import networkx
import condynet.graph
import condynet.plot
import condynet.drawing
import h5obj
import glob
import bundle

I was asked to give a tutorial that may serve also as an introduction to students new to this field of research.

As I am pretty new to the topic myself, I decided to focus on an aspect I started with myself:

The determination of resonance frequencies of LC networks with binary link disorder.

In the case of a binary distribution of edge impedances, a simple scheme can be used that is due to Fyodorov.

Collective Nonlinear Dynamics of
Electricity Networks (CoNDyNet)

---

Determination of resonance frequencies of LC networks with binary link disorder

---

Daniel Jung

Jacobs University Bremen
School of Engineering and Science
Solid State Physics Group

December 10, 2014

This is the outline of this talk.

To give an introduction to interested students new to this topic, I start with the fundamental flow equations.

I continue with some basic knowledge about graph theory that is needed for our analysis.

I then introduce the Small World Model, for which we confirm some results by Huang and coworkers.

We then add impedances to the edges of the graph to complete our network model.

I will proceed with the numerical method to determine the resonance frequencies and the density of resonances which is largely due to Huang et al. and Fyodorov.

Finally we have a look at some results for the 2D grid model and the Small World Model.

Table of contents

1. Motivation
   • LC circuit
   • Flow equations
2. Model
   • Basic Graph Theory
   • The 2D grid model
   • The Small World Model
   • Adding Edge Attributes
3. Numerical method
   • Resonance frequencies
4. Resonance spectra
   • The 2D grid model
   • The Small World Model
5. Summary

As most of you will know, an electric circuit consisting of a capacitor \(C\) and an inductor \(L\) forms an oscillating circuit

(where the energy is periodically stored in the electric field of the capacitor and the magnetic field of the inductor).

The circuit has the well known resonance frequency \(1/\sqrt{LC}\). For an AC current with this frequency, the impedance \(Z\) of the circuit becomes infinite.

The relevance to AC power transmission systems is clear: Operating the system at a resonance frequency prevents power transmission.

maybe motivate by resonance frequencies of a single line,
and how engineers try to stay way below the first
resonance frequency in normal operation [Heuck]

For binary link distribution, there exists a simple scheme to calculate the *resonace frequencies*
by solving a *regular eigenvalue problem*.

Motivation

\[Z = R + i X\]
\[\frac{1}{X} = \frac{1}{i\omega L} + i\omega C\]

A LC circuit has a resonance frequency of \(\omega_0 = \frac{1}{\sqrt{LC}}\)

Frequency \(\omega\) of AC current approaching the resonance frequency: \[\lim\limits_{\omega \rightarrow \omega_0} Z(\omega) = \infty\]

\(\Rightarrow\) No power transmission possible!

Normal operation: \(\omega\) should stay far below the smallest resonance.

Arbitrary LC network

• Coupled LC oscillators
• Set of resonance frequencies \(\omega_n\)

One more slide, reminder of complex conductance and complex impedance? Also polar description (phase angle) etc.?

Flow equations I

An arbitrary one-phasic AC grid
with line impedances \(Z_{ij}\).

Generator: \(I_i > 0\)
Consumer: \(I_i < 0\)

\[\mathbf{Y} = \mathbf{Z}^{-1}\]

Combine Ohm's law

\[V_{ij} = Z_{ij} I_{ij} \quad\text{or}\quad I_{ij} = Y_{ij} V_{ij} \qquad\qquad\]

with Kirchhoff's laws for each node and mesh

\[I_i = \sum_j I_{ij} \qquad V_{ij} = V_i - V_j\]

to derive the current flow equations

\[I_i = \sum_j Y_{ij} (V_i - V_j) \quad\rm.\]

Name	Symbol
Impedance matrix	\(\mathbf{Z}\)
Admittance matrix	\(\mathbf{Y}\)

---

R. Huang, G. Korniss, S. Nayak, 2009

Flow Equations II

An arbitrary one-phasic AC grid.

