aa r X i v : . [ phy s i c s . s o c - ph ] A p r Tenement house model
Wojciech Ganczarek ∗ Institute of Physics, Jagiellonian University, ul. Reymonta 4, 30-059 Krak´ow, PolandInstitute of Mathematics, Jagiellonian University, ul. Lojasiewicza 6, 30-346 Krak´ow, Poland (Dated: November 24, 2012)Most of the common used models of epidemic spreading allow contaminating many neighbors ofa particular node in the network. They are usually analyzed by differential equations on probabilityvectors. We propose a model of epidemic spreading, which restricts to at most one contaminationper time step and analyze it by discrete approach, working on vectors of possible states of thesystem. Theoretical predictions of epidemic treshold, stationary state and time needed to reach itare given and appear to be perfectly consistent with computer simulations. We also point ou thatthe model appears to be well suited to mimic epidemic spreading within student communes.
PACS numbers: 87.23.Ge, 87.19.X-
I. INTRODUCTION
It has been admitted, that the most appropiate mod-els of epidemics spreading are these based on dynamicalprocesses on particular graph models rather than thosedefined by fenomenological differential equations [1, 2].Within this approach the nodes of a network are usu-ally considered as individuals, who are connected witheach other by vertices corresponding to social links. Al-though some authors use continous time simulations (seee.g. [3]), the approach presented commonly (see [4] fora review) is based on the idea that at each discrete timestep a particular node of the network can contaminateeach of its neighbors with some finite probability p . Thewhole set of vertices is being divided into compartments,usually referred to susceptible (S), infected (I) and recov-ered (R) individuals, but the general mechanism staysmore or less unchanged. There has been a broad rangeof methods developed in order to analyze such models.In the most basic approach people assume individualsto be identical and homogeneously mixed (homogeneousassumption, [4]). In order to take into account hetero-genity of the system a kind of block approximation haxbeen used [5], treating nodes with the same degree asstatistically equivalent. This is not always enough, assome real networks manifest degree correlation, mainly:the conditional probability, that two vertices of degree k, k ′ are connected depends on both degrees k, k ′ [6].The next step thus is to take into account correlation[7]. Finally, one can employ whole adjacency matrix de-scribing the graph we analyze [8–10]. The validity of allthese approaches is still under investigation, see e.g. [11].Note however, that all these variations listed above workon equations describing relationships between probabilityvectors. In particular, for the last example, the system isbeing described by p i - probability, that i-th node is in-fected. The problem, however, is that there is not a singlemoment when a particular verte is - say - 0.41 infected. ∗ [email protected] A vertex can be either infected (1) or not (0). This prob-lem has been already noticed by Petermann and De LosRios [12].In this paper we introduce another model of epidemicspreading and analize it with complitely different ap-proach. Let us focus on sexual transmitted diseases. Forthis case the assumption that a particular node is ableto contaminate more then one of its neighbots during atime step seems not to be the most suitable one. Bearingthis idea in mind we develope a single infection epidemicspreading model.This paper is organized as follows. In section II wedescribe proposed model of epidemics spreading with atmost one infection per time step. In Section III the theo-retical analysis of the model: epidemics treshold, station-ary state and mixing time, is being investigated. Simula-tions are presented in section IV. In Section V we drawthe final considerations.
