Percolation in interacting networks

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E. A. Leicht and Raissa M. D'Souza 06jul2009

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Percolation in interacting networks

  1. 1. Percolation on interacting networks E. A. Leicht1 and Raissa M. D’Souza1, 2 1 Department of Mechanical and Aeronautical Engineering, University of California, Davis, CA 95616 2 The Santa Fe Institute, Santa Fe, NM 87501 (Dated: July 6, 2009) Most networks of interest do not live in isolation. Instead they form components of larger systems in which multiple networks with distinct topologies coexist and where elements distributed amongst different networks may interact directly. Here we develop a mathematical framework based on generating functions for analyzing a system of l interacting networks given the connectivity within and between networks. We derive exact expressions for the percolation threshold describing the onset of large-scale connectivity in the system of networks and each network individually. These general expressions apply to networks with arbitrary degree distributions and we explicitly evaluate them forarXiv:0907.0894v1 [cond-mat.dis-nn] 6 Jul 2009 l = 2 interacting networks with a few choices of degree distributions. We show that the percolation threshold in an individual network can be significantly lowered once “hidden” connections to other networks are considered. We show applications of the framework to two real-world systems involving communications networks and socio-tecnical congruence in software systems. PACS numbers: 64.60.aq, 89.75.Fb In the past decade there has been a significant advance works was introduced with the layered network frame- in understanding the structure and function of networks. work of [4]. Yet, the networks in the distinct layers must Mathematical models of networks are now widely used to be composed of the identical nodes (modeling essentially describe a broad range of complex systems, from spread physical connectivity and logical connectivity or flow). of disease on networks of human contacts to interactions Herein we consider systems of l ≥ 2 distinct interact- amongst proteins [1, 2, 3]. However, current methods ing networks and calculate explicitly how the connectiv- deal almost exclusively with individual networks treated ity within and between networks determines the onset of as isolated systems. In reality an individual network is large scale connectivity in the system and in each net- often just one component in a much larger complex sys- work individually. Our mathematical formulation has tem; a system that can bring together multiple networks some overlap with recent works calculating connectivity with distinct topologies and functions. For instance, a properties in a single network accounting for a diversity pathogen spreads on a network of human contacts abet- of node attributes [5, 6] or interactions between modules ted by global and regional transportation networks. Like- within a network [7, 8]. Here we present our formalism wise, email and e-commerce networks rely on the Internet and also applications to real-world systems of interacting which in turn relies on the electric grid. In biological sys- networks coming from telecommunications and software. tems, activated genes give rise to proteins some of which The onset of large-scale connectivity (i.e., the percola- go back to the genetic level and activate or inhibit other tion threshold) corresponding to the emergence of a giant genes. Results obtained in the context of a single isolated connected component in an isolated network has been network can change dramatically once interactions with other networks are incorporated. Consider a system formed by two interacting networks, α and β, Fig 1(a). Network α could be a human contact k1 edges to α network 1 network for one geographic region and network β that for a separated region. When viewed as individual systems, kν t ed wo ·· only small clusters of connected nodes exist, hence, a dis- ne ge rk · sb ν ease spreading in either network should stay contained ac edge traversed within clusters. In reality, a disease can hop from α to β, k µ from ν to µ ··· for instance, by an infected person flying on a airplane, spread in the β network and eventually hop back to the kl edges to α network into new clusters, causing an epidemic out- network l break. Next consider interacting networks that contain β completely different types of nodes. Network α can be (a) (b) a social network, such as an email communication net- work of software developers, while network β can be a FIG. 1: a) Two networks α and β. Nodes interact directly technological network, such as the network of calls be- with other nodes in their immediate network, yet also with tween functions in software code. Here, bi-partite edges nodes in the second network. b) An illustration of the re- connect developers on α to code they author on β. maining edges incident to a node in a network µ reached by An important step towards modeling interacting net- following a random edge between networks ν and µ.
