Economic Dynamics and Complexity

The following notes come from the course Economic Dynamics and Complexity, held by Luca Verginer at ETH Zurich.

For a more practical approach, please refer to the GitHub Reporisoty of the course .

Index

• 1.1 Cobweb Dynamics: a simple model for supply and demand
• 1.2 Basics of Epidemic Models
• 1.3 Supply Chains as Rooted Trees
• 2.1 Growth Models: exponential and logistic
• 2.2 Fishing/Harvesting Model
• 2.3 Growth Models: supply and demand determine price
• 3.1 Common Source Diffusion
• 3.2 Bass Diffusion
• 3.3 SIRS Models
• 3.4 Complex Contagions on a graph: a formal setup
• 4.1 Coupled Dynamics: Lotka-Volterra Predator-Prey Model
• 4.2 Coupled Dynamics: Goodwin's Model (1967), Cycles in Employment and Wages
• 5.1 Discrete Dynamics: Cobweb Plot
• 5.2 Discrete Dynamics: Saturated (Logistic) Growth
• 6.1 Endogenous Cycles
• 6.2 Endogenous Cycles: Kaldor Model (Business Cycle)
• 7.1 Introduction to Networks
• 8.1 Diffusion on Networks
• 9.1 Community Detection: Dirichlet Energy, Rayleigh Quotient and Network Bisection
• 9.2 Network Eigenvector Centralities
• 9.3 Katz and PageRank Centralities
• 10.1 Strategic Network Formation
• 10.2 The Connections Model (1996)
• 10.3 The Co-Author Model
• 11.1 Dynamics of Strategic Network Formation
• 11.2 Spatial Connections Model

1.1 Cobweb Dynamics: a simple model for supply and demand

Consider Supply $(S)$, Demand $(D)$ coupled via the price $(P)$ with linear dependence.

• Demand $D$ responds to price immediately
• Supply $S$ responds with lag (production takes time).

The system can converge, oscillate or diverge depending on how supply and demand are sensitive to price.

1.2 Basics of Epidemic Models

A simple Susceptible-infected (SI) model states:

where $\beta$ denotes the inftected contacts per time unit, $I$ the infected, $S$ the susceptible (healty) individuals. Note that, at the final state everyone is infected.

Assuming that the population is closed and that recovery is not possible, the share $x(t)$ of the population infected increases following a logistic growth:

In the early phase: $\dot{x}\approx \beta x$ (almost a exponential growth), in the long run $x⟹1$.

On the other hand, Susceptible-infected-Removed (SIR) model introduces the possibility of recovery, represented by $\gamma$.

The infected people are a transient state and at the equilibrium (the final stage) the disease cannot spread anymore.

In discrete time, the previous expressions become:

1.3 Supply Chains as Rooted Trees

We can represent a chain as a graph $G=(V,E)$, with $V$ representing the set of verttices (notes) and $E\subseteq V\times V$ the set of edges, that can be directed or not: $(u,v)=\text{or}\ne (v,u)$.

Edges can be weighted ($w_{uv}$) and represent capacity, volume, lead time or reliability of a single link of the supply chain.

We denote by $k_v^{in}$,$k_v^{out}$ the degrees (number of connections) of a node $v$ has in input (suppliers) and output (customers). A directed tree models an isolated supply chain with source suppliers and root assembler.

In reality, firms have multiple interconnected suppliers and customers, leading to a supply network

• Networks evolve: entry, exit, mergers, rewiring.
• The simple mathematical rule behind is: $\mathbb{P}(u\to v)\propto k_v+\alpha$ (the probability that a node $u$ connects to a new node $v$ is proportional to its number of degrees)
• The outcome is a heavily-tailed degree with a few hubs and many small degrees.

Let's denote by $x_v(t)$ the inventory at node $v$.

The flow balance is given by:

Where $f_{uv}$ is the inflow from suppliers, $f_{vw}$ the outflow to customers or other nodes, $s_v$ the production and $d_v$ the demand.

In matrix form: $\dot{X}(t)=-LX(t)+S(t)-D(t)$.

We consider a transition matrix $P$ with entries $P_{uv}=\mathbb{P}(u\to v)$ and we denote by $p(t)$ the probability of good at each node.

