Random walks on weighted networks: Exploring local and non-local navigation strategies

Random walks on weighted networks: Exploring local and non-local navigation strategies

A.P. Riascos aaappprrr@gmail.com    José L. Mateos mateos@fisica.unam.mx Instituto de Física,Universidad Nacional Autónoma de México, Apartado Postal 20-364, 01000 México, D.F., México
July 20, 2019
Abstract

In this paper, we present an overview of different types of random walk strategies with local and non-local transitions on undirected connected networks. We present a general approach to analyzing these strategies by defining the dynamics as a discrete time Markovian process with probabilities of transition expressed in terms of a symmetric matrix of weights. In the first part, we describe the matrices of weights that define local random walk strategies like the normal random walk, biased random walks, random walks in the context of digital image processing and maximum entropy random walks. In addition, we explore non-local random walks like Lévy flights on networks, fractional transport and applications in the context of human mobility. Explicit relations for the stationary probability distribution, the mean first passage time and global times to characterize the random walk strategies are obtained in terms of the elements of the matrix of weights and its respective eigenvalues and eigenvectors. Finally, we apply the results to the analysis of particular local and non-local random walk strategies; we discuss their efficiency and capacity to explore different types of structures. Our results allow to study and compare on the same basis the global dynamics of different types of random walk strategies.

pacs:
89.75.Hc, 05.40.Fb, 02.50.-r, 05.60.Cd

I Introduction

Since their introduction as an informal question posted in the journal Nature in 1905 by Rayleigh, random walks have had an important impact in science with applications in a broad range of fields like biology, physics, chemistry, economy, computation, among many others Klafter and Sokolov (2011); Masuda et al. (2017). The success of random walk models in different applications lies in their simplicity, typically defined as a walker in a particular space that moves randomly without memory of its previous path, making this characteristic a good candidate in the description of processes like the diffusive transport, chemical reactions, fluctuations in the economy and even the foraging of some animal species Klafter and Sokolov (2011); Masuda et al. (2017); Redner (2001); van Kampen (1992); Viswanathan et al. (2011); Wosniack et al. (2017). Despite the mentioned simplicity in the definition, the consequences of the dynamics of a random walker are non-trivial and continue to surprise us with new results and with all the complexity that emerges from its simple rules.
In recent years, much of the interest in random walks have migrated to the study of complex systems described through networks Newman (2010); Barabási and Pósfai (2016); Latora et al. (2017). In this context, the interplay between the topology of the network and the dynamical processes taking place on this structure are of utmost importance Newman (2010); Barrat et al. (2008); Van Mieghem (2011). In particular, random walk strategies that allow transitions from one node to one of its nearest neighbors on the network constitutes a paradigmatic case and are the natural framework to study diffusive transport Barrat et al. (2008); Hughes (1996); Lovász (1996); Mülken and Blumen (2011), navigation and search processes in networks Noh and Rieger (2004); Fronczak and Fronczak (2009); Tejedor et al. (2009); Alessandretti et al. (2017), multiplex networks De Domenico et al. (2016), with applications in a variety of systems like the propagation of epidemics and spreading phenomena Durrett (2010); Pastor-Satorras et al. (2015), the dynamics on social networks Sarkar and Moore (2011), the analysis of information Blanchard and Volchenkov (2011), human mobility Riascos and Mateos (2017), among others Masuda et al. (2017); Barrat et al. (2008).

Figure 1: (Color online) Two types of transitions of a random walker on a network. The walker can hop from to the node that is one of the three nearest-neighbors available for a local transition. Also can make a non-local transition to reach the node . In the non-local displacement, the length of the shortest path is three as indicated by the dashed line.