  2. 2. 2studied extensively, first for random networks with Pois- Consider selecting uniformly at random an edge fallingson degree distributions [9] and later for random networks between a node in network ν and a node in networkwith arbitrary degree distributions [10]. Similar results µ (i.e., a ν-µ edge). The µ node attached to the edgewere then derived using generating functions [11, 12], the is kν times more likely to have ν-degree kν than de-approach we employ herein. Generating functions, simi- gree 1. We can also account for the remaining locallar to the network configuration model [10, 13], evaluate connectivity, to nodes in other networks as shown inthe ensemble of all possible random networks consistent Fig. 1(b). In single isolated networks remaining con-with a specified degree distribution, {pk }, and are most nectivity is called the excess degree of a node [11]. Let µνaccurate in the sparse regime where networks are approx- qk1 ···kν ···kl denote the probability of following a randomlyimately tree-like. Thus in the regime before the emer- chosen ν-µ edge to a node with excess ν degree as showngence of the giant component, generating functions can in Fig. 1(b) (which has total ν-degree of kν + 1). Thenbe used to calculate the distribution of component sizes. qk1 ···kν ···kl ∝ (kν + 1)pµ1 ···(kν +1)···kl , and the generating µν kIn the supercritical regime they can be used to calculate µν function for the distribution, {qk1 ···kl } is,the distribution in sizes of components that are not partof the giant component. ∞ µν For our purposes, a system with l ≥ 2 interacting net- Gµν (x) = qk1 ···kl xk1 · · · xkl 1 l (2)works is described by a set of degree distributions. Each k1 ,...,kl =0individual network µ is characterized by a multi-degree ∞ (kν + 1)pµ1 ···(kν +1)···kl kdistribution, {pµ1 k2 ···kl }, where pµ1 k2 ···kl is the fraction k k = ∞ µ xk1 · · · xkl 1 lof all nodes in network µ that have k1 edges to nodes k1 ,···kl =0 j1 ,...,jl =0 (jν+1 )pj1 ···(jν +1)···jlin network 1, k2 edges to nodes in network 2, etc. The  −1 ∞ ∞multi-degree distribution for each network may be writ- ∂ten in the form of a generating function: = jν pµ1 ···jl  j pµ1 ···kl xk1 · · · xkl k 1 l j1 ,··· ,jl =0 ∂xν k1 ,...,kl =0 ∞ ν Gµ (x1 , . . . , xl ) = pµ1 ···kl xk1 · · · xkl . (1) Gµ (x) k 1 l = k1 ,...,kl =0 ν Gµ (1)To simplify notation in what follows, we now define two ν where Gµ (x) denotes the first derivative of Gµ (x) withl-tuple’s, x = (x1 , . . . , xl ) and 1 = (1, . . . , 1). respect to xν and the denominator is a normalization Our interest is in calculating the distribution of compo- ν ν constant so that Gµν (1) = 1. Also note that Gµ (1) ≡ k µnent sizes, where a component is a set of nodes connected is the average ν-degree for a node in network µ.to one another either directly or indirectly by travers- The distribution of second nearest neighbors foring a path along edges. Clearly such components can that µ node via the ν layer is calculated by us-be composed of nodes distributed among the l different ing Eq. 2 as the argument to Eq. 1, namelynetworks, and our formulation allows us to calculate the Gµ (1, 1, ..., Gνµ (x)|xλ =1,λ=µ , ..., 1). Comparing this dis-distribution of such system-wide components, yet also torefine the focus and calculate the contribution coming tribution calculated via generating functions to thatfrom nodes contained in only one of the l networks. found in real-world interacting networks can reveal in- We begin by deriving the distribution of connectiv- teresting statistical features. Returning to the softwareity forGeneral Availability release randomly chosen edge. example, we have a network of email communication be- First a node at the end of a tween developers, a network of relations between code, Bug and security fix and bipartite edges connecting developers to the code 6 they edit. We would expect that the real system does not 5 resemble a random network, but instead reflects a struc-JSD (norm) 4 ture conducive to project development. For instance, if 3 two developers edit the same code we would like for them 2 to directly communicate via email and thus be first neigh- 1 bors. In a sparse random network these developers would 0 2001 2002 2003 2004 typically be second neighbors, connected indirectly via Time the code they both edit. We analyze the evolution of the Apache 2.0 OpenFIG. 