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Modern Physics Letters B, Vol. 23, No. 10 (2009) 1249–1262 c World Scientific Publishing Company

EPIDEMIOLOGY MODEL ON SHORTCUT AND SMALL WORLD NETWORKS

O. SHANKER Hewlett Packard Company, 16399 West Bernardo Drive, San Diego, CA 92130, USA [email protected] TAD HOGG HP Labs, Palo Alto, CA 94304, USA Received 16 June 2008 Revised 25 June 2008 We show that the behavior of an epidemiology model depends sensitively on the shortcut density in the shortcut network. This is consistent with an earlier work on other processes on the shortcut network. We analytically study the reason for the sensitivity. The shortcut network is similar to the small world network, and it has the advantage that the model dependence on the shortcut density can be analytically studied. The model would be relevant to the spread of diseases in human, animal, plant or other populations, to the spread of viruses in computer networks, or to the spread of social contagion in social networks. It would also be relevant in understanding the variations in the load on routers connecting different computer networks, as the network topology gets extended by the addition of new links, and in analyzing the placement of certain special sensors in a sensor network laid out in a non-random way with some shortcut links. Keywords: Epidemiology; complex networks; fractal dimension.

1. Introduction The properties of complex networks play an important role in epidemiological studies.1 Models of epidemics apply not only to to diseases in human or other populations, but also to the spread of viruses in computer networks, or to the spread of social contagion in social networks. Epidemic models can also be applied in understanding the variations in the load on routers connecting computer networks, as the network topology changes by the addition or deletion of links, and in analyzing the placement of certain special sensors in a sensor network laid out in a non-random way with some shortcut links (e.g. as may be supplied by a few sensors with more powerful communication capability than the others). Network topologies that have been studied2 in epidemiology include random, small world, scale-free and regular lattice networks. In particular, epidemiological model networks have a geographical component to them. The vertices of the network have positions in space. The small world network3–5 starts from this idea by positing a network built on a 1249

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low-dimensional regular lattice and then adding or moving edges to create a low density of shortcuts that join remote parts of the lattice to one another. It has been observed that even for a low density of shortcuts, the small world network transitions from a finite dimension network to a random network. We study a simple modification of the small world network, the shortcut network,6,7 which is tractable and provides insight into the reason for the sensitivity as the shortcut probability is varied. Reviews of complex networks, including the epidemiological applications mentioned above, can be found in Refs. 8–12. In the epidemiology model, the disease spreads from one node to another. Thus, the rate of spread and the pattern of spread of the disease depends on the network connectivity. In particular, it depends on the average rate at which the number of nodes at a given distance from any arbitrary node grows with the distance. For geographical networks, the number of nodes at a given distance often scales as a power of the distance. The exponent for the scaling relation is in fact one of the definitions of network dimension that have been studied in the literature. The best definition to use for the dimension depends on the nature of the process being studied, and this definition is the most suitable for the current application. The effect of network dimension on the pattern and rate of spread has been noted.2,13,14 The networks we study in this work have a sensitive dependence on network dimension on the network parameters. We study this sensitivity analytically in the large system limit in a simple network model. The epidemiology model involves a network search for the carriers of the disease under the simplifying assumptions of rapid spread through a homogeneous population. The model sensitivity to network parameters is reminiscent of abrupt transitions that have been observed in computational search problems.15–19 Section 2 introduces the epidemiological problem and the complex network. Section 3 shows that for finite values of the shortcut probability p, the disease growth rate transitions infinitely sharply. Section 4 derives the scaling behavior in the crossover regime where p → 0 as the graph size tends to infinity. Section 5 studies measures related to the time needed for essentially the whole network to get affected by the disease. Finally, the conclusion is presented. 2. Model We present here the epidemiological model and the network on which we study it. In addition to the study of disease spread and control, the model can also be applied to the spread of viruses on a computer network, or to the spread of social contagion in social networks. The dimension study would also be useful in analyzing the variations in the load on routers connecting computer networks, as the network topology gets extended by the addition of new links. It may also be relevant in analyzing the placement of certain special sensors in a sensor network laid out in a non-random way with some shortcut links.20

