Color-avoiding percolation
CColor-avoiding percolation
Sebastian M. Krause,
1, 2
Michael M. Danziger, and Vinko Zlati´c
1, 4 Theoretical Physics Division, Rudjer Boˇskovi´c Institute, Zagreb, Croatia Faculty of Physics, University of Duisburg-Essen, Duisburg, Germany Department of Physics, Bar Ilan University, Ramat Gan, Israel CNR Institute of Complex Systems UdR Dep. of Physics, University of Rome Sapienza, Rome, Italy
Many real world networks have groups of similar nodes which are vulnerable to the same fail-ure or adversary. Nodes can be colored in such a way that colors encode the shared vulnera-bilities. Using multiple paths to avoid these vulnerabilities can greatly improve network robust-ness. Color-avoiding percolation provides a theoretical framework for analyzing this scenario, fo-cusing on the maximal set of nodes which can be connected via multiple color-avoiding paths.In this paper we extend the basic theory of color-avoiding percolation that was published in[Krause et. al., Phys. Rev. X 6 (2016) 041022]. We explicitly account for the fact that the sameparticular link can be part of different paths avoiding different colors. This fact was previouslyaccounted for with a heuristic approximation. We compare this approximation with a new, moreexact theory and show that the new theory is substantially more accurate for many avoided colors.Further, we formulate our new theory with differentiated node functions, as senders/receivers or astransmitters. In both functions, nodes can be explicitly trusted or avoided. With only one avoidedcolor we obtain standard percolation. With one by one avoiding additional colors, we can under-stand the critical behavior of color avoiding percolation. For heterogeneous color frequencies, wefind that the colors with the largest frequencies control the critical threshold and exponent. Colorsof small frequencies have only a minor influence on color avoiding connectivity, thus allowing forapproximations.
I. INTRODUCTION
In many real-world networks, the vulnerability ofnodes to attack or failure is not uniform [1]. Instead,certain groups of nodes may share the same vulnerabil-ity, enabling a coloring of the nodes of the network interms of their common vulnerabilities [2]. For example,Internet routers run by the same entity or running thesame software version may be subject to the same eaves-droppers [3, 4], or multiple suppliers in a supply chainnetwork may rely on the same critical resource [5–7]. Insuch cases, it may be desirable–or even essential–to find redundant paths [8–15] which avoid every color. Thisanalysis can be translated into a new kind of percola-tion (about percolation see [16–22]), called color-avoidingpercolation which was first introduced in [2]. Previouslypercolation theory on networks was among many otherapplications used to study robustness of complex systems[23–26], epidemic spreading [27–30], opinion spreading[31], and traffic [32, 33].In [2], we presented the theory of color-avoiding per-colation based on probabilities that a randomly chosennode can (or cannot) communicate with a macroscopicfraction of other nodes over a particular link, avoiding acolor c . These probabilities are dependent for differentcolors: If a link is useful for communication avoiding onecolor c , it may be more likely useful for avoiding a secondcolor c (cid:48) as well. These dependencies were treated with aheuristic approximation in [2]. While this approximationworks for all cases discussed there, its limits have notbeen discussed. Further, the dependencies have not beendiscussed in detail. Here we develop a theory treating de-pendencies explicitly. We show that while for few colors, the heuristic approximation of [2] is suitable, when thereare many colors it is not sufficient. For every additionalcolor to be avoided, dependencies of single link probabili-ties affect the reduction of color avoiding connectivity. Inthis way, we can understand the critical behavior of coloravoiding percolation step by step, starting from standardpercolation.Promising generalizations of color avoiding percolationwere introduced and applied to the autonomous systemslevel Internet: Heterogeneous color frequencies, and sce-narios with trusted colors [2]. Here we present a sys-tematic discussion of these generalizations as well as newtheoretical results as well. We employ our theory with aflexible treatment of trust scenarios: Nodes of a certaincolor can be trusted or avoided as senders/receivers , andthey can be trusted or avoided for transmission . Thismakes it possible to compare trust scenarios: Avoidingonly one color (the color which is believed to be mostlikely to fail) for sending/receiving and transmitting isequivalent to standard percolation. Avoiding more andmore colors makes connectivity increasingly robust to-wards correlated failures, but possibly restricts the num-ber of nodes which can participate.We find a surprising and rich critical behavior phe-nomenology. With heterogeneous color frequencies, wefind that the avoided colors with the largest frequen-cies define the critical threshold and exponent. For col-ors with considerably smaller frequency, the difference incolor avoiding percolation can be small, whether thesecolors are trusted or not. As a consequence, color avoid-ing connectivity can be increased by switching node col-ors such that dominant colors are reduced in frequency,or by increasing normal connectivity. The weak effect a r X i v : . [ phy s i c s . s o c - ph ] N ov of small frequency colors on color avoiding percolationfurther allows us to introduce an approximation. Thisis important as, with the new, more detailed theory, thecomputational cost of calculating the largest color avoid-ing component increases exponentially with the numberof avoided colors.The structure of this paper is as follows: in Section IIwe provide a full and precise treatment of color-avoidingpercolation theory and we extend the theory to heteroge-neous color frequencies, arbitrary sets of avoided trans-mitting node colors, and the case where sending nodecolors are also avoided. In Section III we focus on Erd˝os-R´enyi networks and develop a number of analytic resultspertaining to the critical behavior of color-avoiding per-colation including critical exponents and numerical vali-dation of our analytic results. We also present analyticresults for the phase transition which occurs when thecolor frequencies change. In Section IV, we study the casethat each pair of sender and receiver nodes trust nodes oftheir respective colors, leading to a kind of inter-color ad-jacency matrix of color-avoiding connectivity. In SectionV, we describe different approximation approaches forcalculating color-avoiding connectivity and explain thepreviously published approximation. II. THEORYA. Color-avoiding percolation
We consider a network in which there are many dif-ferent design problems or other vulnerabilities, and eachproblem affects nodes of a different color c . This is il-lustrated in Fig. 1a with three colors (cyan, white andblack). Assuming there are n c bad colors (colors whiteand black in the figure) and m c good colors (cyan in thefigure), we might naively opt to restrict connectivity togood colored nodes only. However, this can render theentire system useless, if there are sufficiently many nodeswith bad colors that their effective removal brings thenetwork below the percolation threshold. For instancein Fig. 1a, assuming that black and white nodes arebad, nodes S and R cannot communicate using only cyannodes and are effectively disconnected. Instead of just re-moving all of the bad colors, we can look for redundantpaths where each path avoids one of the bad colors. As wecan see in Fig. 1a, this can restore connectivity: Betweennodes S and R there is a path avoiding black nodes (high-lighted yellow) and a path avoiding white nodes (high-lighted magenta). Because no single of the bad colorsis required for connectivity, we can restore connectivitywhile remaining robust to the failure of any single color.Likewise, if the colors represent eavesdroppers and themessage is split via secret sharing, and transmitted onmultiple paths, no single eavesdropper can decode thewhole message.We define a pair of nodes as color-avoiding connected(CAC), if they have paths between them which avoid all (a) (b)(c) (d)FIG. 1. Color avoiding percolation with redundantpaths for avoiding white and black nodes. (a)