2: Comparison over time of the distribution of the num- Source Software project from mid-2000 thru 2004, withber of developers connected indirectly via co-editing code in data aggregated over three month windows. From this wethe Apache project with the distribution expected in a ran- extracted the multi-degree distribution of the system fordom network with the same multi-degree distribution. Verti- each time-shot, which we then plug into our generatingcal lines mark the first generally available release in 2002, and functions to calculate the expected distribution of seconda significant deviation from random in 2003, when the com- neighbors found by following first a developer-to-codemunication network shrinks and the project seems to become edge then a code-to-developer edge. We then comparemore efficiently organized. this distribution to the real distribution of such devel-
  3. 3. 3 We recognize the form of this equation from Eq. 2, thus µ λ ! ! µ λ µ γ Hµν (x) = xµ Gµν [H1µ (x), . . . , Hlµ (x)]. (5) ! = µ + µ + µ +... We now consider starting from a randomly chosen µ- !"# !"# $"# node, rather than a random ν-µ edge. A topology such as ν µ ν µ ν µ ν µ one from Fig. 3 exists a the end of each edge incident to the µ-node. The generating function for the probabilityFIG. 3: A diagramatical representation of the topological con- distribution of component sizes is,straints placed on the generating function Hµν (x) for the dis- Hµ (x) = xµ Gµ [H1µ (x), . . . , Hlµ (x)]. (6)tribution of sizes of components reachable by following a ran-domly chosen ν-µ edge. The labels attached to each edge While in theory it is possible to solve Eq. 5 for Hµν (x)indicate type or flavor of the edge and summation notation and use that solution in Eq. 6 to solve for Hµ (x), inindicates that we are summing over all possible flavors. practice, even for the case of a single isolated network, as noted in [11] the equations are typically quite difficult to solve. Yet, Eq. 6 allows calculation of average componentoper second nearest neighbors using the Jensen-Shannon size. A component may include multiple node flavors, butdivergence [14], a symmetric measure based on Kullback- we can distinguish between the average number of eachLeibler divergence. The results are shown in Fig. 2 with type. For example, the average number of ν-nodes in thethe JS-score of the real networks normalized by the JS- component of a randomly chosen µ-node isscores from the ensemble of random networks. Valuesgreater or less than unity indicate networks more or less ∂random than average. We indicate two vertical bars sµ ν = Hµ (x) ∂xν x=1where significant difference between the random and real = δµν Gµ [H1µ (1), . . . , Hlµ (1)]networks occurs. The first, in mid-2002 marks the first lgeneral availability release of Apache 2.0, the second, at λ νthe start of 2003, is a bug and security fix [15]. This lat- + Gµ [H1µ (1), . . . , Hlµ (1)]Hλµ (1)ter point, moreover, marks when a substantial purging λ=1of developers from the communication network occurs. l λ νIn any three-month window we observe that only about = δµν + Gµ (1)Hλµ (1) (7)25 developers edit code, yet prior to 2003 the number λ=1of developers in the email network is significantly larger. ν Intuitively Eq. 7 is reasonable because Hγλ (1) representsThus