Analytical results for the distribution of first hitting times of random walks on random regular graphs
aa r X i v : . [ c ond - m a t . d i s - nn ] F e b Analytical results for the distribution of first hittingtimes of random walks on random regular graphs
Ido Tishby, Ofer Biham and Eytan Katzav
Racah Institute of Physics, The Hebrew University, Jerusalem 9190401, Israel.E-mail: [email protected] , [email protected] , [email protected] Abstract.
We present analytical results for the distribution of first hitting times of randomwalks (RWs) on random regular graphs (RRGs) of degree c ≥ N .Starting from a random initial node at time t = 1, at each time step t ≥ c neighbors of its previous node. In some of the timesteps the RW may hop into a yet-unvisited node while in other time steps it mayrevisit a node that has already been visited before. The first time at which the RWenters a node that has already been visited before is called the first hitting time or thefirst intersection length. The first hitting event may take place either by backtracking(BT) to the previous node or by retracing (RET), namely stepping into a node whichhas been visited two or more time steps earlier. We calculate the tail distribution P ( T FH > t ) of first hitting (FH) times as well as its mean h T FH i and variance Var( T FH ).We also calculate the probabilities P BT and P RET that the first hitting event willoccur via the backtracking scenario or via the retracing scenario, respectively. Weshow that in dilute networks the dominant first hitting scenario is backtracking whilein dense networks the dominant scenario is retracing and calculate the conditionaldistributions P ( T FH = t | BT) and P ( T FH = t | RET), for the two scenarios. Theanalytical results are in excellent agreement with the results obtained from computersimulations. Considering the first hitting event as a termination mechanism of the RWtrajectories, these results provide useful insight into the general problem of survivalanalysis and the statistics of mortality rates when two or more termination scenarioscoexist.
Keywords : Random network, random regular graph, random walk, backtracking,retracing, first hitting time, first intersection length. irst hitting times of random walks on random regular graphs
1. Introduction
Random walk (RW) models [1, 2] are useful for the study of a large variety of stochasticprocesses such as diffusion [3, 4], polymer structure [5–7], and random search [8, 9].These models were studied extensively in different geometries, including continuousspace [10], regular lattices [11], fractals [12] and random networks [13]. In the contextof complex networks [14, 15], random walks provide useful tools for the analysis ofdynamical processes such as the spreading of rumours, opinions and infections [16, 17].Consider an RW on a random network. Starting at time t = 1 from a random initialnode x , at each time step t ≥ x → x → · · · → x t → . . . ,where x t is the node visited at time t . In some of the time steps the RW hops intonodes that have not been visited before, while in other time steps it hops into nodesthat have already been visited at an earlier time. Since RWs on random networks mayvisit some of the nodes more than once, the number of distinct nodes visited up to time t cannot exceed t and is typically smaller than t . The mean number h S i t of distinctnodes visited by an RW on a random network up to time t was recently studied [18]. Itwas found that in the infinite network limit it scales linearly with t , namely h S i t ≃ rt ,where the coefficient r < P ( T FH = t ), where T FH is the first time at which the RW stepsinto a node which has already been visited before. The first hitting time is also knownas the first intersection length [20, 21]. The first hitting event may occur either via thebacktracking scenario, in which the RW hops back into the previous node, or by theretracing scenario, in which it hops into a node visited two or more time steps earlier.The first hitting event marks the transition from the first stage to the second stage inthe life cycle of an RW. In the first stage the RW visits a new node at each time step,while in the second stage it combines moves at which the RW visits new nodes andmoves at which it enters nodes that have already been visited before. The second stagein the life cycle of an RW on a finite network can be characterized by the distributionof first passage (FP) times P ( T FP = t ), where T FP is the first time at which an RWstarting from a random initial node i visits a random target node j [22, 23]. Finally, thecover time, at which the RW completes visiting all the N nodes in the network at leastonce marks the transition from the second stage to the third stage in the life cycle of anRW [24]. Beyond the cover time the RW continues to revisit nodes that have alreadybeen visited before.Another type of random walk model is the non-backtracking random walk (NBW).At each time step the NBW steps into a random neighbor of the present node, except for irst hitting times of random walks on random regular graphs P ( T FH > t ), consists of a product of a geometric distribution due tothe backtracking process and a Rayleigh distribution due to the retracing process. Themean and variance of the distribution of first hitting times were also calculated. In thisanalysis we utilized a special property of RWs on ER networks, in which the subnetworkthat consists of the yet-unvisited nodes remains an ER network at all times. Its degreedistribution remains a Poisson distribution, while its mean degree decreases linearlywith the time t . This self-similarity enabled us to calculate the probability that at time t the RW will step into a yet-unvisited node and the complementary probability that itwill step into an already visited node. In the case of other configuration model networks,there is no closed form expression for the time evolution of the degree distribution ofthe sub-network of the yet-unvisited nodes. Therefore, the approach we used for thecalculation of the distribution of first hitting times in ER networks cannot be generalizedto other configuration model networks. However, it turns out that in the special case ofrandom regular graphs there are other simplifying features that can be used to derive aclosed form expression for P ( T FH > t ).In another recent paper we studied the distribution of first hitting times of NBWs onER networks [27]. Apart from the retracing scenario, NBWs on ER networks exhibit anadditional mechanism of first hitting, referred to as the trapping scenario. The trappingtakes place when the NBW enters a leaf node of degree k = 1, which has no otherneighbor except from the previous node. It was found that the tail distribution of firsthitting times of NBWs on ER networks consists of a product of a geometric distributiondue to the trapping process and a Rayleigh distribution due to the retracing process.The mean and variance of the distribution of first hitting times were also calculated.In this paper we present analytical results for the distributions of first hitting timesof random walks on random regular graphs (RRGs) of degree c ≥ N .The first hitting event may take place either by backtracking to the previous node or byretracing, namely stepping into a node which has been visited two or more time stepsearlier. Using the microstructure and statistical properties of RW paths at early times,we calculate the tail distribution of first hitting (FH) times P ( T FH > t ), which is givenby a product of a geometric distribution due to the backtracking process and a Rayleighdistribution due to the retracing process. We also obtain closed form expressions forthe mean first hitting time h T FH i and for the variance Var( T FH ) of the distribution offirst hitting times. The analytical results are found to be in excellent agreement withthe results obtained from computer simulations. We obtain analytical results for the irst hitting times of random walks on random regular graphs P BT and P RET that the first hitting event will occur via the backtrackingor retracing scenarios, respectively. We show that in dilute networks the dominantfirst hitting scenario is backtracking while in dense networks the dominant scenario isretracing. We also obtain expressions for the conditional distributions of first hittingtime, P ( T FH = t | BT) and P ( T FH = t | RET), namely conditioned on the first hittingevent occuring via the backtracking or the retracing scenario, respectively.The paper is organized as follows. In Sec. 2 we briefly describe the random regulargraph. In Sec. 3 we present the random walk model. In Sec. 4 we calculate thedistribution of first hitting times. In Sec. 5 we calculate the mean first hitting time. InSec. 6 we calculate the variance of the distribution of first hitting times. In Sec. 7 weanalyze the interplay between the backtracking and retracing scenarios. In Sec. 8 wecompare the results presented in this paper for RWs on RRGs with previous results forRWs on ER networks. The results are discussed in Sec. 9 and summarized in Sec. 10.