SenderS and receiver R can communicate along the magenta pathavoiding nodes of white color, or along the yellow path avoid-ing black nodes. (b)
The component L +white , where every nodepair can communicate over a path avoiding nodes of whitecolor in between. Links to dangling nodes of white color areshown with dashed lines. (c) Component L +black . (d) Nodesin L color = C white ∩ C black ∩ L +white ∩ L +black are highlightedwith red halos, where C c denotes the set of nodes with colorother then c . Nodes in L color are pairwise color avoiding con-nected. If black and white nodes would be avoided together,connectivity for blue nodes would break down. of the vulnerable node colors, with each path avoidinga different node color. In the case that all of the colorsare vulnerable, (as in [2]), that means that we require asmany paths as colors (though not necessarily distinct). InFig. 1a we see that there exist paths between sender Sand receiver R avoiding only white nodes or avoiding onlyblack nodes, and they are thus CAC. For color-avoidingpercolation, we are interested in L color , the maximal setof nodes which are CAC, and the conditions under which L color occupies a finite fraction of the total network. In[2], the theory of color-avoiding percolation was limitedto the case in which every node was colored and all ofthe colors were trusted as senders/receivers. For trans-mission, the focus was on the case where all colors wereavoided. A generalization with trusted colors for trans-mission was introduced, but the theory was not discussedin more detail. Here we generalize that theory to the casewhere arbitrary sets of colors are avoided or trusted astransmitters or as senders/receivers.Assuming C disjunctive sets of node colors withfrequencies r c such that (cid:80) Cc =1 r c = 1, we definethe set of colors to be avoided as senders/receiversas S = { s , . . . , s R } ⊆ { , . . . , C } , and the setof colors to be avoided as transmitters as T = { t , . . . , t T } ⊆ { , . . . , C } . In Fig. 1a we have colors(1,2,3)=(blue,white,black), and we want to avoid whiteand black nodes for sending and transmitting: S = T = { white , black } . If S = T (cid:54) = { , . . . , C } , the node classeswith colors in T are avoided for transmitting, and forsending and receiving as well. This can be used torepresent the case, where certain node colors are univer-sally mistrusted while other node colors are universallytrusted. For instance, if a node of color c is attempt-ing to communicate with a node of color c (cid:48) , it may trustall nodes of its own color and the receiver color c (cid:48) butnot trust any other colors as either sender/receivers ortransmitters. We return to this special case in detail inSec. IV. For S = ∅ , nodes with problems are allowed tosend or receive, while they are avoided for transmitting.This scenario is useful, if nodes of a class fail with a cer-tain probability less than one. Or if the vulnerabilityof color c does not impair the ability of the nodes tofunction as senders or receivers, but only impairs theirtransmitting abilities. In such a case, even though thesender itself has a vulnerable color, it still makes senseto avoid nodes of its own class on the path, in order toreduce the probability of disconnecting.We define C c as the set of all nodes with color otherthan c , and L + c as in [2]: We remove all nodes with color c and find the largest component in the remaining graph, L ¯ c . We then define L +¯ c as the set of all nodes which have adirect neighbour in L ¯ c . This trivially includes the whole L ¯ c and additionally includes nodes of color c that aredirectly connected to L ¯ c ( L +white in Fig. 1b and L +black inFig. 1c). These “dangling” c -colored nodes represent thenodes which can communicate via L c , without requiringany c -colored nodes aside from themselves.For the general case of avoiding a given set S ofsender/receiver colors and a set T of transmitter colors,we find the color-avoiding giant component L ( S , T )color andits relative size S ( S , T )color as L ( S , T )color = C s ∩ · · · ∩ C s R ∩ L + t ∩ · · · ∩ L + t T (1) S ( S , T )color = P ( C s ∩ · · · ∩ C s R ∩ L + t ∩ · · · ∩ L + t T ) (2)= (cid:32) − (cid:88) s ∈S r s (cid:33) (cid:34) − P (cid:32) (cid:91) t ∈T ¬L + t (cid:33)(cid:35)(cid:124) (cid:123)(cid:122) (cid:125) S T color . (3)With ¬L + t we refer to the network of nodes not belongingto L + t . Notice that negation turns the intersection into aunion. The probability S ( S , T )color describes the case where sending ( S ) and transmitting ( T ) nodes are avoided, andthe probability S T color describes the case where only trans-mitting nodes ( T ) are avoided. In this manner, the gi-ant color-avoiding component S color defined in [2], canbe obtained by letting T = { , . . . , C } and S = ∅ , whereall colors are avoided for transmitting, and no color isavoided for sending and receiving.For Eq. 3, color distribution and connectivity proper-ties are assumed to be independent. The first term inthis equation describes a node property only, while thesecond term describes a property determined by connec-tions and neighbors. We assume the colors to be dis-tributed randomly, regardless of the network structure.This assumption has been verified with simulations, seefor example Fig. 2. In order to use the formalism ofgenerating functions, which are not well suited to unionsof probabilities, we use the inclusion-exclusion principle[34] to rewrite P (cid:32) (cid:91) t ∈T ¬L + t (cid:33) = (cid:88) Q⊆T ( − |Q|− P (cid:92) q ∈Q ¬L + q . (4) Q takes on all possible subsets of T , including T itself,but not the empty set. The term |Q| denotes the numberof elements in the set Q which is a subset of the set ofavoided colors. In words, this equation uses the followingrule: A sample is in the union of events, if it is in thefirst or second or third event etc., but double countingfor pairwise intersections of events has to be subtracted.This procedure over-corrects intersections of triplets ofevents, which is then added back again and so on. We cannow use single link probabilities and generating functionsand rewrite u Q ≡ P (cid:92) q ∈Q ¬L + q , (5) g ( u Q ) = P (cid:92) q ∈Q ¬L + q . (6) u Q is the probability, that a node belongs to ¬L + q for allcolors q ∈ Q , after we destroyed all of its links except forone randomly chosen link. In other words, it gives theprobability that a node is not connected to any of thecomponents L q ( q ∈ Q ) over one particular link. Theset Q is meant as an index of u Q , so we will use u { , } for u Q with Q = { , } , or u T for Q = T etc. In Eq.6, we assume random ensembles of networks with sizegoing to infinity and the locally treelike approximation.The probability for all links to fail to connect a certainnode to components L q can be found with the generatingfunction of the degree distribution g .Finally, we need equations defining probabilities u Q .We use self consistency equations: A link fails to connecta first node with probability u Q , if the other node alongthis link is not connected to any of L ¯ q for q ∈ Q , over anyof its other links. For the first color the usual equationholds: u { c } = r c + (1 − r c ) g ( u { c } ) , (7)where g ( z ) = g (cid:48) ( z ) /g (cid:48) (1) is the generating function ofexcess degree [17]. For two colors c (cid:54) = q the equationreads u { c,q } = r c g ( u { q } ) + r q g ( u { c } )++ (1 − r c − r q ) g ( u { c,q } ) . (8)In equation 8, the first term represents the probabilitythat the node reached through the link in question is ofcolor c , and that at the same time it does not connectto the color avoiding component of color q . The secondterm represents the probability that the reached node isof color q , and that at the same time it does not connectto the color avoiding component of color c . The thirdterm represents the probability that the reached nodeis neither of color c nor q , but it also fails to connectto color avoiding components of colors c and q ( L c , L q ).Clearly we have to plug in the result of the self consistentequations 7 into self consistent equation 8, to be able tocompute the result. This has to be done numerically. Ingeneral, the equation for a joint probability that a nodeconnects to none of the components avoiding colors Q over a particular link reads u Q = (cid:88) q ∈Q r q g ( u Q\{ q } ) + − (cid:88) q ∈Q r q g ( u Q ) (9)where Q \ { q } is defined as the set containing all colorsincluded in Q , except for the color q . For capturing thecase of only one color Q = { c } , we define u ∅ = 1, whichis a consistent definition, as it is a solution of 9 itself( u ∅ = g ( u ∅ )). The only way to compute the joint prob-abilities for larger |Q| is to go step by step from jointprobabilities for subsets of Q (cid:48) ∈ Q using equation 9. Inthis way, all the subsets of T have to be considered, in-cluding T itself. The results have to be plugged intothe equation for S T color . Combining equation 3 and thefollowing equations up to equation 6 together, we find S T color = 1 + (cid:88) Q⊆T ( − |Q| g ( u Q ) . (10)Further using equation 9, this gives us the necessary in-put for calculating the size of the giant color avoidingcomponent, equation 3. The comparison between thistheory and the approximate theory developed in [2] islaid out in section 3.D. B. Color-avoiding percolation as a generalization ofstandard percolation