Generator: \(I_i > 0\)
Consumer: \(I_i < 0\)

\[I_i = \sum_j Y_{ij} (V_i - V_j)\]

To get a standard matrix-vector multiplication, reformulate to

\[I_i = \sum_j L_{ij} V_j \quad\mathrm{or}\quad \mathbf{I} = \mathbf{L} \mathbf{V}\]

with

\[L_{ij} = \delta_{ij} \sum_{k\neq i} Y_{ik} - (1-\delta_{ij}) Y_{ij}\]

Note:

• \(\mathbf{L}\) is defined in analogy to the topological network Laplacian \(\mathbf{G}\).
• \(\mathbf{L}\) is commonly referred to as admittance matrix as well.

---

R. Huang, G. Korniss, S. Nayak, 2009

figsize(4, 1.5)
g = networkx.Graph()
g.add_edges_from([(0, 1), (0, 2), (1, 2), (2, 3)])
networkx.draw(g, node_color='white', with_labels=True)
savefig('pic/small-graph.svg', bbox_inches='tight')
close(gcf())

Clarify basic terms when speaking about graphs and networks.

We represent our electricity grid by a graph.

Mathematically, a graph is given by a set of nodes (or vertices) and a set of edges (or links) connecting the nodes.

The order \(N\) of a graph is the number of its nodes, while the size \(S\) is the number of its edges.

Basic Graph Theory I

Example

\(N = 4\)

A graph is given by

• a set of nodes \(i\)
• a set of edges \((i, j)\)

Properties

Property	Explanation
Order \(N\)	Number of nodes
Size	Number of edges

D = diag(g.degree().values())
print D

[[2 0 0 0]
 [0 2 0 0]
 [0 0 3 0]
 [0 0 0 1]]

import easytable
t = easytable.autotable(D, hc='&', hp=1)
tl = [r+'\\\\' for r in t.split('\n')]
print '\n'.join(tl)

 2 & 0 & 0 & 0 \\
 0 & 2 & 0 & 0 \\
 0 & 0 & 3 & 0 \\
 0 & 0 & 0 & 1 \\

E = networkx.adjacency_matrix(g).todense()
print E

[[0 1 1 0]
 [1 0 1 0]
 [1 1 0 1]
 [0 0 1 0]]

G = networkx.laplacian_matrix(g).todense()
print G

[[ 2 -1 -1  0]
 [-1  2 -1  0]
 [-1 -1  3 -1]
 [ 0  0 -1  1]]

One important property of a node is its degree, which is the number of incident edges.

One can define the degree matrix \(\mathbf{D}\) as the diagonal matrix which lists all node degrees on its diagonal.

Then there is the adjacency matrix \(\mathbf{E}\), whose elements \(E_{ij}\) are only nonzero if there exists an edge from node \(i\) to node \(j\).

An important matrix is the Laplacian matrix \(\mathbf{G}\), which is defined as \(\mathbf{G} = \mathbf{D} - \mathbf{E}\). It has some interesting properties.

Basic Graph Theory II

Example

\[D = \begin{pmatrix} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 1 \\ \end{pmatrix}\]

\[E = \begin{pmatrix} 0 & 1 & 1 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ \end{pmatrix}\]

\[G = \begin{pmatrix} 2 & -1 & -1 & 0 \\ -1 & 2 & -1 & 0 \\ -1 & -1 & 3 & -1 \\ 0 & 0 & -1 & 1 \\ \end{pmatrix}\]

Node Properties

Property	Explanation
Degree	Number of incident edges

Matrices

Name	Explanation
Degree matrix \(\mathbf{D}\)	Diagonal matrix containing all node degrees
Adjacency matrix \(\mathbf{E}\)	Nonzero only if nodes \(i\) and \(j\) are adjacent
Laplacian matrix \(\mathbf{G}\)	\(\mathbf{G} = \mathbf{D} - \mathbf{E}\)

\[G_{ij} = \delta_{ij} \sum_{k\neq i} E_{ik} - (1-\delta_{ij}) E_{ij}\]

evs = networkx.laplacian_spectrum(g)
condynet.plot.pspectrum(evs, limits=[-.2, 4.2], figsize=[6, 1], width=1)
savefig('pic/laplacian-spectrum.svg', bbox_inches='tight')
print around(evs, 6)

[-0.  1.  3.  4.]

