Emergency of Tsallis statistics in fractal networks

Scale-free networks constitute a fast-developing field that has already provided us with important tools to understand natural and social phenomena. From biological systems to environmental modifications, from quantum fields to high energy collisions, or from the number of contacts one person has, on average, to the flux of vehicles in the streets of urban centres, all these complex, non-linear problems are better understood under the light of the scale-free network’s properties. A few mechanisms have been found to explain the emergence of scale invariance in complex networks, and here we discuss a mechanism based on the way information is locally spread among agents in a scale-free network. We show that the correct description of the information dynamics is given in terms of the q-exponential function, with the power-law behaviour arising in the asymptotic limit. This result shows that the best statistical approach to the information dynamics is given by Tsallis Statistics. We discuss the main properties of the information spreading process in the network and analyse the role and behaviour of some of the parameters as the number of agents increases. The different mechanisms for optimization of the information spread are discussed.


Introduction
A large number of problems that are common to modern societies can be addressed in the framework of complex networks. Accurate data and methods were made available by new technologies that are used worldwide, providing for the first time the adequate conditions to the development of scientific approaches to those problems. As a consequence, the last decades witnessed the fast evolution of our knowledge on the behaviour and properties of complex networks [1,2].
In this work, we describe and prove some of the most important characteristics of the flux of information in a fractal network. Information, here, is considered in a broad sense, and can refer to pieces of information locally transmitted, or to people or objects that move from one node to another in the network. We say the information is locally transmitted because we consider only those cases where the information is transmitted from one agent to a limited and small number of agents in the same network. These characteristics are present in several of our socioeconomic activities, and in natural systems. We discuss that the variables describing the

Flux of information in a fractal network
In this section, we define what is meant by information and how it flows in the fractal network. The information is spread in the network from one initial agent that possess that piece of information, which is called an informed agent and transmits it or to the agents that are connected to it, or to its internal agents. When the piece of information reaches an uninformed agent it has a probability τ to received by that agent, and when it happens the uninformed agent becomes an informed one. Information can be transmitted only by informed agents, and only uninformed agents can receive it. When an agent receives the piece of information, its first action is to transmit that piece of information to one of its internal agents. In the case of individual agents, it has a probability τ to change its state from uninformed to informed. When a fraction φ of the internal agents are informed, the parent agent is considered informed.
Below we provide a formal description of the form of information spread in a fractal network and prove some of its characteristics. We also discuss some essential aspects of the scaling symmetry break, which will give rise to the q-exponential function and ultimately to the power-law behaviour. where The parameter q is called q-index, and is completely determined by N. Lemma 2.1.1 The number of informed agents after a period of information transmission at a level of the fractal network is given by Proof: According to Axiom 2.1 and to Axiom 2.4 the information is exchanged by N agents where, initially, one of the agents is informed and all others are uninformed. An uninformed agent can get the information by different modes: it can get it directly from the first informed agent in the network, or it can get it from other agents that got the information from the initial agent, or even by more indirect ways. Mathematically, the number of informed agents after a period of information transmission is given by where α is the number of modes for the transmission of the information in the network. As one agent cannot transfer information to itself, we have that α = N − 1. The combinatorial factor arises because when the information is transferred to an uninformed agent, the order in which the informed agent obtained the information is not important. Eq 5 can be written as the power-law in Eq 4, proving this Lemma. Definition 2. 3 We denominate s l;l l 0 , with l 0 > l, the number of internal agents with size, or scale, λ l 0 that an agent at the level l has. Corollary 2.1.1 The number of informed agents can be expressed in terms of the scale parameter, λ l .
Proof: According to Corollary 1.1.3, each agent in a level l − 1 has exactly s lÀ 1;l l ¼ Nl l internal agents with size λ l . Therefore we can write t ¼ ts lÀ 1;l l =ðNl l Þ. It follows immediately from Eq 4 that Corollary 2.1. 2 The ratio σ l, λl 0 /λ l 0 is scale invariant. Proof: If you multiply λ l 0 by any positive, finite factor, due to Corollary 1.1.3 the number of agents at any level l < l 0 is multiplied by the same factor, hence the ratio above remains invariant.
Corollary 2.1. 3 The symbols σ L and λ L+1 are meaningless. Proof: According to Definition 1.7, the level L corresponds to that where agents are individuals, so they do not present an internal structure, hence there is no meaning in asking about its internal population. The individual size is the minimum size and determines the fundamental scale of the fnet, thereby there is no meaning in asking about scales below λ L . Definition 2.4 When the scale is set to the individual size, that is, λ l 0 = λ L , we use the simplified notation s l ¼ s l;l L . Accordingly, we denote by σ the population of individuals in the fnet, that is, σ = σ 0 . Corollary 2.1.4 At the individual level we have λ L = 1, the scale symmetry is broken and the q-exponential function is obtained.
Proof: Adopting λ l = λ L , Eq 7 becomes where we used, for the sake of clarity, s ¼ s 0;l L as the total population of the network, recognized as a multiple the number of individuals in the network. Using the q-index defined by Eq (3) results that The expression above is not scale invariant, because now λ l = λ L is fixed, and any variation of thefnetpopulation, σ, results in a q-exponential behaviour.
This result proves the Theorem 2.1. Notice that with the introduction of the individual level, indicated by the scale λ L , the scale invariance disappears and we obtain according to the q-exponential function.
An additional comment is necessary at this point. Observe that the argument of the functions in Eqs 8 and 9 are different. This results from the transition from a scale-free network to a fixed scale network. In the first case, the population increases according to the size of λ l 0 and the Corollary 1.1.3 is satisfied. In the second case, due to the symmetry break, the population increases while the scale is fixed. The number of close contacts agent is also fixed, as well as the parameter q. This means that the number of degrees of freedom for the information spread is independent of the population size. This is the main aspect of network for the emergence of non-extensivity, as will be discussed in Section 4.