In standard site percolation, the question is whether alarge part of a network stays connected after a random fraction of nodes is destroyed. To demonstrate how thecolor-avoiding percolation framework can be describedas a generalization of standard percolation, we considerthe following color-avoiding percolation problem which isequivalent to standard percolation. We begin with a net-work composed of two node types: bad ( c = 1) and good( c = 2). We assume the bad fraction is known to havedesign problems (as an extra property beyond the net-work connections), and therefore should be avoided forconnections. Each color occurs with frequency r c suchthat r + r = 1. With S we denote the fraction of nodesin the surviving giant component, or equivalently, theprobability for a single node to belong to the survivinggiant component. That means, all nodes of the bad color c = 1 are excluded as senders/receivers and as transmit-ters. We obtain: S = P ( C ∩ L +1 ) (11)= P ( C ) P ( L +1 ) (12)= P ( C )[1 − P ( ¬L +1 )] . (13)Here we have assumed that C and L +1 are independent,as discussed for color avoiding percolation above. It holds P ( C ) = 1 − r . Equations 12 and 13 are convenientfor calculations using the formalism based on generatingfunctions [17, 35]. Assuming random ensembles of net-works with size going to infinity and the locally treelikeapproximation, we further obtain P ( ¬L +1 ) = g [ P ( ¬L +1 )] , (14) P ( ¬L +1 ) = r + (1 − r ) g [ P ( ¬L +1 )] . (15) P ( ¬L +1 ) is the probability, that a node belongs to ¬L +1 , after we destroyed all of its links except for one ran-domly chosen link. In other words, it gives the probabil-ity that a node is not connected to the giant componentover one particular link. These equations are equivalentto equations that describe random attack on networks[23, 24], as they should be. C. Notes on computation
To summarize the framework of color avoiding perco-lation, we found that 2 |T | − u Q for all subsets Q ⊆ T .Results are plugged into equation 10. For illustratingthe procedure, let us discuss an example with three totalcolors, of which two are avoided for sending and trans-mitting: C = 3, S = T = { , } . u { } = r + (1 − r ) g ( u { } ) ,u { } = r + (1 − r ) g ( u { } ) ,u { , } = r g ( u { } ) + r g ( u { } ) + r g ( u { , } ) ,S ( S , T )color = r (cid:2) − g ( u { } ) − g ( u { } ) + g ( u { , } ) (cid:3) . (16)For ten avoided colors, 1023 subsets of T have to beconsidered, which is numerically still easy to do. If fre-quencies of all avoided colors are identical, the numberof different transcendent equations reduces drastically to |T | . However, for tens of avoided colors, the limited pre-cision of numerical results for u Q (especially due to thelimited precision numerics for the generating function g )and the combination of many terms limits feasibility ofthe straight forward evaluation. In the general case ofheterogeneous color frequencies, we observed that sum-ming over the sets needed is an easy numerical task forour computer as long as |T | ≤
20. For twenty avoidedcolors, about one million subsets have to be considered.We provide Python code in the appendix which worksfast and precise enough for all examples discussed in thefollowing (for 20 disjoint colors, it needs about a minuteto calculate S T color on one core of an Intel R (cid:13) Core TM i7-4770 CPU (3.40GHz)), even if there is still much roomfor optimization. Below we discuss an approximation forreducing the number of colors, if many color frequenciesare small compared to the largest color frequency. D. Criticality
The critical behavior of color avoiding percolation hasa rich, multifaceted nature, depending on graph topol-ogy, number of avoided colors and color frequencies. Wewill discuss a number of phenomena using special casesbelow. However, the transition point can be understoodwithin the general theory framework, by referencing stan-dard percolation. Without loss of generality, we assumefor the avoided colors T = { , , . . . , T } , and the totalfrequency of color c = 1, r , is larger or equal comparedto the frequencies of all other avoided colors. We havefor c ∈ T : u { c } = r c + (1 − r c ) g ( u { c } ). From stan-dard percolation we know, that if color c = 1 can beavoided ( u { } < c = 1 is the first to have vanishing connec-tivity, u { } = 1. In this latter case, we have S ( S , T )color = 0,as all terms g ( u Q ) in equation 10 cancel out pairwise.First this is true for 1 − g ( u { } ). Equation 9 for u { , } reduces with g ( u { } ) = 1 to the defining equation for u { } , thus g ( u { , } ) − g ( u { } ) = 0. This generalizes to u Q = u Q∪{ } . Accordingly, the critical point is deter-mined by the avoided color with the largest frequency, inthe same way as for standard percolation. With the re-sult of Cohen [23] we find as a condition for non-vanishingconnectivity (with expected degree ¯ k ) r c < r crit = 1 − ¯ k (cid:104) k (cid:105) − ¯ k . (17) E. Dual variables for simultaneous probabilities
The probabilities u Q are hard to interpret, as they de-scribe a negative statement: The probability of a link to not connect to several L ¯ c at the same time. This makesit confusing to discuss the meaning of dependencies, forexample in the form u { , } (cid:54) = u { } u { } . Furthermore, itturns out that for larger sets Q with more than one ele-ment, it is hard to find good approximations for u Q . Thiscomplicates understanding the critical behavior. There-fore, let us define new positive variables, describing theprobability that a link connects to all L q for q ∈ Q at thesame time: v Q = P (cid:92) q ∈Q L + q (18)= 1 + (cid:88) P⊆Q ( − |P| u P , (19) u Q = 1 + (cid:88) P⊆Q ( − |P| v P . (20)The conditions between u Q and v Q hold with the com-plementary event and the inclusion-exclusion principle.As the transformation from u Q to v Q can be reversedwith the same transformation (it is an involution), and itpreserves all information, variables v Q are dual variablesto u Q . If probabilities were independent, to connect toseveral L q for q ∈ Q at the same time over the same link,it would hold v Q = (cid:81) q ∈Q v { q } . As we will discuss below,conditional probabilities for further colors can both besuppressed and increased.It is useful to have equations for calculating v Q directly,instead of first calculating u Q and then transforming to v Q . Approximations are easier found for calculating v Q directly. For Q (cid:54) = { , . . . , C } it holds: g ( u Q ) = g (cid:88) P⊆Q ( − |P| v P (21)= 1 + (cid:88) P⊆Q ( − |P| v P − (cid:80) q ∈P r q . (22)In the first of these equations, only u Q was plugged inusing equation 20. Equation 22 can be understood asfollows: Starting with Q = { c } , replace the left handside of Eq. 9 using Eq. 20, and solve for g ( u { c } ). Forincreasing from Q\ { q } to Q , replace the left hand side ofEq. 9 using Eq. 20. Then plug in the lower order resultsalready obtained to the right hand side of equation 9,and solve for g ( u Q ). Finally, it holds v { ,...,C } = 0 . (23)This reflects the fact that a single link can never be usedfor avoiding all colors at the same time. The node reachedover this link has a certain color, consequently this colorcannot be avoided. (a) (b) (c)FIG. 2. Color avoiding connectivity depends on topology and color distribution.