The Laplacian matrix of a graph has some fascinating properties:

All eigenvalues of \(\mathbf{G}\) are positive or zero, making it a positive semi-definite matrix.

There is always at least one eigenvalue zero, so \(\mathbf{G}\) is a singular matrix and can only be inverted using the generalized inverse (or pseudo-inverse).

Interestingly, the number of eigenvalues zero corresponds to the number of connected components of the graph. If all nodes are connected, there is exactly one eigenvalue zero.

The second eigenvalue is the algebraic connectivity.

Lowest non-zero EV: *Spectral gap*

Basic Graph Theory III

Example

Laplacian spectrum:
[0.  1.  3.  4.]

\[G = \begin{pmatrix} 2 & -1 & -1 & 0 \\ -1 & 2 & -1 & 0 \\ -1 & -1 & 3 & -1 \\ 0 & 0 & -1 & 1 \\ \end{pmatrix}\]

Laplacian spectrum

Certain eigenvalues (EVs) of the Laplacian matrix \(\mathbf{G}\) have special properties:

• All EVs are non-negative
  (as \(\mathbf{G}\) is positive semi-definite).
• At least one eigenvalue is \(0\).
• Number of eigenvalues equal
  to \(0\): Number of connected subgraphs.
• Second-smallest EV:
  Algebraic connectivity.

g = networkx.grid_2d_graph(3, 3, periodic=True)
figsize(4, 4)
condynet.drawing.draw_grid2d_periodic(g, node_color='white', with_labels=True, node_size=2000)
savefig('pic/grid2d.svg')
close(gcf())

The regular 2D grid graph

\(N=9\), periodic boundary conditions

g = condynet.graph.small_world(12, p=1.5)
figsize(4, 4)
networkx.draw_circular(g, node_color='white', with_labels=True, node_size=500, width=3)
savefig('pic/small-world.svg', bbox_inches='tight')
close(gcf())

Now I want to introduce a simple disordered graph, known as the Small World Model:

The nodes form a closed ring, i.e. every node is connected to its neighbor.

Then, additional random shortcuts are created. The density of shortcuts can be defined as \(S/N\), where \(S\) is the number of shortcuts. There exist different definitions for the it; we are using the parameter \(p\) here which is equal to twice the shortcut density, i.e. the density of nodes that "participate" at a shortcut.

\(p\) is a parameter influencing directly the disorder in the system. For \(p=0\), there is no disorder at all and we have a closed ring topology. The maximum value of \(p\) is \(N-3\), where we arrive at the complete graph with \(N\) nodes. Thus, the parameter \(p\) allows us to tune the connectivity of our graph.

Following the formulas, there should be 9 shortcuts in the example (on average).

The Small World Model

Example

\(N=12 \qquad p=1.5\)

Closed ring with \(N\) nodes, plus \(S\) randomly chosen shortcuts.

Shortcut density \(\frac{p}{2} = \frac{S}{N}\)

Number of shortcuts \(S = \frac{Np}{2}\)

Two limiting cases:

Limiting case	Resulting graph
\(p_{\rm min} = 0\)	Closed ring
\(p_{\rm max} = N - 3\)	Complete graph

Maximum number of shortcuts:

\[S_{\rm max} = \frac{N(N-3)}{2}\]

---

J. Travers, S. Milgram, 1969; M. Newman, D. Watts, 1999; R. Huang, G. Korniss, S. Nayak, 2009

q = .5
hvals = choose(rand(g.number_of_edges()) < q, [-1, 1])
networkx.set_edge_attributes(g, 'hval', dict(zip(g.edges(), hvals)))

edgelist, hvals = zip(*networkx.get_edge_attributes(g, 'hval').items())
edgecolors = choose(array(hvals) == 1, array(['r', 'g']))
networkx.draw_circular(g, edgelist=edgelist, edge_color=edgecolors, width=3,
                       node_color='white', node_size=500, with_labels=True)
savefig('pic/small-world-binary.svg', bbox_inches='tight')
close(gcf())

To describe an AC electricity network, we put a property on the edges: The impedance \(Z_{ij}\).