On the other hand, in different cases full or partial knowledge of the network structure is available to define a random walk capable to use this information to increase the capacity to visit nodes with hops to the nearest neighbors but also long-range transitions beyond this local neighborhood. In Fig. 1 we illustrate local and non-local transitions in a network. In this case, the walker can visit one of the nearest-neighbors with a local transition but also there is the option of a non-local transition. By using this long-range dynamics the random walker can contact directly long-distance nodes without the intervention of intermediate nodes and without altering the topology of the network. As we will see, some non-local random walk strategies consider the shortest path connecting two nodes whereas others include quantities that contain all the possible paths between nodes.
In addition, it is important to mention that random walks with a non-local character have been explored in the literature. This is the case of Lévy flights on networks where random transitions occur to non-nearest neighbors with a probability that decays as a power law of the distance separating two nodes Riascos and Mateos (2012); the capacity of this strategy to explore networks has been studied in Riascos and Mateos (2012); Zhao et al. (2014); Huang et al. (2014); Weng et al. (2015, 2016); Guo et al. (2016); Zheng et al. (2017). Lévy flights on networks were generalized by Estrada et. al. by using a random multi-hopper model defined in terms of decaying functions of the shortest-path distances; this approach is explored in detail in Estrada et al. (2018). Furthermore, we also found non-local dynamics in the fractional transport on networks defined in terms of the fractional Laplacian of a graph Riascos and Mateos (2014). In this case, long-range displacements on the network emerge from a formalism that is introduced as the equivalent of the fractional diffusion equation on networks Riascos and Mateos (2014, 2015a). This strategy is studied in the context of transport in networks and lattices Riascos and Mateos (2014, 2015a); Michelitsch et al. (2016a, 2017a, 2017b); de Nigris et al. (2016, 2017a), in connection with information analysis de Nigris et al. (2017b) and quantum transport on networks Riascos and Mateos (2015b). The fractional transport is a particular case of a series of strategies that can be defined in terms of functions of the Laplacian of a network Riascos et al. (2018). In general, the study of random walks with long-range displacements on networks opens several questions regarding the way in which these large displacements can appear or be induced in different applications. Moreover, it is necessary the introduction of new methods and quantities that allow us to compare in the same background the efficiency to visit the nodes of a network through random walk strategies.
In this paper, we explore different types of local and non-local random walks on networks. We present a general approach to study these processes on the same basis by using the information contained in a symmetric matrix of weights used to define the probability of transition between nodes. We model the dynamics as a discrete time Markovian process. In the first part, we describe the matrices of weights that define local random walk strategies: traditional random walks, biased random walks, random walks in the context of digital image processing and, maximum entropy random walks. In the same way, examples of non-local random walks are described: Lévy flights on networks, fractional dynamics and applications in the context of human mobility. In all these cases, explicit relations for the stationary probability distribution of the random walker are obtained in terms of the elements of the matrix of weights that defines each strategy. After analyzing the transition matrix for these different processes, in a second part of the paper, a general formalism to calculate the mean first passage time and global times to characterize the dynamics is presented. Analytical expressions in terms of eigenvalues and eigenvectors of the transition matrix are obtained for all these quantities. Finally, we apply the results to the analysis of local and non-local random walk strategies to discuss and compare their efficiency and capacity to explore networks.

Ii Random walks on weighted networks

In this section, we introduce different concepts about random walks on weighted networks and the notation implemented to describe this process. We introduce a general random walker with probabilities of transition defined in terms of a network and a matrix of weights, the respective temporal evolution is modeled as a discrete time Markovian process for which we find an analytical result for its stationary probability distribution.
We consider undirected weighted networks with nodes . The topology of the network is described by an adjacency matrix with elements if there is an edge between the nodes and and otherwise; in particular, to avoid lines connecting a node with itself. The degree of the node is the number of neighbors that this node has and is given by . Additionally to the network structure, we have a symmetric matrix of weights with elements and . The matrix can include information of the structure of the network or incorporate additional data describing characteristics of links and nodes. By definition, the strength of the node is given by and represents the total weight of the node .
In the following, we study discrete time random walks on connected weighted networks with transition probabilities between nodes determined by the elements of the matrix of weights . The occupation probability to find the random walker in the node at time starting from at is given by and obeys the master equation Hughes (1996); Weiss (1994)

(1)

where the transition probability between the nodes and is given by

(2)

The transition matrix , with elements , in the general case is not symmetric; however, as a consequence of Eq. (2) and the symmetry of the matrix , we obtain , a result that establishes a connection between the transition probabilities and . On the other hand, iterating the master equation (1), the probability takes the form

(3)

and, using Eq. (3), we obtain

(4)

In this way, the detailed balance condition

(5)

is deduced as a direct consequence of the symmetry of . The relation in Eq. (5) allows to obtain the stationary probability distribution , that gives the probability to find the random walker in the node when . We have

(6)

showing that the stationary distribution of the node is directly proportional to its strength . The stationary distribution in Eq. (6) is a general result that characterizes the global behavior of the random walker. As we will see in the next section, this quantity allows to rank and classify the nodes of the network with a measure that combines the topological characteristics of the network structure with their capacity of transport modeled by the master equation (1) and the transition matrix . Furthermore, it is well known in the context of Markovian processes that the value is the average number of steps required for the random walker to return for the first time to the node Zhang et al. (2013); Condamin et al. (2007).

Iii Random Walk Strategies

Diverse types of random walk strategies can be explored in terms of the matrix of weights formalism described before. The only restrictions to this approach are the symmetry of the elements of the matrix of weights , the condition and . In this section, we present particular cases of navigation strategies that can be described by using this method. We divide our discussion into local strategies, for which the transitions of the random walker are restricted to adjacent sites on the network, and long-range strategies, for which the walker can hop with displacements beyond its nearest neighbors.

iii.1 Local random walks

In local random walk navigation strategies, the random walker always hops from a node to one of its nearest neighbors on the network. As a consequence, the elements of the matrix of weights take the form , where, as we explain in the following part, the value is related to quantities assigned to each node or to the weight of the link that connects the nodes and .

iii.1.1 Normal random walk

In this case, the weights coincide with the elements of the adjacency matrix; therefore . As a consequence, from Eq. (2), the transition matrix is given by Noh and Rieger (2004)

(7)

By definition, the normal random walker hops with equal probability from a node to one of its nearest neighbors in the network. In addition, from Eq. (6), the stationary distribution is . Normal random walks have been extensively studied in different contexts with applications in diverse types of networks; in particular, lattices Hughes (1996); Weiss (1994), general graphs Telcs (1989); Lovász (1996), complex networks Yang (2005); Sanders (2009); Tejedor et al. (2009); Kishore et al. (2011), fractal and recursive structures Meyer et al. (2012), among others Blanchard and Volchenkov (2011).

