# On principal eigenpair of temporal-joined adjacency matrix for spreading phenomenon

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## Abstract

This paper reports a framework of analysis of spreading herbivore of individual-based system with time evolution network \(\widetilde{A}(t)\). By employing a sign function \(\theta _1 \left( x \right)\), \(\theta _1 \left( 0 \right) =0\), \(\theta _1 \left( x \right) =1\)\(x \in {\mathbb {N}}\), the dynamic equation of spreading is in a matrix multiplication expression. Based on that, a method of combining temporal network is reported. The risk of been-spread and the ability to spread can be illustrated by the principal eigenpair of temporal-joined matrix in a system. The principal eigenpair of post-joined matrix can estimate the step number to the farthest agent \(S_i\) in a non-time evolution network system \({\widetilde{A}}\left( t\right) ={\widetilde{A}}\) as well.

## Keywords

Agent model Epidemic Contact network## Introduction

Various applications of network are applied crossing social and natural disciplinaries [1]. As a mathematical abstraction method, network describes the interactions among elements inside of system as links among nodes. The non-direction and static network gets the highest abstraction level and gives us uncountable results of explaining the dynamic properties of system. Temporal network [2] reduced the level of abstraction to contain essential dynamic information of system. Spectral method and eigensystem decomposition of network adjacency matrix, or of Laplacian matrix, are for the word abstraction method, which reveals the topology properties of network [3] and dynamic properties of system [3, 4]. But we do not have a good framework to combine spectral method and temporal network nowadays. Spreading on network is a scientific problem that wants such framework most. A query on the Thompson Web of Science database, more 1500 papers for the year 2017, shows the importance of spreading in complex network. But the situation of eigensystem explanation for spreading problem does not go well. Valdano et al. reviewed past works and left a negative comment for applying eigenvector centralities method from static contacting network in year 2015 [5], they used statistics of contacting data.

This paper is organized as follows: the next section shows the dynamic equation of spreading. Following that the numerical result of non-time evolution network is shown; followed by the spreading speed of periodical repeating temporal network. Future work is followed by the discussion section. Derivation of the dynamic equation is arranged in the “Appendix”.

## Matrix representation formula for spreading via network

*i*th agent at time

*t*. While the value of \(( \overrightarrow{H} ( t)) _i\) is zero it indicates the situation that the

*i*th agent is in free-form been-spread state, and value one for been-spread state, respectively. The only condition of turning to be been-spread state of each agent is the existence of been-spread neighbour. Since an agent is in been-spread state, this been-spread agent will remain in this state forever. We formulate this spreading dynamic of system with

*N*agents on time as an evoluting network:

*t*, \({\widetilde{A}} \left( t\right) _{i,j}=1\) means that

*i*th and

*j*th agent is connected with a unidirectional link at time

*t*, the value zero \({\widetilde{A}} \left( t\right) _{i,j}=0\) means not linked, respectively. \({\widetilde{I}}\) is a \(N \times N\) identity matrix. The symbol \(\overset{\curvearrowleft }{\prod }\) is left-matrix-product notation, \(\overset{\curvearrowleft }{\prod \nolimits _{l=1}^{m}}\widetilde{A_l}=\widetilde{A_m}\cdots \widetilde{A_2}\widetilde{A_1}\). Where function \(\theta _1 \left( x\right)\) is a simplified form denoting unit step function, \(\theta _1 \left( x+1\right) = \theta \left( x\right)\). The “Appendix” shows the properties of \(\theta _1 \left( x\right)\). The derivation of Eq. (1) is shown in the “Appendix” with these \(\theta _1 \left( x\right) \hbox {s}\) properties.

## Non-time-evolution matrix

*i*th agent is the unique spreading source. In the non-evolution network case, the Eq. (1) can be simplified as \(\overrightarrow{H} ( t) = \theta _1 ( ( {\widetilde{I}} + {\widetilde{A}} ) ^t \overrightarrow{H} ( 0) )\), by employing Eq. (13). The been-spread state of

*j*th agent at time

*t*will be \(\theta _1 ( ( ( {\widetilde{I}} + {\widetilde{A}} ) ^t ) _{ij} )\). This condition is the 100\(\%\) been-spread starting from

*i*th agent as a unique spreading origin:

*k*th eigenpair of matrix \({\widetilde{A}}\) is \(\lambda _k\) and \(\overrightarrow{W}_k\) is, such eigenpairs satisfy the equation \({\widetilde{A}}\overrightarrow{W}_k =\lambda _k \overrightarrow{W}_k\). The

*i*th agent’s eigenvector component in

*k*-eigenvector is \(w_{i,k}\), \(w_{i,k}=( \overrightarrow{W}_1 ) _k\) The eigenpair indexes

*k*are arranged as a descending order: \(\lambda _1 \ge \lambda _2 \ge \cdots \ge \lambda _N\). While \(k =1\), \(\lambda _1\) and \(w_{i,1}\) are called the principal eigenpair. The three similar matrices, \(( {\widetilde{I}} + {\widetilde{A}} ) ^t\), \({\widetilde{I}} + {\widetilde{A}}\) and \({\widetilde{A}}\) share the same eigenvector set. While condition:

### Case 1: Spanning tree

#### Case 2: Circular loop and it with and without an radius link

*N*will partition as four groups, within the edge node of group, the \(S_i\)s have \(N/4+1\) values. Therefore, the agents in the same set share the same value of \(S_i\): {1,9}, {2,16,10,8}, {3,15,11,7}, {4,14,12,6}, {5,13}. The results of Fig. 2 state that: \(S_i=3+i \quad i=1{-}5,\) we can understand them from the following easy examples. In this network, the distance to the farthest agent of 5th and of 13th agent is the same as them in the loop network without radius link \(A_{1,N/2+l}\). The value of \(S_i\) for 5th and of 13th agent remains the same \(S_5=S_{13}=8\). For the 4th agent, the distance to the farthest agent, 12th, gets one step smaller by shifting to the route with the radius link, from the route \(\{ 4 \rightarrow 3 \rightarrow 2 \rightarrow 1 \rightarrow 16 \rightarrow 15 \rightarrow 14 \rightarrow 13 \rightarrow 12\}\) to the route \(\{4\rightarrow 3\rightarrow 2\rightarrow 1\rightarrow 9\rightarrow 10\rightarrow 11\rightarrow 12\}\). These symmetric properties can also be found in our estimator because they are in the principal eigenvector, that can be revealed by calculating higher order perturbations. These correspondences of symmetry are shown in Fig. 3 as five \(\{x,y\}\) points, otherwise it will show more than five points. Comparing to the relation of \(S_i\) and its lower bound estimator \(E^\text {lower}( S_i )\), our estimator has the network symmetric properties and the monotonic relation to \(S_i\).

## Temporal evolution network

The following will show the \({\widetilde{P}}\) and its principal eigenpair for two artificial cases, \(N=3\) and \(N=60\), respectively. The results show that principal eigenpair of \({\widetilde{P}}\) carries the dynamic information during the period \(\tau\).

### \(N=3\), \(\tau =2\) case

### \(N=60\), \(\tau =5\) case

*i*th agent with this condition: \(\mod \left( i,N/N_{\mathrm{group}}\right) =1\), or \(\mod \left( i,N/N_{\mathrm{group}}\right) =0\). For example, first, 6th and 60th agents are the edge agents. We discuss two kind of permutation orders in a \(\tau =5\) window, \(\widetilde{A^{\#2}}\) first and \(\widetilde{A^{\#2}}\) last. In the first kind of permutation order, \(\widetilde{A^{\#2}}\) first, network \(\widetilde{A^{\#2}}\) appears in the first step \({\widetilde{A}}\left( 0\right) = \widetilde{A^{\#2}}\), and \(\widetilde{A^1}\) appears in the following next four steps, \({\widetilde{A}}\left( t\right) = \widetilde{A^{\#1}}\)\(t=1{\text{-- }}4\). This permutation order repeats every \(\tau =5\) steps: \({\widetilde{A}}\left( t\right) ={\widetilde{A}}\left( t-\tau \right)\). The beginning position of permutation order of network \(\#2\), \({\widetilde{A}}\left( 0\right) = \widetilde{A^{\#2}}\), helps the edge agents spread to his neighbourhood group. Other non-edge can spread to his neighbourhood group since next \(\tau\) is repeating. The stronger spread ability to other agents of edge agents has been confirmed by the magnitude ratio principal eigenvector components shown in Fig. 4 as blue points. In the second permutation order, the inter-group link was placed in the last \({\widetilde{A}}\left( 4\right) = \widetilde{A^{\#2}}\), the ability of spreading of edge agents is suppressed. This result can also be understood from a perspective of principal eigenvector components in Fig. 4 as orange points. In summary, the principal eigenvector components of matrix \({\widetilde{P}}\) or \({\widetilde{P}}^\mathrm{T}\) contain the information we need, the ability of spreading and the risk of been-spread, respectively. This eigenvector representation is highly compressed. Post the normalization of eigenvector components, \(\sim w^2_{i,1}=1\), we use \(N-1\) numbers to represent the information among \(\tau N/2\) binary numbers.

## Conclusion

Including highly transitive disease, a generalized formalism of dynamic equation of spreading phenomena in a matrix multiplication expression is shown in Eq. (1). That dynamic equation also states the importance of principal eigenpair. In a non-time evolution network system \({\widetilde{A}}\left( t\right) ={\widetilde{A}}\), principal eigenpair can estimate the step number to the farthest agent \(S_i\). In a time evolution network system \({\widetilde{A}}\left( t\right)\), the risk of been-spread and the ability to spread is illustrated by the principal eigenvector of matrix \({\widetilde{P}}\) and of its transposed one \({\widetilde{P}}^\mathrm{T}\), respectively.

We find the asymptotic degeneracy for principal eigenvalues in “Derivation of the formula”. How the other eigenpair and degeneracy impact spreading phenomena is arranged in our recent studies. We also will apply this method for studying the various epidemic model besides traditional compartmental epidemic agent models [6] and for super-spreading phenomena and target vaccine problem.

## Notes

### Acknowledgements

This research was supported by Japan MEXT as Exploratory Challenges on Post-K computer (Studies of multi-level spatiotemporal simulation of socioeconomic phenomena).

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