Here, we put random impedances on the edges, following a binary distribution: With probability \(q\), the edge shall carry a capacitance \(C\), otherwise, it shall carry an inductance \(L\).

Split this slide?

Adding Edge Attributes

Example

\(N=12\)
\(p=1.5 \qquad q=0.5\)

In order to describe LC networks, we attribute an impedance \(Z_{ij}\)
to each edge.

Here, we consider random impedances, using a binary distribution:

Edge type	\(Z_{ij}\)	Chance
Capacitance	\((i\omega C)^{-1}\)	\(q\)
Inductance	\(i\omega L\)	\(1-q\)

---

R. Huang, G. Korniss, S. Nayak, 2009

Here, I have extended the table a bit, also including the corresponding admittance values, which I would like to call \(y_1\) and \(y_2\) in the following. Further, I would like to introduce the matrix \(\mathbf{h}\), which takes values \(-1\) or \(1\), if there is a capacitance or an inductance on the link \((i, j)\), respectively, or \(0\) otherwise.

Edge type	\(Z_{ij}\)	Chance	\(Y_{ij}\)	\(h_{ij}\)
Capacitance	\((i\omega C)^{-1}\)	\(q\)	\(y_1 = i\omega C\)	\(-1\)
Inductance	\(i\omega L\)	\(1-q\)	\(y_2 = (i\omega L)^{-1}\)	\(1\)

Introduce matrix \(\mathbf{h}\):

\[h_{ij} = \begin{cases} -1 & \rm,\quad \text{edge $(i,j)$ carries capacitance $C$} \\ 1 & \rm,\quad \text{edge $(i,j)$ carries inductance $L$} \\ 0 & \rm,\quad \text{no edge between $i$ and $j$.} \end{cases}\]

\(\Rightarrow\) Similar to \(\mathbf{E}\), but also \(-1\) allowed.

---

R. Huang, G. Korniss, S. Nayak, 2009

Also include a drawing here.

Now that we have the current flow equations, we want to consider the resonance case, so the case when there is no incoming or outcoming current.

In this way, we can calculate the resonance frequencies of the whole circuit, which can be seen as a system of coupled oscillators.

Resonance frequencies I

Remember:

Flow equations:

\[I_i = \sum_j L_{ij} V_j \quad\mathrm{or}\quad \mathbf{I} = \mathbf{L} \mathbf{V}\]
with "Laplacian matrix" \(\mathbf{L}\).

• Consider resonance case, \(I_i = 0\)
  
• The system can be seen as a system of coupled LC oscillator circuits
• The system has \(N\) resonance frequencies \(\omega_n\)

\[\sum_j L_{ij}(\omega)\, V_j = 0 \qquad\text{or}\qquad \mathbf{L} \mathbf{V} = \mathbf{0}\]

---

R. Huang, G. Korniss, S. Nayak, 2009

I present here the method described by Huang et al. and Fyodorov, which allows for the determination of resonance frequencies of random LC networks with binary link disorder by solving a regular eigenvalue problem.

To do so, we have to define several quantities:

Resonance frequencies II

Remember: \[\begin{align} y_1 &= i\omega C \\ y_2 &= (i\omega L)^{-1} \end{align}\]
\[h_{ij} = \begin{cases} -1 &\rm,\quad Y_{ij} = y_1 \\ 1 &\rm,\quad Y_{ij} = y_2 \\ 0 &\rm,\quad Y_{ij} = 0 \end{cases}\]

Define \(\mathbf{H}\),

\[H_{ij} = \delta_{ij} \sum_{k\neq i} h_{ik} - (1-\delta_{ij}) h_{ij} \quad\rm,\]

and

\[\lambda = \frac{y_1 + y_2}{y_1 - y_2} \quad\rm,\]

so that the flow equations for the resonance case can be rewritten as

\[\mathbf{L} \mathbf{V} = \mathbf{0} \qquad\Rightarrow\qquad (\mathbf{H} - \lambda \mathbf{G}) \mathbf{V} = \mathbf{0} \quad\rm.\]

But this is not yet a regular eigenvalue problem...