iii.1.2 Preferential navigation

In the preferential navigation strategy, a random walker hops with transition probabilities that depend of the quantity assigned to each node of the network. The value can represent a topological feature of the respective node (e.g., the degree, the betweenness centrality, the eigenvector centrality, the clustering coefficient, among other measures Newman (2010)) or a value, independent of the network structure, that quantifies an existing resource at each node. We define preferential random walks with local information by means of the weights , where the exponent is a real parameter. Then, from Eq. (2), we have

(8)

In Eq. (8), describes the tendency to hop to neighbor nodes with large values of , whereas for this behavior is inverted and the walker tends to hop to nodes with lower values of . On the other hand, for the normal random walk is recovered. By means of Eq. (6), the stationary distribution for the preferential random walk is

(9)

As we will see in the next part, the general preferential strategy defined by Eq. (8) determines different types of local random walks depending of the election of the quantities .

iii.1.3 Degree biased random walks

Figure 2: (Color online) Stationary distribution as a function of for degree biased random walks. The values are obtained by direct evaluation of Eq. (10). We use three values of the parameter and we study two types of networks with nodes. (a) An Erdös-Rényi network (ER) with an average degree ; the dashed lines represent the results obtained by the mean field approximation. (b) A scale-free network (SF) with .

This type of random walk is a particular case of the preferential navigation with in Eq. (8). The resulting strategy is known as degree biased random walks Wang et al. (2006); Fronczak and Fronczak (2009). For this particular case, the stationary distribution takes the form

(10)

Degree biased random walks have been studied extensively in the literature in different contexts as varied as routing processes Wang et al. (2006), chemical reactions Kwon et al. (2010), extreme events Kishore et al. (2012); Ling et al. (2013), among others Fronczak and Fronczak (2009); Lambiotte et al. (2011); Battiston et al. (2016). Additionally, mean field approximations have been explored for diverse cases Fronczak and Fronczak (2009); Kwon et al. (2010); Zhang et al. (2011a). For example, in networks with no degree correlations is valid the approximation . In Fig. 2 we present the values of the stationary distribution for degree biased random walks on an Erdös-Rényi network (ER) and, a scale-free network (SF) of the Barabási-Albert type, in which each node has a degree that follows asymptotically a power-law distribution Newman (2010); Barrat et al. (2008). We calculate the stationary distribution by direct evaluation of the Eq. (10) and we depict as a function of the degree . The results reveal that in the ER network is valid the mean-field approximation whereas in a SF network, this is only valid for nodes with high degrees Fronczak and Fronczak (2009).

iii.1.4 Maximal entropy random walks

Maximum entropy random walks (MERW) are a particular strategy derived from Eq. (8) for which the random walker uses information of the neighbor nodes. In this case, the transition probability is defined in terms of the components of the eigenvector centrality of the node . The value is determined by the -th component of the normalized eigenvector of the adjacency matrix that satisfies , where is the maximum eigenvalue of . In the study of topological features of networks, the components of the eigenvector centrality quantify the global influence of the node in the whole structure Newman (2010).
In this way, MERW are defined in the formalism of weighted networks with the election of weights . Then, the value of the strength is

(11)

where the last result is a consequence of the relation that satisfy the components of the eigenvector centrality. In this way, by using Eq. (2), the transition rule is given by

(12)

relation that defines a maximal entropy random walk Burda et al. (2009). Additionally, by using the Eq. (6), the stationary distribution of the maximal entropy random walk is

(13)

It is worth to mention that, the MERW defined by the transition probabilities in Eq. (12) maximizes the entropy rate production of the process given by Burda et al. (2009)

(14)

Combining this expression with Eqs. (12) and (13), Burda et al. (2009). In this case, the trajectories that follow the random walker are maximally random Burda et al. (2009); Sinatra et al. (2011). Diverse variations of the MERW and applications of this process have been explored in Sinatra et al. (2011); Ochab (2012); Frank and Galinsky (2014); Lin and Zhang (2014).

iii.1.5 Random walks for image segmentation

Figure 3: (Color online) Strength for random walks in the context of image segmentation. The values are obtained evaluating the sum with the weights given by Eq. (15). In (a) we present the original image (#3096 from the Berkeley segmentation database BSD300 Martin et al. (2001)). In (b)-(d) we present the obtained results for different values of the parameter , the colorbar denotes the scale of values for . Regions with present little variations in the intensity of a pixel in relation with its nearest neighbors, in these regions the random walker behaves as a normal random walker.