We then have the following dynamics:

2.1 Growth Models: exponential and logistic

We initially study a simple growth dynamic: the compounding interest.
We consider a initial deposit $x_0$ and a nominal annual rate $r$, with interest credited every $\Delta t$.

with $t=n\Delta t$, we then obtain: $x(t)=x_0(1+r\Delta t{)}^{t/\Delta t}$.

As $t\to 0$, we can switch to continuous time:

The analytical solution of this simple ODE is obtained via separation of variables:

and then, solving for x:

If we instead want to solve the initial value problem with a numerical approximation (Euler's method) we choose a step size $\Delta t$ and set up a time grid.

At each step, we obtain the gradient $f(x_n,t_n)$ at the current point and take a step $\Delta t$ in that direction. This leads to the Euler update rule:

In python, for example:

def euler(f, x0, t0, T, dt):
   t, x = [t0], [x0]
   while t[-1] < T:
       slope = f(x[-1], t[-1])
       x.append(x[-1] + dt * slope)
       t.append(t[-1] + dt)
return t, x

We now analyze a saturated growth model. Let $x$ be the population size, $r$ the intrinsic growth rate, $K$ the carrying capacity.

We start from the per-capita growth rate (proportional growth):

We then add a limited resources constraint: per-capita rate decreases with $x$.

Intuition: the term $-x/K$ limits the exponential growth up to $K$.

The close form solution brings to:

In the logistic growth, if we consider $f(x)=\frac{dx}{dt}=rx(1-x/K)=rx-r/K{x}^2$, we obtain two equilibria:

• $x_1^{eq}=0$: unstable. $(\frac{df}{dx}|x_1^{eq}>0)$
• $x_2^{eq}=K$: stable $(\frac{df}{dx}|x_1^{eq}<0)$

The MSY (maximum sustainable yield) is obtained where $\frac{df}{dx}=0⟹$ $x^{\ast }=K/2$.

We now consider the Allee Effect, modelling the fact that, a low density, individuals struggle to grow. The model then becomes:

where the term $(1-x/A)$ models the critical density. The growth rate is then:

In the Allee dynamics, there are three equilibrium:

• $0$ and $K$ are stable
• $A$ is an unstable equilibrium

Indeed, if $x(0)<A⟹$ the population declines to $0$, otherwise it grows to $K$.

2.2 Fishing/Harvesting Model

We now consider a harvesting function $h(x,e)$, representing the rate at which a resource is removed from the population.:

• $x$ represents the stock(fish population)
• $e$ represents the effort (boats, manpower)
• $h(x,e)$ represents the outflow of resources (catch per unit time)

Then we have:

At the equilibrium:

If we set $h(x,e)=h$, the equilbrium is given by $f(x)=h$.

Multiple cases possible:

• $h>f(x^{\ast })$: above the MSY $⟹$ extinction.
• $h=f(x^{\ast })⟹$ maximum yieldl, but not maximum risk.
• $0<h<f(x^{\ast })⟹$ two intersections:
  • the lower one is unstable, the upper one is stable.

If we instead consider the harvesting function proportional to effor and resources (intuition: If the stock $x$ or $e$ drops, your harvest $h$ naturally drops - assuming effort the other stays the same- preventing accidental extinction better than a fixed quota because the "drain" slows down automatically as the population decreases):

the two equilibria are:

• $x=0$
• $x^{\ast }=K(1-\frac{ae}{r})$ (positive and then stable only if $ae<r$)

2.3 Growth Models: supply and demand determine price

In this case, the harvesting function $h$ is not fixed, but market driven.

• demand is decreasing in price $p$
• supply is increasing in price $p$ and stock $x$.

The market clearing is obtained as: $q^d=q^s=h^{eq}(x)$ and the equilibrium price $p^{eq}(x)$:

The harvesting function becomes then a function of stock:

Then, since the stock is "consumed" by supply, that is equal to the harvesting function, We can the model the stock dynaimics as:

We now consider the profit function $\pi$ of a firm, given by:

with $TC=we$, $w$ wage effort; and revenue $TR=ph$, with $h=aex$.

If we plug the previous equilibrium $x^{\ast }$ in the profit function and we set $\frac{d\pi }{de}=0$, we obtain:

• market entry as long as $\pi >0,⟹e$ increases
• $e=e_2⟹TR(e_2)>TC(e_2)$: excess profit, entry
• $e_2<e<e^{\ast }$: profit further increases
• $e^{\ast }<e<e_1$: proif decreasing but still $\pi >0$.
• $e=e_1$: no profit, not entry.