The dynamics of the information spread
In this section, we describe how to describe the dynamics of the information spread by including the time evolution of the number of informed agents. Definition 2. 5 We define the information spread time interval, Δt λ , and the rate of transmission of information, κ λ , such that τ = κ λ Δt λ . Theorem 2.2 The time interval Δt λ depends on the population size, σ λ as a power-law function, that is, Dt l ¼ s b Dt l L , and the transmission rate depends on the population size according Proof: Consider an agent at the scale λ l 6 ¼ λ L . According to Definition 1.1 this agent has N internal agents. The elapsed time for the information transmission to an agent, Dt l l , depends on the time its internal agents will demand to get the information, Dt l lþ1 .
Due to Axiom 1.2 all agents at the same hierarchic level corresponding to λ l+1 have similar values for Dt l lþ1 , so the maximum value for interval for the parent agent is Dt l l ¼ NDt l lþ1 in the case of the information is spread among the internal agents sequentially. The minimum value is Dt l l ¼ Dt l lþ1 , in the case of simultaneous transmission of the information among the internal agents. In the gernal case we write Dt l l ¼ N b Dt l lþ1 , with 0 � β � 1. The equalities correspond to the two special cases mentioned above. As τ is a parameter, according to Definition 1.6 we must have κ λ = N −β κ λ 0 .

Corollary 2.2.1
The parameter β is independent of the agent level. Proof: It follows from the Axiom 1.2.

Corollary 2.2.2
The variables Dt l 1 of an agent at level l 1 is related to the variable Dt l 2 at the level l 2 > l 1 by Dt l 1 ¼ N bðl 2 À l 1 Þ Dt l 2 , and the variable k l 1 is related to k l 2 by k l 1 ¼ N À bðl 2 À l 1 Þ k l 2 . Proof: Applying recursively the result of Lemma 2.2.1 we get The result for k l 1 can be obtained in the same way or by considering that τ is constant, as done in Section 3.1.
Considering the result of Corollary 2.2.2 for an agent at level l 1 = l and another agent at level l 2 = L, we get : Setting l = 0 we have N L = σ, that is the number of individuals in the fnet, these results prove Theorem 2.2. Theorem 2.3 For a randomfnetβ = 0.5.