2. The random regular graph
A random network (or graph) consists of a set of N nodes that are connected by edgesin a way that is determined by some random process. For example, in a configurationmodel network the degree of each node is drawn independently from a given degreedistribution P ( k ) and the connections are random and uncorrelated [30].An important example of a configuration model network is the Erd˝os-R´enyi (ER)network [31–33]. The ER network, denoted by ER ( N, p ), consists of N nodes such thateach pair of nodes is connected with probability p . The degree distribution of an ERnetwork is a binomial distribution, B ( N, p ). In the limit N → ∞ and p →
0, wherethe mean degree c = ( N − p is held fixed, it converges to a Poisson distribution. Inthe asymptotic limit ( N → ∞ ), the ER network exhibits a phase transition at c = 1 (apercolation transition), such that for c < c > c = ln N , there is a second transition,above which the entire network is included in the giant cluster and there are no isolatedcomponents.The RRG is a special case of a configuration model network, in which the degreedistribution is a degenerate distribution of the form P ( k ) = δ k,c , namely all the nodesare of the same degree c . Here we focus on the case of 3 ≤ c ≤ N −
1, in which fora sufficiently large value of N the RRG consists of a single connected component. Inthe infinite network limit the RRG exhibits a tree structure with no cycles. Thus, inthis limit it coincides with a Bethe lattice whose coordination number is equal to c . Incontrast, RRGs of a finite size exhibit a local tree-like structure, while at larger scalesthere is a broad spectrum of cycle lengths [36]. In that sense RRGs differ from Cayleytrees, which maintain their tree structure by reducing the most peripheral nodes to leafnodes of degree 1. irst hitting times of random walks on random regular graphs
5A convenient way to construct an RRG of size N and degree c is to prepare the N nodes such that each node is connected to c half edges or stubs [15]. At each step of theconstruction, one connects a random pair of stubs that belong to two different nodes i and j that are not already connected, forming an edge between them. This procedure isrepeated until all the stubs are exhausted. The process may get stuck before completionin case that all the remaining stubs belong to the same node or to pairs of nodes thatare already connected. In such case one needs to perform some random reconnectionsin order to complete the construction.RRGs provide a useful benchmark for the study of dynamical processes on regularlattices. This is due to the fact that for any regular lattice structure in any spacedimension, one can construct an RRG whose degree c is equal to the coordinationnumber z of the regular lattice. For example, a simple cubic lattice in a d dimensionalspace corresponds to an RRG with degree c = 2 d . Some special cases in two dimensionsinclude the honeycomb lattice that corresponds to an RRG with c = 3 and thetriangular lattice that corresponds to an RRG with c = 6. However, all the otherstructural properties of RRGs are completely different from those of the correspondingregular lattice. Therefore, comparing the behavior of dynamical processes taking placeon regular lattices and on the corresponding RRGs, one can distinguish between theproperties that depend on the coordination number and those that depend on otherstructural properties such as the lengths of short cycles. A famous example is the Bethe-Peierls approximation of the Ising model [37], which is actually an exact result on anRRG with the same coordination number as a d -dimensional lattice. In 2 dimensionsthe critical exponents are wrongly identified as the mean-field ones, but the actualpredictions for the critical temperature and the heat capacity are highly insightful. Indimensions higher than 4, even the critical exponents are correct.
3. The random walk model
Consider an RW on an RRG of degree c ≥ N . At each time step the RWhops from its current node to one of its neighbors, such that the probability of hoppingto each neighbor is 1 /c . For sufficiently large N the RRG consists of a single connectedcomponent, thus an RW starting from any initial node can reach any other node inthe network. In the long time limit t ≫ N the RW visits all the nodes with the samefrequency, namely on average each node is visited once every N steps. However, overshorter periods of time there may be large fluctuations such that some nodes may bevisited several times in a given time interval while other nodes are not visited at all.In some of the time steps an RW may visit nodes that have not been visited beforewhile in other time steps it may revisit nodes that have already been visited before.For example, at each time step t ≥ /c . In the infinite network limit the RRG exhibits a tree structure.Therefore, in this limit the backtracking mechanism is the only way in which an RWmay hop from a newly visited node to a node that has already been visited before. irst hitting times of random walks on random regular graphs Figure 1.