Results for Poisson graphs, wherethe first two colors with frequencies r + r . r = 0 . (a) The black line shows the connectivity S { } color avoiding thedominating color c = 1 with frequency r = 0 .
5. The blue line shows results S { , } color , if first and second color are avoidable, wherethe second color has a smaller frequency r = 0 .
3. The fraction of connected nodes reduces only slightly, even if many morenodes have to be avoided. Numerical results confirm the theory (blue crosses, averages over 100 networks of size N = 10 ).If nodes of colors one and two would be avoided at the same time, as shown with the red line, connectivity would reduceconsiderably. (b) For ¯ k = 4, color one cannot be avoided at all as long as r > r crit = 3 /
4. Therefore, in a network with onedominating node class, it can be beneficial to replace colors of some nodes, thus reducing r and increasing r (as outlined inthe inset). While standard connectivity stays suppressed (red line), color avoiding connectivity (blue line) increases almost asmuch as connectivity avoiding only the larger color (black line). (c) S { , } color is shown for varying topology and color frequencies.The solid white line indicates the critical manifold. Dashed white lines highlight the parameter manifolds of (a) and (b), andthe phase transition along these lines has critical exponent β k = β r = 1. The phase transition along the dotted white line is ofdifferent type, with exponent β k = 2 (see figure 3). This is connected to a sharp bend in the critical manifold. III. PHASE TRANSITION FOR POISSONGRAPHS
We start with equations 16, where the first two out ofthree colors are avoided. In figure 2, results are shownfor Poisson graphs. Colors one and two are avoided forsending and for transmitting as well, and connectivity iscalculated as the fraction of CAC nodes among the nodesof color three, S ( { , } , { , } )color /r = S { , } color . For this figure,we fixed r + r = 0 . r = 0 .
2, and varied (a) con-nectivity with constant color frequencies ( r = 0 . r = 0 . k = 4, and (c) both together. In (a)and (b), additional to the CAC results (blue lines) weshow results similar to standard percolation, where onlyone color is avoided for transmission. With black lineswe show results for avoiding only the dominant color fortransmission, S ( { , } , { } )color /r = S { } color . As above, the frac-tion among all nodes of color c = 3 is shown. With redlines we show the situation where nodes of colors one andtwo are both together totally avoided for transmission, S ( r + r ) /r . Here we use the notation S ( φ ) for stan-dard percolation, for a random fraction φ of destroyed /avoided nodes (fraction 1 − φ of surviving nodes respec-tively, compare equation 13). We see that CAC is almostas high as standard connectivity avoiding the dominant color, while avoiding the problematic nodes of colors oneand two all together, reduces the connectivity remark-ably. In (b), we further see that color avoiding connec-tivity increases only slowly, when color frequencies comeclose, r ≈ r . This observation can help for finding thebest cost-benefit trade-off of replacing nodes colors, ifthere is a dominant color in the system. With (c) we seethat for color avoiding connectivity, both graph topologyand color distribution are crucial. Therefore, a combina-tion of improving topology and color heterogeneity canbe most efficient for increasing color avoidability.To prepare for approximate results, let us switch tovariables v Q . For Poisson graphs, we have g ( x ) = g ( x ) = exp[¯ k ( z − k . Therefore, we have to solve equations9 for g ( u Q ), allowing to plug in g ( u Q ) = g ( u Q ) directlyinto equation 10. This is possible for Q (cid:54) = { , . . . , C } . In-serting results formulated with positive variables v Q (Eq.22) to equation 10, we find for T (cid:54) = { , . . . , C } that S ( S , T )color = (cid:32) − (cid:88) s ∈S r s (cid:33) v T − (cid:80) t ∈T r t . (24)This is a surprisingly simple result, reducing to singlelink probabilities (the same holds for normal percolationon Poisson graphs, being captured with a single avoidedcolor: S = S ( { } , { } )color = 1 − u { } ). Finally, for Q = { , . . . , C } we find the condition on v { ,...,C } = 0, insteadof a replacement of g , which is not available here. A. Critical behavior
For Poisson graphs, the conditions for critical param-eters read r c < r crit = ¯ k − k , c ∈ T (25)and solving for ¯ k , we have¯ k crit = 11 − max c ∈T r c . (26)Critical behavior according to varying topology or vary-ing color frequencies is, as usual, described using the crit-ical exponent β . For clarity we use two exponents: S ( S , T )color ∝ (cid:0) ¯ k − ¯ k crit (cid:1) β k , (27) S ( S , T )color ∝ (cid:18) r crit − max c ∈T r c (cid:19) β r . (28)This can be dominated by the avoided color with thelargest frequency, but in general the interplay of differentcolors can lead to new effects which are not present instandard percolation.The critical behavior of color avoiding percolationshows interesting features, as we see in Fig. 2c. While thephase transition along the dashed white lines has criticalexponent β k = β r = 1, the situation is different, alongthe dotted white line, with β k = 2. The behavior of β k can be summarized as follows: The critical exponent β k is determined by the degeneracy of the maximal colorfrequency, i.e., the number of colors which are tied formost common, β k = n deg = (cid:88) c ∈T δ r c , max t ∈T r t . (29)Below we will confirm this result using an approxi-mate scheme, explicitly presenting results for up to threeavoided colors. In [2] it was reported that β k = C , underthe assumption that all C colors are avoided for trans-mission, and they all have the same frequency. The moredetailed analysis here confirms these results and showsthe correct extension to cases where the colors have dif-ferent frequencies.The increased critical exponent β k = 2 along the dot-ted line in Fig. 2c is connected to a sharp bend in thecritical manifold, shown with a straight white line. Italso is connected to the fact, that color connectivity ismost suppressed compared to only avoiding the domi-nant color, if r = r (see (b)). Between both kinds ofphase transition, with β k = 1 and β k = 2, there has to besome kind of crossover. This is important to understand FIG. 3.