Proof:
The time interval, Δt, for the information spread for an agent with a number σ of internal agents if δt, is formed by the superposition of the intervals δt for the information transmission to each of the internal agents.
If the interval of transmission for n − 1 of the agents is Δt n−1 , the inclusion of an additional agent might increase the total time spent for information transmission only if the transmission in the nth agent starts at the instant t such that Δt − δt < t < Δt. This condition is satisfied with a probability δt/Δt n , and when it happens the increase in the total interval of time is δt/2, on average, otherwise, the increase is null. Therefore we have, for σ sufficiently large, If η = n/σ, we have 0 � η � 1, and the equation above becomes Integrating from η = 0 to η = 1 we have what proves that β = 0.5.

Differential equations for the information spread
In this section we derive the differential equations governing the dynamics of the information spreading. In what follows we assume that, at any time, the number of agents being informed is much smaller than the total population, therefore the variation of the uninformed population during the elapse of time necessary to the newly informed agents change their states from uninformed to informed is negligible. This can be expressed mathematically by assuming that _ ut � u at any time. Definition 2. 6 The number of uninformed individuals in a population of individuals, u(t), varies along time as more individuals receive the information and become informed agents. The uninformed population at any time is given by u(t) = σ − ν(t), where σ is the population in the fnet, which is considered constant. Theorem 2.4 Given a small time interval Δt, it is always possible to find an agent for which the elapsed time to spread the information is dt < Δt.
Proof: Consider an arbitrary agent at a level l 1 whose spreading time is Dt l 1 . If Dt l 1 < dt, the condition is satisfied and the theorem is proved. If Dt l 1 > dt, using Eq (10), one can find a level l 2 > l 1 at which the agents have a spreading time interval Dt l 2 such that Definition 2. 7 We call smooth information spreading dynamics the process for which the individual spreading time is sufficiently small, so that for any reasonably small time interval Δt, the elapsed time for an individual receive a piece of information, once the individual is reached by the spreading dynamics, is Dt l L < dt.
In what follows we assume the spreading dynamics is smooth. Theorem 2.5 If at time t, measured in an appropriate scale for the dynamics of information at the individual level, afnethas u(t) uninformed individuals, the rate of increase in the number of informed individuals, i(t), is represented by Eq 7, which we write as Proof: When a piece of information reaches an agent at the level L − 1, it is passed to its N internal individuals. In a population, u, of uninformed individuals, the number of agents at this level is because of Corollary 1.1.3. The number of those groups that receive the piece of information in the interval dt is For each group reached by the information, the number of individuals turning to the state informed is given by Eq 4, thus the number of individuals changing their state fron uninformed to informed is given by Using Eq (3) and Definition 2.6 we obtain Eq (16), proving the theorem.
Proof: Deriving Eq (20) and using the assumption € ut � u, we obtain the differential equation given in Eq (16), proving the theorem. Corollary 2.5.2 If a network is formed by N fnets with independent spreading dynamics, the number of informed individuals is iðtÞ ¼ where if t > t oj and i j (t) = 0 if t < t oj . Here, t oj is the time when the information is received by the agent. Proof: It follows directly form Corollary 2.5.1. Theorem 2. 6 The spread of information in an agent can be described by the two coupled differential equations below (For the sake of clarity, we do not use the index j when we refer to the process in a single agent): where t o is the instant when the spread of information starts. Proof: By differentiating Eq (21) we obtain the first equation above. Considering that the uninformed population is determined according to Definition 2.6, the second equation is obtained.
Theorem 2.7 The set coupled differential equations can be written in terms of constant parameters in the case where the function u(t) can be linearized, that is, the coupled equation can be written as where with jh _ uij being the modulus of the average rate of decrease of the uninformed population during the information spreading. is the instant when the spread of information starts.
Proof: Considering the second term on the righ-hand side of the first equation, we have We also have and we identify τ = κΔt. Considering that the distribution of new informed agents is practiacally symmetric with respect to the peak position, we also can approximate Using these results we prove the theorem. Theorem 2.8 When the variation in the total population can be disregarded, the spread of information in a scale-free network can be approximately described by the two coupled differential equations below: where t o is the instant when the spread of information starts. Proof: If jh _ uij � 0, then κ † = 0, and the result is evident from the last theorem. Theorem 2.9 The solution to the coupled equations are

Theorem 2.10 An approximate analytical solution for u(t) can be obtained, resulting in
Proof: An approximate solution can be easily obtained by noticing that, in most cases of interest, we have u(t)/λ � 1. In this case we can approximate the equation for u(t) by that is a separable equation resulting in duðt l L Þ u 1=ð1À qÞ ¼ À Integrating both sides we get This equation results in and can be rearranged to obtain Observe that with the definitions given in the Eq (32) the equation above can be conveniently written as Eq (31), proving the theorem.