II. MODEL DESCRIPTION
Consider a connected, unweighted graph with n ver-tices enumerated by indices i = 1 , . . . , n , described bytransition matrix { P ij } , P nj =1 P ij = 1. The model willbe of SIS kind: all the individuals are at the beginningconsidered as susceptible (S). After contamination theybecome infected (I) but they still have a chance to recoverand be susceptible again.We start thus with the all but one nodes susceptible.The one which is infected is chosen at random. The wholeprocess consists of 4 actions which we repeat at each dis-crete time step. At each time step we choose randomly,with identical probability n , a node - say - i -th one. Thenwe choose its neighbor according to the transition matrix { P ij } , i.e. there is P ij chance that we point j -th vertex.If one of these two individuals i, j is infected, it contam-inates the second one with probability z . At the end ofeach time step we recover each infected node with prob-ability r .Note, that this method restricts not only each infectednode to contaminate at most one of its neighbor. In factwe restrict all the dynamics to at most one contaminationper time step. One can say it is not realistic approach.However, from the one hand we can say that this couldbe a basis for further generalisation. From the other:we can imagine, and even find in reality, systems thatfulfill assumption described above. In large, academiccities there are often big flats situated in old tenementhouses, settled by quite large amounts of students, wholive with 3-4 roommates per chamber. As there is nospace for privacy in this way of living, they sometimesdevote one room in the flat to be a so-called sexroom , socontamination by sexually transmitted diseases can takeplace at most once per time step (say: per hour). Thisseems to be a good example of a system which can bedescribed by our model. III. MODEL ANALYSIS
In order to mathematically describe the model we de-fine X j ( t ) which takes the value 1 if the node j is beingcontaminated by one of its neighbors at the time step t , and 0 otherwise. Furthermore, we denote the set ofall infected nodes at the time step t by I ( t ). We willbe interested in the expectation value of X j ( t ) with acondition that the set of infected nodes consists of someparticular vertices.There are two independent ways of contaminating j -th node during one time step. Either we choose j -thnode (with probability n ) and then one of its infectedneighbor (with probability P k ∈ I ( t ) P jk ) or we choose j -th node’s neighbor (with probability n for each one) andthen we pick j -th node (it happens with probability P kj for a particular node k , so P k ∈ I ( t ) P kj for all of them).Summing up we obtain: E ( X j | I ( t )) = zn (cid:16) X k ∈ I ( t ) P jk + X k ∈ I ( t ) P kj (cid:17) , (1)where both terms are multiplied by contamination prob-ability z . We are, however, interested in the behaviourof whole system, not one node only.Let us thus define D ( t ) - expectation value of changeof the number of infected nodes. Due to additivity ofexpectation value we can write: D ( t ) = E ( | I ( t +1) |−| I ( t ) | | I ( t )) = X j / ∈ I ( t ) E ( X j | I ( t )) − r | I ( t ) | , (2)where, apart from adding all E ( X j | I ( t )) terms, we sub-stract the term responsible for healing: number of in-fected nodes multiplied by recovery probability r . UsingEq. (1) we immediately conclude: D ( t ) = zn (cid:16) X k ∈ I ( t ) , j / ∈ I ( t ) P jk + X k ∈ I ( t ) , j / ∈ I ( t ) P kj (cid:17) − r | I ( t ) | . (3) The equation above defines the dynamics of the model:by solving it one could provide the complete informa-tion about the process. Unfortunately, in general sums P k ∈ I ( t ) , j / ∈ I ( t ) P jk , P k ∈ I ( t ) , j / ∈ I ( t ) P kj are not preciselyknown as they strongly depend on the shape of the set I ( t ). We will show, however, that we are able to de-rive exact result for epidemic treshold for any graph andstationary state for some special cases. A. Epidemic treshold
