On principal eigenpair of temporal-joined adjacency matrix for spreading phenomenon Abstract. Keywords: 1 Introduction

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Adrian McKenzie
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1 1,3 1, r A ptq pxq p0q 0 pxq 1 x P N S i r A ptq r A

2 ÝÑH ptq i i ÝÑH t ptq i N t1 ÝÑ ź H ptq θ1 t 1 0 i `t1 ri ` A r ÝÑ H p0q ra ptq t ra ptq i,j 1 i j t A r ptq i,j 0 I r ś N ˆ N mś ĂA l A Ą m... A Ă 2A1 Ă pxq l 1 px ` 1q θ pxq pxq pxq A r ptq A r ÝÑH p0q δ i,j i ÝÑ H ptq ˆrI ` A t r ÝÑ H p0q j t ˆrI ` A t r % ij j

3 i ˆrI ` r A t ij ě ri ` A t r ř k pλ k ` 1q t w i,k w j,k k A r λ k ij ÝÑ W k A rýñ ÝÑ W k λ kw k i ÝÑW k w i,k w i,k 1 k λ 1 ě λ 2 ě... ě λ N k 1 λ 1 w i,1 ri ` r A t ri ` ra r A k ppλ 1 ` 1q { pλ 2 ` 1qq t " 1, S i E ps i q log `w i,1 w j,1. log pλ 1 ` 1q i 1 A r ij A r ij 1 i 1 S 1 1 i 1 S i 2 N Ñ 8 1{α 2 N 1 A r λ 1 ÝÑ W 1 A rýñ W 1 ÝÑ ÝÑW λ 1W 1 w i,1 1 ą 0 w j,1 w l ą 2 ř i w2 i,1 1 λ 1 1{α i w i,1 #? 1 α4`1? α α4`1 i 1. E ps i q $ & E ps i q % log α α 4`1 ˆ 2 log? α α 4`1 logp 1 α `1q «1 ` logp 1 α `1q «2 ` 2α α logpαq ` O `α2 logpαq ` O `α 2 i 1. N Ñ 8 α Ñ 0

4 i 1 i 2 N S 1 1 i 1 S i 2 N 16 w i,1 1{? N S i N{2 8 ri ` A t r Si E ps i q log N log 3 « λ 1 2 lim NÑ8 E ps i q S i lim NÑ8 λ 1 {λ 2 lim NÑ8 1{ pcos p2π{nqq 1 E ps i q N{2 ` 1 λ 1 w 1,1 w 1,N{2`1 S i S 1 S N{2`1 S i S i S 1 S n{2`1

5 Principal eigenpair of temporal network # S1`l Sn1l Sn{2`1`l Sn{2`1`n1 l 1, 2,..., N {2. l 0, 1..., N {2 5 (9) Combined the three symmetric properties, the system of N will partition as four group, within the edge node of group, the Si s have N {4`1 values. Therefore, the agents in the same set share the same value of Si : {1,9}, {2,16,10,8}, {3,15,11,7}, {4,14,12,6}, {5,13}. The results of Fig (3) states that :Si 3 ` i i 1 5, we can understand them from the following easy examples. In this network, the distance to the farest agent of 5-th and of 13 th agent is the same as them in the loop network without radius link A1,N {2`l. The value of Si for 5-th and of 13 th agent remains the same S5 S13 8. For the 4-th agent, the distance to the farest agent,12-th, get one step smaller by shifting to the route with the radius link, the route t4 Ñ 3 Ñ 2 Ñ 1 Ñ 16 Ñ 15 Ñ 14 Ñ 13 Ñ 12u to the route t4 Ñ 3 Ñ 2 Ñ 1 Ñ 9 Ñ 10 Ñ 11 Ñ 12u. These symmetric properties can also be found in our estimator because they are in the principal eigenvector, that can be revel by calculating higher order perturbations. These correspondence of symmetric is shown in Fig (3) as five tx, yu points, otherwise it will shown more than five points. Comparing to the the relation of Si and its lower bound estimator E lower psi q, our estimator have the network symmetric properties and the monotonic relation to Si. Fig. 2. Matrix multiplication of adjacency matrix. Using this multiplication matrix post mapped by the function θ1 pxq, shown as black and white, the step number to the furthest agent can be got. In the left panel as a network without radius link, all columns r ` Iq r t qij 1 while t 8. Before that, at least on element in or rows turn to black θ1 ppa r ` Iq r t qij 0 means that j-th agent can not row is white. White matrix element θ1 ppa be accessed by i-th agent. Therefore, Si 8 for all agent in the loop network without radius link. There are two white matrix element in the loop network with a radius link r ` Iq r 7 q5,13 θ1 ppa r ` Iq r 7 q13,5 0. It makes at t 7. The two matrix elements are θ1 ppa thatsi 8 for i 13 and 5. Other agents Si is shown in section 3. Pre mapped matrix by the function θ1 pxq, shown as gray level, do not show Si information clearly.