Strategies for optimization of the information diffusion
One of the most important results of the investigation of flux of information through networks is the possibility to understand the optimization of the information spread dynamics, what is important both for increasing the efficiency of communication and for formulating the best methods to avoid the information spread.
The main characteristic of the dynamics of information spread in the fractal network studied here is the local transmission of information by a small number of agents with close contact. The question that arises is the following: what is the best way to increase the efficiency of the information spread?
Two mechanisms could be devised to increase the efficiency: improve the probability of transmission, described by the parameters τ or by κ, or increasing the number of contacts between agents, given by q. We will see that the second option, when available, is the most effective.

Theorem 2.11 The increase of the rate of information transmission by increasing the efficiency of transmission is given by
Proof: Using Definition 2.5 in Eq (20) and deriving with respect to τ we have therefore the infinitesimal variation in the number of informed agents when the transmission probability varies from τ to τ+Δτ is Hence, when the transformation τ ! τ 0 = τ + δτ is performed, the number of informed agents transformation is Corollary 2.11.1 In the limit τu/λ L � 1/(1 − q), the transformation of τ leads to a logarithmic increase in the number of informed agents.
Proof: In this limit we have Substituting the result above in Eq (41) we obtain the logarithmic increase of the number of informed agents, i.e., Theorem 2. 12 The increase of the rate of information transmission by increasing the number of links per agent is Proof: Using Definition 2.5 in Eq (20), and deriving with respect to N we have From Eq (3) it follows that From the results above we obtain that, under the transformation q ! q − δq, the number of informed agents transforms as

Discussion of the results
The characteristics of the network presented in Section 1 lead to the formation of a hierarchical scale-free structure typical of scale-free, or fractal, networks. The scaling parameter, λ, is given in terms of the number of internal agents. At some points the agents are considered as individuals with no internal structure, and at this point the scaling symmetry is broken. The locally transmitted information and its spreading dynamics is defined in Section 2. The information is always shared among a number of connected agents in the same network or with the parent agent. This number is limited and constant throughout the network. This characteristic of the information spread dynamics and the broken scaling symmetry of the network structure give rise to a q-exponential function that describes the flux of information in the network. The power-law behaviour is obtained asymptotically, as the number of agents in the network increases. The results obtained here contributes to the discussion about the ubiquity of scale-free networks, since we obtain a heavy-tailed distribution that is not, in general, a powerlaw.
The fact that the number of informed agents is described in terms of a q-exponential function, with the power-law behaviour obtained in the asymptotic limit, is an interesting result and deserves some additional comments. The q-exponential function results from the fact that the number of degrees of freedom of the spreading dynamics [5] is uncorrelated to the number of agents in the network. This aspect of the fractal structure allows the number of individuals increase without any change in the number of modes by which an arbitrary agent can exchange information with another in the fnet. It is easy to understand, from the results obtained here, why fnets can describe so many aspects of natural and social systems: in many cases the information is transmitted locally among a small group of agents, and this number will be the same, no matter how many individuals one are in that population of the network.
The Theorems proved in Section 2 show that any variable describing some quantity related to the information spread must appear in a scale-free form. The time interval for the spread of information in the network, for instance, increases with the squared-root of the number of individuals in the network. This result is in agreement with the works in Refs. [12,13], where the close connections between power-law distributions and scale-free networks is observed. In the present case, we show that any fractal network will depend only on power-law variables.
The rate of information spread given by the q-exponential function shows that the spread dynamics results in a slower transmission of information than one would expect in an exponential spread. But as q ! 1 the number of links among agents increases and the exponential behaviour is recovered. This corresponds to broadcast information, with chaotic transmission of information to all the individuals in the network. We verified that the most efficient way to increase the number of informed agents is not by increasing the transmission probability, but by increasing the number of connections among agents.
This result is interesting in many aspects, but here we would like to emphasize one of them with an example of application in an epidemic spread of a virus. The piece of information being transmitted is the virus, and the transmission happens in close contact between an infected individual and a susceptible individual, that is, one that does not carry that piece of information, the virus. Observe that the coupled equations in Theorem 2.7 are very similar to those found in the standard SI and SIR models [40], except that we did not consider in those equations the recovered population, what can be straightforward done by considering that the total population is the sum of each kid of population, infected, susceptible and recovered, and that this population is constant. In Fig 2 we show plots of the informed and uninformed population along time, as well as the number of individual that receives the information at each instant.
We can understand the aspects related to the best strategies for the spreading dynamics also in the context of contagious diseases. A virus undergoes random mutation, and the dominant strain will be more likely the one that can be transmitted more effectively. The way mutagenesis of virus can lead to a more effective spread is not by increasing its probability of transmission, which we associate with the parameter τ, but by increasing the number of susceptible individuals in contact with the infected individuals, which we associate with N. Those strains that succeed to increase N will be more effective in transmission, and therefore will be dominant. Thus, viruses will increase the multiplication factor more efficiently if they succeed to provide a longer transmission time before the symptoms of its associated disease become evident.
As mentioned in the introduction, the definitions given in Section 1 and in Section 2 can be relaxed in many ways. For instance, the number of internal agents can be set as variable, but must follow the same distribution whatever is the fractal level of the parent agent, and the corresponding variable must be scale-free, i.e., it must appear as fractions of the scaling parameter. The same reasoning applies to the number of edges linking the agents. The information spread can follow an arbitrary distribution instead of being completely random, as far as the distribution is scale-free. Even the number of modes, or degrees of freedom, by which the agents can obtain information from the others in the same network do not need to be constant. If these modifications are introduced in the scale-free network presented here, as far as the scale symmetry is preserved, our conclusions should hold. Even if different numbers of edges among the nodes are used throughout the network, a multifractal network may be obtained. In all these cases, however, the general results obtained here will remain valid.