Our first aim is to find out the epidemic treshold forthe process described above. We are interested in somerelation of model parameters n, z, r that defines a bor-der between two situations: dropping and rising of thenumber of infected nodes in the beginning of the process.We are going to analyze Eq. (3). We have there twosums that look similar, so the first idea would be toadd them somehow. But in general P k ∈ I ( t ) , j / ∈ I ( t ) P jk = P k ∈ I ( t ) , j / ∈ I ( t ) P kj , so we cannot that easily simplifythis equation (except for { P ij } - bistochastic). How-ever, in order to find epidemic treshold, we are inter-ested in the behaviour of the system in the vicinityof t = 0. Let us thus take | I ( t ) | = 1 then, as ithappens at the very beginning of the evolution, anddenote the only infected neighbor by index l . Then P k ∈ I ( t ) , j / ∈ I ( t ) P kj = P j / ∈ I ( t ) P lj = 1 (as { P ij } - stochas-tic) and P k ∈ I ( t ) , j / ∈ I ( t ) P jk = P j / ∈ I ( t ) P jl . Now we usethe fact that at the beginning the first infected node isbeing chosen uniformly at random. Therefore the lastterm should be averaged over all possibilities of choos-ing l : n P l ∈ V P j / ∈ I ( t ) P jl = n P j / ∈ I ( t ) f racn − n ,where we used once again the fact, that { P ij } is stochas-tic. Finally we write the condition D ( t ) ≥ D ( t ) ≤ zn × (1 + n − n ) − r ≤ . (4)For large n the term n − n can safely be substituted by n . The condition for epidemic treshold for the model weproposed is thus: zr = n . (5) B. Stationary state
Let us now turn to stationary state problem. Themodel being analized is by definition a purely Marko-vian one and above the epidemic treshold we anticipateour system to stay at some non-zero stationary state, i.e.we expect that the number of infected nodes will, in longtimes, oscillate about a fixed value. Practically however,due to statictical flucutation in finite real or simulationalsystem, the epidemy may die out even above the treshold.The stationary fraction of infected nodes in generalcase (not specifying any particular shape of the graph) isnot as easy reachable as the treshold calculated in the lastsection. What we basically have to do is to use once againall the formalism presented above and find the solutionfor the equation D ( t ) = 0 without the constraint | I ( t ) | =1. The problem is to compute the sum P k ∈ I ( t ) , j / ∈ I ( t ) P jk - a task which is not trivial. We will thus estimate onlystationary state for general case. In later subsections wegive exact solutions for special cases of complete graphand uncorrelated homogenous graph.In order to perform estimation of the stationary state,we introduce the notion of graph conductance [16]: Definition 1.
Conductance of a given graph G describedby a stochastic matrix { P ij } is: Φ( P ) = min S ⊂ V P j ∈ S, k / ∈ S P jk min {| S | , | V − S |} , (6) where V is the set of vertices of a graph G. This quantity measures how well-connected a givengraph is. Due to the definition above we will analyzeseparately cases with the stationary fraction of infectednodes i s = | I s | n smaller and greater than .Consider first i s ≥ . Then also | I s | ≥ n − | I n | and,using Eq. (3), we lowerbound D(t): D ( t ) ≥ zn Φ( P )( n − | I | ) − r | I | . (7)Bounding the latter expression in Eq. (7) from zero wefind that D ( t ) is positive for ≤ i ≤ rn z Φ( P ) , thereforethe stationary fraction i s must be higher than this: i s ≥