Schematic illustrations of possible events taking place along the path ofan RW on an RRG: (a) a path segment in which at each time step the RW enters anode that has not been visited before; (b) a path segment that includes a backtrackingstep into the previous node (step no. 4); (c) a path that includes a retracing step(step no. 5) in which the RW hops into a node that was visited a few time stepsearlier. Retracing steps are not possible in the infinite network limit and take placeonly in finite networks, which include cycles. Note that in this illustration the RRG isof degree c = 4. However, in finite networks the RW may utilize the cycles to retrace its path and hopinto nodes that have already been visited two or more time steps earlier. Once the RWstepped into a node that has been visited before, it may continue to hop back and forthalong the path of its previously visited nodes, until it eventually leaves the path andenters a newly visited node.In Fig. 1 we present a schematic illustration of the backtracking and the retracingevents which may take place along the path of an RW on an RRG. While backtrackingis independent on the network size, retracing takes place only in finite networks and isnot possible in the infinite network limit, in which the RRG exhibits a tree structure. irst hitting times of random walks on random regular graphs
4. The distribution of first hitting times
Consider an RW on an RRG of a finite size N and degree c ≥
3. Starting from a randomnode at time t = 1, the RW hops randomly between nearest neighbor nodes. At earlytimes t ≪ N all the nodes it enters are likely to be visited for the first time. Thefirst time at which the RW enters a node that has already been visited before is calledthe first hitting time. The first hitting process may take place either by backtracking(BT) or by retracing (RET). In the backtracking scenario the RW moves back into theprevious node, while in the retracing scenario it hops into a node that has already beenvisited two or more time steps earlier. Below we calculate the distribution of first hittingtimes. In case that the RW has not returned to any previously visited node up to time t −
1, the first hitting time satisfies the condition T FH > t −
1. Given that T FH > t − P ( T FH > t | T FH > t − P ( T FH > t | T FH > t −
1) = P t ( ¬ RET |¬ BT) P t ( ¬ BT) , (1)where P t ( ¬ BT) is the probability that the RW will not backtrack to the previous nodeat time t . Given that the RW has not backtracked at time t , the conditional probability P t ( ¬ RET |¬ BT) is the probability that it will also not retrace its path at time t , namely,that it will not hop into a node that has already been visited two or more time stepsearlier.Since all the nodes in the RRG are of degree c , at each time step (apart from t = 1and 2) the probability of backtracking into the previous node is 1 /c . Therefore, theprobability that a backtracking step will not occur at any given time t ≥ P t ( ¬ BT) = 1 − c . (2)Provided that the RW has not backtracked at time t , we will now evaluate the probability P t ( ¬ RET |¬ BT) that it will also not retrace its path. Apart from the current node attime t − t − N − t − i and j canbe considered as a combination of two links, one from i to j and the other from j to i . Thus, a node of degree c is connected to c incoming links and c outgoing links. Thenumber of yet-unvisited nodes at time t − N − t + 1. Each one of these nodes is ofdegree c .Since at each time step the RW enters a node via one edge and leaves it via anotheredge, up to time t − t − t − c − t (apart from the initialnode that can be reached via c − irst hitting times of random walks on random regular graphs t is L v = ( c − t −
3) + 1 . (3)The number of incoming links that may lead the RW to a yet-unvisited node at time t is L u = ( N − t + 1) c. (4)Summing up L v and L u we obtain the total number of incoming links that may leadthe RW at time t to either a visited node (by retracing) or to a yet-unvisited node. Itis given by L T = ( N − c − t −
3) + 1 . (5)Given that the possibility of backtracking into the previous node was eliminated, the RWselects randomly one of the c − L v /L T .Therefore, the probability of retracing at time t under the condition of no-backtrackingis given by P t (RET |¬ BT) = L v L T = ( c − t −
3) + 1( N − c − t −
3) + 1 . (6)In Fig. 2 we present analytical results, obtained from Eq. (6), for the probability P t (RET |¬ BT) that an RW on an RRG will retrace its path for the first time at time t under the condition that it has not backtracked its path at time t (or earlier). The RRGsused in Fig. 2 are of size N = 1000, where the nodes are of degree c = 3 (solid line), 5(dashed line) and 10 (dotted line). The analytical results are in excellent agreement withthe results obtained from computer simulations (circles). These results are qualitativelydifferent from the corresponding results for RWs on ER networks, which are given by [26] P ER t (RET |¬ BT) = c ( t − c + 1)( N − , (7)where c is the mean degree.Using Eq. (6), the probability that the RW will not retrace its path at time t ≥ P t ( ¬ RET |¬ BT) = 1 − ( c − t −
3) + 1( N − c − t −
3) + 1 . (8)Inserting the probabilities P t ( ¬ BT) and P t ( ¬ RET |¬ BT) from Eqs. (2) and (8),respectively, into Eq. (1), we obtain P ( T FH > t | T FH > t −
1) = (cid:18) − c (cid:19) (cid:20) − ( c − t −
3) + 1( N − c − t −
3) + 1 (cid:21) . (9) irst hitting times of random walks on random regular graphs Figure 2.