Conclusion
In this work we studied the spreading dynamics of information locally transmitted through nodes, or agents, in self-similar, or fractal, network. The fractal network is defined by its fine internal structure that is scale-free. The self-similarity is a consequence of the the scaling property of the network and of its fine internal structure. However, at some point the scale symmetry is broken, and as a result the flux of information follows a q-exponential function, typical of the Tsallis' statistics. The pure power-law behavior results in the asymptotic limit, when the number of informed agents in the network is large. The exponential behavior, on the other hand, is obtained in the asymptotic limit of the number of connection of each agent increasing indefinitely.
The locally transmitted information, which goes from one agent to its neighbours and involves a limited number of nodes, independent of the total number of agents in the network, was studied. Its spreading dynamics reveal that the number of informed agents increases according to a q-exponential function. From the statistical point of view, this result indicates that the Tsallis Statistics is the correct framework to investigate the scale-free network. The constraint on the number of contact of each agent implies in an increasing number of levels in the network as the population increases. This is contrary to the small-word hypothesis [1,2], where the number of levels in the network is fixed and the number of contacts increases.
The time interval for the transmission increases as the squared-root of the number of individuals in the network. The scaling properties, establishes a power-law condition to the probability of transmission of a piece of information by agent in the network. Our results can be easily tested in real or simulated data by checking the characteristics of the distributions.
Differential equations describing the information spread are derived. We discussed the different strategies one can take to increase the information spread efficiency. These strategies can be formulated by increasing the transmission efficiency or by the number of connected agents. We show that the most effective strategy is the second one.
The results obtained in the present work have an impact on the formulation of the best strategies for the information spread. In practice, it may have implications on Environment Sciences [41], since modifications in the local environment may evolve by locally transmitted effects to larger and distant areas [42][43][44]; Epidemiology, since viruses may evolve, by mutagenesis, to variants that optimize its transmission, a track that would prefer increasing the number of degrees of freedom by extending the period of virus transmission rather than increasing its transmission rate [45,46]; Sociology [8], since communication among individuals in the society can be made more or less effective by controlling the mechanisms of spreading, what may have an impact in policies and strategies to, e.g., combat fake-news and other irrational behaviours in social media [47][48][49][50]. In Computer Sciences [15,33], Biology [14], Physics [21], Economics [9], and Machine Learning [51], to name just a few.
The model of fractal network studied here can be modified in some aspects without changing the conclusions.