3.1 Common Source Diffusion

We start by analyzing a simple model with:

• $x(t)\in [0,1]$ representing the fraction adopted
• only external push
• No initial adopters: $x(0)=0$.
• $p$ adoption hazard
• $(1-x)$ share of non-adopters

The analytical solution is:

We clearly that this model is linear in $x$ (the logistic is instead quadratic) and it is always concave, with adoption slowing down as $x\to 1$.

If we instead want to find the time $t(x)$ to reach a target, we simply set:

3.2 Bass Diffusion

In this model, each nonadopter is influenced by twho channels: adoption (as in the commono source one) and imitation or world of mouth, expressed by the term $qx$, increasing in $x$:

Two special cases in this model:

• $q=0:$ commono source model
• $p=0:$ pure imitation, needs $x(0)>0$ to start.

However, note that the Bass Model requires strong assumptions:

• adoption only driven by direct persuasion ($p$ and $q$)
• homogeneous population, no spatial or network effects
• no threshold dynamics and no critical mass modeled
• no economic ingredients (prices, preferences etc etc)

and strong limitations too:

• at the final state: $x\to 1$ with no reversal
• the model cannot capture competition, substitution or churn.

3.3 SIRS Models

Bass is an example of spreading model, epidemic models (SIRS) are the canonical toolkit for this scenarios.

We start by analyzing a SI model :

if we add a recovery rate $\gamma$ and infection duration $1/\gamma$, we obtain a SIR model:

where $r$ is the recovery rate. We can define the Basic Reproduction Number $R_0$ as the number of infected from one person in a fully subscettible population as:

In this case, we have an early growth if and only if $R_0>1$.

The effective reproduction number is then $R(t)=R_{0}s(t)$. The spread slows down and eventually stops when:

(since $s+i+r=1$ and $i\to 0$).

Intuition: once $r^{\ast }$ individuals are immune (recovered or vaccinated), each case generates less than one new case on average and infections dies out.

Finally, a SIRS model introduces a waning immuunity that vanishes at rate $w$.

The end state is non trivial and it is given by the vector $(s^{\ast },i^{\ast },r^{\ast })$.

3.4 Complex Contagions on a graph: a formal setup

We consider a network $G=(V,E)$, with noted $v\in V$ and edges $(u,v)\in E$.

• each node $x_v\in \{0,1\}$ ($0$ non-adopter, $1$ adopter)
• adoption threshold $\theta$
• neighborhood $N(v)=\{u:(u,v)\in E\}$

The threshold rules says:

Intuition: node $v$ adopts if $\theta$ or more neighbors are adopters.

4.1 Coupled Dynamics: Lotka-Volterra Predator-Prey Model

Coupled dynamics are needed when two (or more) variables evolve together, leading to cycles, attractors and repellors. Common examples are predator and prey in biology, susceptible and infected in epidemiology, employment and wages in economics. In formulas, we can write:

The analysis of this systems comes from the classification of fixed points and stability, both analytically and numerically.

We start by analyzing the Lotka-Volterra Predator-Prey Model, representing the interaction between two species:

• $x(t)$: preys (hares)
• $y(t)$: predators (lynx)

In isolation we know that:

• prey reproduces exponentially, $\dot{x}=\alpha x$
• predators die out: $\dot{y}=-\gamma y$.

But, if together in the same ecosystem:

• interaction remove prey at rate $\beta xy$
• interaction yields predators growth at rate $\delta xy$.

resulting in the following coupled system:

We now show the procedure determine fixed points. We start by setting the following to find the equilibrium (fixed points $x_k^{\ast },y_k^{\ast })$:

In the case of the LV model we have:

resultig in:

In general, the number of roots will depend on $f$ and $g$:

• Zero: no equilibria
• One: unique equilbrium
• Finite number $>1$: several isolated equilibria
• inftyinitely many: whole x-axis equilibria

Note that, in higher dimensions, understanding stability and trajectories become more complicated that in the simple 1D case.

To visualize the trajectory of the system, a common approach is to numerically solve the ODEs with initial condition $(x_0,y_0)$ chosen and then run a series of simulations. From the plot, we should then be able to clearly see if the trajectories are converging, diverging or staying in rotation around the fixed points.