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We consider an infectious disease which has an incubation time. We assume the disease spreads fast within the incubation time. Because of the incubation time, when the first case is identified, one is not sure how many people may have been exposed to the disease in geographically dispersed locations (e.g. different towns), due to normal travel between the locations by infected but not yet symptomatic citizens. When the first case of the disease is reported, one must assume that the disease may have spread to any of the towns, so one would need to check all nodes. Because of the fast spread, it is not feasible to locate all the potential “vectors” who might have carried the infection, and to locate all their contacts, in a recursive fashion. Thus, it becomes necessary to search for the disease in the whole population. The population search approach has proven quite useful in the control and eradication of virulent diseases like smallpox, cholera, polio and plague. The nodes of the complex network represent individuals and the links represent contacts which allow the disease to spread. When applied to the spread of viruses in a computer network, the nodes would represent hosts, and the links would represent the connections. The small world networks and regular lattice networks have been used to abstract the behavior of such networks. We are interested in studying the spread of the epidemic between the nodes, and its dependence on the links between the nodes. The spread between the nodes can be represented by one of the standard disease spread models.1,2,13,14 For example, in the SIR model, one divides the population into three classes: susceptible (S), meaning that they do not have the disease but may catch it if exposed to an infected person, infective (I) meaning that they have the disease and can pass it on, and recovered (R) which includes all those who have developed permanent immunity to the disease. One also assumes transition rates, e.g. β is the probability per unit time that a susceptible individual catches the disease from an infective one, and γ is the probability per unit time that an infective person recovers and becomes immune. There are variations on the basic model, e.g. the recovered individual may not develop perfect immunity. Different choices for β and γ lead to different cycles and patterns of disease spread within the population. For the purposes of our study, we assume that the spread between the nodes takes place in a time shorter than the time in which the infective persons recover, so the disease, once it affects a node, will be active for that node for all the time that it is spreading to other nodes. For the spread between the nodes, we divide time into discrete intervals, and assume that in one time interval, all nodes which neighbor an affected node will also get affected by the disease. Thus the disease spreads steadily outward from the initial affected nodes. In particular, the rate of spread between nodes will be proportional to the surface area of the network, measured by the average number of nodes at a given distance from any node, averaged over all the nodes. If the recovery time implied by γ is shorter than we assume, and if the immunity is not complete, then more complex patterns for the spread of the disease, including spread in waves and with oscillatory behavior, become possible.

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There is another possible interpretation of the network. The nodes of the complex network could represent population centers and the links could represent travel routes (e.g. roads). There are two aspects to the growth rate for the spread of the disease. One aspect is the spread within any population center, represented by the nodes. The second aspect is the spread between population centers. For the large recovery time that we are assuming, the disease, once it gets established in a population center, would persist in that population center in the time that it takes to spread to other population centers. For this situation, the conclusion of the study would remain the same. The rate of growth of the number of nodes at a given distance from a node, averaged over all nodes, plays an important role in the epidemiological model. For example, the higher the rate of growth, the faster will be the rate of spread of the epidemic. That motivates the study of the growth rate of volume with distance as the density of shortcuts is varied. In the literature, the volume function for a complex network has been defined as the number of nodes j within a distance r(i, j) of node i, averaged over i, and the exponent which determines the scaling behavior of the volume with distance has been studied as one of the possible definitions for the dimension of the complex network. The small-world network widely used in such studies starts with a onedimensional lattice of N vertices with periodic boundary conditions. Each vertex is joined to its neighbors on either side, which results in a system with N edges. The network is extended by taking each node in turn and, with probability p, adding an edge to a new location which is at a random distance from the original node. To get a tractable model for the dimension transition, we study a modification, the shortcut network,6,7 in which the shortcuts connect to a node at a fixed distance m in the starting network, instead of to an arbitrary node. The network has fractal dimension as defined√by the complex network zeta function.7 We require that N  m  1, say m = N. The shortcut network is parametrized by size = N, shortcut distance = m, and shortcut probability = p .

(1) (2) (3)

When the shortcut probability p = 0, the network is a one-dimensional regular lattice of size N . Figure 1 shows the network. The nodes are connected by edges represented by the sides of the polygon. On such a network, the disease spreads at a uniform rate, with two new nodes getting affected in each time interval. Thus, the time for the whole network to get affected is of the order of the network size. When p = 1, every node is connected by a shortcut edge to a new location. Each node now has edges in two directions, the first along the original direction (on the outside polygon), and the second along the shortcut edges. The graph maps to a square lattice with m nodes in the original direction and N/m = n nodes in the shortcut direction. Figure 2 shows the network for p = 1. On this network,

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√ Fig. 2. Shortcut network for p = 1 (disease coverage time O( N )). The nodes are now numbered by the mapping to a square lattice.

the disease spreads at a linearly increasing rate, √ and thus the whole network gets affected in a much shorter timeframe, of order N . For the square lattice node numbering in the figure, the x-coordinate represents the distance along the original edges, and the y-coordinate represents the distance along the shortcut edges. For

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Fig. 3.

