Identification of a current-carrying subset of a percolation cluster using a modified wall follower algorithm
Renat K. Akhunzhanov, Andrei V. Eserkepov, Yuri Yu. Tarasevich
aa r X i v : . [ c ond - m a t . d i s - nn ] N ov Identification of a current-carrying subset of apercolation cluster using a modified wall followeralgorithm
Renat K Akhunzhanov, Andrei V Eserkepov, and Yuri Y Tarasevich
E-mail: [email protected]
Laboratory of Mathematical Modeling, Astrakhan State University, Astrakhan 414056, Russia
Abstract.
We have proposed and implemented a modification of the well-known wall followeralgorithm to identify a backbone (a current-carrying part) of the percolation cluster. Theadvantage of the modified algorithm is identification of the whole backbone without visitingall edges. The algorithm has been applied to backbone identification in networks produced byrandom deposition of conductive sticks onto an insulating substrate. We have found that (i) forconcentrations of sticks above the percolation threshold, the strength of the percolating clusterquickly approaches unity; (ii) simultaneously, the percolation cluster is identical to its backboneplus simplest dead ends, i.e., edges that are incident to vertices of unit degree.
1. Introduction
An approach describing the composites is the percolation theory [1]. Percolation, i.e., theemergence of a connected subset (a cluster) that spans opposite boundaries in a disorderedmedium, has attracted the attention of the scientific community for several decades [1, 2, 3, 4, 5].The occurrence of a percolation cluster drastically changes the physical properties of the medium,e.g., an insulator–conductor phase transition can be observed when the disordered medium isa mixture of conductive and insulating substances. However, only a fraction of the percolationcluster takes a part in the electrical conductivity [6, 7]. When the percolation cluster is treatedas a random resistor network (RRN), a set of current-carrying bonds of such RRN is called the(effective) backbone [8]. The rest of the percolation cluster is dead ends [8] (tag ends [9], tanglingends [10]) and so-called perfectly balanced bonds (Wheatstone bridges). The electrical currentthrough a perfectly balanced bond is absent because potential difference between its ends isequal to zero [11]. The geometrical backbone is the union of all the self-avoiding walks (SAWs)between the two given points [12]. SAW or a simple path is a path that contains no vertextwice. An algorithm for finding simple paths in a graph is based on depth-first search [13]. Thegeometrical backbone is the effective one plus the ideally balanced bonds. Thus, the effectivebackbone is defined as the set of bonds that carry a current, while the geometrical backbone isthe set of bonds that either carry a current, or are perfectly balanced [14]. Another definitionof the backbone says that it is the largest biconnected component of the graph [9]. Such thedefinition may be confusing since a set of vertices is said to be biconnected, if each pair ofvertices can be linked by at least two distinct paths. Hence, the two vertices connected by onlyone SAW can form no backbone according this definition. Although this definition is true wheneriodic boundary conditions (PBCs) are applied to the plane, i.e., a percolation on torus isconsidered, an examination of the electrical conductivity on a torus looks somewhat artificial.Some of the bonds belonging to the backbone may carry the total current. These bonds arecalled red bonds or singly connected bonds; when they are cut, the current flow stops [15].There are two different approaches for identification of backbones. On the one hand, one canuse Ohm’s law or Kirchhoff’s rules to calculate potentials and currents in the RRN [16, 17, 11, 18].However, direct calculations of electrical potentials and currents are based on floating-pointarithmetic and, hence, produce round-off errors. Due to these round-off errors, some ghostcurrents may arise which impedes the backbone extraction. Moreover, these calculations dealwith huge systems of linear equations and require a lot of computer memory. Only relativesmall systems can be treated in these approaches because number of equations to be solved isproportional to the square of linear size of the system under consideration.On the other hand, one can apply search algorithms on graphs [19, 20, 21, 8, 22, 10, 23, 24,25, 26, 27, 28, 29]. In fact, some of the algorithms of backbone identification belong to mazesolving algorithms (such as “Ariadne’s clew algorithm” [30]), which, in particular, are appliedto wire routing on chips [31]. Graph theory algorithms are sometimes difficult to understandor/and to realize. Some of them require storing not only original network but its dual [20].Moreover, some algorithms can produce stack overflow because of recursion. All of the availablegraph-based algorithms remain storage limited, as some information at each node of the graphremains necessary [32]. In fact, application of these algorithms also is restricted to the RRN ofmoderate size.Each of these two approaches have both advantages and disadvantages. A comparison andanalysis of the algorithms devoted to identification of current-carrying part of the RRN can befound in Ref. [33].In this conference paper, we present a modification of a wall follower algorithm for a mazesolving. The rest of the paper is constructed as follows. Section 2 describes some technical detailsof simulation and our modification of the wall follower algorithm that extracts a geometricalbackbone of a percolating cluster if any. Section 3 presents our main findings. Section 4summarizes the main results.