6 S i S i S i N 16 S i S i

7 A r ptq τ A r ptqa r pt ` τq ÝÑ H ptq P r ÝÑ ˆ t{τ ÝÑ H ptq rp H p0q, τ1 ź rp t 1 0 `t1 ri ` A r. r P N 3 N 60 r P τ N 3 τ 2 ra p0q ra p0q 1 ra p0q 0 1,2 2,1 i,j ti, ju ra p1q 2,3 ra p1q 1 ra p1q 0 ti, ju 3,2 i,j A r pt 1 q P r rp ra p1q ` I r ra p0q ` I r Ñ P r P r?? ( 1 2 `3 ` 5, 1 2 `3 5, 0 t `? ( , 1, 1, 1? ( 2 `1 5, 1, 1, t1, 1, 0uu tt , 1., 1.u, t , 1., 1.u, t1., 1., 0uu w 1,1 A r pt 1 q P r A r pt 1 q P r P r P r w 1,1 {w 3,1 w 2,1 {w 3,1 1.61

8 N 60 τ 5 Ą A #1 Ą A #2 N 60 N 10 N N #2 A Ą#2 `N N #1 ĄA #1 A Ą#1 ` ĄA #2 A Č i mod `i, N{N 1 mod `i, N{N 0 τ 5 A Ą#2 A Ą#2 A Ą#2 A Ą#2 A r p0q A Ą#2 A Ă1 A r ptq A Ą#1 t 1 4 τ 5 A r ptq A r pt τq #2 A r p0q A Ą#2 τ A r p4q A Ą#2 P r P r wi,1 2 1 N 1 τn{2 pxq r A ptq r A S i r A ptq r P r P

9 r P Ć A #2 τ i 1 i 2 r P

10 pxq θ pdq pdq " 1 d ě 1 pdq θ pd 1q 0 pdq A r ij ÝÑH ptq i pdq d e ÝÑ D ÝÑ E " * θ pd ˆ e 1q θ pd 1q ˆ θ pe 1q θ pd ` e 1q θ pθ pd 1q ` θ pe 1q 1q Ñ " * θ1 pdeq pdq peq pd ` eq p pdq ` peqq Ñ pd 1 e 1 ` d 2 e 2 q p pd 1 e 1 q ` pd 2 e 2 qq p pd 1 q pe 1 q ` pd 2 q pe 2 qq ÝÑD ÝÑD i,m i,m $ & ÝÑD T ÝÑ E Ñ % ÝÑD ÝÑ ` E ÝÑD T ÝÑE ÝÑD,. ÝÑE ` - pxq p0q 0 p1q 1 ÝÑ B ÝÑ B ÝÑB ÝÑ B. pdq ÝÑD ÝÑD.

11 ÝÑ ÝÑ ÝÑ H pt ` 1q H ptq ` θ1 ra ptq H ptq Ñ ÝÑ H pt ` 1q ÝÑH ptq ` ra ptq ÝÑ H ptq Ñ ÝÑ H pt ` 1q ÝÑH ptq ` r A ptq ÝÑ H ptq ri ` r A ptq ÝÑH ptq ri ` r A ptq ÝÑH ptq Ñ ÝÑ H pt ` 2q ri ` r A pt 1q ri ` r A ptq ÝÑH ptq Ñ ÝÑ t1 ź `t1 H ptq ri ` A r ÝÑ H p0q t 1 0 Ñ ÝÑ t1 ź H ptq t 1 0 Ñ ÝÑ t1 ź H ptq t 1 0 `t1 ri ` A r `t1 ri ` A r ÝÑ H p0q ÝÑ H p0q

REAL ANALYSIS II TAKE HOME EXAM. T. Tao s Lecture Notes Set 5

REAL ANALYSIS II TAKE HOME EXAM CİHAN BAHRAN T. Tao s Lecture Notes Set 5 1. Suppose that te 1, e 2, e 3,... u is a countable orthonormal system in a complex Hilbert space H, and c 1, c 2,... is a sequence