11 + rn z Φ( P ) . (8)Let us now focus on the opposite case, mainly i s ≤ , | I s | ≤ n − | I n | . We again lowerbound D(t) using Eq. (3): D ( t ) ≥ zn Φ( P ) | I | − r | I | ≥ zn Φ( P ) | I | − r ( n − | I | ) . (9)Bounding right hand side of Eq. (9) from zero, we con-clude analogically to the situation above: i s ≤
11 + z Φ( P ) rn . (10)This result, however mathematically correct, appears tobe quite useless: the value of Φ( P ) is usually much lowerthan the sums that it approximates ( P k ∈ I ( t ) , j / ∈ I ( t ) P jk , P k ∈ I ( t ) , j / ∈ I ( t ) P kj ) during the process. Let us thus workout exact results for some special cases. C. Special cases
1. Complete graph
For complete graphs, i.e. graphs with all possible linkspresent, we easily find the exact solution of stationarystate problem. Note, that for this special case: X k ∈ I ( t ) , j / ∈ I ( t ) P jk = X k ∈ I ( t ) , j / ∈ I ( t ) P jk = | I ( t ) | ( n − | I ( t ) | ) n − , (11)as each of | I | infected nodes is linked to each of ( n − | I | )susceptible nodes by an edge chosen with probability n − as each node has ( n −
1) neighbors. We can thus findexplicit and exact condition for D ( t ) = 0. From Eq. (3)we get: i ( t ) s = 1 − r ( n − z . (12)
2. Uncorrelated homogenous graph
Let us consider now hypothetical uncorrelated homoge-nous graph. The term ”uncorrelated” stands for the fea-ture that the probability that an edge departing from avertex of degree j points on a vertex of degree k is inde-pendent from the degree of vertex j . By ”homogenous”we mean that average number of connections betweensets of vertices of some fixed sizes depends only on thesesizes, not on the actual constituents of those sets.Bearing these assumptions in mind let us compute ex-pectation values of the two sums from Eq. (3): E ( P j ∈ I ( t ) , l/ ∈ I ( t ) P lj ) = E ( k ) n − P j ∈ I ( t ) , l/ ∈ I ( t ) E ( k | k ≥ E ( k ) n − E ( k | k ≥ | I ( t ) | ( n − | I ( t ) | ) , (13)where we put E ( k ) / ( n −
1) for the expectation value ofexistence of link between two vertices. We substract 1from n as a node cannot be connected with itself. Thestationary infected nodes density comes to be: i s = 1 − r ( n − z h k ih k i , (14)where we denote h k i = E ( k ) and h /k i = E (1 /k ). Specif-ically, for G ( n, p ) random graph (with the well-knownbinomial degree distribution) the product of h k ih k i goesto 1. In this case the latter result (14) recovers the solu-tion for complete graphs (12). Moreover, G ( n, p ) graphsare indeed uncorrelated in the limit of large n [13], so weexpect G ( n, p ) behaving like complete graphs for large n . D. Mixing time
In this chapter we will be interested in mixing timedescribed in this article, i.e. the time needed by the pro-cess to reach the stationary state. Strictly speaking, thisis kind of meta-stable stationary state, as in simulationson finite networks the only absorbing, stable state is thesituation when the number of infected nodes is zero. Itis clearly visible on Fig. 2, that we can distinguish tworegimes with different behaviour: the regime of rapid in-crease in the number of infected nodes and the regimeof stabilization. Let us state and prove a general the-orem restricting mixing time for any graph. The proofis inspired by related considerations for gossip spreadingdone by Shah [14].
Theorem 1.
Let P be a stochastic transition matrix ofa graph G of the size n . Then the mixing time T for theprocess described above fulfills: T ( ǫ ) = O (log n + log ǫ − ) . Proof.