√ Shortcut network in intermediate case (disease coverage time O( N )).

p between 0 and 1 (Fig. 3), we have a graph which interpolates between the one and two dimensional systems. For this network, the disease initially spreads at a linearly increasing rate, similar to the p = 1 case. After a time period equal to m/2 time intervals, the spread rate becomes uniform, but the number of nodes affected per time period is much√larger than two. The timeframe for the whole network to get affected is of order N . The faster spread of the disease for the network in Fig. 2, can be understood by the fact that it can be mapped to a square lattice, as shown in Fig. 4. The links along the outside polygon are represented by the horizontal and the slanting edges. The shortcut edges are the vertical edges in Fig. 4. The chief difference of the shortcut network from the usual square lattices is in the links at the right side of the lattice. Usually they are either free, or connect back to the node at the left end of the same row. For the shortcut network, the unusual aspect is that the right node connects to the left node one level above instead of to the node on the same row. The difference between the shortcut network and the small-world network of Watts and Strogatz is that in the small world network, the shortcuts can connect to any randomly chosen node instead of being constrained to connect to a node a fixed distance ahead. As a result, the small world network tends to a random graph rather than a two-dimensional graph as the shortcut probability is increased. The rate of spread of the disease and the search for infected persons depends sensitively on the scaling of volume V (r) with distance r. For random graphs, the scaling is exponential for the distance. For the shortcut network, the scaling is a

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power law. The scaling relation is V (r) = krd ,

(4)

where d is not necessarily an integer for complex networks. k is a constant which depends on the complex network. If the scaling relation Eq. (4) holds, then we can also define the surface area S(r) as the number of nodes which are exactly at a distance r from a given node, and S(r) scales as S(r) = kdrd−1 .

(5)

The scaling exponent in Eq. (4) has been shown to be the value at which the complex network zeta function7 transitions from non-convergence to convergence in the infinite system limit. This relation allows us to handle the growth rate of the volume analytically in the shortcut network. The complex network zeta function ζG (α) is defined as ζG (α) :=

1 X X −α rij , N i

(6)

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where N is the graph size measured by the number of nodes and rij in the distance between nodes i and j. The distance is measured by the number of edges along the shortest path connecting two nodes. One could generalize this by considering

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weighted edges. The definition Eq. (6) can be expressed as the Dirichlet series expression, X ζG (α) = S(r)/rα . (7) r

When the exponent α tends to infinity, the sum in Eq. (6) gets contributions only from the nearest neighbors of a node. The other terms tend to zero. Thus, ζG (α) tends to the average vertex degree for the complex network. When α is zero the sum in Eq. (6) gets a contribution of one from each node. This means that ζG (α) is N − 1, and hence tends to infinity as the system size increases. Furthermore, ζG (α) is a decreasing function of α. Thus, if it is finite for any value of α, it will remain finite for all higher values of α. If it is infinite for some value of α, it will remain infinite for all lower values of α. Thus, there is at most one value of α, αtransition , at which ζG (α) transitions from being infinite to being finite. This is reminiscent of the behavior of Hausdorff dimension.21 The value of the exponent α at which ζG (α) transitions from being infinite to being finite is the same as the volume scaling exponent.7 For a one-dimensional regular lattice, the graph surface function S1 (r) in Eq. (5) is exactly two for all values of r. This is because there are two nearest neighbors, two next-nearest neighbors, etc. The graph surface function Sd (r) for a lattice of dimension d is a polynomial in r of order d − 1 for a given d. It can be written as    k X d r−1 Sd (r) = 2i , (8) i i−1 i=0

for any k > d. Equation (8) is a polynomial in d and r, and is valid for all integer r when the lattice dimension d < k and for all integer values of lattice dimension d when r is a positive integer less than k. The graph surface function scales aymptotically as Sd (r) → 2d rd−1 /Γ(d) for large r. r → ∞ corresponds to α → αtransition . Thus, ζG (α) → 2d ζ(α − d + 1)/Γ(d) as α → αtransition . The largest pole of ζG (α) occurs for α = d. In the next section, we study finite values of the shortcut probability p, for which the spread rate of the disease transitions infinitely sharply. 3. Sharp Transition in Disease Growth Rate The growth of the number of nodes affected with time has a sharp transition from linear growth when p = 0 (one-dimensional limit) to quadratic growth when the shortcut probability p becomes non-zero, even for very small values of p.22,23 In this section, we review the results. The sharp transition is related to the large increase in the available number of paths connecting a pair of points as the path length is increased.24 Since the disease spreads to the neighboring nodes of an affected node in one time period, the graph surface function S(r) is equal to the number of new nodes