---

R. Huang, G. Korniss, S. Nayak, 2009; Fyodorov 1999

explain definitions

Resonance frequencies III

Remember: \[\lambda = \frac{y_1 + y_2}{y_1 - y_2}\]
\[\begin{align} y_1 &= i\omega C \\ y_2 &= (i\omega L)^{-1} \end{align}\]

Define

\[\tilde{\mathbf{H}} = \mathbf{G}^{-1/2}\, \mathbf{H}\, \mathbf{G}^{-1/2}\]

(real symmetric) and

\[\tilde{\mathbf{V}} = \mathbf{G}^{1/2} \mathbf{V} \quad\rm.\]

Notes:

1. \(\mathbf{G}\) is real and positive semi-definite, so its matrix square root \(\mathbf{G}^{1/2}\)
   is uniquely defined.
2. \(\mathbf{G}\) is positive semi-definite, i.e. it has at least one eigenvalue \(0\) and hence is always singular. So its pseudo-inverse \(\mathbf{G}^{-1}\)
   has to be considered.

---

Fyodorov 1999

When applying this mapping, it is ensured that \(\tilde{\mathbf{H}}\) is real and symmetric, which can help with the numerical method used to solve the eigenvalue problem.

Resonance frequencies IV

Remember: \[\lambda = \frac{y_1 + y_2}{y_1 - y_2}\]
\[\begin{align} y_1 &= i\omega C \\ y_2 &= (i\omega L)^{-1} \end{align}\]

Define

\[\tilde{\mathbf{H}} = \mathbf{G}^{-1/2}\, \mathbf{H}\, \mathbf{G}^{-1/2}\]

(real symmetric) and

\[\tilde{\mathbf{V}} = \mathbf{G}^{1/2} \mathbf{V} \quad\rm.\]

Then, we can rewrite

\[(\mathbf{H} - \lambda \mathbf{G}) \mathbf{V} = \mathbf{0} \qquad\Rightarrow\qquad \mathbf{\tilde{H}}\, \mathbf{\tilde{V}}_n = \lambda_n \mathbf{\tilde{V}}_n\]

So we are facing a regular eigenvalue problem, with a known relationship between the eivenvalues \(\lambda_n\)
and the resonance frequencies \(\omega_n\):

\[\omega_n = \frac{1}{\sqrt{LC}} \sqrt{\frac{1 + \lambda_n}{1 - \lambda_n}}\]

---

Fyodorov 1999

# load data
data2d = {}
filenames = glob.glob('/home/djung/work/condynet/dorg2d/l????/q????.h5')
for filename in filenames:
    f = h5obj.File(filename)
    try:
        reson = f['reson']
        dens = f['dens']
        stderr = f['stderr']
        param = f['param']
    except:
        continue
    shape = param.shape
    shape0, shape1 = shape
    if shape0 != shape1:
        raise ValueError
    q = param.q
    if shape0 not in data2d:
        data2d[shape0] = {}
    if q in data2d[shape0]:
        raise ValueError('data for L=%i and q=%03f already exists' % (shape0, q))
    data2d[shape0][q] = bundle.Bundle(reson=reson, dens=dens, stderr=stderr, param=param)

figsize(8, 5)
L = 30
for q in [.2, .3, .4, .5]:
    d = data2d[L][q]
    errorbar(d.reson, d.dens, yerr=d.stderr, label='q=%.1f' % q)
axis([-1, 1, 0, None])
legend(loc=1)
title('Resonance spectrum of 2D grid with binary link disorder,\n'
      + r'$N\, =\, 30 \times\, 30\, =\, 900$, for different composite ratios $q$', fontsize=18)
xlabel(r'$\lambda$', fontsize=24)
ylabel(r'$\rho(\lambda)$', fontsize=24)
savefig('pic/ador-q05.svg', bbox_inches='tight')
close(gcf())

Resonance spectra I: 2D grid

Define the density of resonances (DOR):

\[\rho(\lambda) = \frac{1}{N} \sum_{n=1}^{N_{\mathrm{R}}} \delta(\lambda - \lambda_n) \qquad\qquad\]

\(N_{\mathrm{R}}\rm:\)
Number of "true" resonances
\((-1 < \lambda_n < 1)\)

Obtain ensemble average (arithmetic mean) of the DOR over many disorder realizations (ADOR).