We devide the proof into two parts, consideringseparately two stages of the process evolution: | I ( t ) | ≤ n and | I ( t ) | ≥ n . • | I ( t ) | ≤ n We recall first the general result for i s ≤ stated in Eq.(9): D ( t ) ≥ zn Φ( P ) | I ( t ) | − r | I ( t ) | . Denote now by Λ the smallest time t such that the num-ber of infected nodes exceeds n :Λ = inf { t : | I ( t ) | > n } , Λ ∧ t = min(Λ , t ) . Note, that as long as | I ( t ) | ≤ n , we have Λ ∧ ( t + 1) =Λ ∧ t + 1. Recall now the general feature for any convexfunction g , x , x ∈ R : g ( x ) ≤ g ( x ) + g ′ ( x )( x z − x ) . (15)Let us take: g ( x ) = x , x = | I ( t + 1) | and x = | I ( t ) | ,then: 1 | I ( t + 1) | ≤ | I ( t ) | − | I ( t + 1) | (cid:16) | I ( t +1) |−| I ( t ) | (cid:17) . (16)By construction of the process we have: | I ( t + 1) | ≤ | I ( t ) | + 1 = d | I ( t ) | , where 1 ≤ d ≤
2, but as | I ( t ) | = O ( n ) for n big enoughthe constant d can be arbitrarily close to 1. Now wecontinue with Eq. (16): | I ( t +1) | ≤ | I ( t ) | − d | I ( t ) | (cid:16) | I ( t + 1) | − | I ( t ) | (cid:17) ≤ | I ( t ) | − d | I ( t ) | (cid:16) zn Φ( P ) | I ( t ) | − r | I ( t ) | (cid:17) ≤ (17) | I ( t ) | (cid:16) − ( zn Φ( P ) − r ) d − (cid:17) ≤ | I ( t ) | exp( − d ( zn Φ( P ) − r )) , where in the second line we used Eq. (9) and the defini-tion of D ( t ), Eq. (3). In the last line we used the factthat 1 − x ≤ exp( − z ). Let us now define: ζ ( t ) = exp( at ) | I ( t ) | , (18) where a = d ( zn Φ( P ) − r )We show that ζ ( t ) is a supermartingale, i.e. E ( ζ ( t ) |{ ζ ( s ) : s ≤ t ′ } ] ≤ ζ ( t ′ ) ∀ t ′ ≤ t . As the only component of ζ ( t )which is a random variable is I ( t ) and as the process weanalyze is Markovian and as Λ ∧ ( t + 1) = Λ ∧ t + 1, it isenough to show that E ( ζ (Λ ∧ ( t + 1)) | I (Λ ∧ t )) ≤ ζ (Λ ∧ t ).We do it using Eq. (17): E ( ζ (Λ ∧ ( t + 1)) | I (Λ ∧ t )) =exp((Λ ∧ t ) a ) exp( a ) E ( | I (Λ ∧ t +1) | | I (Λ ∧ t )) ≤ (19)exp((Λ ∧ t ) a ) exp( a ) | I (Λ ∧ t ) | exp( − a ) = ζ (Λ ∧ t ) . As ζ ( t ) is a supermartingale we conclude that E ( ζ (Λ ∧ t )) ≤ E ( ζ (Λ ∧ | I ( t ) | ≤ n : ζ (Λ ∧ t ) ≥ n exp((Λ ∧ t ) a ) , (20)and directry from it we conclude that: E (exp((Λ ∧ t ) a )) ≤ n E ( ζ (Λ ∧ t )) ≤ n , (21)where in the last step we used the supermartingale prop-erty. Moreover, as exp((Λ ∧ t ) a ) ↑ exp(Λ a ) as t → ∞ , wehave also: E (exp(Λ a )) ≤ n . (22)Finally, let us recall the Markov inequality: P ( | X | ≥ c ) ≤ E ( | X | ) c (23)and choose t = a (ln( n ) − ln( ǫ )). Then we straightfor-wardly get: P (Λ > t ) = P (exp( λa ) > nǫ ) ≤ E (exp( λa )) nǫ ≤ ǫ . (24) • | I ( t ) | ≥ n For this case we perform exactly the same procedure,but starting from Eq. (7) instead of Eq. (9), which westarted with in the previous case. Following the samesteps as above we only change constant a in Eq. (18)into b = d ( z | I s | Φ( P ) − zn Φ( P ) − r ), where explicitelyappears the number of infected nodes at the stationarystate. Second thing that has to be changed is Eq. (20)where, instead of n we can put n . Resulting time for thisstage is: P (Λ > t ) = ≤ E (exp( λb )) nǫ ≤ ǫ, (25) where t = b (ln( n ) − ln( ǫ )) . From this general theorem we conclude, that the closerwe are with chosen parameters to the zero–stationarystate (i.e. the smaller is the stationary density of infectednodes), the slower is the first phase of rapid increase:
Corollary 1.
Mixing time is linear with inverse of thedistance η from the epidemics treshold, i.e.: T ( η, ǫ ) = O (cid:16) η (log n + log ǫ − ) (cid:17) . Proof.