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Table 1. Number of affected nodes versus time (N = 16384). Time

p = 0.0

p = 0.05

p = 1.0

32 64 96 128 256 512 700 8192

64 128 192 256 512 1024 1400 16384

263 1086 1985 2852 6350 13076 16384

2112 8320 14272 16384

affected by the disease in the next time interval when the disease has already spread a distance r. When p = 0, the graph surface function for the shortcut network is constant as the path length is varied. When p = 1, the graph surface function increases linearly with the path length r, S(r) = 4r for r < m 2 and then it decreases m linearly for 2 < r < m. When p is between 0 and 1, the graph surface function lies between the two limiting cases. Table 1 shows the number of nodes affected versus time, for N = 16384, m = 128 and different values of p. The sharp speed-up in the rate of disease spread even for small values of p can be discerned from the table. Consider a pair of points connected by a possible path with i shortcut edges
r and r − i normal edges. There are i possible different paths connecting the two points, corresponding to all the possible ways of choosing the i shortcut edges in the total path of length r. This is a very large number when r and i are large. Consider a particular path from among this large set. When p lies between 0 and 1, the probability that this particular path does not have all the required shortcut edges present is (1 − pi ). Thus, any one particular path will have a small probability that all the required shortcut edges are present. This small probability is offset by the large number of available paths. We first estimate the probability pno-paths that none of the possible paths have the required shortcuts. The probabilities for different paths are not independent since different paths share edges. Neglecting r the correlations, one can estimate pno-paths to be (1 − pi )( i ) , which can be written as r    1 pf p , (9) pno-paths = exp − f f (1 − f )1−f 2πrf (1 − f ) for large r and i (where f := i/r). pno-paths goes to zero when f < fp where fp is the solution of the transcendental equation p = fp (1 − fp )

1−fp fp

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(10)

Because of the large value of the exponent pno-paths rapidly goes to zero for a finite fraction fp of nodes when p lies strictly between 0 and 1. For small p, the solution

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of Eq. (10) is p = fp /e. The probability ppath-exists that there is at least one path which has all the i required shortcuts, ppath-exists = 1 − pno-paths ,

(11)

is close to one for a finite fraction of the nodes. The dependence of the graph surface function on distance becomes essentially the same as for the case p = 1, i.e. it becomes linear with slope greater than 4fp . The dimension transitions abruptly when p increases from 0. In the next section, we investigate more closely the nature of the transition. 4. Scaling Behavior in Crossover Regime In this section, we study the crossover regime where p → 0 as the graph size tends to infinity. The study is based on determining the single unique value of αtransition at which the complex network zeta function transitions from non-convergence to convergence. In the previous section, we saw that for p = 0, αtransition is one, and for any p > 0, αtransition is two. The crossover regime (which becomes vanishingly small in the infinite system limit) is studied by considering  2−β  2 , (12) p=Θ m for 1 < β < 2, as m → ∞. For β < 1, the network behaves like a one-dimensional lattice, and β > 2 is not meaningful because in that case the probability increases with size. Since the disease spreads to the neighboring nodes of an affected node in one time period, the graph surface function S(r) is equal to the number of new nodes affected by the disease in the next time interval when the disease has already spread a distance r. The graph surface function consists of a region which is linear m for r up to r = m 2 , a small linearly decreasing region for r > 2 and a region in which the surface function is flat and which extends much further out (to r = O(m/fp ) where fp is given by Eq. (10)). The graph surface function is O(fp m) in the flat region. Since m → ∞, the transition point αtransition will be determined by the linear increasing region of the graph surface function, r < m 2 . Figure 5 shows the graph surface function for N = 16384, m = 128, and p = 0.05. For these values, the slope of the graph surface function is 0.59, which is not far from the lower bound 4fp = 0.51. From Eq. (10), it follows that for small p the slope of the graph surface function is proportional to p. If p remains finite as the graph size increases to infinity, then from Eq. (7), we see that the leading part of the complex network zeta function is proportional to ζ(α − 1), and the transition point is αtransition = 2. If p = 0, then S(r) = 2, and the complex network zeta function is proportional to ζ(α), and the transition point is αtransition = 1. αtransition will be between 1 and 2 if p → 0 as m → ∞. The sum over r in Eq. (7) 2 2−β is Θ(m2−α ). Since p = Θ(( m ) ), the slope of the graph surface function S(r) is

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2 2−β ) ). The complex network zeta function has a contribution from the also Θ(( m linear part of the graph surface function, and a contribution from the flat part of the graph surface function. The contribution from the linear part is given by the product of the slope of the graph surface function and the sum over r, and is of order  2−β  2 , (13) Θ(m2−α )Θ m

i.e. Θ(mβ−α ) .

