputs": [],
"tabbable": null,
"tooltip": null
}
}
},
"version_major": 2,
"version_minor": 0
}
}
},
"nbformat": 4,
"nbformat_minor": 5
}