added hierarchical design approach in specs

Author: debsankha manik

spec/hierarchical.md

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+# Hierarchichal Design Overview
+
+# User Story 1: Scaling of Basin of attraction volumes for Kuramoto Networks
+## Setting: 
+A Kuramoto network $$F$$ is completely specified by it's graph topology $$G(V, E)$$ and its set of natural frequencies $$\vec{\omega})$$ for all the nodes. Here we are only intersted in ring topology. All fixed points of a Kuramoto ring are uniquely identified by an integer called the **winding number** $$k$$. 
+
+## Study:
+We are interested in the probability that a random initial condition evolves to the steady state with $$k = k'$$, i.e. $$V_{k, N} = P(k | len(V) = N)$$. And we want this over all possible distributions of natural frequencies $\vec{\omega}$ (subject to some constraints, that make it a finite space). 
+
+## Sequential pseudocode would be:
+```python
+N_range = np.arange(10,100)
+V = []
+
+# Outermost loop: obver many ring sizes
+for N in N_range:
+    V_N = 0
+    # loop over all omega combinations
+    V_array = []
+    for idx, omega in enumerate(all_omegas(N)):
+        # loop over many initial conditions
+        k_list = []
+        for initcond in get_initconds(num_initconds):
+            k = find_k(N, omega, initcond)
+            k_list.append(k) 
+        # Now bin the data to find the frequencies
+        V_omega = scipy.stats.itemfreq(k_list)
+        V_array.append(V_omega)
+    # Do the averaging over all omegas
+    V_N = np.average(V_array, axis = 2)
+    V.append(V_N)
+```
+
+## Can we transform this to this?
+```python
+basin_singlering = Experiment(find_k, initcond_range = initcond_iter(...), aggregator =\
+                                <#something that counts final states and computes probabilities>)
+# outout is of the form: {0: 0.78, 1: 0.11, -1: 0.14}
+basin_all_omegas_for_single_size = Experiment(basin_singlering, omega_range = omega_iter(100),\
+                                             aggregator = <#averages over all omega distributions,\
+                                             computes some sort of central tendency>)
+# outout is of the form: {'averages': {0: 0.78, 1: 0.11, -1: 0.14}, 'variances': {0: 0.1, 1: 0.2, -1:0.3}
+basin_scaling_with_size = Experiment(basin_all_omegas_for_single_size, n_range \=
+                                     np.arange(1, 100, 10), aggregator = <# concatenates results for all n>)
+# outout is of the form: {n0: {'averages': {}, 'variances': {}}, n1: {'averages': {}, 'variances': {}}, ....}
+```
+
+### Advantages:
+-  Completely Hierarchichal
+### Disadvantages:
+-  Fill it in!
+
+## Design idea:
+Here I outline how to acheive the user interface that I outlined in the last subsection. 
+So here I present a code snippet.
+
+### Disclaimer: 
+#### What these code snippets are:
+1.  A *minimum working example (MWA)* that is capable of supporting a hierarchichal user interface. 
+2.  I do this by writing some classes and their attributes and methods. 
+3.  Everything apart from the object structures (classes) and call signature (functions) are demonstrative only.  
+
+#### What they are NOT:
+1. Taking into account the whole problem of scheduling. 
+2. The aspect of chunking and parallelizing. 
+
+```python
+class Experiment(object):
+"""
+Experiment
+=========
+An object that specifies how to compute a function *func* on a range of *arguments* and process
+all the outputs  from single runs to compute an *output*. 
+
+Parameters
+----------
+func :  a function/callable
+        func must take a single argument that we call *input*
+param : a parameter that, along with the function, completely specifies what the *output*
+            should be.
+input_generator :
+        a callable taking a mandatoryargument, *parameter*, that generates the range of *inputs*
+        to *func*. it can also take optional parameters. See draft implementation of *run* below. 
+"""
+    def __init__(self, func, input_generator, input_generator_args = None):
+        """
+        """
+        pass
+    def run(self, param):
+        """
+        """
+        return [func(input) for input in input_generator(param, *input_generator_args)]
+```
+
+## Design used in test case: basin volume of Kuramoto rings
+Let me try to make it clearer by giving an example use case, our very own problem of Kuramoto networks. 
+```python
+def compute_k((omega, initcond)):
+    """Evolves a ring network with frequencies *omega* from initial condition
+    *initcond*
+    Returns winding number of final state k
+    """
+    pass
+
+def generate_initconds(omega, nrepeat):
+    """generates nrepeat vectors of random phase angles, each same size as omega"""
+    for initcond in np.random.uniform(0, 2*np.pi, size = (nrepeat, len(omega))):
+        yield omega, initcond
+
+def gen_all_vectors_with_len(len):
+    """
+    Generates all vectors with elements ±1 and length len
+    """
+    pass
+    
+# Experiment that computes the basin volumes of a single ring. Note how teh number of initial conditions is specified as input_generator_args. 
+E_onering = Experiment(compute_k, input_generator=\
+                        generate_initconds, input_generator_args = (1000,)) 
+
+# Experiment that computes the basin volumes of all rings of a given size
+E_onesize = Experiment(E_onering.run, input_generator=\
+                        gen_allvectors_with_len, input_generator_args = None)
+
+# Experiment that computes the basin volumes of rings of size in certain range
+E_basin_scaling = Experiment(E_onesize.run, input_generator = lambda x:x)
+# Finally, call the run function of E_basin_scaling causes all child experiments to run recursively. 
+E_basin_scaling.run(param = np.arange(min_size, max_size, 10))
+```
+OK. So this design can do hierarchical experiments. But let's see how it works in some other scenario, for sake of being thorough:
+## Design used in another test case: find out the critical coupling of a kuramoto network
+```python
+def order_param(Network, initcond):
+    """Computes the order parameter by simulating the network from given initial condition"""
+    pass
+def generate_initconds(nrepeat):
+    """
+    generates args for func:`order_param`
+    """
+    <a name="ugly"></a>
+    G = nx.Graph()
+    # code to set up your graph here
+    for thetas in np.random.uniform(0, 2*pi, size = nrepeat):
+        yield G, thetas
+E_one_k = Experiment(order_param, input_generator = generate_initconds, input_generator_args = (1000,))
+E_orderparam_scaling = Experiment(E_one_k.run, input_generator =\
+                                lambda mink, maxk: np.arange(mink, maxk, (maxk - mink)/100))
+E_orderparam_scaling.run((0, 1))
+```
+
+## Problems:
+1. It is plain ugly how we [link text](#ugly). 
+
+