Crossover between critical behavior with β k = 1 (black line) and β k = 2 (red line). We fix thefrequency of the dominating avoided color r = 0 .
2, thus fix-ing ¯ k crit = 1 .
25 on Poisson graphs. Results are shown for r = r (red line), r = r − .
03 (green line), r = r / r = 0 (blue line, identical to standard per-colation avoiding first color). For the third color, trusted forsending and transmitting, we have r = 1 − r − r . Sym-bols of according colors are averages over 100 networks of size N = 10 . in order to determine the behavior of finite size networkswith r ≈ r . Before we turn to an analytic examina-tion of S color in the crossover regime, let us discuss thephenomenology with figure 3. We use logarithmic scal-ing on both axis, thus power law dependencies show asstraight lines. We set r = 0 . k crit fixed. Resultsare shown with straight lines for the cases r = r (redline), r = r − .
03 (green line), r = r / r = 0 (blue line). Numerical results are in goodagreement with the theory, only limited by finite size ef-fects for small size of the largest CAC component. If thefrequency of the second color is far below the frequencyof the first color (black line), we see that color connec-tivity is overall reduced compared to one avoided color(blue line), but the critical behavior is not affected. If thefrequency of the second color is close to that of the firstcolor (green line), the behavior is close to this with iden-tical color frequencies for large connectivity (red line).Therefore, the increased critical exponent β k = 2 is ofpractical relevance in finite size graphs, if color frequen-cies are close to each other. Closer to the critical point,the deviation of color frequencies results in a critical ex-ponent of β k = 1.For small deviations of color frequencies r ≈ r (and r ≥ r ), we can analyze the critical behavior analyti-cally. We start with one avoided color. It is well known,that for Poisson graphs the critical exponent of standardpercolation is β k = 1. We reproduce this result using thealternative formulation Eq. 22 with an expansion of thegenerating function for small v { c } ( c = 1 , v Q replaces u Q as defined in Eq. 19. We have g (1 − v { c } ) = 1 − v { c } − r c (30) ≈ − ¯ kv { c } + (¯ kv { c } ) / v { c } ≈ k (cid:18) ¯ k − − r c (cid:19) ≡ k (cid:0) ¯ k − ¯ k c (cid:1) . (32)We define ¯ k c = 1 / (1 − r c ). For ¯ k < ¯ k = ¯ k crit we have v { } = 0. With Eq. 24 we have S ( { } , { } )color = v { } , andtherefore we found β k = 1. How does the critical behav-ior generalize to two colors, v { , } ? Still, color c = 1 isthe color with higher frequency, and the critical behav-ior is connected to small v { } , increasing from zero. Aswe are interested in color frequencies with small devia-tion, we assume small v { } as well. We develop the selfconsistency equation 22 accordingly, g ( u { , } ) ≈ − ¯ kv { } + (¯ kv { } ) − ¯ kv { } + (¯ kv { } )
2+ ¯ k v { } v { } + ¯ kv { , } (33)= 1 − v { } − r − v { } − r + v { , } − r − r , (34)¯ k v { } v { } ≈ v { , } (cid:18) − r − r − ¯ k (cid:19) , (35) v { , } ∝ (¯ k − ¯ k )(¯ k − ¯ k ) (36)= (¯ k − ¯ k crit )[(¯ k − ¯ k crit ) + (¯ k − ¯ k )] . (37)Most of the terms canceled out with approximate resultsfor v { } and v { } from above. The largest correction inEq. 35 is of order v { } v { } , therefore the approximationrequires v { } (cid:28)
1. With ¯ k ≈ ¯ k and S ( T , T )color = v T , wefind that color connectivity scales with (¯ k − ¯ k crit ) , aslong as (¯ k − ¯ k crit ) (cid:29) (¯ k − ¯ k ). For r = r , we thus find β k = 2. For r > r , the critical exponent β k = 1 is onlyvisible for (¯ k − ¯ k crit ) (cid:28) (¯ k − ¯ k ), what can be dominatedby finite size effects, if color frequencies are close to eachother.With three avoided colors, we can analyze how the crit-ical behavior generalizes to more colors. Figure 4 showsresults for disjoint colors on Poisson graphs, where threecolors are avoided. We fix the frequency of the first colorto r = 0 .
3. Again, S { , , } color can be seen as the fractionof nodes in the largest CAC component, among all nodesof color four. Compared to the black line, showing re-sults where the other frequencies of avoided colors aresmaller ( r = r = 0 . r = 0 . r = r = 0 .
3, andthe exponent increases to β k = 2. We saw this exponentalready for two avoided colors with degenerated frequen-cies. Now we see, that with a third avoided color with FIG. 4.
Influence of the dominating colors.
Resultsfor Poisson graphs. The first three colors are avoided. (a)
Compared to a single dominating color (black line, r = 0 . r = r = 0 .
12, trusted fourth color with r = 0 . r = r = 0 . r = 0 . r = 0 .
28; red line: r = r = r = 0 . r = 0 . (b) with logarithmic scaling. We see that the critical ex-ponent β k is identical to the degeneration of the highest colorfrequency. smaller frequency r = 0 .
12, the exponent stays the same.The crossover as described for two avoided colors applieshere as well, and the two-dimensional critical manifoldin the parameter space ( r , r , r ) has a sharp bend aswell (results not shown). The red line shows results fortriple degenerate highest color frequency of avoided col-ors r = r = r . This is connected with critical exponent β k = 3. This exponent can be extracted analytically, inthe same way as it was done for two avoided colors. Using v { } = v { } etc. for identical frequencies of avoided col-ors, we develop Eq. 33 to third order (instead of secondorder), plug into an equation 22 for g ( u { , , } ), developto third order and solve the leading terms for v { , , } : v { , , } (cid:18) − r − ¯ k (cid:19) ≈ k v { } v { , } + ¯ k v { } ) , (38) v { , , } ∝ (¯ k − ¯ k crit ) , (39)where we have used the fact that v { } ∼ ( k − k crit ) fromEq. 32 and v { , } ∼ ( k − k crit ) from Eq. 36. A similarprocedure can be applied for higher numbers of avoidedcolors and heterogeneous color frequencies, which wouldrequire higher order expansions in the generating func-tions, and many terms of same order to be considered.The only case, in which this procedure is not applica-ble, is for avoiding all colors for transmission. It holds v { ,...,C } = 0, while S color is finite. For this case, theapproximation of [2] can be useful, as discussed in thesection about approximations. B. Dependencies among variables v Q and criticalexponents FIG. 5.