Analytical results for the probability P t (RET |¬ BT) that an RW willretrace its path at time t under the condition that it has not backtracked its path attime t , on an RRG of size N = 1000 and c = 3 (solid line), 5 (dashed line) and 10(dotted line). The analytical results, obtained from Eq. (6), are in excellent agreementwith the results obtained from computer simulations (circles). Focusing on large networks, where N ≫
1, and assuming that t ≪ N c , the probability P t ( ¬ RET |¬ BT) can be written as a series expansion in powers of ( t − / ( N c ). It isgiven by P t ( ¬ RET |¬ BT) = 1 − ( c − t − N c − c − t − ( N c ) + O "(cid:18) t − N c (cid:19) . (10)Using the expansion ln(1 − x ) = − x − x / O ( x ) for x ≪
1, we obtainln [ P t ( ¬ RET |¬ BT)] ≃ − ( c − t − N c − c − t − ( N c ) − (cid:20) ( c − t − N c + 2( c − t − ( N c ) (cid:21) . (11)Rearranging terms up to second order in ( t − / ( N c ), we obtain P t ( ¬ RET |¬ BT) ≃ exp (cid:20) − ( c − t − N c − ( c − c + 2)( t − N c ) (cid:21) . (12)As a result, Eq. (9) is replaced by P ( T FH > t | T FH > t − ≃ (cid:18) − c (cid:19) exp (cid:20) − ( c − t − N c − ( c − c + 2)( t − N c ) (cid:21) , (13) irst hitting times of random walks on random regular graphs t ≥ P ( T FH > t ) that the first hitting event will not take place duringthe first t time steps is given by the product P ( T FH > t ) = t Y t ′ =3 P ( T FH > t ′ | T FH > t ′ − . (14)Inserting P ( T FH > t ′ | T FH > t ′ −
1) from Eq. (13) into Eq. (14), we obtain the taildistribution of first hitting times P ( T FH > t ) = P BT ( T FH > t ) P RET ( T FH > t ) , (15)where P BT ( T FH > t ) = ( t = 1 , (cid:0) − c (cid:1) t − t ≥ , (16)and P RET ( T FH > t ) ≃ ( ≤ t ≤ Q tt ′ =3 exp h − ( c − t ′ − Nc − ( c − c +2)( t ′ − Nc ) i t ≥ . (17)Note that the contribution of the backtracking process, given by Eq. (16), depends onlyon the mean degree c , while the contribution of the retracing process, given by Eq. (17)depends on the mean degree c and on the network size N . In the infinite network limitthe retracing scenario becomes irrelevant and Eq. (15) is simplified to P ( T FH > t | N → ∞ ) = ( t = 1 , (cid:0) − c (cid:1) t − t ≥ . (18)Taking the logarithm of P RET ( T FH > t ), as expressed in Eq. (17) for t ≥
4, we obtainln[ P RET ( T FH > t )] ≃ − t X t ′ =3 ( c − t ′ − N c − t X t ′ =3 ( c − c + 2)( t ′ − N c ) . (19)Carrying out the summations in Eq. (19), we obtainln [ P RET ( T FH > t )] ≃ − ( c − t − t − N c − ( c − c + 2)( t − t − t − / N c ) . (20)Inserting P BT ( T FH > t ) from Eq. (16) and P RET ( T FH > t ) from Eq. (20) into Eq. (15),we obtain irst hitting times of random walks on random regular graphs P ( T FH > t ) ≃ t = 1 , (cid:20) − ( t − t − α − ( t − t − t − / ( c − c +2 ) α − β ( t − (cid:21) t ≥ , (21)where the parameters α and β are given by α = s(cid:18) cc − (cid:19) N , (22)and β = ln (cid:18) cc − (cid:19) . (23)Using a Taylor expansion of the second term in the exponent in Eq. (21), we obtain P ( T FH > t ) ≃ t = 1 , (cid:20) − ( t − t − t − / ( c − c +2 ) α (cid:21) exp h − ( t − t − α − β ( t − i t ≥ , (24)where the first term in the second line is the leading term and the second term is acorrection term, which is small for sufficiently small values of t .It turns out that for large networks the second term in the second line of Eq. (24)is extremely small and its effect on the tail distribution P ( T FH > t ) can be neglectedin most cases. The only case in which it was found to make a small but noticeabledifference is in the calculation of the second moment h T i . For the sake of consistency,we use Eq. (24) in the calculations of both the mean first hitting time h T FH i and thesecond moment h T i . In all the other calculations presented in this paper we use asimplified form of the tail distribution, which is given by P ( T FH > t ) ≃ ( t = 1 , h − ( t − t − α − β ( t − i t ≥ , (25)The tail distribution P ( T FH > t ), given by Eq. (25), is thus a product of a geometricdistribution, associated with the backtracking process and a Rayleigh distribution,associated with the retracing process [38].In Fig. 3 we present (on a semi-logarithmic scale) analytical results for the taildistributions P ( T FH > t ) of first hitting times (solid lines) of RWs on RRGs of size N = 1000 in which the nodes are of degree c = 3 (left), 5 (middle) and 10 (right). Theanalytical results, obtained from Eq. (25) are in excellent agreement with the resultsobtained from computer simulations (circles). It is found that as the degree c is increasedthe first hitting event tends to occur at a later time. This can be attributed to the factthat the probability of backtracking decreases as c is increased. irst hitting times of random walks on random regular graphs -5 -4 -3 -2 -1 -5 -4 -3 -2 -1 -5 -4 -3 -2 -1 Figure 3.
Analytical results for the tail distributions P ( T FH > t ) of first hitting times(solid lines) of RWs on RRGs of size N = 1000, in which the nodes are of degree c = 3(left), 5 (middle) and 10 (right). The analytical results, obtained from Eq. (25) are inexcellent agreement with the results obtained from computer simulations (circles). The excellent agreement observed in Fig. 3 between the analytical results andthe results obtained from computer simulations indicates that Eq. (25) is valid for abroad range of parameters. However, Eq. (25) combines the backtracking and retracingmechanisms. Thus, for parameters in which the backtracking mechanism is dominant,Fig. 3 is not sufficient to establish the validity of the first term in Eq. (25), associatedwith the retracing scenario. This is due to the fact that in this regime the retracingprocess is effectively screened by the backtracking process.In order to establish the validity of the retracing term in Eq. (25) we considerthe distribution of first hitting times of NBWs on RRGs. The suppression of thebacktracking process in NBWs leaves the retracing scenario as the only possible scenarioof first hitting in these systems. This is unlike the case of NBWs on ER networks, inwhich the trapping scenario emerges once backtracking is suppressed. This contrast isdue to the fact that RRGs do not include leaf nodes of degree k = 1, in which an NBWmay become trapped. The suppression of the backtracking mechanism implies that thedistribution of first hitting times of NBWs on RRGs is expressed by the right hand sideof Eq. (25), where α is given by Eq. (22) and β = 0, namely P NBW ( T FH > t ) ≃ ( t = 1 , h − ( t − t − α i t ≥ . (26)In Fig. 4 we present (on a semi-logarithmic scale) analytical results for the taildistributions P NBW ( T FH > t ) of first hitting times (solid lines) of NBWs on RRGs ofsize N = 1000 in which the nodes are of degree c = 3 (left), 5 (middle) and 10 (right).The analytical results, obtained from Eq. (26) are in excellent agreement with theresults obtained from computer simulations (circles). irst hitting times of random walks on random regular graphs -6 -5 -4 -3 -2 -1 -6 -5 -4 -3 -2 -1 -6 -5 -4 -3 -2 -1 Figure 4.