Then, at a fixed point, a non-linear system can be approximated by a linear one. Define perturbations $u,v$ as:

and, thanks to Taylor expansions, we end up in the dynamics for small perturbations:

The eigenvalues of Jacobian tell us whether perturbations grow, decay or oscillate. In the 1D case, we had that:

• if $\lambda <0$: perturbations decay (stable)
• if $\lambda >0$: perturbations grow (unstable)

The same applies to the 2D case. We consider $z(t)=p{e}^{\lambda t}$, where $p$ is a costant vector.
Then we plug $\dot{z}=Jz$, resulting in:

We can clearly see that the eigenvalues of $J$ describe how small perturbations evolve.

Recall that to compute the eigenvalues of $J$ we can simply write and solve the characteristic polynomial:

leading to the following solution:

The classification of the fixed points can be done analytically depending on the characteristic polynomial:

• (a): Saddle: $\Delta <0$. ${\lambda }_i$ are real and have opposite single
• (b,c): Stable and Unstable Nodes: $\Delta >0$ and ${\tau }^2\ge 4\Delta$ (stable with $\tau <0$). Also known as attractors and repellors, ${\lambda }_i$ are real and have the same sign.
• (d,e): Stable and Unstable Spiral: $\tau \ne 0$ and ${\tau }^2<4\Delta$ (stable with $\tau <0$). Also known as attractive focus and repulsive focus, ${\lambda }_i$ are complex.
• (f): Center: $\tau =0$ and $\Delta >0$. ${\lambda }_i$ are complex with zero real part.

Example: Stable/Unstable Spirals or Center in the Lotka-Volterra Model.

We consider the following model:

Explicitily computing the Jacobian:

we then find fixed points:

and we evaluate them at:

4.2 Coupled Dynamics: Goodwin's Model (1967), Cycles in Employment and Wages

This model comes from a simple observation: economies often show recurrent cycles.
Booms cause high employment, so workers gain bargaining power, then wages rise, wage share of outpus increases as well. As a consequence, profits fall, investment slows and growth stalls.
Employment then falls, weakening workers' position, wages stagnate or declinde, profits recover and growth resumes.

Goodwin's idea is to model class struggle dynamics with similar equations used for predator-prey cycles.

We then consider two state variables:

• $q_t$: employment ratio (jobs over the population) - analogy with the prey population
• $w_t$: wage share (workers' share of output) - analogy with the predator population

The qualitative cycle logics says that:

• $q\uparrow ⟹$ bargaining power $\uparrow ⟹w\uparrow$
• $w\uparrow ⟹$ profits $\downarrow ⟹$ investment $\downarrow ⟹q\downarrow$
• $q\downarrow ⟹$ wage pressure $\downarrow ⟹w\downarrow ⟹$ profits $\uparrow ⟹q\uparrow$

Formally, if we consider the following variables:

• $q_t=l_t/n_t$ employment ratio
• $w_t=w_t{l}_t/Y_t=w_t/a_t$ wage share
• $Y_t$ total output (GDP), with capital-output ratoi $\sigma$ such that: ${\dot{Y}}_t=\frac{1-w_t}{\sigma }{Y}_t$
• $k_t$ capital stock (invested at: ${\dot{k}}_t=(1-w_t)Y_t$)
• $l_t$ employed labor
• $n_t$ total labor force, growing at rate $\beta$.
• $a_t=Y_t/l_t$ labor productivity, growing a rate $\alpha$.
• $w_t$ real wage, following the Philipps Curve with: $\frac{{\dot{w}}_t}{w_t}=-\gamma +\kappa q_t$

and parameters $\alpha ,\beta ,\gamma ,\kappa ,\sigma$.

We then differentiate the employment ratio and the wage share:

Putting toghether, we have the Goodwmin's System:

Simplifying, we get:

The intuition, very similar to the previous predator-prey model, is that workers and capitalists are two interacting groups. Workers indeed seek higher wages, capitalists seek higher profits. These opposing objectives create a self-regulating cycle, neither side "wins" permanently but they co-move in economic cycles.

From an economic point of view:

• Growth-recessions cycles are endogenous without external shocks
• Booms (high $q$) are the beginning of slowdowns (rising $w$) and viceversa
• The neutral center is not robust, small modifications destroy cycles.

5.1 Discrete Dynamics: Cobweb Plot

There are several reason behind why we should discrete time dynamics:

• decision timing: firms and individuals set prices/output quarterly or at fixed intervals
• institutional calendars: central banks and governments meet on fixed dates
• contracts and billing: loans, rent and utilities are paid monthly or similarly
• observed data: GDP, monthly unemployments, annual reports etc etc...