Simultaneous probabilities for avoiding manycolors give a qualitative understanding of critical ex-ponents. (a)
With two avoided colors and r = 1 / r = 1 /
5, the critical connectivity is ¯ k = 4 /
3. Close to thecritical point, v { } grows with exponent of one (black line),saturating to the probability of having color c (cid:54) = 1 for large ¯ k . v { } (solid green line) has a value of about 1 /
10 at the crit-ical point. The product of both probabilities v { } v { } (reddashed line) can be seen as an approximation of independentprobabilities for v { , } ∝ S { , } color (red solid line). This helpsto qualitatively understand the critical exponent β k = 1. (b) Three avoided colors with r = r = r = 1 /
4. Probabilities v { } (black solid line), v { } (red dashed line) and v { } (bluedashed line) are useful for a qualitative understanding of thequadratic behavior of v { , } (red solid line) and cubic behaviorof v { , , } ∝ S { , , } color (blue solid line). In this way, the criticalexponent β k = 3 is comprehensible. Asymptotic for large ¯ k :1 − r − r true result, (1 − r )(1 − r ) = 1 − r − r + r r with independence assumption. Here we consider the probability of a link simultane-ously avoiding all colors in a set Q , or in other wordssimultaneously connecting to all sets L ¯ c for the colors c ∈ Q . This probability is calculated exactly in equation(19) and is related directly to the overall connectivity(Eq. (24)). When considering the phase transition, wefound that this probability is proportional to the productof probabilities for each color c ∈ Q separately. Althoughthese probabilities are not independent, we found abovethat the critical point and scaling exponent are consis-tent with the assumption that they are independent (cf.equation (35)). To understand what impact the depen-dencies between colors have on the overall color-avoidingconnectivity, we compare the assumption that the con-nections are independent for different colors with the fullsolution obtained above. We define the assumption of independent probabilities (AIP) by taking v AIP Q ≡ (cid:89) q ∈Q v { q } . (40)As we found for Poisson graphs S ( S , T )color ∝ v T , this canhelp us to understand the critical behavior. Apart fromthat, comparing v AIP Q with results v Q , where all depen-dencies are explicitly included, can teach us about de-pendencies.In Fig. 5a we see that for two avoided colors withdifferent frequencies, v T and v AIP T have the same qualita-tive behavior. This way, the critical exponent β k can beunderstood as follows: The probability v { } for avoidinga first color has a linear onset starting from the criti-cal point. As the probability v { } already has reached apositive value, the probability for both at the same timehas a linear onset as well. With Eq. 35, we can esti-mate dependencies close to the critical point (as long as v { } (cid:28)
1) to be v { , } v { } v { } ≈ − r − r (1 − r ) r . (41)We find that the conditional probability that the samelink helps to avoid a second color can be increased ascompared to the probability for the first color (as in thefigure close to the critical point). The conditional prob-ability can also be suppressed (in the figure for large ¯ k ,or for r = 0 .
45 and r = 0 . k ,results not shown). In Fig. 5b we see how v AIP Q can helpto understand larger critical exponents for degeneratedlargest color frequencies.If all colors are avoided, we always have v { ,...,C } =0 < v AIP { ,...,C } . This reflects the fact that in this case nosingle link can be used to avoid all colors, as the nodereached over the link in question has one color, whichcannot be avoided. Accordingly, only nodes in the two-core can be in the giant CAC component [2]. This is inline with our analysis of the limit where C → ∞ ( N ) [2].There we showed that, rather than recovering standardpercolation in this case, we instead recover k -core perco-lation with k = 2 [10, 11, 36]. On the other hand, colorconnectivity can be expressed in terms of variables v Q using Eq. 10 and 20. In Sec. V we discuss that the con-dition v { ,...,C } = 0 must be fulfilled by any appropriateapproximation for color avoiding percolation, and howthis helps for the heuristic approximation of [2]. IV. SENDER AND RECEIVER TRUST THEIRCOLORS
The theory of color-avoiding percolation has been de-veloped to answer the question, what is the maximalset of nodes that can mutually communicate using color-avoiding paths? Until now, we have examined this ques-tion globally, without considering the colors of the sender0
FIG. 6.
Sender and receiver trust their own colors.
Having four colors with frequencies r c = c/
10, we choose pairs( c , c ) ⊂ { , , , } of trusted colors for sending and receiv-ing. Shown is the fraction of sender nodes with color c whichare able to reach nodes of the receiver color c . With denot-ing c and c for the two remaining colors, it reads S { c ,c } color (for sender and receiver color being the same c , we use c , c and c denoting the other colors, and S { c ,c ,c } color ). For thesmallest ¯ k = 1 .
4, we see that all nodes of colors one and twoare excluded from communication, but they are needed forcolors three and four to connect them. For increased ¯ k = 1 . k = 2 .
0, nodes of all color pairs can communicate. or receiver nodes. This is a reasonable assumption formany scenarios, but there are cases where the answerto this central question will depend on the color of thesender and receiver nodes. For example, if a person incountry A wants to communicate with a person in coun-try B, it may be that they both trust the routers of theirrespective countries, but want to avoid all other coun-tries [2]. In such a case, the CAC giant component variesdepending on the sender color and receiving color. Thisgives rise to a new concept of inter-color connectivity andan inter-color adjacency matrix, as shown in Fig. 6. Wehave four colors c = 1 , , , r c = c/
10. This situation is illustrated in the upper left ofthe figure. On the upper right, we see the fraction amongthe sender nodes of a color specified on the x-axis, whichcan reach a macroscopic part of receiver nodes of anothercolor, specified along the y-axis. Results are for Poissongraphs with small ¯ k = 1 .
4. For different trusted colors ofsender and receiver nodes, this is S { c ,c } color with c , c beingthe two colors which are not present on sender and re-ceiver nodes. For example, lets discuss whether nodes ofcolor three can communicate to nodes of color four. The according fraction of nodes having color three is S { , } color ,and this is identical to the fraction among nodes of colorfour which can communicate to color three. We see, thatnodes of these both colors are the only ones being CAC.If for example nodes of color two would like to connect tonodes of color four, they need to avoid color three. This isnot possible for the small ¯ k chosen here. With this resultwe found, that whole classes of nodes (here with colorsone and two) can be excluded from CAC, while they areneeded for other nodes to provide connectivity. This isin sharp contrast to standard percolation and other vari-ants of percolation questions. It makes sense to allowfor transmitting nodes which themselves cannot benefitfrom CAC. These nodes are connected in the normal gi-ant component and thus functional, only excluded fromthe more robust color-avoiding connectivity.Interestingly, even nodes of color four are not CACto each other for ¯ k = 1 .
4, as they cannot avoid colorthree. The according fraction was calculated as S { , , } color ,as trusted color of sender and receiver are both c = 4, andall other colors have to be avoided. For increased ¯ k = 1 . k = 2 (lower right of the figure), nodes of all colors canbe CAC, while the fractions for different trusted colorsof senders and receivers are still highly heterogeneous.This is different to the case where all colors of a certainset T always have to be avoided for transmission, evenif they are present on the sender and receiver nodes. Asa consequence, the fraction of nodes in the largest CACcomponent is S T color , being the same for all colors.Let us finally discuss graphs with broad degree distri-butions with p k ∼ k − α ( k >
0) and generating functions g ( z ) = Li α ( z ) /ζ ( α ) and g ( z ) = Li α − ( z ) / [ zζ ( α − α ( z ) is the polylogarithm function. In [2] it was pointedout that for such graphs color avoiding connectivity issuppressed when avoiding all colors. This is the case, asthe largest CAC component can only be a subset of thetwo-core. The situation is different, if sender and receivercolors are trusted. This is illustrated in Fig. 7. V. APPROXIMATIONS
By discussing the critical behavior for Poisson graphs,we already saw that approximations in calculating v Q arepossible, as long as the defining transcendent equationsare developed until sufficient orders. It helps that lowerorders cancel out in the defining equations for v Q . Con-trarily, for defining equations for u Q , lower order termsdo not cancel out, and polynomials of high order have tobe solved. While variables v Q facilitate approximationsconsiderably, there are still some problems left: For de-generated highest frequencies of avoided colors, v Q hasto be approximated up to order β k in order to reflectthe critical behavior. For heterogeneous frequencies, the1 FIG. 7.