Recall Eq. (3): we demand D ( t ) ≥ znr ≥ | I ( t ) | (cid:16) P k ∈ I ( t ) , j / ∈ I ( t ) P jk + P k ∈ I ( t ) , j / ∈ I ( t ) P kj (cid:17) , (26)which boils down to equality for stationary state. We de-note right hand side of this equation by p c for the smallestpossible situation, i.e. for epidemics treshold. Now let ustake values of parameters z , n and r such that: znr = p c (1 + η ) , (27)where η ≥
0. Now we recall some parts of the proofof Theorem 1. Actually, all we have to do is to rewritecondition for D ( t ) in parametrization given in Eq. (27)and notion of p c : D ( t ) ≥ r (cid:16) znr p c | I ( t ) | − | I ( t ) | (cid:17) = rη | I ( t ) | . (28)We put this result into Eq. (17) obtaining:1 | I ( t + 1) | ≤ | I ( t ) | exp( − rηd ) , (29)end then we proceed in the same way as in the proof ofTheorem 1. The result is P (Λ > t c ) = ≤ E (exp( λ rηd )) nǫ ≤ ǫ, (30) where t c = d rη (ln( n ) − ln( ǫ )) . IV. SIMULATION
Here we present simulations for stationary state of var-ious types of networks, i.e. complete graph, G ( n, p ) ran-dom graph [15], Watts-Strogatz small world graph [17]and graphs with power law degree distribution (scale-freenetwork, see e.g. [2]). Computer-simulational investiga-tions focus on the topics described theoretically in thelast section, i.e. epidemic treshold, stationary state andmixing time.
20 40 60 80 100 120 1400.00.20.40.60.81.0 n z c FIG. 1. Epidemic treshold for four different type of graphs:dots stay for simulational results, lines present theoreticalprediction, Eq. (5). Starting from the bottom we have re-sults for G ( n, p ) random graph (green line, r = 0 . r = 0 . r = 0 .
01) and small world graph (orange line, r = 0 . A. Epidemic treshold
We check here the behaviour of the process in the verybeginning, i.e. exactly at the first time step. Four kindsof networks are being examined: complete graph, G ( n, p )random graph with p = 0 .
5, small world graph with k = 6 neighbors on the circle and rewiring probability p = 0 . α = 2 .
5. We vary sizes of networks n and for eachtype of the graph we choose different recovery probability r . Looking for the critical value of contamination prob-ability z c we change parameter z and check for whichvalue the fraction of infected nodes starts to increase.This procedure is being repeated 100000 times. Resultsare presented in Fig. 1. Visibly, simulations follow thetheoretical prediction Eq. (5) prefectly for all four kindsof graphs being examined. B. Stationary state
Results for stationary state are obtained by perform-ing many runs (typically 1000), finishing each of themat a fixed, long time step (10 000 - 100 000), cutting thebeginning phase of rapid increase and fitting a line to thepoints oscillating about the stationary state. There aretwo types of results which we can end up with after a sin-gle run: epidemics either dies at a certain point (i.e. num-ber of infected nodes, due to fluctuations, reaches zeroand - by construction of the model - stays zero, usuallyit happens at the very beginning of the process) or num-ber of infected nodes increases rapidely in the first stage,and then oscillates over some fixed value (see Fig.2). Wecall this value stationary state (presicely, as we have al-ready noted in Sec. III D, meta-stable state). In order to t i H t L FIG. 2. An example of a single run for z =1, r =0.005, randomgraph G ( n, p ) of the size n =100 and p =0.5.
50 100 150 2000.40.50.60.70.80.91.0 n i s FIG. 3. Plot of stationary state value of infected nodes denstiy i s for complete graphs versus network size n : simulation (bluedots) and theoretical result (12) (green line). We fix here z =1, r =0.005. compute average stationary state we neglect all the runswhere there exist such a time step, when the number ofinfected nodes equals zero.First we examine complete graphs, as in the last sectionwe provided the exact result for them (12). In Fig. 3 weshow how stationary infected nodes density i s depends onnetwork size n . Then, in Fig.3, we show dependence oncontamination probability z . Both figures show perfectagreement between simulation and theory, Eq. (12).As we have already seen the behaviour of completegraphs and how they relate to the theory described above,let us compare stationary state i s for four different kindsof graphs. In Fig. 5 we show the results for completegraph, G ( n, p ) random graph with p = 0 .
1, small worldgraph with k = 10 neighbors on the circle and rewiringprobability p = 0 . α = 2 .
5. Sizes of the graphs are fixed, n = z i s FIG. 4. Plot of stationary state value of infected nodes denstiy i s for complete graphs versus contamination probability z :simulation (blue dots) and theoretical result (12) (green line).We fix here n =100, r =0.005. æ æ æ æ æ æ ææ æà à à à à à àà àì ì ì ì ì ì ìì ìò ò ò ò ò ò òò ò z i s FIG. 5. Plot of stationary state value of infected nodes den-stiy i s for complete graph (blue circles), G ( n, p ) random graphwith p = 0 . k = 10neighbors on the circle and rewiring probability p = 0 . α = 2 . z and fixed network size n = 100. G ( n, p ) and small world graphs only.In Sec. III C 2 we concluded, that G ( n, p ) graphs forlarge n should resemble like complete graphs. It is in-structive to see that in the limit of large n , epidemics,not only on G ( n, p ), but also on small world graphs be-haves the same as on complete graphs, see Fig. 6. C. Mixing time
In this section we examine mixing times of the process,i.e. we check how long does it take to reach stationary æææææ æ æ æ æ æ æ æ æàà àà à à à à à à à à n i s FIG. 6. Plot of stationary state value of infected nodes den-stiy i s for G ( n, p ) random graph with p = 0 . p = 0 . n . Number of neighbors on thecircle k = 2 n/
10 is chosen such that the edges density k n stays fixed. Red line shows theoretical prediction for com-plete graphs (12). We fix here z = 1 and n × r = 1. æ æ æ æ æ æ æ æ æ æà à à à à à ààì ì ì ì ì ì ì ì ì ì ln H n L Τ FIG. 7. Average mixing time τ for complete graph (bluedots), G ( n, p ) random graph with p = 0 . p = 0 . ln ( n ). Number ofneighbors on the circle k = 2 n/
10 is chosen such that theedges density k n stays fixed. Simulational results are depictedby blue dots and red line shows theoretical prediction for com-plete graphs (12). We fix here r = 0 .
001 and n/z = 1000 inorder to have stationary state not changed. Lines are plottedto guide the eye. state. Fig. 7 depicts how does average mixing time de-pend on ln ( n ), where n is network size, as usually. Thisis done for complete graph, G ( n, p ) random graph with p = 0 . p = 0 .
5. For the same graphs we check average mixingtime dependence on inverse of distance from epidemicstreshold η (see Corrolary 1). It is shown in Fig. 8.These result show actually much more than Theoremand Corollary from Sec. III D. We examine here average ææææ ææ ææ àààà àà àà ìììì ìì ìì (cid:144) Η Τ FIG. 8. Average mixing time τ for complete graph (bluedots), G ( n, p ) random graph with p = 0 . p = 0 . k = 20neighbors on the circle (yellow rotated squares) versus inverseof distance from epidemics treshold η . Simulational results aredepicted by blue dots and red line shows theoretical predictionfor complete graphs (12). We fix here r = 0 .
001 and n = 100.Lines are plotted to guide the eye. mixing time and show, that they are linear with ln ( n )and 1 /η , as theory in Sec. III D suggest by bounds ofprobability of mixing time proportional to ln ( n ) and 1 /η . V. CONCLUSIONS
We have proposed model of epidemics spreading withat most one infection per times step. Starting from thegeneral formula for the change of the number of infectednodes (3) we provided condition for epidemics tresholdfor any kind of graph. Simulational results for epidemicstreshold follow the theoretical predictions perfectly. Fur-ther more, stationary density of infected nodes for com-plete and uncorrelated homogenous graphs has been de-rived and bounds for this density, using the notion ofgraph conductance, have been obtained. Complete graphsimulations show agreement with the theory. Epidemyon G ( n, p ) random graphs, according to no correlation inlarge n limit [13], as well as on small world graphs, in thelarge n limit, behave like epidemy on complete graphs.We have stated and proven theorem and corollary thatbouds the probability of mixing time by values propor-tional to ln ( n ) and 1 /η , where n and η are size of the net-work and distance form epidemics treshold respectively.Simulations on complete, G ( n, p ) and small world graphsshow even more, mainly that the average mixing time islinear with ln ( n ) and 1 /η . ACKNOWLEDGEMENTS
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