Analytical results for the tail distributions P NBW ( T FH > t ) of first hittingtimes (solid lines) of NBWs on RRGs of size N = 1000, in which the nodes are ofdegree c = 3 (left), 5 (middle) and 10 (right). The analytical results, obtained from Eq.(26) are in excellent agreement with the results obtained from computer simulations(circles).
5. The mean first hitting time
The moments h T r FH i , r = 1 , , . . . , of the distribution of first hitting times of RWs onRRGs can be obtained from the tail-sum formula [39] h T r FH i = N X t =0 [( t + 1) r − t r ] P ( T FH > t ) . (27)In particular, the mean first hitting time h T FH i can be obtained by inserting r = 1 inEq. (27), which yields h T FH i = N X t =0 P ( T FH > t ) . (28)Inserting P ( T FH > t ) from Eq. (24) into Eq. (28), shifting the summation index from t to t −
1, and and replacing the sum in Eq. (28) by a sum of two integrals, using theformulation of a middle Riemann sum, we obtain h T FH i ≃ I + I , (29)where I = 3 + e − β + N − / Z / exp (cid:20) − t ( t − α − βt (cid:21) dt, (30)and I = − N − / Z / t ( t − t − / (cid:0) c − c +2 (cid:1) α exp (cid:20) − t ( t − α − βt (cid:21) dt. (31) irst hitting times of random walks on random regular graphs I ≃ e − β + r π α exp " (2 α β − α erf (cid:20) α ( β + 1 − /c ) √ (cid:21) − erf (cid:18) α β + 1 √ α (cid:19)(cid:27) , (32)where erf( x ) is the error function, also called Gauss error function [40]. The errorfunction erf( x ) is a monotonically increasing function, defined for −∞ < x < ∞ . It isan odd function, namely erf( − x ) = − erf( x ), and exhibits a sigmoidal shape. For | x | ≪ x ) ≃ x/ √ π while for | x | > x ) → sign( x ). The parameter β obtains its largest value at c = 3, where β = ln(3 / c is increased. Thus, for any value of c it satisfies β < c ≫
1, it can be approximated by β = 1 c + 12 c + O (cid:18) c (cid:19) . (33)Since the degree c satisfies c < N −
1, the parameter β is bounded from below by β > / ( N − β → / ( N − αβ thus takes values in the range 1 / √ N − < αβ ≤ ln(3 / √ N − I ≃ e − β + r π α exp (cid:18) − β (cid:19) exp (cid:18) α (cid:19) exp "(cid:18) αβ √ (cid:19) × (cid:20) erf (cid:18) α (1 − /c ) √ αβ √ (cid:19) − erf (cid:18) αβ √ √ α (cid:19)(cid:21) . (34)In the large network limit, where N ≫
1, the term exp[1 / (8 α )] satisfies exp[1 / (8 α )] ≃ O (1 /N ) and thus it can be neglected. Similarly, in the large network limit theargument of the first erf function in Eq. (32) is always very large. Therefore, onecan safely set the first erf function to be equal to 1. The second erf function can beapproximated byerf (cid:18) αβ √ √ α (cid:19) ≃ erf (cid:18) αβ √ (cid:19) + √ √ πα exp " − (cid:18) αβ √ (cid:19) (35)Making these approximations, we obtain I ≃ e − β − e − β/ + r π αe − β exp "(cid:18) αβ √ (cid:19) − erf (cid:18) αβ √ (cid:19)(cid:21) . (36)Carrying out the integration in Eq. (31) and taking the large network limit, weobtain irst hitting times of random walks on random regular graphs I ≃ r π (cid:18) c + 2 c − (cid:19) αβ (cid:0) α β + 3 (cid:1) e ( − α β ) α (cid:20) − erf (cid:18) α β + 1 √ α (cid:19)(cid:21) − (cid:18) c + 2 c − (cid:19) (cid:0) α β − β + 2 (cid:1) e − α − β . (37)In the dilute-network limit, where 3 ≤ c ≪ . √ N , the argument of the errorfunction in Eq. (36) satisfies the condition αβ/ √ ≫
1. In this limit one can use theapproximation [40] 1 − erf( x ) ≃ e − x x √ π , (38)which is the first term in the asymptotic expansion of the complementary error function,and obtain h T FH i ≃ c + 2 + O (cid:18) c (cid:19) . (39)This result reflects the fact that in the dilute network limit the first hitting time isdominated by the backtracking mechanism.In the dense-network limit, where 1 . √ N ≪ c ≤ N the argument of the errorfunction in Eq. (36) satisfies αβ/ √ ≪
1. In this case, one can use the approximationerf( x ) ≃ x , which is the leading term in the Taylor expansion of erf( x ) around x = 0.Using this approximation, we obtain h T FH i ≃ r πN . (40)In this limit the first hitting time is dominated by the retracing mechanism.In Fig. 5 we present analytical results for the mean first hitting time h T FH i (solidline) of RWs on RRGs as a function of the degree c . The analytical results, obtainedfrom Eqs. (29), (36) and (37) are in excellent agreement with the results obtained fromcomputer simulations (circles). In the dilute-network limit h T FH i quickly increases as c is increased, reaching saturation in the dense-network limit.
6. The variance of the distribution of first hitting times
Inserting r = 2 in Eq. (27) we obtain the second moment h T i of the distribution offirst hitting times, which is given by h T i = N X t =0 (2 t + 1) P ( T FH > t ) . (41)Inserting the tail distribution P ( T FH > t ) from Eq. (24) into Eq. (41), shifting thesummation index from t to t − irst hitting times of random walks on random regular graphs Figure 5.