For scale-free graphs, trusting colors can in-crease color avoiding connectivity remarkably.
Graphswith broad degree distribution ( p k ∼ k − α , k >
0) and thesame color frequencies r c = c/
10 as in Fig. 6. The redline shows S { , , , } color , where no color is trusted for transmis-sion (the circles indicate numerical results, averages over 100graphs of size N = 10 ). This is restricted by the two-core(dashed black line). With trusting colors, color-avoidable con-nectivity is increased above the size of the two-core. Resultsshown for trusting color four ( S { , , } color , green line and circles)and trusting colors three and four ( S { , } color , blue line and cir-cles). The latter case, where sender and receiver nodes ofcolors three and four trust their colors, is close to full stan-dard connectivity (black line). number of sets Q can be large. Therefore, we discuss twoother ways of approximating color avoiding percolation.The first is a heuristic approximation developed in [2],the second works by redefining avoided colors in order toreduce the number of avoided colors. A. Previously published heuristic approximation
In order to compare with the approximate results pre-sented in [2], we avoid all C colors for transmission andnone for sending. All colors have identical frequencies r c = r = 1 /C . We have u { c,q } = u { , } etc. We cansimplify u { } = r + (1 − r ) g ( u { } ) , (42) u { , ,...,j } = jr g ( u { , ,...,j − } ) + (1 − jr ) g ( u { , ,...,j } ) j = 2 , . . . , C. (43) S color = 1 + C (cid:88) j =1 ( − j (cid:18) Cj (cid:19) g ( u { , ,...,j } ) . (44)This reduces the number of quantities u Q to be calculatedfrom 2 C − C . This only reduces the computationtime, as there are still numerical problems. Combina-torial factors (cid:0) Cj (cid:1) are large for large C , and results for u { , ,...,j } can only be numerical, with limited precision. FIG. 8.
Comparison with the approximate theory of[2].
Left: C = 3 colors of the same frequency r c = 1 /C areavoided for transmission, all colors trusted for sending. Re-sults with the approximate theory of [2] (AICP, red dashedline) are close to results with the theory presented here (blacksolid line). The deviation (positive values with the blue dot-ted line and negative values shown with the green dash-dottedline) is small. Right: For C = 10 colors, the deviation islarger, and close to the critical point, the theory of [2] givespoor results, even if the critical behavior is qualitatively cap-tured. Black circles show numerical results (averages over 100networks of size N = 10 ). Especially limited precision for calculating the generatingfunctions causes problems for small S color .In [2], u { , ,...,j } was estimated using the approxima-tion u { ,...,j } ≈ u + (1 − u ) (cid:20) jC ( U { } ) j − + C − jC ( U { } ) j (cid:21) (45) ≡ u AICP { ,...,j } , (46) U { } = 1 − − u { } (1 − u )(1 − r c ) . (47) u is the probability that a node is not connected to thegiant component over one particular link and is computedas the solution of u = g ( u ), where g ( z ) = g (cid:48) ( z ) /g (cid:48) (1)is the generating function of excess degree [17]. U { } denotes the conditional probability that a link fails toconnect to L ¯1 given that it does connect to the normalgiant component via a node having a color c (cid:54) = 1. Wedefine U { } = 1 if u = 1. The probability u { } that asingle link does not connect to a giant L ¯1 is calculatedwith Eq. 7.The approximation for u { ,...,j } is motivated as follows:With probability u a link does not connect to the giantcomponent, so u { ,...,j } ≥ u . If it connects to the giantcomponent (probability (1 − u )), then the node on theother side has either one of the j colors (probability j/C ),2 FIG. 9.
A fake approximation, closer to the theorythen AICP but violating constraint Eq. 49, gives poorresults.
Left: For C = 5 colors, we see results of the the-ory presented here (black solid line), and results with AICP(red dashed line). Results of a fake approximation are poor(solid green line), because this approximation violates theconstraint, residual term ∆ app1 shown with a dashed greenline. Right: The error δ AICP5 (red line) made in describing u { ,..., } is quadratic in ¯ k − ¯ k crit close to the critical point.For the fake approximation, δ app15 is constructed to be smaller(green line). leaving simultaneous failure probability of ( U { } ) j − , orit has a different color with probability ( C − j ) /C , leav-ing simultaneous failure probability of ( U { } ) j . In thisapproximation, we neglect dependencies among the con-ditional probabilities U { } , U { } etc. Therefore, we callthis approximation in the following as the approximationof independent conditional probabilities (AICP).A comparison of AICP and the theory presented hereis shown in Fig. 8. We see that for C = 3 colorsAICP works well (upper panel). For as many as C = 10colors, results are still good for large S color , while for S color < − there are strong deviations between thetheory presented here and AICP. However, the criticalbehavior in terms of the critical point ¯ k crit , and the factthat the critical exponent β k is large, are still captured.In order to understand AICP better, let us define δ AICP j ≡ u { ,...,j } − u AICP { ,...,j } . (48)In Fig. 9 on the right, δ AICP5 is shown with a red solidline, for C = 5. We see that δ AICP5 is a quadratic func-tion in ¯ k − ¯ k crit for small values of this variable, therefore u AICP { ,...,C } behaves like a linear expansion around the crit-ical point. This is true for j < C as well (results notshown). However, final results for S color as shown on theleft of the figure (black line for the theory, red dashed linefor AICP) are compatible with an exponent β k = C = 5.To understand better, how a linear expansion in ¯ k − ¯ k crit of variables u { ,...,j } can finally reproduce such a steep FIG. 10.
Uniting colors of small frequency.