this time seems to indicate when the Apache project the average number of ν-nodes in the component foundbecomes more efficiently organized, eliminating noise of by following a µ-λ edge towards a λ-node, and the ex-spurious emails to inactive developers. pected number of µ-λ edges incident to an initial µ-node We are now in position to consider component sizes. λ λ λAssume we follow a randomly chosen ν-µ edge to a µ node is Gµ (1) (recall, Gµ (1) = k µ ). The product of the two(Fig. 1(b)), and consider the distribution in sizes of the terms summed over all λ networks produces the num-component found by following the additional outgoing ber of ν-nodes in a component connected to a randomlyedges. Let Hµν (x) denote the associated generating func- chosen µ-node, sµ ν .tion. Fig. 3 illustrates all the types of connectivity possi- The preceding results regarding components hold inble for the µ-node, and summing over all these possibili- the sub-critical regime where no giant connected compo-ties leads to the self-consistency equation for Hµν (x): nent exists. Once a giant component emerges, generating functions allow us to calculate properties of components µν Hµν (x) = xµ q0···0 (3) not belonging to it. The giant component will span mul- 1 l tiple networks and calculating its size requires accounting + xµ δ1,Pl µν Hγµ (x)kγ for the contribution from each network. Let Sµ be the kλ qk1 ···kl k1 ...kl =0 λ=1 γ=1 fraction of µ-nodes belonging to the giant component. 2 l The probability that a randomly chosen µ-node is not µν part of the giant component must then satisfy the fol- + xµ δ2,Pl q Hγµ (x)kγ + ··· λ=1 kλ k1 ···kl lowing equation, k1 ,...,kl =0 γ=1 ∞δij denotes the Kronecker delta, used here to account for 1 − Sµ = pµ1 ,...,kl uk1 · · · ukl = Gµ (u1µ , . . . , ulµ ), (8) k 1µ lµall combinations of flavors of edges connected to the µ- k1 ,...,kl =0node leading to specified excess degree i. Reordering theterms, Eq. 3 becomes where uνµ is the probability that an µ-ν edge is not part of the giant component. In addition, for all µ, ν ∈ l, uνµ ∞ must satisfy, µν Hµν (x) = xµ qk1 ···kl H1µ (x)k1 · · · Hlµ (x)kl . (4) k1 ...kl =0 uνµ = Gνµ (u1ν , . . . , ulν ), (9)
  4. 4. 4derived using the same self-consistency arguments that 1resulted in Eq. 5. Though all the equations above hold for a system ofl ≥ 2 interacting networks, we now give a concrete ex- 0.8 Fraction of nodesample for l = 2, with the networks indexed as α and β.Consider first the simplest of systems, where the inter- 0.6nal connectivity of α and β each has a distinct Poissondegree distribution, and the inter-network connectivity 0.7is described by a third Poisson degree distribution, for 0.4 α α β β 0.6instance, pαα kβ = (k α )kα e−kα /kα ! (k α )kβ e−kα /kβ ! . k 0.5 0.4 ν(Recall k µ denotes the average ν-degree for a node in 0.2 0.3 0.2network µ.) Then, from Eq. 1, 0.1 01 α β 10 κ 100 Gα (xα , xβ ) = ekα (xα −1) ekα (xβ −1) (10) 00 1 2 3 4 5 β α β Gβ (xα , xβ ) = ekβ (xα −1) ekβ (xβ −1) . (11) kβUsing Eq. 7, the average number of α-nodes in a compo- FIG. 4: Numerical simulations of connectivity in a systemnent reachable from a randomly chosen α-node is, of two interacting Poisson degree distributed networks, α and α β α α β β, with inter-network connectivity also Poisson distributed, as kα + kα kβ − kα kβ connectivity on β increases. Each network has 100,00 nodes, sα α =1+ α β β α . (12) α β α with kα = 0.4 and kα = kβ = 0.5. Shown are the fraction of (1 − k α )(1 − k β ) − k α k β α nodes, Sα (circles), β nodes, Sβ (squares), and all nodes, S (triangles) in the system-wide giant component, with theThe average component size diverges for α β β α dashed curves giving the analytic results, Eqns. (13) and (14).