Analytical results for the mean first hitting time h T FH i of RWs on RRGsof size N = 1000 (solid line), as a function of the degree c . The analytical results,obtained from Eqs. (29), (36) and (37) are in excellent agreement with the resultsobtained from computer simulations (circles). h T i = J + J , (42)where J = 9 + 7 e − β + Z N − / / (2 t + 5) exp (cid:20) − t ( t − α − βt (cid:21) dt, (43)and J = − Z N − / / (2 t + 5) t ( t − t − (cid:0) c +2 c − (cid:1) α exp (cid:20) − t ( t − α − βt (cid:21) dt. (44)Carrying out the integral in Eq. (43) and taking the large network limit, we obtain J ≃ e − β + 2 α exp (cid:18) − α β + 158 α (cid:19) − α √ π ( α β − e (2 α β − α (cid:20) − erf (cid:18) α β + 1 √ α (cid:19)(cid:21) . (45)Carrying out the integration in Eq. (44) and taking the large network limit, we obtain irst hitting times of random walks on random regular graphs J ≃ (cid:18) c + 2 c − (cid:19) (cid:2) α β − α β (4 β −
5) + 15 β − (cid:3) e − α − β − r π (cid:18) c + 2 c − (cid:19) α (cid:2) α β − α ( β − β − β − β + 12 (cid:3) × e ( − α β ) α (cid:20) − erf (cid:18) α β + 1 √ α (cid:19)(cid:21) . (46)In the limit of dilute networks, where 3 ≤ c ≪ . √ N , the product αβ satisfies αβ ≫
1. In this limit the second moment can be approximated by h T i ≃ e − β − √ πα βe (cid:16) αβ √ (cid:17) (cid:20) − erf (cid:18) α β + 1 √ α (cid:19)(cid:21) + 2 α e − β/ . (47)In the dense network limit, where 1 . √ N ≪ c < N , the parameter β satisfies β ≃ /c and αβ ≪
1. As a result, the error function on the right hand side of Eq. (45)becomes negligible and the second moment converges towards h T i ≃ N. (48)The variance of the distribution of first hitting times is given byVar( T FH ) = h T i − h T FH i . (49)In the dense network limit, the variance converges towardsVar( T FH ) ≃ (cid:16) − π (cid:17) N. (50)In Fig. 6 we present analytical results for the variance of the distribution of firsthitting times of RWs on RRGs of size N = 1000 (solid line), as a function of the degree c . The analytical results, obtained from Eq. (49), where h T i is given by Eqs. (42),(45) and (46) and h T FH i is given by Eqs. (29), (36) and (37), are in excellent agreementwith the results obtained from computer simulations (circles).
7. Analysis of the backtracking and retracing mechanisms
The first hitting process consists of two competing scenarios, backtracking and retracing.In each instance of an RW trajectory the first hitting time is determined by the scenariothat occurs first. We denote by P BT the probability that the first hitting event of an RWstarting from a random initial node will take place by backtracking. Similarly, we denoteby P RET the probability that the first hitting event will take place by retracing. Sincethese are the only possible mechanisms of first hitting of RWs, these two probabilitiesmust satisfy P BT + P RET = 1. The conditional probability that the first hitting event irst hitting times of random walks on random regular graphs Figure 6.
Analytical results for the variance Var( T FH ) of the distribution of firsthitting times of RWs on RRGs of size N = 1000 (solid line), as a function of the degree c . The analytical results, obtained from Eq. (49), are in excellent agreement with theresults obtained from computer simulations (circles). will take place at time t , given that it occurs via the backtracking scenario, is denotedby P ( T FH = t | BT). Similarly, the conditional probability that the first hitting eventwill take place at time t given that it occurs via the retracing scenario is denoted by P ( T FH = t | RET). The overall distribution of first hitting times can be expressed as aweighted sum of the two conditional distributions, in the form P ( T FH = t ) = P BT P ( T FH = t | BT) + P RET P ( T FH = t | RET) . (51)The first term on the right hand side of Eq. (51) can be written in the form P BT P ( T FH = t | BT) = P ( T FH > t − − P t ( ¬ BT)] , (52)namely as the probability that the first hitting event will not take place up to time t − t due to backtracking. Inserting P t ( ¬ BT) from Eq. (2) into Eq.(52) and summing up over t , it is found that the overall probability that the first hittingevent will take place via the backtracking scenario is given by P BT = 1 c N +1 X t =3 P ( T FH > t − . (53)Comparing with the tail-sum formula (28), the probability P BT can be expressed interms of the mean first hitting time, namely P BT = h T FH i − c . (54)Thus, the complementary probability that the first hitting will occur via the retracingmechansim is given by irst hitting times of random walks on random regular graphs Figure 7.
Analytical results for the probabilities P BT and P RET that the first hittingevent will occur via the backtracking or the retracing scenarios, respectively. Theanalytical results, obtained from Eqs. (54) and (55), are in excellent agreement withthe results obtained from computer simulations (circles). P RET = 1 − h T FH i − c . (55)Note that Eqs. (54) and (55) are exact. However, inserting h T FH i from Eq. (29) with I and I given by Eqs. (36) and (37), respectively, the results become approximate dueto the replacement of the sum in Eq. (28) by integrals.In Fig. 7 we present the probability P BT that the first hitting event will occur viathe backtracking scenario and the complementary probability P RET that it will occur viathe retracing scenario, as a function of the degree c . As expected, the probability P BT decreases as c is increased while the probability P RET increases. The crossover from thebacktracking-dominated regime of dilute networks to the retracing-dominated regime ofdense networks occurs where P BT = P RET = 1 /
2. Using Eqs. (54) and (55) we find thatat the crossover point h T FH i = ( c + 4) /
2. Comparing the right hand side of Eq. (36) to( c + 4) /
2, it is found that the crossover point occurs at c ≃ . √ N .In Fig. 8 we present a diagram describing the first hitting process of RWs onRRGs. In the regime of dilute networks, where c . . √ N the first hitting process isdominated by backtracking, while in the regime of dense networks, where c & . √ N the first hitting process is dominated by retracing.The conditional probability P ( T FH = t | BT), for t ≥
3, can be written in the form P ( T FH = t | BT) = P ( T FH > t − − P t ( ¬ BT)] P BT . (56)Inserting P t ( ¬ BT) from Eq. (2) and P BT from Eq. (54), we obtain irst hitting times of random walks on random regular graphs BacktrackingRetracing
Figure 8.
A diagram describing the first hitting process of RWs on RRGs. In theregime of dilute networks, where c . . √ N the first hitting process is dominatedby backtracking, while in the regime of dense networks, where c & . √ N the firsthitting process is dominated by retracing. P ( T FH = t | BT) = P ( T FH > t − h T FH i − . (57)Focusing on the retracing mechanism, the second term on the right hand side of Eq.(51) can be expressed in the form P RET P ( T FH = t | RET) = P ( T FH > t − P t ( ¬ BT)[1 − P t ( ¬ RET |¬ BT)] . (58)Inserting P t ( ¬ BT) from Eq. (2), P t ( ¬ RET |¬ BT) from Eq. (8), and P RET from Eq. (55)into Eq. (58), we obtain P ( T FH = t | RET) = (cid:18) c − c − h T FH i + 2 (cid:19) ( c − t −
3) + 1( N − c − t −
3) + 1 P ( T FH > t − . (59)In Fig. 9 we present the probabilities P ( T FH = t | BT) and P ( T FH = t | RET) thatthe first hitting process will take place at time t , under the condition that it occurs viathe backtracking and the retracing scenarios, respectively.The mean first hitting time under the condition that the first hitting occurs via thebacktracking scenario is given by E [ T FH | BT] = N +1 X t =3 tP ( T FH = t | BT) . (60)Inserting P ( T FH = t | BT) from Eq. (57) into Eq. (60), we obtain irst hitting times of random walks on random regular graphs Figure 9.
Analytical results for the distribution P ( T FH = t | BT) and P ( T FH = t | RET) of first hitting times, conditioned on the first hitting to occur via thebacktracking or the retracing scenarios, respectively. The analytical results, obtainedfrom Eqs. (57) and (59), are in excellent agreement with the results obtained fromcomputer simulations (circles). E [ T FH | BT] = 1 h T FH i − N +1 X t =3 tP ( T FH > t −
1) (61)Using the tail-sum formula [39], it is found that E [ T FH | BT] = h T i + h T FH i − h T FH i − . (62)The mean first hitting time h T FH i can be expressed as a weighted sum of the form h T FH i = E [ T FH | BT] P BT + E [ T FH | RET] P RET . (63)Inserting E [ T FH | BT] from Eq. (62) into Eq. (63) and solving for E [ T FH | RET], we obtain E [ T FH | RET] = h T i − (2 c − h T FH i − h T FH i − c − . (64)In Fig. 10 we present analytical results for the conditional expectation values E [ T FH | BT] and E [ T FH | RET] of the first hitting time, vs. the degree c given that the firsthitting event occurs via the backtracking or the retracing scenario, respectively. Theanalytical results, obtained from Eqs. (62) and (64), are in excellent agreement withthe results obtained from computer simulations (circles). irst hitting times of random walks on random regular graphs Figure 10.
Analytical results for the conditional expectation values E [ T FH | BT] and E [ T FH | RET] of the first hitting time, vs. the degree c given that the first hitting eventoccurs via the backtracking or the retracing scenario, respectively. The analyticalresults, obtained from Eqs. (62) and (64), are in excellent agreement with the resultsobtained from computer simulations (circles).
8. Comparing P ( T FH > t ) between RWs on RRGs and RWs on ER networks The distribution P ( T FH > t ) of first hitting times of RWs on RRGs is given by Eq.(25). It can be expressed as a product of a geometric distribution, associated with thebacktracking process, and a Rayleigh distribution, associated with the retracing process.The Rayleigh distribution is parameterized by the parameter α , while the geometricdistribution is parameterized by β . It is interesting to compare the results obtainedabove for the distribution P ( T FH > t ) of RWs on RRGs of size N and degree c withthe corresponding results for RWs on ER networks of the same size and mean degree h K i = c [26]. Another interesting comparison is between the distributions of first hittingtimes of NBWs on RRGs and ER networks. In all these cases, the tail distribution offirst hitting times is described by the same functional form as in Eq. (25), but theexpressions for α and β are different. In Table 1 we present a 2 × α and β for RWs on RRGs (upper-left cell), NBWs on RRGs(upper-right cell), RWs on ER networks (lower-left cell) and NBWs on ER networks(lower-right cell).One difference between RRGs and ER networks is that ER networks include isolatednodes of degree k = 0. Since an RW starting from an isolated node cannot make evena single move, the initial node of an RW on an ER network is chosen randomly fromall the nodes of degree k ≥
1. Moreover, isolated nodes are not accessible to the RWat times t >
1. As a result, for RWs on ER networks the probability of backtrackingat any time t is given by P t (BT) = (1 − e − c ) /c , compared to 1 /c in the case of RWson RRGs [26]. This is reflected in the parameter β for RWs on ER networks, which irst hitting times of random walks on random regular graphs Table 1.
The parameters α and β of the distribution P ( T FH > t ) [Eq. (25)] for RWsand NBWs on RRGs and on ER networks RW NBW α = q(cid:0) cc − (cid:1) N α = q(cid:0) cc − (cid:1) N RR G β = ln (cid:0) cc − (cid:1) β = 0 α = q(cid:0) c +1 c (cid:1) N α = q(cid:0) c +1 c (cid:1) N E R β = ln (cid:0) cc − e − c (cid:1) β = − ln(1 − e − c )includes the e − c term.The effect of the retracing process on the distribution of first hitting times isdescribed by a Rayleigh distribution parametrized by α . In fact, the probability that atany time t the RW will hop into any one of the nodes that were previously visited up totime t − /α . Naively, one expects that the probability to enter a specificnode at time t is 1 /N . However, each visit of a node exhausts two of its edges, whichcannot be used to revisit the node in the retracing scenario. Therefore, the probabilitythat at a given time an RW will revisit a specific node that has already been visited twoor more time steps earlier is smaller than 1 /N . In the case of RWs on RRGs it is givenby (1 − /c ) /N , while in the case of RWs on ER networks it is given by [1 − / ( c + 1)] /N .These corrections are reflected in the values of α in Table 1.In Fig. 11 we compare the analytical results for the distributions P ( T FH > t ) ofRWs on RRGs (circles) and ER networks (+) of size N = 1000 and (a) c = 3 and (b) c = 5. For c = 3, where the first hitting process is dominated by the backtrackingscenario, there is a slight difference at short times due to e − c term in the expression for β in the ER network. For c = 5 the results obtained for the two networks are found tobe similar. This is due to the fact that as c is increased the e − c term in the expressionfor β of the ER network becomes negligible and it converges towards the expression for β of the RRG. This convergence occurs within the dilute network regime, in which thefirst hitting process is dominated by backtracking.In order to take a closer look at the retracing process, we consider the distribution offirst hitting times in NBWs, in which the backtracking process is suppressed. In the case irst hitting times of random walks on random regular graphs Figure 11.