Theblack line shows results, where eight colors are avoided, withfrequencies ( r c ) c = (4 / , / , / , / , / , / , / r c ) c =(4 / , / , / r c ) c =(4 / , / , / critical behavior, let us discuss the constraints v { ,...,C } = 1 + C (cid:88) j =1 ( − j (cid:18) Cj (cid:19) u { ,...,j } = 0 , (49)1 + C (cid:88) j =1 ( − j (cid:18) Cj (cid:19) u AICP { ,...,j } = 0 . (50)The first of these equations represents the fact that anode can never be CAC via a single link, if no color istrusted. This constraint is also respected with AICP:In [2] it was shown that the largest CAC is always asubset of the 2-core. This constraint also implies that S color , as calculated with equation 10, cannot easily betruncated after terms with a certain |Q| = Q truncate . Toshow that the constraint equation 49 is crucial for AICP,let us define a fake approximation denoted as “app1”,which is closer to the theory, but violates the constraint.We set u app1 { ,...,j } = u { ,...,j } − δ app1 j = u { ,...,j } − ( δ AICP j ) . . (51) δ app15 is shown with a green solid line in the right panelof the figure. As δ AICP j (cid:28) j , u app1 { ,...,j } is a slightlybetter approximation of the variables u { ,...,j } . However,the resulting S app1color fails to describe the critical behav-ior of S color , as shown on the left of the figure with a3solid green line. Instead we find a result compatible with S app1color ∝ (¯ k − ¯ k crit ) , what reflects the fact that we havea linear expansion in ¯ k − ¯ k crit . Indeed, the result is dom-inated by 1 + C (cid:88) j =1 ( − j (cid:18) Cj (cid:19) δ app1 j ≡ ∆ app1 (cid:54) = 0 , (52)shown on the left of the figure with a dashed green line.If the constraint would be fulfilled, this residual wouldbe zero. B. Approximation for many small color frequencies
With the last subsection we saw that the equationsdefining S color as presented in this paper are hard to ap-proximate. On the other hand, the complexity for homo-geneous color frequencies grows exponentially with thenumber of avoided colors (for transmission). If there is alarge number of colors with marginal frequencies, we canuse the theory as presented in this paper, and manipu-late the set of avoided colors, or combine all colors withsmall frequencies to one color. As can be seen in Fig. 10,this allows to give upper and lower bounds to S color . VI. CONCLUSION
Here we developed a theory for calculating the size ofthe giant color avoiding connected component, for ran-domly distributed colors on random network ensembles.We used dependent simultaneous probabilities, that acertain link does or does not enable to avoid several col-ors at the same time. The conditional probability, thata link helps avoiding a second color c (cid:48) , after it alreadyhelps avoiding a first color c , can be enhanced or sup-pressed compared to independent probabilities. An opentask for future work would be to understand the mechan-ics behind these dependencies. Further we found that aclear understanding of simultaneous probabilities helpsanalyzing the critical behavior. It also helps assessinga previously published heuristic approximation. In gen-eral, it is an interesting finding that dependent probabil-ities for the same link to fulfill different functions can becalculated simultaneously, within the framework of theconfiguration model. To our knowledge, this is a new di-rection in percolation theory, with possible applicationsalso beyond color avoiding percolation.We developed the theory in a way such that it allowsfor flexible trust scenarios, where nodes of a certain colorcan be trusted or avoided for sending/receiving, and fortransmission. This allowed us to compare different trustscenarios among each other, and directly with standardpercolation. We found that trusting colors for transmis-sion, can remarkably increase color-avoiding connectivity,especially for scale free graphs. A sender node trustingits own color and the color of potential receiver nodes for transmission, can increase its color-avoiding connec-tivity the most if its own color is dominating in the net-work. Colors with small frequencies as compared to thedominating color, have a small impact on color-avoidingconnectivity. This allowed us to introduce an approxima-tion, where all colors of small frequencies are united intoone color. This idea could also be helpful for designingrouting algorithms, without keeping track of too manycolors. ACKNOWLEDGMENTS
We acknowledge financial support from the Euro-pean Commission FET-Proactive project MULTIPLEX(Grant No. 317532) and the Italy-Israel NECST project.M.D. is grateful to the Azrieli Foundation for the awardof an Azrieli Fellowship. V.Z. acknowledges support bythe H2020 CSA Twinning Project No. 692194, RBI-T-WINNING, and Croatian centers of excellence Quan-tixLie and Center of Research Excellence for Data Scienceand Cooperative Systems.
APPENDIX ””” This Python s c r i p t i s f o r c a l c u l a t i n g c o l o ra v o i d i n g c o n n e c t i v i t y , as d e s c r i b e d in t h epaper : Color − a v o i d i n g p e r c o l a t i o n , byS e b a s t i a n M. Krause , Michael M. Danziger ,and Vinko Z l a t i c . See e q u a t i o n s (9) and(10) . The s c r i p t c o n t a i n s one exapmle , howt h e f u n c t i o n S c o l o r can be used . ””” from pylab import ∗ import s c i p y . o p t i m i z e as opt 56 def s u b s e t s l i s t ( a s e t ) : 7 ””” Returns a l l p o s s i b l e s u b s e t s o f a s e t asl i s t o f t u p l e s ( i n c l u d i n g a s e t i t s e l f ,e x c l u d i n g empty s e t ) . Ordered by numbero f elements ””” ∗ ∗ N,N) , dtype=i n t ) 10 for i in range ( 1 ,N+1) : 11s l i s t [ : , i −
1] = 1 ∗ ( ( arange (2 ∗∗ N) %(2 ∗∗ i ) ) > ( 2 ∗ ∗ ( i − −
1) ) 12r e s o r t=a r g s o r t (sum( s l i s t , a x i s =1) ) 13s l i s t = s l i s t [ r e s o r t , : ] 14s l i s t =[] 15 for q in range ( 1 , l e n ( s l i s t [ : , 0 ] ) ) : 16Q=t u p l e ( a s e t [ i ] for i in range ( l e n ( s l i s t [ 0 , : ] ) ) i f s l i s t [ q , i ] ) 17s l i s t . append (Q) 18 return s l i s t 1920 def gen 0 ( z , k ) : 21 ””” Generating f u n c t i o n o f degree f o rPoisson graph . ””” return exp( − k ∗ (1. − z ) ) 2324 def gen 1 ( z , k ) : ””” Generating f u n c t i o n o f e x c e s s degree f o rPoisson graph . ””” return exp( − k ∗ (1. − z ) ) 2728 def f i x p o i n t u (Q, u , r , k ) : 29 ””” C a l c u l a t e s u [Q] , g i v e n v e c t o r o f c o l o rf r e q u e n c i e s r and a l l v a l u e s u [R] withR s u b s e t o f Q ( e x c e p t R=Q) . k i s t h eaverage c o n n e c t i v i t y . See Eq . (9) . ””” for q in range ( l e n (Q) ) : 33sum r+=r [Q[ q ] ] 34Q =t u p l e ( q2 for q2 in Q i f q2!=Q[ q ] ) 35s u m r t i m e s g+=r [Q[ q ] ] ∗ gen 1 ( u [ Q ] , k ) 36g= lambda x , sum r , sum r times g , k :s u m r t i m e s g +(1. − sum r ) ∗ gen 1 ( x , k ) 37u = opt . f i x e d p o i n t ( g , 0 . 5 ,a r g s =[sum r , sum r times g , k ] ) 38 return u 3940 def S c o l o r ( r ,A, k ) : 41 ””” C a l c u l a t e S c o l o r , g i v e n c o l o rf r e q u e n c i e s r , f i r s t A c o l o r s avoided .k i s t h e average c o n n e c t i v i t y . See Eq .(10) . ””” { () : 1 }
43S c o l o r =1. 44 for Q in s u b s e t s l i s t ( arange (A) ) : 45u [Q]=1. 46u [Q]= f i x p o i n t u (Q, u , r , k ) 47S c o l o r +=( − ∗∗ l e n (Q) ∗ gen 0 ( u [Q] , k ) 48 return S c o l o r 4950
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