To start with discrete time dynamics, we define the difference equation, representing how the next state depends on the previous:

The evolution if then given by the repeated application of $f$:

To visualize this, we usually use Cobweb Plot, showing the evolution of the discrete rule.

• Start with a initial value $x_0$.
• Move vertically till the blue line representing $f(x)$ and find $f(x_0)$.
• Move horizontally to the diagonal $x=y$ and find $x_1$.
• Repeat with $x_1$ and iterate.

Each update defined a sequence:

when $f(x)$ crosses $y=x$ we have fixed points $x^{\ast }=f(x^{\ast })$.

Note that the sequence may cycle set of points (periodi orbits) or may be aperiodic (chaotic) deterministic, but not repeating.

To determine the fixed points, we have to find $z_t^{\ast }$ such that $z_{t+1}=f(z_t)$.

We then consider, as usual, the deviation from the fixed point:

and we linearize around $z^{\ast }$:

if we then iterate $t$ times we obtain:

To determine if it is stable or not, we simply look at:

• $|f^{\prime }(z^{\ast })|<1⟹:$ attracting
• $|f^{\prime }(z^{\ast })|>1⟹:$ repelling
• $0<f^{\prime }(z^{\ast })<1⟹:$ monotone
• $-1<f^{\prime }(z^{\ast })<0⟹:$ alternating

5.2 Discrete Dynamics: Saturated (Logistic) Growth

We previously analyzed the continuous-time logistic growth as:

if we apply the Euler discretization as:

we can then write:

if we define $z_t=x_t\frac{r}{1+r}$:

We now determine fixed points:

we now apply the stability condition on the first derivative $f^{\prime }(z^{\ast })$:

• if $1<R<3$ then $|2-R|<1$ and the fixed points attract
• at $R=3:f^{\prime }(z^{\ast })=-1$: flip bifurcation (fixed points becomes repelling: stable 2-cycles).

Intuition: As long as the growth rate $R$ is between 1 and 3, the population settles down to $z^{\ast }$. At exactly $R=3$, the slope becomes $-1$. This is the "tipping point": The system can no longer settle at a single number.

In the second period, if $R>3$, the population starts jumping back and forth between two values, $z_1$ and $z_2$, every period. The period-2 points solve $f(f(z))=z$, resulting in:

As in the first cycle, in the $k$-cycle, we obtain stability is reached when: $|(f^{(k)}{)}^{\prime }(z^{\ast })|<1$, that, for the $2$-cycle rewrites as:

At $R=1+\sqrt{(}6)$ the $2$-cycle does another flip (stable 4-cycle, the population repeats a pattern of 4 different values)

For many periods, we now consider $R_n$ the value of $R$ where a $2^n$-cycle first appears:

The sequence converges geometrically to $R_{\infty }=3.569$

We then define the Feigenbaum constant as:

resulting in the Scaling Law for $R_n$:

• For $R_{\infty }<R<4$ we can see a mixture of order and chaos: periodic windows reappe within chaotic bands.
• At $R=4$ we see fully developed chaos.

This is called deterministic chaos: once $x_0$ and $r$ are fixed, each step is predetermined, but long term behavior becomes unpredictable, because tiny differences in $x_0$ grow rapidly.

A deterministc system is then called chaotic when it shows:

• aperiodic long-run behavior (no fixed points or repeating patterns)
• determinism
• sensitivity to initial conditions

Note that, as we've seen, deterministic $\ne$ predictable. In non-linear systems, tiny errors or deviations can explode over time, therefore long term forecasts lose meaning.

6.1 Endogenous Cycles

In the economy, markets are rarely isolated (producers can switch between goods, farmers can sell domestically or abroad, energy and manufacturing costs increase etc, etc...) We start now from the simple Cobweb Model but allow producers to switch markets.

Each market $i\in \{A,B\}$ has a ficed supplier share $W_i$, with $W_A=1-W_B$.

if we set the market clearing condition $D_i=W_i{S}_i$:

then we can find the Price-Update Rule:

At the equilibrium:

Note that larger $W_i⟹$ faster response to equilibrium and then less stability.