---

R. Huang, G. Korniss, S. Nayak, 2009

w, h = 8, 5

# load data
# data is saved in the form data[N][p]
data = {}
filenames = glob.glob('/home/djung/work/condynet/dorswe/n*/p*h5')
for filename in filenames:
    f = h5obj.File(filename)
    try:
        reson = f['reson']
        dens = f['dens']
        stderr = f['stderr']
        param = f['param']
    except:
        continue
    size = param.size
    p = param.p
    if size not in data:
        data[size] = {}
    if p in data[size]:
        raise ValueError('data for N=%i and p=%02f already exists' % (size, p))
    data[size][p] = bundle.Bundle(reson=reson, dens=dens, stderr=stderr, param=param)

figsize(w, h)
start, stop, step = 4, None, 10
for (N, color, marker) in zip([100, 200, 400, 1000], ['red', 'green', 'blue', 'magenta'],
                              ['o', 's', 'D', '^']):
    for (p, facecolor) in zip([N/2, N-3], ['white', color]):
        d = data[N][p]
        errorbar(d.reson[start:stop:step], d.dens[start:stop:step], yerr=d.stderr[start:stop:step], label='N=%i, p=%g' % (N, p),
                 marker=marker, markeredgecolor=color, color=facecolor, markersize=12, linestyle='--')
axis([-1, 1, 0, None])
legend(loc=1)
title('density of resonances of the small world model with binary link disorder, \n' +
      'with system size $N$, shortcut density $p$ and composite ratio $q=1/2$', fontsize=12)
xlabel(r'$\lambda$', fontsize=24)
ylabel(r'$\rho(\lambda)$', fontsize=24)
savefig('pic/fig1c.svg', bbox_inches='tight')
close(gcf())

figsize(w, h)
start, stop, step = 4, None, 10
N = 1000
for (p, color, marker) in zip([.01, .1, 1, 2, 4, 10, 100], ['purple', 'red', 'blue', 'cyan', 'magenta', 'brown', 'lightgreen'], 
                              ['o', 's', 'D', '^', '<', 'v', '>']):
    d = data[N][p]
    plot(d.reson[start:stop:step], d.dens[start:stop:step], label='p=%g' % p,
         marker=marker, markeredgecolor=color, color=color, markersize=12, linestyle='--')
axis([-1, 1, 0, None])
legend(loc=1)
title('density of resonances of the small world model with binary link disorder, \n' +
      'with system size $N=1000$, shortcut density $p$ and composite ratio $q=1/2$', fontsize=12)
xlabel(r'$\lambda$', fontsize=24)
ylabel(r'$\rho(\lambda)$', fontsize=24)
savefig('pic/fig1b.svg', bbox_inches='tight')
close(gcf())

Here are some results for the Small World Model.

On the left side we see the DOR for a fixed system size \(N=1000\), but changing shortcut density \(p\). The shortcut density is tuned from the low-density regime (\(p < 1\)) all the way up to the high-density regime (\(p > 1\)).

The right plot compares -- for different system sizes -- the DOR of the small world model with half the possible shortcuts (\(p \approx p_{\rm max}/2\)) with the DOR of the complete graph (\(p = p_{\rm max}\)).

Resonance spectra II: Small World Model

Large-\(p\) limit

\(\Rightarrow\) Confirming results by Huang et al.

---

R. Huang, G. Korniss, S. Nayak, 2009

figsize(w, h)
start, stop, step = 4, None, 10
p = .01
for (N, color, marker) in zip([100, 200, 400, 1000], ['red', 'lightgreen', 'blue', 'magenta'], 
                              ['o', 's', 'D', '^']):
    d = data[N][p]
    plot(d.reson[start:stop:step], d.dens[start:stop:step], label='N=%i' % N,
         marker=marker, markeredgecolor=color, color=color, markersize=12, linestyle='--')
axis([-1, 1, 0, None])
legend(loc=2)
title('density of resonances of the small world model with binary link disorder, \n' +
      'with system size $N$, shortcut density $p=0.01$ and composite ratio $q=1/2$', fontsize=12)
xlabel(r'$\lambda$', fontsize=24)
ylabel(r'$\rho(\lambda)$', fontsize=24)
savefig('pic/fig2a.svg', bbox_inches='tight')
close(gcf())

figsize(w, h)
start, stop, step = 4, None, 10
p = .1
for (N, color, marker) in zip([100, 200, 400, 1000], ['red', 'lightgreen', 'blue', 'magenta'], 
                              ['o', 's', 'D', '^']):
    d = data[N][p]
    plot(d.reson[start:stop:step], d.dens[start:stop:step], label='N=%i' % N,
         marker=marker, markeredgecolor=color, color=color, markersize=12, linestyle='--')
axis([-1, 1, 0, None])
legend(loc=2)
title('density of resonances of the small world model with binary link disorder, \n' +
      'with system size $N$, shortcut density $p=0.1$ and composite ratio $q=1/2$', fontsize=12)
xlabel(r'$\lambda$', fontsize=24)
ylabel(r'$\rho(\lambda)$', fontsize=24)
savefig('pic/fig2b.svg', bbox_inches='tight')
close(gcf())

Resonance spectra III: Small World Model

Small-\(p\) limit

\(\Rightarrow\) Largely confirming results by Huang et al.

\(\Rightarrow\) Difference: Peak at \(\lambda = 0\).

---

R. Huang, G. Korniss, S. Nayak, 2009

figsize(w, h)
start, stop, step = 4, None, 10
for (N, color, marker) in zip([100, 200, 400, 1000], ['red', 'green', 'blue', 'magenta'],
                              ['o', 's', 'D', '^']):
    for (p, facecolor) in zip([N/2, N-3], ['white', color]):
        d = data[N][p]
        errorbar(d.reson[start:stop:step]*sqrt(p), d.dens[start:stop:step]/sqrt(p), yerr=d.stderr[start:stop:step],
                 label='N=%i, p=%g' % (N, p), marker=marker, markeredgecolor=color, color=facecolor,
                 markersize=12, linestyle='--')
axis([-6, 6, 0, None])
legend(loc=1)
title('density of resonances of the small world model with binary link disorder, \n' +
      'with system size $N$, shortcut density $p$ and composite ratio $q=1/2$', fontsize=12)
xlabel(r'$\lambda\sqrt{p}$', fontsize=24)
ylabel(r'$\rho(\lambda)/\sqrt{p}$', fontsize=24)
savefig('pic/fig1ci.svg', bbox_inches='tight')
close(gcf())

Resonance spectra IV: Scaling law

Test scaling law, valid in the large-\(p\) limit:

\[\rho(\lambda) = p^{\frac{1}{2}} \phi(p^{\frac{1}{2}} \lambda)\]

---

R. Huang, G. Korniss, S. Nayak, 2009

Show more results?

Summary & Outlook

Summary

• Description of LC networks
  • simple graphs
  • binary distribution of edge impedances
• Calculation of resonance frequencies
  and the density of resonances

Outlook

• Different topologies (triangular grid, honeycomb grid, and realistic network topologies).
• Other impedance distributions (also continuous distributions), also including ohmic resistances.
• Beyond the resonance case (current and power flow calculations).

Thank you for your attention!

References

• J. Travers, S. Milgram, 1969
• M. Newman, D. Watts, 1999
• R. Huang, G. Korniss, S. Nayak, 2009
• Fyodorov 1999

Jacobs University Bremen
School of Engineering and Science
Solid State Physics Group

Collective Nonlinear Dynamics of
Electricity Networks (CoNDyNet)