An important application of random walks on networks emerges in the context of the processing and segmentation of digital images Grady (2006). In this case, the statistical description of the diffusive transport from seed regions to specific pixels allows to detect and differentiate objects and structures in a digital image Grady (2006). The network is a square lattice where each node represents a pixel and the normalized intensity of is a quantity associated to the norm of the vector that contains the values RGB (red, green, blue) of the respective pixel, . In terms of a matrix of weights , a local random walker is defined by Grady (2006)

(15)

Here, the real parameter satisfies and the values give the elements of the adjacency matrix of a square lattice associated to the pixels positions and interactions between nearest neighbors. The resulting random walker follows a strategy given by Eq. (2) to visit the pixels; this transition probability gives high probability to the pass to pixels with the same intensity and determines the interaction between the pixels establishing a characteristic scale for the differences of intensity in the model controlling the capacity to hop to sites with a different color. In Fig. 3 we plot the strength for each pixel, this quantity is proportional to the stationary probability distribution for a random walker in a digital image that follows a strategy determined by the weights given by Eq. (15). It is observed how with this strategy, takes high values in regions with uniform color and low values in the boundaries of the object. In this way, the random walker propagates uniformly in regions with the same color and with low probability passes through the boundary of the object. This property makes this type of weights good candidates for image segmentation algorithms.
In addition to the local strategy mentioned before, it is worth mentioning that exists different variations of these models; this is the case of the topological biased random walks for which Zlatić et al. (2010), where the quantity describes the properties of the edge that connects with . A similar idea is explored for image segmentation in Sinop and Grady (2007), showing the vast applicability of random walks in different scenarios.

iii.2 Non-local random walks

Non-local random walks on networks are motivated by the possibility of hopping from one node to sites on the network beyond the neighbor nodes in cases where the total structure of the network is available. Random walk strategies with long-range displacements have shown an unprecedented applicability in the context of web searching. The PageRank introduced to classify pages on the Web Brin and Page (1998) and variations of this non-local strategy have been explored to rank the importance of nodes in a broad range of systems. In the following part, we present diverse non-local strategies on undirected networks that can be expressed in terms of a matrix of weights that includes information about the whole structure of the network to define the dynamical process. As particular examples of this case we have the Lévy flights on networks Riascos and Mateos (2012), the fractional diffusion on networks Riascos and Mateos (2014, 2015a), the dynamics of agents moving visiting specific locations in a city Riascos and Mateos (2017) and different strategies in the context of the random multi-hopper model Estrada et al. (2018). The study and possible applications of non-local dynamical processes on networks are relatively new and open questions related with the exploration of the effects that non-locality introduces as well as the search of global quantities that allow us to compare the performance of non-local against local dynamics.

iii.2.1 Lévy flights on networks

Figure 4: (Color online) Monte Carlo simulations of two different types of random walks on a plane. (a) Brownian motion. (b) Lévy flights. We depict steps for each realization of the dynamics. The colorbar codifies the number of each step that in this case is a measure of a discrete time.

The term Lévy flights makes reference to a random walk with displacements of length that appear with a probability distribution that asymptotically is described by an inverse power-law relation Metzler and Klafter (2004); Zaburdaev et al. (2015). For Lévy flights in the -dimensional space , and if for . With this definition, the variance of the displacements diverges; this characteristic differentiates Lévy flights from the Brownian motion for which the variance is finite Weiss (1994). In Fig. 4 we present Monte Carlo simulations for Brownian motions and Lévy flights in a plane. Lévy flights have a fractal behavior consisting of trajectories that alternate between groups described by local movements (similar to the observed in the Brownian motion) interrupted by long-range jumps; this structure is repeated at all levels. In this way, Lévy flights combine local movements, that appear with high probability, with long-range displacements that emerge with low but non-null probability. These characteristics are illustrated in Fig. 4(b). Lévy flights constitute an active area of research in different complex systems. For example, Lévy flights are encountered in the modelling of animal dynamics and foraging Ramos-Fernández et al. (2004); Boyer et al. (2006, 2012); Viswanathan et al. (2011); Wosniack et al. (2017), human mobility Brockmann et al. (2006); Brown et al. (2007); Rhee et al. (2011), among many others Metzler and Klafter (2004, 2000); Zaburdaev et al. (2015).

In the context of networks, Lévy flights are introduced in reference Riascos and Mateos (2012). In this case, the transitions are defined in terms of the distance that gives the number of lines in the shortest path connecting the nodes and . All the information about the distances between nodes is contained in the distance matrix with elements for . The distance matrix contains more information about the structure of the network than the adjacency matrix , but can be calculated efficiently from using different algorithms Newman (2010). In Fig. 5 we depict the relative frequency of distances in the entries of the matrix for large-world networks (square lattice and tree) and small-world networks (Erdős–Rényi network and scale-free network of the Barabási-Albert type). The histograms reveal the marked difference between the distances in these two types of structures.

Figure 5: (Color online) Frequencies of the non-null distances in the entries of the distance matrix . We analyze networks with . (a) Square lattice with dimensions . (b) Tree. (c) Erdős–Rényi network with probability of connection . (d) Scale-free network of the Barabási–Albert type. The results are expressed as a fraction of the value that gives the total number of different non-null entries in the distance matrix .
Figure 6: (Color online) Stationary distribution in terms of the degree for Lévy flights on networks with . The strategies with and use long-range displacements in the network. The result for the normal walker (limit is also depicted.