( 1 − k α ) ( 1 − k β ) = k α k β ; the point at The horizontal dashed line is the asymptotic value to whichwhich the giant component emerges. (Ref. [8] recently Sα approaches. (Inset) Analogous results when α has Poisson αpresented an alternate method for deriving similar per- distribution with kα = 0.5, inter-network edges follow a Pois- β αcolation thresholds and connectivity properties, but in a son distribution with kα = kβ = 0.4, but β has a power-lawsingle network with multiple interacting communities.) distribution with exponent τ = 2.5 and an exponential cutoffNote, following Eq. 7, we can show sβ α , sα β , and that we vary between 1 ≤ κ ≤ 300. The solid curve is the sβ β also all diverge at this point, marking when a giant result for network β when viewed in isolation.component emerges in each network and throughoutthe system. Further simplifying, by assuming the two β α α β −1interacting networks have the same degree distribution, shown that as k β increases Sα → α W kα −k α e−kα −kα +1 α β β αk α = k β = k intra and k α = k β = k inter , then the giant (dashed horizontal line in Fig. 4), where W is the Lam-component emerges when, k inter + k intra = 1, recovering bert W function, also known as the product log.the standard result for a single network (which, by We next consider more complex degree distribu-definition, has k inter = 0) that emergence occurs for tions, where α is still described by a Poisson dis-k intra = 1. tribution, but the internal connectivity of β is de- Once the giant component emerges the uνµ which sat- scribed by a power-law distribution with an exponen-isfy Eq. 9 are uαα = uαβ = 1 − Sα and uββ = uβα = tial cutoff. While power-law degree distributions have1 − Sβ , while Sα and Sβ , respectively, the number of α- attracted considerable attention as a model for nodenodes and β-nodes in the giant component of the system, degree distributions in many types of networks [16],satisfy a power-law with an exponential cutoff may be a better model for real-world networks [17]. Here α β α α Sα = 1 − e−(kα Sα +kα Sβ ) (13) pβα kβ k = (k β )kα e−kβ /kα ! (kβ )τ e−kβ /κ /Liτ (e−1/κ ) α β Sβ = 1 − e−(kβ Sα +kβ Sβ ) . (14) where Lin (x) is the nth polylogarithm of x and serves as a normalizing factor for the distribution. Thus, we To observe the change in connectivity of one network can write our basic generating function for network β,precipitated by an increase in connectivity of a secondnetwork attached to the first, we simulated a system of α Liτ (xβ e−1/κ ) Gβ (xα , xβ ) = ekβ (xα −1) . (15) α β αtwo interacting networks and fixed k α , k α , and k β while Liτ (e−1/κ ) β βvarying k β from 0 to 5 (Fig. 4). As k β increases the The generating function for α is still given by Eq. 10.β-network becomes a single connected component (the We simulate the impact on the connectivity of the α-traditional behavior for a single network) and Sβ → 1. network as the exponential cutoff and hence the averageHowever, the connectivity of α remains limited. It can be degree of network β increases, inset of Fig. 4. Again
  5. 5. 5 1 tooth connectivity between individuals from raw data of Bluetooth sightings by 41 attendees at the 25th IEEEE International Conference on Computer Communications Fraction of nodes (S) 0.8 (INFOCOM) [19]. We initially partition the raw data into discrete 20 minute windows and consider that a 0.6 communication edge exists between any two devices so α long as they are within contact for at least 120 seconds. 0.4 Each network has approximately a Poisson degree dis- tribution of connectivity. We choose two arbitrary 20 minute snapshots as proxies for two distinct networks, 0.2 β α and β, representing, for instance, two separate rooms at the conference. We calculate how adding long-range 0 0.2 0.4 0.6 0.8 1 connections between α and β (for instance via text mes- β α sages or email) enhances overall connectivity in the sys- kα and kβ ] tem. In other words, we calculate how many long-rangeFIG. 5: Inset are two sample networks of Bluetooth connec- connections would be needed between two isolated localtivity. The main figure shows the increase in participation Bluetooth networks to create the desired large scale con-in the giant component as connectivity between α and β in- nectivity, potentially allowing many users to share infor- β αcreases, starting from kα = kβ = 0.1. Points are obtained by mation. Figure 5 shows the size of the giant componenttaking the empirical data and simulating inter-network edges obtained via numerical simulations using the real data β αwith the appropriate kα and kβ , averaged over 100 realiza- (points) and the analytic calculations obtained via gen-tions. The solid line is from analytic calculations. erating functions (dashed line). The analytic calculations slightly overestimate connectivity, yet there is remarkable agreement with empirical data even though the actuallythe dashed curves are the analytic results obtained by networks are quite small.solving Eqns. 8 and 9. The solid red line is the behavior In summary, we have introduced a formalism for cal-for the β network considered in isolation, showing that culating connectivity properties in a system of l interact-even the percolation threshold for β is lowered through ing networks. We demonstrate the extreme lowering ofconnectivity with network α. the percolation threshold possible once interactions with Finally we consider an application of connectivity to other networks are taken into account. This frameworkcommunications networks, building on the increasing in- for calculating connectivity and statistics of interactingterest in using Bluetooth connectivity between individ- networks should be broadly applicable, and we show po-uals to transmit data [18]. For instance, rather than tential applications to software and communications sys-downloading a webpage (such as the CNN homepage) tems.by connecting to the Internet, a copy could be obtained Acknowledgements We thank Christian Bird forfrom a close-by individual already in possession of this providing data on the Apache project and for useful con-data. We construct prototypical networks of local Blue- versations. [1] S. N. Dorogovtsev and J. F. F. Mendes, Advances in Rev. E 64, 026118 (2001). Physics, 51, 1079-1187 (2002). [12] D. S. Callaway, M. E. J. Newman, S. H. Strogatz, and [2] M. E. J. Newman, SIAM Review 45, 167 (2003). D. J. Watts, Phys. Rev. Lett. 85, 5468 (2000). [3] S. Boccaletti, V. Latora, Y. Moreno, M. Chavez and D.- [13] B. Bollob´s, European Journal of Combinatorics 1, 311 a U. Hwang, Physics Reports, 424, 175-308 (2006). (1980). [4] M. Kurant and P. Thiran, Phys. Rev. Lett. 96, 138701 [14] J. Lin, IEEE Trans. Information Theory, 37 (1) 145-151 (2006). (1991). [5] B. Bollob´s, S. Janson and O. Riordan, Random Struc- a [15] http://www.apacheweek.com/features/ap2. tures and Algorithms 31, 3-122 (2007). [16] A.-L. Barab´si and R. Albert, Science 286, 510-512 a [6] A. Allard, P-A No¨l, L. J. Dub´ and B. Pourbohloul, e e (1999). Phys. Rev. E 79, (3) 036113 (2009). [17] A. Clauset, C. R. Shalizi, and M. E. J. Newman, SIAM [7] S. N. Dorogovtsev, J. F. F. Mendes, A. N. Samukhin, Review, in-press (2009), (arXiv:0706.1062). and A. Y. Zyuzin, Phys. Rev. E 78, 056106 (2008). [18] S. Ioannidis, A. Chaintreau, and L. Massouli´. Proceed- e [8] M. Ostilli and J. F. F. Mendes, arXiv:0812.0608 (2008). ings of IEEE INFOCOM, 2009. [9] P. Erd˝s and A. R´nyi, Publicationes Mathematicae 6, o e [19] J. Scott, R. Gass, J. Crowcroft, P. 290 (1959). Hui, C. Diot and A. Chaintreau,[10] M. Molloy and B. Reed, Random Structures and Algo- http://crawdad.cs.dartmouth.edu/cambridge/haggle/ rithms 6, 161 (1995). imote/infocom (2006).[11] M. E. J. Newman, S. H. Strogatz, and D. J. Watts, Phys.

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