We now consider costs and profits in each market. Each supplier earn ${\pi }_i$, depending on price and total quantity:

assuming that cost increases with ouput:

we result in the following profit funtion:

Now we can see the coupled system as a sequence of events:

• The suppliers produce $W_{i,t}(\frac{P_{i,t-1}-g_i}{d_i})$, with $W_{i,t}$ is the probability of choosing a market depending on the profits:

where $f\ge 0$ is the intensity of choice (rationality).

• Prices adjust to clear each market:

• Profits are realized
• Suppliers update market choice for the next period

Be aware that, in the following plots, the x-axis is different!

To understand if it is chaos or just complex, we can perform a FFT test.
A time series can be indeed seen as a mix of pure waves. We then apply the Fourier Transform to understand how much of each frequency is present.

Plotting then $|X(\text{frequency})|$ gives a spectrum:

• peaks in the spectrum represents cycles (not chaotic)
• no clear peaks in the spectrum implies that there's not a dominant rhythm.

In general, we can conclude about endogenous cycles that:

• Volatility is structural and it is amplified by a large supplier share in the market, faster switching intensity $f$ and excessive supply response (low $d$).
• Cycles create deadweight loss: mis-times investment and wasted inventory/resources.
• Coupling transmits fluctuations across markets.

6.2 Endogenous Cycles: Kaldor Model (Business Cycle)

We now model a 2-player model with:

• Household that can either consume or save: $Y=C+S$
• Capicalists that can either consume or invest: $Y=C+I$
• Savings fund investments: $S\to I$ via banks
• At equilibrium: $S=I$.

Starting from the setup we have:

• Households consume $C$ and save $S$.
• Firms plan investment $I^p$ based on the expecteded consumption $C^e$
• in reality, $C^e\ne C^{acqual}$.
• The gap shows up as:

• The firm reacts:
  • If $\Delta \text{Inv.}>0⟹$ overproduction, cut output
  • If $\Delta \text{Inv.}>0⟹$ underproduction, expand output
• Leading to the dynamic rule:

we can then incur in two scenarios:

• if $S$ has a steeper slope than $I$ (at the right of the equilibrium $\dot{Y}<0$ and at the left $\dot{Y}>0$): stable equilibrium.
• if $I$ has a steeper slope than $S$ (at the right of the equilibrium $\dot{Y}>0$ and at the left $\dot{Y}<0$): unstable equilibrium.

Kaldor's idea (1940) is that the saving and investment curves are not linear:

• For middle ranges of income, $S(Y)$ apperas linrar, but with larger income people save a larger proportion of their investment and at lower incomes they save drastically less .
• Similarly, $I(Y)$ is linear only in the middle range: at lower $Y$ there's little incentive to invest and at a very high $Y$ there are rising costs and borrowing constraints (so $I$ grows slower).

• Note that both $I,S$ can be "S" shaped.
• Note also that, at low income levels $(Y)$, the savings curve $S$ drops below the horizontal axis, representing negative savings, rather than approaching zero asymptotically.

So far, we treaded investment and savings as a function of the output, in reality, they also depend on the capital stock $K$, that accumulates through investment and declines thorugh depreciation, reflecting past investment decisions. We can then write (Kaldor Model):

In this model, if we assume that $K$ is set externally, we have 3 equilibria: two are stable (A,C) and one unstable (B)

$K$ then introduces a hysteresis cycle: the equilibrium state is path dependent.

• an economy starts on the high $Y$ branch (high income).
• as $K$ increases, the economy moves to the right along the top red line.
• eventually, it reaches the "knee" or the edge of the cliff.
  • if $K$ increases past this point, the economy crashes down to the low $Y$ branch.
• once it falls to the bottom level, it remain stuck in the low-income state
• to get back to the high-income state, the economy jas to decrease $K$ to move way to the left on the graph and to find the other "knee" where the system jumps up.

Note that, since no equilibrium satisfies $\dot{Y}=\dot{K}=0$, the economy follows a closed cycle.

If we then allow $K$ to have its own dyncamics, we have the full Kaldor Model.

The economy then moves around the unstable equilibrium: the interaction of fast $Y$ and slow $K$ creates endogenous cycles. Note that $\alpha$ represents the speed of the economy, while $\delta$ the speed of capital and they both shape the curvature of the cycles.

