Moving average network examples for asymptotically stable periodic orbits of monotone maps

Barnab´asGaray — Judit V´ardai

Faculty of Information Technology, P´azm´anyCatholic University, Budapest, Hungary

Hatvani 75 - Szeged, June 26, 2018

Examples for asymptotically stable periodic orbits of monotone maps The talk is based on

• the mathematical analysis of Andr´asSimonovits’ model of tax evasion in a joint paper with him and J´anosT´oth • as well as on a recent work with Judit V´ardai

0B.M. Garay, A. Simonovits, J. T´oth: Local interaction in tax evasion, Economics Letters 115(2012), 412–415. 0B.M. Garay, J. V´ardai: Moving average network examples for asymptotically stable periodic orbits of monotone maps, EJQTDE Special Issue HATVANI 75 (June 26, 2018) Examples for asymptotically stable periodic orbits of monotone maps added in November

Presented also at the MIKLOS´ FARKAS Seminar

Budapest - 8th of November, 2018

TISZTELETTEL EMLEKEZVE´ A SZEMINARIUM´ NEVAD´ OJ´ ARA´

0free access to all EJQTDE papers including Moving average network examples for asymptotically stable periodic orbits of monotone maps, EJQTDE Special Issue HATVANI 75 (June 26, 2018) Examples for asymptotically stable periodic orbits of monotone maps Introduction

Examples for asymptotically stable periodic orbits of monotone maps Basic notation 1.)

• Graph G with vertices V (G) = {1, 2,..., N} and edges E(G)

• Adjacency matrix AG of G

• Degree matrix DG = diag(d1, d2,..., dN ) of G • −1 Transition matrix PG = DG AG of the random walk on G • Eigenvalues 1 = ν1 ≥ ν2 ≥ ... ≥ νN of matrix PG N • Column eigenvector 1N = col (1, 1,..., 1) ∈ R belonging to ν1 = 1

Examples for asymptotically stable periodic orbits of monotone maps Basic notation 2.) and Basic assumption 1

• The adjacency matrix AG of graph G is primitive, i.e., k∗ ∗ (A1) AG > 0 for some integer k ≥ 1 As a consequence, 1 = ν1 > |νi | for each i = 2, 3,..., N

• Dominant eigenvalue, dominant right/left eigenvector of matrix PG : 1 = ν1, 1N and wN = (w1, w2,..., wN ) Entries of the left eigenvector wN are all positive Eigenvectors 1N and wN are normalized by letting wN 1N = 1

Examples for asymptotically stable periodic orbits of monotone maps Basic notation 3.) and Basic assumption 2

• A finite interval [ω, Ω] • A C ∞ function f :[ω, Ω] → [ω, Ω] satisfying (A2) f 0(x) > 0 for each x ∈ [ω, Ω] • A mapping F :[ω, Ω]N → [ω, Ω]N defined by  (F(x))i = f (PG x)i for each i = 1, 2,..., N

Note that 1 X (PG x)i = xj , di {j | (i,j)∈E(G)}

the local average of the neighboring xj ’s at vertex i, i = 1, 2,..., N.

Examples for asymptotically stable periodic orbits of monotone maps Basic consequences

• k (A1) implies: PG → 1N wN as k → ∞ k In particular, given a probability vector w, we have wPG → wN • (A2) implies: F is monotone (in the sense of Hirsch) • (A1)–(A2) imply: F is eventually strongly monotone

Existence versus nonexistence of nontrivial, asymptotically stable periodic orbits is one of the most striking differences between discrete–time and continuous–time monotone dynamics.

Examples for asymptotically stable periodic orbits of monotone maps Recall the definitions of monotonicity

By letting x ≤ y if and only if xi ≤ yi for each i, a closed partial order on [ω, Ω]N is introduced.

We write x ≺ y if xi < yi for each i. Mapping F is monotone if F(x) ≤ F(y) whenever x ≤ y. Monotonicity is strong if F(x) ≺ F(y) whenever x ≤ y and x 6= y. If only F k (x) ≺ F k (y) for some integer k∗ > 1, then F is eventually strongly monotone.

Examples for asymptotically stable periodic orbits of monotone maps Assuming (A1), (A2) : Modeling Tax Evasion

• Normalized income of each taxpayer: 1 • Tax rate: θ ∈ (0, 1) • Years: t = 0, 1, 2 ...

• Reported income vector in year t: xt = (x1,t , x2,t ,..., xN,t ) N • Reported income vector in year 0: x0 6= 0 ∈ [0, 1] • In year t, taxpayer i computes the average income reported by his acquaintances: 1 P (P x ) = xj,t G t i di {j | (i,j)∈E(G)} In year t + 1, taxpayer i’s reported income:  xi,t+1 = (F(xt ))i = f (PG xt )i where [ω, Ω] = [0, 1], f (0) = 0, f 0(0) > 1 and f 00(x) < 0 for each x

• Then xi,t → x∗ (i = 1, 2,..., N) as t → ∞ where 0 6= x∗ = f (x∗)

Examples for asymptotically stable periodic orbits of monotone maps On the history of order relations - time series analysis

Order relations as well as ordinal patterns (essentially Parsons code for melodic contours) have always been an important tool in the analysis of music. All great composers were aware of extremely demanding and mostly dramatic tone combinations: • Gesualdo’s music is full of sharp dissonances expressing pain, grief, and conflict. He used a disorganized and disintegrated musical language not heard again until the late 19th century • Schuetz, Bach, Haydn, ... : Passion music for Good Friday services • Beethoven’s late string quartets – even in the slow movement of the Quartet, which Beethoven called ”Holy song of thanks (’Heiliger Dankgesang’) to the divinity, from one made well.”

Examples for asymptotically stable periodic orbits of monotone maps RESULTS

Examples for asymptotically stable periodic orbits of monotone maps Periodic Orbits of Period Two — Sketch Function f is constructed on a finite set of real numbers first. For several classes of graphs, the function obtained will be strictly increasing. Then a C ∞ extension on an suitably chosen interval [ω, Ω] follows. While doing this, the derivative can be kept positive. Finally, we set  N N f(x) = col f (x1), f (x2),..., f (xN ) ∈ [ω, Ω] for each x ∈ [ω, Ω] . Period Two Orbits will be constructed according to the scheme

PG f PG f p −→ a = PG p −→ q −→ b = PG q −→ p . Observe that 0 0 0  F (x) = diag f ((PG x)1),..., f ((PG x)N ) PG for each x ∈ [ω, Ω]N . It follows that asymptotic stability can be guaranteed by making the derivative of function f , on a discrete set of at most 2N real numbers on the interval [ω, Ω], small enough. Examples for asymptotically stable periodic orbits of monotone maps Periodic Orbits of Period Two on various classes of graphs

K6 K3,3,1

  G7 / G8 / G9

  K4,4\{e} P

Figure: The Petersen Family. Each arrow represents a ∆–Y transform

Five Frucht Graphs, the Petersen Family, the Seven Symmetric Generalized Petersen Graphs, Hypercubes, a Technical Class of Circulant Graphs (containing Paley graphs of prime order), Complete Graphs. Some proofs are based on Elementary Eigenstructure Results in Algebraic Graph Theory, other proofs require Ad Hoc Argumentation. Examples for asymptotically stable periodic orbits of monotone maps A Periodic Orbit of Period 3 – the Underlying Graph

38 20 140 1 P = 89 104 5 G∗ 198   71 74 53

Note that PG∗ is the transition matrix of the random walk on a directed graph G∗ on 3 vertices with 594 edges. Note also that 1 P p = p where κ = 66 and p = col (8, −7, −1) ∈ 3 . G∗ κ R

1 The remaining two eigenvalues are 1 and µ = − 33 , with the respective eigenvectors 1 = 13 and col (−5, 4, 1).

0 Graph G∗ has multiple and multiple loop edges. One can say G∗ has only 9 weighted edges. Examples for asymptotically stable periodic orbits of monotone maps A Periodic Orbit of Period 3 – the Construction

3 With κ = 66, p = col (8, −7, −1) ∈ R and PG∗ as above, it is readily checked that  21κ   6  PG∗ f PG∗ f PG∗ f κp −→ p −→ 2κp −→ 2p −→ −18κ −→ −6 −→ κp −3κ 0

defines a periodic orbit of period 3. The crucial fact is of course that function f : {−14, −7, −6, −2, −1, 0, 6, 8, 16} → R given by f (−14) = −18κ, f (−7) = −14κ, f (−6) = −7κ, f (−2) = −3κ, f (−1) = −2κ, f (0) = −κ, f (6) = 8κ, f (8) = 16κ, and f (16) = 21κ is strictly increasing.

Examples for asymptotically stable periodic orbits of monotone maps Periodic Orbits of Period 4 + r for r = 0, 1,... Variants of the Period Three Example above

The second map p −→f 2κp in the six–arrow–scheme above is replaced by a chain of 2r + 3 maps.

In an alternating order, r + 2 maps are “nonlinear”, and r + 1 maps are “linear”.

The particular form of f plays a pivotal role here. Actually, the second map is subdivided into 2r + 3 maps.

They can be defined by a homogeneous interpolation process.

Examples for asymptotically stable periodic orbits of monotone maps KEDVES LACI, ISTEN ELTESSEN´

BOLDOG SZULET¨ ES–´ ES´ TOBBI¨ NAPOKAT

Examples for asymptotically stable periodic orbits of monotone maps