2. Modified wall follower algorithm
Consider an embedding of an undirected simple planar graph G = G ( E, V ) in two-dimensionalEuclidean space R . We are looking for all simple paths (self-avoiding walks, SAWs) between“entry” vertex, V in , and “exit” one, V out . Three kinds of edges and vertices are distinguished inthe algorithm. Initially, all edges and vertices are supposed to be “black”. During the executionof the algorithm, the edges and vertices are colored in yellow or green. Namely, “green” ones areclassified to be a part of the geometric backbone, “yellow” ones are classified to be dead ends.Before the algorithm starts working, the two “green” ghost edges should be added to the graphin the way depicted in Fig. 1. After identification of all SAWs between V in and V out , the twoghost edges have to be removed.Recursive procedure NG ( E, V , V , G ) is looking for SAWs between V and V .The procedure uses the two functions, viz., NextEdge ( V, E, G, color, direction ) and
AdjacentVertex ( V, E, G ) (hereinafter, V is a vertex of G incident on edge E ). The function NextEdge ( V, E, G, color, direction ) returns an edge of the graph G incident on the vertex V ; thereturned edge is the first one following the edge E in the traversal direction indicated by pa-rameter direction and has the color indicated by the parameter color . The parameter direction may be clockwise ( (cid:8) ) or counterclockwise ( (cid:9) ). The parameter color may be green , yellow or any color . The function AdjacentVertex ( V, E, G ) returns the vertex of the graph G thatis incident to the edge E and differs from the vertex V . The procedure NG looks through all theedges incident to the vertex V , starting from the green edge E , and then counterclockwise, to igure 1. Transformation of the system underconsideration for application of the modified wallfollower algorithm.the next green edge E ′ . Let us denote the next vertex of E ′ as V ′ (see Fig. 2a). Then, startingfrom the green edge E ′ clockwise, we are looking for the nearest edge of any color. If the nearestedge is E (see Fig. 2b), one should call NG ( E ′ , V ′ , V , G ). Otherwise (see Fig. 2c), one shouldcall lines 9–14 of the below pseudocode. Figure 2. (a) The edges incident to the vertex V . Edges between the edge E and the nextgreen edge E ′ counterclockwise are under consideration (solid lines). Other edges (dashed lines)are not under consideration. (b) The case when E is the nearest edge of any color clockwise of E ′ . (c) The case when E ′′ is the nearest edge of any color clockwise of E ′ . procedure NG ( E, V , V , G ) if V = V then return end if E ′ ← NextEdge ( V , E, G, green, (cid:9) ) V ′ ← AdjacentVertex ( E, V , G ) E ′′ ← NextEdge ( V , E ′ , G, any color, (cid:8) ) if E ′′ = E then NG ( E ′ , V ′ , V , G ) else V ′ ← WF ( E, V , E ′′ , G ) if V ′ = V then NG ( E ′ , V ′ , V , G ) else NG ( E ′ , V ′ , V ′ , G ) NG ( E, V , V , G ) end if end if end procedure To find all SAWs between V in and V out , one should call the procedure as NG ( E , V in , V out , G ).A modified wall follower algorithm is presented as a pseudocode of the function WF . function WF ( E , V, E , G ) V.color ← black : V ′ ← V E ′ ← E loop E ′ .color ← yellow V ′ ← AdjacentVertex ( E ′ , V ′ , G ) if V ′ .color = green then V ′′ ← V ′ loop E ′ .color ← green V ′ ← AdjacentVertex ( E ′ , V ′ , G ) V ′ .color ← green if V ′ = V then return V ′′ end if E ′ ← NextEdge ( V ′ , E ′ , G, yellow, (cid:8) ) end loop end if E ′ ← NextEdge ( V ′ , E ′ , G, any color, (cid:9) ) if E ′ = E then V.color ← green return V end if end loop end function Zero-width sticks of unit length were randomly deposited onto a substrate of size L × L with PBCs until the desired number density was reached. Their centers are assumed to beindependent and identically distributed (i.i.d.) on the substrate, while their orientations areassumed to be equiprobable. Hence, a homogeneous and isotropic network is produced. Forbasic computations, we used the system of size L = 32. Finite-size effect has been additionallytested via variation of the system size.Consider an arbitrary 2D network produced by random isotropic deposition of equally-sizedzero-width sticks. Each stick is treated as a resistor with a specified electrical conductivity, σ .When this network is a subject to a potential difference, there are two natural possibilities [34],viz., • the “bus-bar geometry”, when two parallel (super)conducting bars (buses) are attached tothe opposite borders of the network; a potential difference (say, V and 0, V >
0) is appliedto these buses [20, 10, 29], • the “two-point geometry”, when a potential difference is applied to the two distinct sites, sothat an electrical current, I , injected into one site (source) and the same current withdrawnfrom the other (sink) [8].In the case of superconducting buses, “bus-bar geometry” can be turn to “two-point geometry”by addition of ghost vertices.To detect the spanning cluster, the Union–Find algorithm [35, 36] modified for continuoussystems [37, 38] was applied. When the spanning cluster was found, all other clusters wereremoved since they cannot contribute in the electrical conductivity. All edges of the spanningcluster incident on vertex of unit valence were cut off, since, obviously, they are simplest deadends. According to Ref. [39], we denote such the preprocessed spanning cluster as “approximatebackbone”. To detect the backbone of the spanning cluster, the modified wall follower algorithmwas used. When the geometrical backbone has been identified, an adjacency matrix was formedfor it. With this adjacency matrix in hands, Kirchhoff’s current law was used for each junctionf sticks, and Ohm’s law for each circuit between two junctions.The computer experiments were repeated 100 times. The error bars in the figures correspondto the standard deviation of the mean. When not shown explicitly, they are of the order of themarker size.A particular case of a planar graph is N zero-width sticks of length l which centers areassumed to be i.i.d. within a square domain D of size L × L with periodic boundary conditions( D ∈ R ), i.e., x, y ∈ [0; L ], where ( x, y ) are the coordinates of the center of the stick. Therelation L > l is assumed. All intersections of sticks with the lines x = L and x = 0 aresupposed to be vertices (“entries” and “exits”, respectively). To apply the above algorithm, wetransform a bus-bar geometry to a two-point geometry by adding two ghost vertices, viz., V in isadjacent to all vertices belonging to “entries”, while V out is adjacent to all vertices belonging to“exits”. In such a way, the problem of geometrical backbone identification for bus-bar geometryis transforms into the one for two-point geometry.In many cases, modified wall follower algorithm can identify the backbone without visitingall edges. In a graph produced by random isotropic deposition of zero-width sticks onto a plane,near the percolation threshold, a fraction of unvisited edges after complete identification of thebackbone approaches to 0.5.
3. Results
Figure 3 demonstrate how the quantities of interest depends on the shifted number density, n − n c .Solid symbols present our results, while the open symbols represents the results extracted fromRef. [39]. The strength of the percolation cluster approaches unit reflecting the fact that almostall sticks belong to the percolation cluster when n ' n c . This observation is quite consistentwith the previously published results [39]. At the large number density, the backbone and theapproximate backbone [39] are indistinguishable within simulation accuracy. This fact validatesthe assumption [39] that the percolation cluster is identical to its geometrical backbone plussimplest dead ends, i.e., edges incident on the vertices of the unit degree. Solid line correspondsto theoretical estimate of the approximate backbone offered in Ref. [39]. P n n c Figure 3.
Dependencies of the strength of thepercolation cluster ( and ⊓⊔ ), of the strengthof the approximate backbone ( • and ◦ ),and of the strength of the backbone ( △ ) onthe shifted number density, n − n c . Fullsymbols correspond to our results, while theopen symbols represents the results extractedfrom Ref. [39]. Solid line (——) corresponds toanalytical estimation from Ref. [39].
4. Conclusion
We have proposed and implemented a modified wall follower algorithm for backboneidentification. The algorithm was applied to backbone identification for different system sizesand concentrations of conductive sticks. We have found that (i) for concentrations of sticks abovethe percolation threshold, the strength of the percolating cluster quickly approaches unity asconcentration of sticks increase; (ii) simultaneously, the percolation cluster is almost identical toits backbone plus simplest dead ends, i.e., edges that are incident to the vertices of degree one. cknowledgments
Y.Y.T. and A.V.E. acknowledge the funding from the Foundation for the Advancement ofTheoretical Physics and Mathematics “BASIS”, grant 20-1-1-8-1. The authors would like tothank I.I.Gordeev for stimulating discussions.
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