Lévy flights on networks can be described in terms of the weights for and . Here is a real parameter in the interval . For the elements of the transition matrix, we have and by using Eq. (2) for is obtained Riascos and Mateos (2012)

(16)

The dynamics inspired in Lévy flights allows long-range transitions on the network. For a finite non-null value of , the transitions to the nearest neighbors appear with high probability, but hops beyond these nodes are allowed generalizing the dynamics observed in the normal random walker in Eq. (7). In the limit we have , then and the Lévy strategy recovers the normal random walk. Another interesting limit case is obtained when , in this case if and the dynamics induces the possibility to reach with equal probability any node of the network Riascos and Mateos (2012).
Once defined Lévy flights in terms of the elements for ; for this particular model we denote the strength as and by using the Eq. (6) we obtain for the stationary distribution

(17)

This result establishes that is proportional to the quantity . In addition, the value , can be expressed as Riascos and Mateos (2012)

(18)

where is the number of nodes at a distance of the node ; in particular, . In this way, by means of the expression in Eq. (18) we observe that is a generalization of the degree that combines all the information about the structure of the network. This long-range degree emerges from the study of Lévy flights on networks and was introduced in Riascos and Mateos (2012).
In Fig. 6 we depict the stationary distribution obtained from the analytical result in Eq. (17) for an Erdős Rényi network and a scale-free network. Also calls the attention that, compared to the normal strategy, Lévy flights represent a democratic strategy in the sense that the probability of visiting sites with many connections decreases and for sites with a lower degree, this probability increases. Being able to easily reach any node on the network can offer advantages if the goal is the exploration of the entire structure. This aspect is discussed in detail later when the efficiency of the random walker is analyzed.
Different aspects of Lévy flights and their capacity to explore networks have been studied in Zhao et al. (2014); Huang et al. (2014); Weng et al. (2015, 2016), as well as in the context of multiplex networks Guo et al. (2016). A general approach to study the random walker in Eq. (16) and other strategies defined in terms of a function of the distances in a network are analyzed in detail by Estrada et.al. in Estrada et al. (2018). In this context is introduced the exponential strategy that in terms of our matrix of weights formalism is defined by for and using . By following a similar approach to the presented in Eqs. (17)-(18), can be deduced analytical expressions for the stationary probability distribution of the exponential strategy.

iii.2.2 Gravity law, spatial networks, and human mobility

In diverse situations networks are embedded in a metric space, then, spatial locations are assigned to each node. This is the case of spatial networks that describe several real systems like social networks, airports and transportation networks, among others Barthélemy (2011); Huang et al. (2014); Barbosa et al. (2018).
On the other hand, it has been suggested that migration and human movements are well described in terms of a “Gravity Law” that models the number of trips from a location to the location as . Here and denote the population of the respective locations, is the geometric distance between the nodes , is a constant and is a free parameter. This type of model suggests a similar algorithm for a random walker on networks described by the weights

(19)

for . Here the value is a quantity associated to the node in the network and is the topological distance in the network. The general formalism in terms of weighted networks also applies to the model presented in Eq. (19), but with geometric distances . In this model, the structure of the network is absent and it is assumed as a complete graph.
In the gravity law model, the resulting random walker contains characteristics of the biased random walks determined by Eq. (8) and the Lévy flights on networks with transition probabilities given by Eq. (16). The random walk defined in Eq. (19) has been explored in order to characterize co-occurrences of words on web pages Liu et al. (2014). In addition, there are different variations of the gravity law in spatial networks (see Barthélemy (2011) and references therein). Some of these models are described in the weighted network approach by weights proportional to a positive function of the distance Barthélemy (2011).
As an example of random walks that take place in a continuous space but can be modeled with the formalism of random walks defined in terms of a matrix of weights, in reference Riascos and Mateos (2017) is introduced a strategy to visit randomly specific locations in a spatial region modeling characteristics of human mobility in urban settlements. In this case, points are located in a plane and integer numbers label each location. In addition, the coordinates of the locations are known and we denote as the distance between the places and . The distance can be calculated by different metrics; for example, in some applications could be appropriated the use a Euclidean metric, whereas, in other contexts, a Manhattan distance could be more useful. In order to define a discrete time random walker that at each step visits one of the locations, the transition probability to hop from site to site is given by Riascos and Mateos (2017)

(20)

where the weights are defined by the relation Riascos and Mateos (2017)

(21)
Figure 7: (Color online) A schematic illustration of the random walk strategy as defined in Eq (20). In (a) we depict random locations on the plane (represented by triangles); the probability to go from location to a different site is determined by two types of transition probabilities: First, to a site inside a circular region of radius centered in the location , , which is a constant; and second, a transition to a site outside the circle of radius , that considers long-range transitions with a power-law decay proportional to , where is the distance between sites and . In (b) we show Monte Carlo simulations of a discrete-time random walker that visits specific locations in the region in following the random strategy defined by the transition probabilities in Eq (20), with . We depict the results for and . The total number of steps is and the scale in the color bar represents the time at each step.

Here and are positive real parameters. The radius determines a neighborhood around which the random walker can go from the initial site to any of the locations in this region with equal probability; this transition is independent of the distance between the respective sites. That is, if there are sites inside a circle of radius , the probability of going to any of these sites is a constant. Additionally, for places beyond the local neighborhood, for distances greater than , the transition probability decays as an inverse power law of the distance and is proportional to Riascos and Mateos (2017). In this way, the parameter defines a characteristic length of the local neighborhood and controls the capacity of the walker to hop with long-range displacements. In particular, in the limit the dynamics becomes local, whereas the case gives the possibility to go from one location to any different one with the same probability. In this limit, we have . This model is then a combination of a rank model Liben-Nowell et al. (2005); Noulas et al. (2012); Pan et al. (2013) for shorter distances and a gravity-like model for larger ones Simini et al. (2012); Barbosa et al. (2018). It is important to mention that in the strategy defined by the weights in Eq. (21), we choose , in this way the walker also can stay in the node with non-null probability. All the results presented are also valid for this case whenever the value of is such that the random walker can reach any of the sites used in the definition of the transition matrix.
In Fig. 7(a) we illustrate the model for the random strategy introduced in Eq (20). In Fig. 7(b), we present Monte Carlo simulations of the random walker described by Eqs (20)-(21). We generate random locations (points) on a 2D plane on the region in and, for different values of the exponent , we depict the trajectories described by the walkers. In the case of , it is observed how the dynamics is local and only allows transitions to sites in a neighborhood determined by a radius around each location. In this case, all the possible trajectories in the limit form a random geometric graph Dall and Christensen (2002); Estrada and Sheerin (2015); we can identify features of this structure in our simulation. On the other hand, finite values of model spatial long-range displacements such as the dynamics illustrated in Fig. 7(b) for the case . We observe how the introduction of the long-range strategy improves the capacity of the random walker to visit and explore more locations in comparison with the local dynamics defined by the limit Riascos and Mateos (2017).

iii.2.3 Fractional transport

Figure 8: (Color online) Monte Carlo simulation of a discrete-time fractional random walker on a tree with transition probabilities given by Eq. (26). The dynamics starts at from an arbitrary node. We show three discrete times for three values of the parameter . The case corresponds to a normal random walker whereas the cases with correspond to a fractional random walk leading to anomalous transport. We represent with different colors the unvisited nodes, visited nodes and the position of the random walker at time .

The fractional transport on networks is defined in terms of a power of the Laplacian matrix with elements given by , where denotes the Kronecker delta; in particular, . The Laplacian matrix is introduced in graph theory and in the modeling of dynamical processes on networks Barrat et al. (2008); Arenas et al. (2008); Estrada (2015); Mohar (1991, 1997); Mülken and Blumen (2011); McGraw and Menzinger (2008); Estrada et al. (2012); Fouss et al. (2016). In addition, the matrix is interpreted as a discrete form of the Laplacian operator Newman (2010); Mohar (1991, 1997). In the context of the fractional diffusion on networks is introduced the fractional Laplacian matrix , where is a real number (). The resulting process models the fractional dynamics on general networks Riascos and Mateos (2014, 2015a).
Since the Laplacian matrix is a symmetric matrix, by using the Gram-Schmidt orthonormalization of the eigenvectors of , we obtain a set of eigenvectors that satisfy the eigenvalue equation for and , where are the eigenvalues, which are real and nonnegative. For connected networks, the smallest eigenvalue is and for Van Mieghem (2011). We define the matrix with elements and the diagonal matrix . These matrices satisfy , therefore , where denotes the conjugate transpose of . Therefore Bellman (1960)

(22)

where . It is worth noticing that the diagonal elements of the fractional Laplacian matrix defined in Eq. (22) introduce a generalization of the degree to the fractional case. In this way, the fractional degree of the node is given by Riascos and Mateos (2014)

(23)

The fractional random walk is the random walk associated to the fractional diffusion in networks Riascos and Mateos (2014). In the formalism of weighted networks is defined by the elements and, for

(24)

with . On the other hand, the elements of the Laplacian matrix satisfy and, in the fractional case we have . As result the strength of the node is given by

(25)

then, by using Eq. (2), the transition probability is given by

(26)

In the limit , the normal random walk strategy is recovered. In addition, by using the Eq. (6), the stationary distribution is

(27)

This is a generalization of the result for normal random walks discussed before and recovered from Eq. (27) when .
The fractional random walk is the process associated to the fractional diffusion on networks and the transition probabilities in Eq. (26) define a navigation strategy with long-range displacements on the network Riascos and Mateos (2014). The case of infinite -dimensional lattices with periodic boundary conditions has been addressed in different in contexts Michelitsch et al. (2016b, a, 2017a, 2017b, 2018). For this type of periodic structures, it is obtained the analytical relation Michelitsch et al. (2017a)

(28)

The result in Eq. (28) establishes a connection between Lévy flights on networks Riascos and Mateos (2012) and the fractional strategy defined by Eq. (26). On the other hand, in networks with constant degree , the fractional Laplacian can be expressed as Riascos and Mateos (2015a)

(29)

where and denotes the Gamma function Abramowitz and Stegun (1970). The result in Eq. (29) relates the fractional Laplacian matrix with the integer powers of the adjacency matrix for for which the element is the number of all the possible trajectories connecting the nodes , with links Godsil and Royle (2001). In this way, the fractional strategy defined by the transition matrix with elements in Eq. (26) incorporates global information about all the possible trajectories connecting the nodes and Riascos and Mateos (2015a).
In order to illustrate the effect of the fractional dynamics of a random walker on a network, in Fig. 8 we present Monte Carlo simulations of discrete-time random walks on a tree. The discrete time denotes the number of steps of the random walker as it moves from one node to the next node on the network. Given the topology of the network, we calculate the adjacency matrix and the corresponding Laplacian matrix of the network. Then we obtain its eigenvalues and eigenvectors that allow us in turn to get the fractional Laplacian matrix . Finally, using Eq. (26), we determine the transition probabilities for different values of the parameter . The dynamics starts at from an arbitrary node. We show three discrete times for three values of the parameter . Here, we depict one representative realization of a random walker as it navigates from one node to another randomly. The case corresponds to normal random walk leading to normal diffusion. In this case, the random walker can move only locally to nearest neighbors and, as can be seen in the figure, the walker revisits very frequently the same nodes and therefore the exploration of the network is redundant and not very efficient. The cases with correspond to a fractional random walk leading to anomalous diffusion. In this case, the random walker can navigate in a long-range fashion from one node to another arbitrarily distant node. This allows us to explore more efficiently the network since the walker does not tend to revisit the same nodes; on the contrary, it tends to explore and navigates distant new regions each time. All this can be seen in the figure for different times, and allow us to make a comparison between a random walker using regular dynamics and a fractional dynamics Riascos and Mateos (2015a). A detailed analysis of the fractional Laplacian of graphs and its relation with long-range navigation on networks and applications is presented in references Riascos and Mateos (2014, 2015a); Michelitsch et al. (2016b, a, 2017a); de Nigris et al. (2016, 2017a); Michelitsch et al. (2017b).
The introduction of the fractional random walks is motivated by the search of an equivalent on networks of the fractional diffusion and its relation with Lévy flights. Recently, other types of functions of matrices with local information have shown interesting properties associated with long-range dynamics and the global structure of networks; this is the case of the concept communicability Estrada et al. (2012) and the accessibility random walk introduced in de Arruda et al. (2014). Particular functions of matrices can be used to define different types of long-range strategies and characterized with the formalism reviewed in this work.
As a generalization of Eq. (26), other functions of the Laplacian can be applied to define random walk strategies on networks. The functions to define random walk strategies should satisfy the following conditions Riascos et al. (2018)

  • Condition I: The matrix must be positive semidefinite, i.e., the eigenvalues of are restricted to be positive or zero. In this way, the property of the Laplacian eigenvalues for is preserved by the function .

  • Condition II: The elements of the matrix denoted as , for , should satisfy

    (30)

    Therefore, the function maintains the property associated to the elements of the Laplacian matrix.

  • Condition III: All the non-diagonal elements of must satisfy

    (31)

For this type of functions, transition probabilities are defined by the relation

(32)

where we use the generalized degrees defined by the diagonal elements of that satisfy Riascos et al. (2018)

(33)

Examples of functions that satisfy the conditions I-III are the fractional Laplacian of a graph with , the logarithmic function for and the function with . In all these cases is observed that the random walker hops with long-range displacements on the network Riascos et al. (2018).
In terms of the formalism of the matrix of weights, the generalized random walk strategy in Eq. (32) can be analyzed by using the weights for and . In this way, as a consequence of the condition in Eq. (31), the weights satisfy ; also, the strength of each node is given by the generalized degree allowing us to write the stationary probability distribution of the process as

(34)

In the general case described in Eq. (32), the values of can be obtained by using the spectral methods described before for the fractional Laplacian (see Riascos et al. (2018) for details).

LOCAL STRATEGIES
Strategy Weights , . Parameters References
1. Normal random walk Lovász (1996); Hughes (1996); Noh and Rieger (2004)
2. Biased random walk ,
3. Degree biased random walk Fronczak and Fronczak (2009)
4. Maximal entropy random walk Burda et al. (2009); Sinatra et al. (2011); Lin and Zhang (2014)
5. Random walks for image segmentation Grady (2006)
6. Topologically biased random walk , Zlatić et al. (2010); Sinop and Grady (2007)
NON-LOCAL STRATEGIES
1. Lévy Flights Riascos and Mateos (2012); Lin and Zhang (2013); Zhao et al. (2014); Estrada et al. (2018)
2. Exponential strategy Estrada et al. (2018)
3. Gravity Law González et al. (2008); Barthélemy (2011); Liu et al. (2014)
4. Fractional Diffusion Riascos and Mateos (2014, 2015a)
5. General functions of the Laplacian Riascos et al. (2018)
6. Communicability Estrada et al. (2009, 2012)
Table 1: Diverse types of random walks described in terms of a symmetric matrix of weights with transition probabilities defined by Eq. (2). For all the random walk strategies and, the non-diagonal elements are presented in the table with a short description of the quantities and parameters involved in the definition. The detailed description of each random walk strategy is presented in Section III.

We conclude this section with a compilation of the types of random walk strategies represented by specific types of weighted networks. In Table 1 we summarize the matrices of weights that define the local and non-local strategies analyzed in this section. Each model is presented with the respective parameters that define the random walker and key references to works analyzing these strategies.

Iv Mean first passage time and global characterization

Once described a general formalism that allows us to define different types of local and non-local random walks strategies on networks and analytical results for their respective stationary distributions; in this section, we explore the mean first passage time (MFPT) Redner (2001), that gives the average number of steps needed by the random walker to reach a specific node for the first time. We also study global times to quantify and compare the capacity of local and non-local random walks to explore different types of networks.

iv.1 Mfpt

In order to calculate the MFPT for strategies defined in terms of weighted networks, we use a similar approach to the formalism presented in Hughes (1996); Noh and Rieger (2004) where normal random walks are studied. We start representing the probability in the master equation in Eq. (1) as

(35)

The first term in Eq. (35) represents the initial condition and is the probability to start in the node and reach the node for the first time after steps, by definition . Now, by using the discrete Laplace transform , the relation in Eq. (35) takes the form

(36)

By definition, using the quantity , the MFPT for a random walker that starts in the node and reach for the first time the node is given by Hughes (1996)

(37)

Now, by means of the moments of the probability defined as

(38)

the expansion in series of is

(39)

Introducing this result in Eq. (36), the MFPT is obtained

(40)

In Eq. (40) there are three different terms: the mean first return time , the quantity

(41)

which is a time independent of the initial node and the time that depends on and . Furthermore, from the detailed balance condition is obtained , as consequence

(42)

relation that describes the asymmetry of navigation Noh and Rieger (2004). The time is interpreted as the average time needed to reach the node from a randomly chosen initial node of the network; on the other hand, the quantity is the random walk centrality introduced for the analysis of random walks with local information Noh and Rieger (2004). The centrality combines information of the network and the random walk strategy implemented to visit nodes and gives a high value to nodes easy to reach and small values to nodes for which the random walker takes, in average, many steps to hit the node for the first time starting from any node of the network Noh and Rieger (2004); Riascos and Mateos (2012).
Additional to the times and , from Eq. (40) we have

(43)

The quantity in the context of stochastic processes is denominated Kemeny’s constant Kemeny and Snell (1960); Zhang et al. (2011b). As result of the relation in Eq. (43)

(44)

equation that establishes a connection between the Kemeny’s constant of Markovian processes and the global time obtained by averaging the mean first passage times weighted with the stationary distribution .

iv.2 Linear algebra approach

Once defined general quantities that characterize the performance of a random walk strategy to explore a network, it is important to have an algorithm that allows us to calculate these values by using the information consigned in the transition probability matrix in Eq. (2), defined in terms of the matrix of weights and that essentially contains all the information about the random walker. Therefore, in the following part we deduce expressions for the MFPT , the time and the Kemeny’s constant in terms of the eigenvalues and eigenvectors of the transition matrix .
In order to calculate and is necessary to find . We start with the matrical form of Eq. (1)

(45)

Here is the probability vector at time . Using Dirac’s notation

(46)

where represents the canonical base of .
Due to the existence of a detailed balance condition, the matrix can be diagonalized and its spectrum has real values van Kampen (1992). For right eigenvectors of we have for , where the set of eigenvalues is ordered in the form and . On the other hand, from right eigenvectors we define a matrix with elements . The matrix is invertible, and a new set of vectors is obtained by means of , then

(47)

and

(48)

where is the identity matrix.
In different cases, especially when it is necessary to calculate numerically the eigenvalues and eigenvectors of the transition matrix, it is convenient to use the symmetry of the matrix of weights . In this way, the eigenvectors and can alternatively be deduced from the analysis the symmetric matrix with elements

(49)

From an orthonormal set of eigenvectors that satisfy for , it is obtained and where is the diagonal matrix .
Once obtained the spectrum and the left and right eigenvectors of the transition matrix, we can deduce different analytical expressions for quantities that characterize the random walker. By using the diagonal matrix is obtained , therefore Eq. (46) takes the form

(50)

From Eq. (50), the stationary probability distribution , where the result makes independent of the initial condition. Now, by means of the definition of , we have

(51)

Therefore, the time is given by

(52)

and, for in Eq. (40), the MFPT is

(53)

whereas . Finally, from Eqs. (44) and (51) is obtained the Kemeny’s constant

(54)

result that only depends on the eigenvalues of the transition matrix .

iv.3 Global characterization

Figure 9: (Color online) Time vs. for degree biased random walks on networks obtained using Eq. (52). (a) Erdös-Rényi (ER) network with and . (b) Scale-free (SF) network with . We use three values of the parameter .

In this part we define global times that quantify the capacity of a random walk to reach any site of the network; by using these global times is possible to compare the efficiency of the different strategies defined through Eq. (2). Global quantities like entropy rates Gómez-Gardeñes and Latora (2008); Burda et al. (2009), the global mean first passage time Tejedor et al. (2009), and the cover time Barrat et al. (2008); Hughes (1996) have been used to study random walks on networks. We use the global quantity Riascos and Mateos (2012)

(55)

that gives an estimate of the average time to reach any site of the network. The values can present a huge dispersion due to the fact that in some irregular networks there are nodes easily accessible to the random walker and other sites that are hardly reached; despite this fact, the mean value of the times is an important quantity that characterize the capacity of a random walker to visit the nodes of a network. In the following section we explore the time for different random walk strategies.
On the other hand, in the particular case of random walks on weighted networks for which the value is constant, the stationary distribution given by Eq. (6) is