Comparison between the analytical results for the tail distributions P ( T FH > t ) of first hitting times of RWs on RRGs ( ◦ ) and RWs on ER networks (+).The network size is N = 1000. The degree is (a) c = 3 and (b) c = 5. In both cases,the tail distribution takes the form of Eq. (25), with the parameters α and β givenin Table 1. Note that the agreement between the analytical results and computersimulations was established in Fig. 3 (for RWs on RRGs) and in Ref. [26] (for RWs onER networks). of NBWs on RRGs, upon suppression of the backtracking process, the retracing scenarioremains unchanged and the distribution of first hitting times becomes a Rayleighdistribution. In contrast, in the case of NBWs on ER networks the suppression ofthe backtracking process gives rise to a new mechanism of first hitting, referred toas the trapping scenario. This scenario occurs when the NBW enters a leaf node ofdegree k = 1. In the following time step it becomes trapped in the leaf node becausethe backtracking move into the previous node is not allowed. The distribution of firsthitting times of NBWs on ER networks was studied in Ref. [27] and the correspondingexpressions for α and β are shown in the lower-right cell in Table 1. irst hitting times of random walks on random regular graphs Figure 12.
Comparison between the analytical results for the tail distributions P NBW ( T FH > t ) of first hitting times of NBWs on RRGs ( ◦ ) and NBWs on ERnetworks (+). The network size is of size N = 1000. The degree is (a) c = 3 and(b) c = 5. In both cases, the tail distribution takes the form of Eq. (25), with theparameters α and β given in Table 1. Note that the agreement between the analyticalresults and computer simulations was established in Fig. 4 (for NBWs on RRGs) andin Ref. [27] (for NBWs on ER networks). In Fig. 12 we present a comparison between analytical results for the distributions P NBW ( T FH > t ) of NBWs on RRGs (circles) and ER networks (+) of size N = 1000 and(a) c = 3 and (b) c = 5. The first hitting times of NBWs on ER networks are foundto be much shorter than those obtained for NBWs on RRGs. This contrast is mostpronounced for small values of c . This is due to the emergence of the trapping scenariowhich is most effective in the limit of dilute networks. irst hitting times of random walks on random regular graphs
9. Discussion
Beyond the specific problem of first hitting times of RW on networks, the analysispresented here provides useful insight into the general context of the distribution of lifeexpectancies of humans, animals and machines [41, 42]. It illustrates the combinationof two lethal hazards, where one hits at a fixed, age-independent rate, while the otherincreases linearly with age. The first hazard may be considered as an external causesuch as an accident while the second hazard involves some aging related degradationwhich results in an increasing failure rate.In a more specific context of survival problems, the RW model that terminates uponits first hitting event can be cast in the language of foraging theory as a model describinga wild animal, which is randomly foraging in a random network environment [43]. Eachtime the animal visits a node it consumes all the food available in this node and needs tomove on to one of the adjacent nodes. The model describes rather harsh conditions, inwhich the regeneration of resources is very slow and the visited nodes do not replenishwithin the lifetime of the forager. Moreover, the forager does not carry any reservesand in order to survive it must hit a vital node at every time step. More realisticvariants of this model have been studied on lattices of different dimensions. It was shownthat under slow regeneration rates, the forager is still susceptible to starvation, whileabove some threshold of the regeneration rate, the probability of starvation diminishessignificantly [43]. The case in which the forager carries sufficient resources that enableit to avoid starvation even when it visits up to S non-replenished nodes in a row, wasalso studied [44, 45].In the dense network limit the first hitting process is dominated by the retracingscenario. As a result, the distribution of first hitting times becomes insensitive tothe degree c . In this limit the mean first hitting time scales like √ N . This can beunderstood as follows. In this limit, the backtracking probability is very low and thus thebacktracking-induced first hitting events become negligible. Instead, retracing becomesthe dominant scenario. Due to the very high connectivity, the hopping between adjacentnodes can be considered as the simple combinatorial problem of randomly choosing onenode at a time from a set of N nodes, allowing each node to be chosen more than once.In this limit the statistical properties of first hitting times become analogous to thoseof the birthday problem [46, 47]. More specifically, in this limit the probability that P ( T FH > t ) in a network that consists of N = 365 nodes is equal to the probabilitythat in a party of t participants there will not be even one pair who share the samebirthday [46].
10. Summary
We presented a statistical analysis of the first hitting times of RWs on RRGs, which maytake place either via backtracking or via retracing. The tail distribution P ( T FH > t )of first hitting times was calculated. It can be expressed as a product of a geometric irst hitting times of random walks on random regular graphs h T FH i and for the variance Var( T FH ) of the distribution of first hittingtimes. The analytical results are found to be in excellent agreement with the resultsobtained from computer simulations. We obtained analytical results for the probabilities P BT and P RET that the first hitting event will occur via the backtracking or retracingscenarios, respectively. We showed that in dilute networks the dominant first hittingscenario is backtracking while in dense networks the dominant scenario is retracing.We also obtained expressions for the conditional distributions of first hitting time, P ( T FH = t | BT) and P ( T FH = t | RET), in which the first hitting event occurs via thebacktracking or the retracing scenario, respectively. These results provide useful insightinto the general problem of survival analysis and the statistics of mortality rates whentwo or more termination scenarios coexist. We also analyzed the distribution of firsthitting times in non-backtracking random walks (NBWs), in which the backtrackingprocess is suppressed and compared the results obtained here for RWs and NBWs onRRGs to earlier results for RWs and NBWs on Erd˝os-R´enyi networks.This work was supported by the Israel Science Foundation grant no. 1682/18.
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