(14)

The contribution from the linear part to ζG (α) will be zero if α > β and will diverge if α < β. Thus, ζG (α) will be finite if α > β and will diverge if α < β. We find that the transition point αtransition = β, which lies between 1 and 2. The contribution from the flat part will also be finite for α > β. The transition point in the crossover regime depends on the exponent with which p → 0 as the graph size tends to infinity. In the next section, we study measures related to the time needed for essentially the whole network to get affected. 5. Time for Complete Disease Coverage for Different Networks In this section, we study some measures which are related to the time for the disease to spread completely throughout the network. The first measure is the

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Network diameter

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maximum distance between any pair of nodes in the network (the network diameter), which gives an upper bound on the time taken for the disease to spread completely throughout the network. Table 2 shows the maximum distance between any pair of nodes for different values of the shortcut probability and for N = 16384, m = 128. The network diameter is large for p = 0 and drops drastically as soon as p becomes non-zero. When √ p = 0, the network diameter is N/2. When p = 1, the network diameter is m = N . For non-zero p, the network diameter is O(m/fp ) where fp is given by Eq. (10). Once again, we see that the large increase in the number of available paths causes a sharp speed-up in the disease spread rate even for a small number of shortcuts. Another measure for the disease coverage time is the mean path length for the network. If the mean path length is large, the disease takes a long time to spread completely. For a d-dimensional system, the mean path length ` scales with network size N as N 1/d . In both the shortcut and small world networks, the initial scaling exponent for the variation of the mean path length with system size is 1 since one starts with a one-dimensional regular lattice. In both networks, it drops sharply as the probability of shortcuts p increases from zero. For the shortcut network, the scaling exponent drops to 0.5, while for the small world network, it drops to a low value.25 (This is because the scaling for the small world network with system size is logarithmic.) We normalize the mean path length using the number of nodes N in the denominator rather than the maximum possible number of edges N (N − 1)/2. We do this merely for the convenience of working with positive scaling exponents. The difference between the shortcut and small world networks may be viewed as a difference in the probability distribution for the shortcut hop distance, once a node is chosen to have a shortcut. For the small world network, the shortcut hop can randomly connect to any other node, giving a uniform probability distribution for the shortcut hop distance between 1 and N/2. In the shortcut network, the shortcut hop distance is fixed to a single value m. To interpolate between the two networks, we study the mean path length when the shortcut hop distance has a 1 for r between 1 and N/2, with 0 < α < 1. probability distribution given by r1+α α = 0 gives a behavior similar to the small world network, while α = 1 gives a behavior similar to the shortcut network.

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When the shortcut probability p = 0, the mean path length scales linearly. We calculated the mean path lengths for different p for N varying from 1000 to 10000. Table 3 shows the scaling exponent for ` for the shortcut and small world networks, and for different values of α in the interpolating case where the shortcut hop distance 1 has a probability distribution r1+α . Figure 6 shows a plot of the scaling exponent against the shortcut probability in the shortcut network. Note, in particular, the rather sharp drop in the exponent from the value 1 as p increases from 0. The scaling of the mean path length with system size shows the sharp change in behavior as the shortcut probability increases from 0. In the next section, we present the conclusion. 6. Conclusion In this work, we studied the nature of the transition in the disease spread rate in the shortcut network in the crossover region where the shortcut probability p → 0

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as N → ∞. The phenomenon of the sharp transition would be relevant in the study of the spread of diseases in human or other populations, of the spread of viruses in computer networks, or of the spread of social contagion in social networks. The proper locations in the network to initiate response also depend on the growth rate of the number of nodes with distance. The sensitive dependence of the network topology on network parameters would be helpful in understanding the variations in the load on routers which connect different computer networks, as the addition of new links extends the network topology. It would also be useful in analyzing the placement of certain special sensors in a sensor network laid out in a non-random way with some shortcut links. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

21. 22. 23. 24. 25.

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