7.1 Introduction to Networks

A network (graph) is tuple $G=(V,E)$:

• $V$: the set of nodes (vertices)
• $E\subseteq V\times V$: set of links (edges)
• $V\times V$: all possible ordered pairs

We use the direction convention:

A network can be stored as a adjacency matrix $A$:

• Rows represent the sources (outgoing link)
• Column represent the targets (incoming links)

A network is directed if $A_{ij}=A_{ji}\forall i,j$

In a weighted network links have a numerical strength. A function $w:E\to \mathbb{R}$ assigns a weight to each link and the adjacency matrix becomes real-valued.

A link of the form $(i,i)$ is called self-loop and it appears in the diagonal entry $A_{ii}$ (economically, it represents the self-dependence or internal feedback).

A node's degree is the number of links connected to it (economically: they represent connectivity of a node). Also, in a directed network, we can distinguish:

• indegree $d_{in}(i)$: number of incoming links $(j,i)$
• outdegree $d_{out}(i)$: number of incoming links $(i,j)$

In a weighted network, a node's strength sums the weights of its connections:

In a unweighted network, a node's strength is a simple in-and-out degree count.

A path between nodes $v$ and $w$ is a sequence $(p_0,p_1,p_2...p_l)$ such that:

• $p_0=v,p_l=w$
• each pair $(p_i,p_{i+1})\in E$

The path length is, of course, the number of links.

The distance between two nodes is the length of the shortest path connecting them. If no path exists: $\text{dist}(v,w)=\infty$.

The network diameter is the longest shortest path in the network.

The average shortest path length measures typical separation.

A network is connected if $\text{dist}(v,w)<\infty$ for all $v,w\in V$.

Connected components are connected subgraphs $G^{\prime }=(V^{\prime },E^{\prime })\subseteq G$.

The largest component is the giant connected component (GCC) if $|V^{\prime }|/|V|\approx 1$.

Degree centrality of a node is the number of its direct links (economically it measures the connectivity as exposure/access to partners, lenders, clients or other nodes)

Betweeness centrality counts how often node $i$ lies on the shortest paths between others:

where $n_i(s,t)$ is the shortest path $s\to t$ through $i$. In the normalized form, we also write:

where $N_{st}$ is the total number of shortest shortest paths $s\to t$.

8.1 Diffusion on Networks

We define diffusion as flow across links (in the two nodes case). Suppose we have two noes $i,j$ connected by one link: the flow is then proportional to the difference between two states. We can then write an ODEs for the nodes:

with the conservation constraint: $\frac{d}{dt}(x_i+x_j)=0$ and parameter $C$ with unit ${\text{time}}^{-}1$.

The previous ODEs bring the convergence to stationarity:

If we generalize this to a graph and define an ODE for every node:

with the conservation contraint: $\frac{d}{dt}{\sum }_i{x}_i=0$ (average preserved).

From the aggregate network, we can study the node-level dynamics. Consider a node $i$:

(For example, we could choose $g(\dot{)}$ modelling a Bass diffusion and then the rest to model the network)

To study better the behavior of the network, we proceed by computing the Laplacian Matrix.
Consider degrees $k_i={\sum }_j{A}_{ij}$ and $K=\text{diag}(k_1,k_2...k_n)$.

The Laplacian Matrix is $\mathcal{L}=K-A$ and we can then rewrite the ODEs compactly:

We now proceed with the spectral decomposition of the Laplacian Matrix.

Recall: the spectral decomposition of a generic diagonalisable matrix $S$ is $S=U^{-1}{D}_{\lambda }U$, where $U$ is the eigenvector matrix and $D_{\lambda }$ the eigenvalues diagonal matrix.

Define $\{w_i\}$ as the eigenvectors, forming an orthonormal basis. Then we write: $\mathcal{L}{w}_i={\omega }_i{w}_i$ with $0<{\omega }_1<{\omega }_2...<...$.

Define then the projections ${\alpha }_i(0):=w_i^{T}x(0)$ (intuition: ${\alpha }_i$ are the time-dependent coefficients showing how "strong" each pattern/eigenvector is at time $t$) and then:

Now plug $x(t)={\sum }_i{\alpha }_i(t)w_i$ into $\dot{x}=-C\mathcal{L}x$. Since $w_i$ is an eigenvector, multiplying it by the matrix $\mathcal{L}$ is simple: $\mathcal{L}{w}_i={\omega }_i{w}_i$, resulting in:

Because eigenvectors are linearly independent, we can splits the big matrix problem into $N$ separate problems: