Scaling relations emerged from message diffusion in directed dynamical small-worldnetworks

Chengping Zhu (1)(2), Shi-Jie Xiong(2), Yingjie Tian(3),Nan Li(3), Ke-Sheng Jiang(3)

(1)Department of Applied Physics, Nanjing University of Aeronautics and Astronautics, Nanjing, China.(2)Solid State Microstructure Laboratory and Department of Physics, Nanjing, China.(3)College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing, China.

Background

• We live in networks, actually complex networks. Here we use the word “network” in its most general meaning.

• In a complex network, sites (nodes) represent agents, and links (edges) represent correlation between them.

• Depending on problems discussed , complex network cancover various entities at different levels , such as individualpersons , companies , airports , ecological species , www , even proteins or genes.

• Small-world Network (SWN) reproduced by D. J. Wattz and S. H. Strogatz provides us with an proper platform for understanding processes in many natural and social systems.• Examples of SWN (1D):

Fig. 1. (a) a one-dimensional circular lattice: each site has links with 2zadjacent neighbors. (b) a rewired lattice : a fraction p of links in the graph (a) is randomly chosen , moving one end of each to a randomly selected new site , thus to make up a SWN . (c) another type of SWN, with randomly added links to the original lattice.

We concentrate on rewired SWN of type (b) .

• Previous research on SWN mainly concentrates on static models. People pay less attention to either dynamical updated relations among sites or randomness of site-evolution , and most work related with un-directional relation between sites , except for a few researchers investigating food webs or Internet.

Directed SWN: [1] J. M. Montoya et al. J. Theo. Biol. 214, 405 (2002) [2]M. E. J. Newman et al. Phys. Rev. E 66, 035101(R), (2002)

• Small-world Effect in complex networks is characterized by two quantities, average path length l(N,p) and clustering coefficient C.

Where, )/(),( ** NNFNPNl

1ln

1)u(F

uifu

uifu

d/1* p~Nand d is the dimension.

• For reviews, see: [3]R. Albert and A-L. Barabasi, Rev. Mod. Phys. 74, 47 (2002) [4]S. N. Dorogrovtsev and J. F. F. Mendels, Adv. Phys. 51, 1079(2002) [5]M. E. J. Newman, SIAM Review 45, 167 (2003) or, cond-mat/0303516.

Motivations

• When we think about real systems, we noticed that many problems can be accounted as message diffusion process in which there show three prominent features: dynamically changed relations, directed links ,random responses.

• Interactions in social or biological systems are not static but evolve in time. • Suppose an exclusive shop sells out a new brand of portable commodity without advertisement. The message of it can be spread out by showing it in the population, and every new buyer can be a source of the message. After purchasing the commodity, he/she changed the role in sales problem, from a receiver to a propagator of the message. Meanwhile, relations among people change continuously, they are not fixed due to ones’ consumptive activities.

(1) Dynamical

• Consider epidemics, a contact-infectious disease with all possible incubation time propagates through an acquaintance network starting from an index patient. By incubation time we mean the period one is infected- but-not-infective. When an infected one finally becomes infective, he changes his role to all others. In some cases, people continue their ordinary activities even they are not aware of their infection promptly. • In most information or disease diffusion processes, effective interactions

are uni-directed. e.g. infection can only passes from infectives to susceptibles, all other types of links related with persons are inactive to an epidemic problem.

• In food webs, prey-predator relations are directional, too. If the environment was polluted, the effect can be transported along trophic links from basalest species to upper ones.

(2) directed links ：

(3) random responses ：

• Incubation time of people to the same infectious disease varies depending on ones’ resistance to it. In parallel, the delaying interval from getting a message of the new commodity to practically buying it also varies according to the spending potential of particular buyers. In one word, there always exists random responses in such dynamical processes.

The model(1)We put a spin-like variable at every site of a SWN. A site is

randomly chosen as the seed of the spin-up state, all other sites are in their primitive state--spin-down state. Any site in spin-up state can send out message demanding its receivers to convert their spin up.

(2)Consider the random relaxation of every site. Let{ } follow certain distribution and keep unchanged with time. We have

tested with uniform distribution and discrete Poisson distribution.

(3)Let connections in SWN be updated at every moment, and keep the rewiring fraction p of links the same.

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(4)Spins at every site evolves their states in parallel. Only when a site of spin-down state is at the end of its relaxation time after receiving the message, it flips up and keeps the state, sending out the same message; otherwise, it remains down.

Message in this dynamical SWN with random responses can only be transmitted from spin-up state to down state.Therefore we have a Directed Dynamical Small-World Network (DDSWN) model .

Numerical results and discussion

(1) To investigate how fast and how wide the message spreading

out in each moment, we use average spreading length <L>

and average spreading time <T>, which are arithmetic average of

path length and time to get every receivers at moment t, we

did average over at least 50 configurations of { } and sites of

seed. Numerical results for uniform distribution are plotted in

Fig.2 and Fig.3.

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Fig.2

In upper-left inset of Fig.2, we see that <L> increases monotonically towards its P-dependent saturation value <L>s, which shows the finite-size effect of p. Since<L>s are independent on time, they can be used to probe the properties of globalDDSWN. In main panel of it we plot log(p<L>s) versus log(PN) to check the validityof original relation of for undirected static SWNs. The straight lines indicate that ɭ<L>s versus p in a power law. The lower right inset says,the lnN relation is kept well. Therefore we have

NpL Ls ln~

And, Fig.3 says, we have the same form for <T>s only with the power index changed.

Fig.3

Fig.4

(2) We take a spin-like variable S(t) to count the ratio of all sites with their spin flipped. It is found to behave in a duple scaling form

where q is the external parameter to represent the diversity of distribution { }, and its average value. The scaling behaviors are illustrated in Fig.4 and Fig.5.

)~(~)( tqpftS

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The derivation of s(t), s’(t) is meaningful, it reflects the probabilityfor any site to flip up at moment t. Simulations revealed that s’(t)follows Gaussian distribution approximately. It diffuse along the t-axi just as a wave-packet diffuses along x-axi with time increased.

Fig.5

Actually, partial evidences have been found in real food webs [6],and quantum diffusion simulations [7]

[6] D. Garlachelli et al. Nature, 423, 165 (2003)[7] B. J. Kim et al. Phys. Rev. B 68, 014304(2003)

(3) The distribution function of { } is actually phenomenological , it should be tuned to fit for practical problems in real systems. We have tested our scaling relations with several distributions, just yield corresponding power exponents. That implies the scaling forms are quite robust in certain variation of distributions of response time. It provides us with large potential for its interdisciplinary applications in many fields.

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Fig.6

Fig.7

Thank you!